Characterisation of polarised supercontinuum generation and its focal field

Transcription

1 Characterisation of polarised supercontinuum generation and its focal field A thesis submitted for the degree of Doctor of Philosophy by Brendan James Chick Centre for Micro-Photonics Faculty of Engineering and Industrial Sciences Swinburne University of Technology Melbourne, Australia

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6 Declaration I, Brendan James Chick, declare that this thesis entitled : Characterisation of polarised supercontinuum generation and its focal field is my own work and has not been submitted previously, in whole or in part, in respect of any other academic award. Brendan James Chick Centre for Micro-Photonics Faculty of Engineering and Industrial Science Swinburne University of Technology Australia Dated this day, June 3, 21 i

7 Abstract Since the first investigation of supercontinuum generation in microstructured optical fibre almost a decade ago, an enormous interest has developed in its application. Supercontinuum generation, the construction of broadband light from nonlinear and dispersive optical processes is a unique type of radiation that has the design functionality to enhance a broad range of applications. The temporal and spectral characteristics of a supercontinuum make it an ideal source in microscopy, as these features can provide a means to simultaneously optically image with different carrier frequencies or simultaneously optical record in a spectrally selective storage medium. These applications all involve the diffraction and interference of the supercontinuum field and what needs to be understood is how such a field behaves under these conditions. The investigation in this thesis identifies the supercontinuum characteristics which are important to the diffraction by a lens and how these characteristics will affect the measurement of the optical properties in microscopic applications. To achieve this goal there are two major areas of investigation; supercontinuum generation and optical diffraction theory. A theoretical and experimental investigation into supercontinuum generation is first presented, which investigates the polarisation properties of supercontinuum generation in highly birefringent photonic crystal fibre with two zero dispersion wavelengths. It is shown that the polarisation state of the incident ultrashort optical pulse maintains its polarisation state as it propagates through the optical fibre. The temporal and spectral properties of the principal axes are determined not only by the phase mismatch and the group velocity mismatch between the two fundamental linear polarised modes, but are affected by the different higher order dispersion coefficients. The balance between i

8 the nonlinearity induced by the Kerr effect and the second order dispersion initiates the formation of a high order soliton which then shifts spectrally toward the infrared frequencies. This formation sets the condition for the emission of dispersive waves which shifts toward the lower visible frequencies. However, the dispersion parameters associated with the two fundamental modes produces different high order solitons and phase matching conditions, which determine the wave-numbers for the dispersive waves. The larger of the two dispersion terms enlarges the initial compression of the ultrashort pulse creating a high order soliton with a significantly smaller temporal width, which under conditions of Raman scattering the shift of the soliton is further. Experimentally, it is confirmed that the two fundamental modes of the photonic crystal fibre have different spectral and temporal features. The degree of polarisation also confirms that the supercontinuum spectrum is highly polarised with the degradation attributed to the depolarisation caused by the objective lens. The processes of nonlinearity and dispersion act as phase shifts onto an ultrashort pulse. When superimposed through the diffraction by a lens of low numerical aperture, the temporal phase associated with the field couples with the spatial phase incurred by the lens. This coupling changes the way the field correlates, which is analysed through the degree of coherence of the field. Fluctuations occur in the temporal coherence of the field because of enlarged variations in spatial phase, which are associated with the conditions of destructive interference, which imposes zero intensity locations in the focal region of the lens. These variations are quantified through the coherence time of the field and is most dramatic for a nonstationary observation frame which is affected by the path difference between the rays at the extremities of the lens and the rays along the optical axis. The significant phase contribution that affects the temporal coherence of the SC field is the initial formation of the high order soliton. The compression of the ultrashort pulse and the formation of the high order soliton increases the bandwidth of the field altering the coherence time. After this point in the evolution the coherence is constructed by the interference from dispersive waves and the fission of the high order solitary waves. The two dominant processes which influence the temporal coherence in the focal region are the third order dispersion effect and ii

9 the Raman scattering. However, the interference of the temporal phase from these effects and the other higher order dispersion and third order nonlinear effects couple with the spatial phase from the diffraction by the lens increasing the complexity of the degree of coherence. Specifically, the coherence time in the case of a nonstationary observation frame can be enhanced by a factor of 3 and occurs at the zero intensity locations within the focal region. Furthermore, it is shown that such an enhancement in the degree of coherence can be controlled by the pulse evolution through the photonic crystal fibre, in which nonlinear and dispersive effects such as the soliton fission process provides the key phase evolution necessary for dramatically changing the coherence time of the focused electromagnetic wave. An extension to this theory can be developed by an investigation into vectorial effects in polarisation, which are achieved through vectorial diffraction theory. This theoretical treatment gives insight into the coherence fluctuations introduced by a supercontinuum in a high numerical diffraction system. Under such conditions and due to the increased refraction at the extremities of the lens the incident polarisation state rotates to transfer energy from this state to the orthogonal transverse field and the longitudinal field, which is known as depolarisation. For a supercontinuum with a horizontal polarisation state the coherence times along the x, y and z axes are different and change with increased numerical aperture. The coherence time for the x axis increases with numerical aperture and the y axis decreases with numerical aperture, which is due to the transfer of energy because of depolarisation. The influence of numerical aperture is evident along the optical axis (z), which shows the most significant change in coherence time. The mean coherence time as a function of numerical aperture decreases by an order of magnitude and is due to the superposition conditions no longer forming points of destructive interference. Since the field is a vector field containing three polarisation components, the theory for the degree of coherence is extended to incorporate cross correlation effects within these vectorial components which is calculated through a coherency matrix. The use of this matrix provides insight into interesting correlation effects between co-propagating vector fields such as the coupled modes (linear polarised modes) iii

10 of a photonic crystal fibre. An investigation is presented on the coherence times for the supercontinuum field generated by cross coupling into the photonic crystal fibre. The coherence times under cross coupling conditions show that the degree of coherence of the two coupled modes from the fibre are different, which is due to the difference in phase associate with each mode. The effect of temporal phase from a supercontinuum and the spatial phase inherent from diffraction by a lens, are important to many experimental applications of supercontinuum generation. The manifestation of these temporal and spatial phase effects result in a modification of the focal region and the bandwidth of the field. Applications involving supercontinuum generation must first understand the generation of the supercontinuum and the modification imposed by the optical system. iv

11 Acknowledgements At the start of 26 I was given the opportunity to enrol in a PhD at Swinburne University or the University of Newcastle. At the time, I think the major reason for coming to Swinburne was my supervisor Dr. John Holdsworth. However, John made it clear to me that coming to Swinburne and working for Prof. Min Gu would not be easy, but then I suppose a PhD is never simple. First I would like to thank my three supervisors Prof. Min Gu, Dr. James Chon and Prof. Richard Evans for all the effort that they have put into my research development. Prof. Min Gu s tireless contribution has developed my ability to conduct and convey scientific research in a professional manner. I thank Dr. James Chon for imparting his valuable guidance and knowledge throughout my research. To Prof. Richard Evans for scientific contribution in the initial stages and the difficulties of my PhD. A PhD would never run smoothly without the help of administrative staff and technicians. I would like to thank Ms. Johanna Lamborn and Ms. Katherine Cage with all the administrative issues associated with my research. I thank Mr. Mark Kivinen for all the custom made opto-mechanics and the enlightening conversations every now and then of a morning. I would like to thank Dr. Daniel Day and Dr. Dru Morrish for their endless support and for imparting their scientific knowledge. During my PhD I have gained a valuable group of colleagues but within that a valuable group of friends. I would like to thank both Dr. Peter Zijlstra and Dr. Michael Ventura for their scientific contributions in knowing what to do and more importantly what not do during my PhD. I would also like to thank Ms. Elisa Nicoletti and Dr. Joel Van Embden for their extensive insight, which may v

12 not always be scientific but has contributed. To my colleagues in the Centre for Micro-Photonics I would like to thank you all for providing a constructive and open environment to conduct research. Finally, I would like to thank my family and friends. To my parents for their encouragement and their guidance throughout my PhD. Thanks go to my twin brother Joel for helping me through my PhD and for finishing your PhD before me. Most importantly I would like to thank my best friend and partner, Skye. I could not have done my PhD without you. Thank you for understanding my complicated mind and for all your support. Brendan James Chick June 3, 21 vi

13 Contents Declaration i Abstract i Acknowledgements v Contents vii List of Figures x List of Tables xviii 1 Introductory Literature Review Introduction to Supercontinuum Generation Nonlinear photonic crystal fibre Dispersion Birefringence Nonlinearity Self phase modulation (SPM) Cross phase modulation (XPM) Self steepening Raman scattering The nonlinear Schrödinger equation Discussion Introduction to Diffraction Theory Fresnel Diffraction Vectorial Diffraction Applications vii

14 1.4 This thesis Theory Introduction Nonlinear pulse propagation Maxwell s equations Slow varying envelope equation Optical properties of photonic crystal fibre Nonlinear Schrödinger equation Coupled mode nonlinear Schrödinger equation Diffraction theory: low numerical aperture Introduction Huygen-Fresnel principle Fresnel approximation Fresnel diffraction by a circular lens Diffraction theory: high numerical aperture The Debye integral Evaluation of the vectorial diffraction formula Coherence Pulse Propagation in Nonlinear Photonic Crystal Fibre Introduction Photonic crystal fibre characteristics Nonlinear and dispersion effects Supercontinuum generation Experimental study Conclusion Fresnel Diffraction Introduction Numerical methodology Ultrashort hyperbolic secant pulse Nonlinear and dispersive phase Supercontinuum generation viii

15 4.6 Conclusion Vectorial Diffraction Introduction Three-dimesional coherence matrix Vectorial diffraction of a supercontinuum Linear Polarisation Coupled mode propagation Conclusions Conclusion Thesis conclusion Future work Bibliography 17 Appendix A: Numerical Code for the CMNLS A 1 A.1 Split step Fourier method A 1 A.2 Matlab Script A 2 Appendix B: Numerical Code for Diffraction Theory B 1 B.3 Diffraction theory B 1 B.4 Matlab Script - Scalar diffraction B 1 B.5 Matlab Script - Scalar diffraction B 2 Appendix C: Supplementary movies C 1 Author s Publications ix

16 List of Figures Illustration depicting the structure of a PCF. Λ is the pitch or periodicity, d is the hole diameter and φ is the core diameter A comparison between the effective refractive index of two PCF s with different d to Λ ratios and the material dispersion of fused silica. The parameters in the calculation were d =.6 µm and Λ = 1.2 µm (dashed line), d = 1 µm and Λ = 1.2 µm (dot dashed line) A comparison between the dispersion of two PCF s with different d to Λ ratios and the material dispersion of fused silica. The parameters in the calculation were d =.6 µm and Λ = 1.2 µm (dashed line), d = 1 µm and Λ = 1.2 µm (dot dashed line) The effect of β 2 dispersion on an optical pulse after propagation for 1 m The effect of β 3 on an ultrashort pulse propagating along a 3 m PCF. Except for where specified the coefficient β 3 = ps 3 /m Illustration depicting the introduction of birefringence into a PCF structure Raman response function modelled using Eqs. (1.1.8), (1.1.9) and (1.1.1). (a) Raman transfer function, (b) The real (R(ω)) and imaginary (S(ω)) components of the Raman response function in the frequency domain Diffraction by a lens of NA =.1 for an incident wavefront with λ =.78 µm. S/S is the normalised intensity, z and r are the axial and radial dimensions, respectively The geometric illistration of vectorial diffraction 53 of a incident electric field (Ex) i in the x direction. Er i and Eφ i are the polar components of Ex. i x

17 1.2.3 Diffraction by a lens of NA = 1 for an incident wavefront with λ =.78 µm. (a) the xy plane; (b) the xz plane and (c) the yz plane The formation of the third order soliton. The parameters in the simulation were (a) β 2 =.5 ps 2 /m and γ =.95 W/m; (b) β 2 =.5 ps 2 /m, β 3 = ps 3 /m and γ =.95 W/m; and (c) β 2 =.5 ps 2 /m, γ =.95 W/m and R (t) determined by Eq. (1.1.7). All other terms were neglected Ultrashort ( t =.5 ps) pulse propagation using the CMNLSE. (a) y polarised mode and (b) the x polarised mode. The parameters used in the simulation were β j2 =.5 ps 2 /m, β k2 =.5 ps 2 /m, γ =.95 W/m and a β 1 = ps/m Ultrashort ( t =.5 ps) pulse propagation using the CMNLSE. (a) y polarised mode and (b) the x polarised mode. The parameters used in the simulation were β j2 =.5 ps 2 /m, β k2 =.5 ps 2 /m, γ =.95 W/m and a β 1 = 2 ps/m Illustration of mutual interference caused by the superposition of the primary wavefront and secondary spherical waves Illustration of the geometry of vectorial diffraction The geometry as defined in the simulation using a refractive index profile resolution for the PCF of pixels and a supercell size 1 1 unit cells The dispersion coefficients related to the mode propagation constant β. (a) shows the first- and second-order dispersion coefficients for the two fundamental modes. (b) shows the phase mismatch ( β ) and the group velocity mismatch ( β 1 ) between these modes The effects of TOD originating from a PCF pumped with an ultrashort pulse with a pulse duration of.1 ps and a peak power of 1 W. (a) the time domain and (b) the frequency domain The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a power of 1 W. (a) the time domain and (b) the frequency domain xi

18 3.3.3 The effects of TOD originating from a PCF pumped with an ultrashort pulse with a pulse duration of.1 ps and in each mode of power of 75 W. The coupled polarisation state was 45. (a) the time domain of the y polarised mode (b) the frequency domain of y polarised mode (c) the time domain of x polarised mode and (d) the frequency domain of x polarised mode The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and in each mode of power of 75 W. The coupled polarisation state was 45. (a) the time domain of the y polarised mode (b) the frequency domain of y polarised mode (c) the time domain of x polarised mode and (d) the frequency domain of x polarised mode The effects of TOD originating from a PCF pumped with an ultrashort pulse with a pulse duration of.1 ps and a fibre length of.3 m. (a) the time domain and (b) the frequency domain The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a fibre length of.3 m. (a) the time domain and (b) the frequency domain The effects of TOD originating from a PCF pumped with an ultrashort pulse with a pulse duration of.1 ps and a fibre length of.3 m. The coupled polarisation state was 45. (a) the time domain of the y polarised mode (b) the frequency domain of y polarised mode (c) the time domain of x polarised mode and (d) the frequency domain of x polarised mode The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a fibre length of.3 m. The coupled polarisation state was 45. (a) the time domain of the y polarised mode (b) the frequency domain of y polarised mode (c) the time domain of x polarised mode and (d) the frequency domain of x polarised mode xii

19 3.4.1 Theoretically obtained spectra of propagation within a 13 mm NL- PCF with a 87 fs pulse. Figures (a), (b) and (c) are the spectra for the y polarised output field with (d), (e) and (f) for the x polarised output field. θ is the input polarisation angle with respect to the y axis Modulation instability gain for the y and the x modes Theoretically obtained spectral and temporal profile of 87 fs pulsed propagation within a 13 mm nonlinear PCF. Figures (a), (b) and (c) are the spectra for the y polarised output field with (d), (e) and (f) for the x polarised output field Optical arrangement used in this study. GT - Glan Tomson, WP - Wave Plate, Spec - Spectrograph and SA - Spectrum Anaylser Spectral properties of the polarised modes of the nonlinear PCF. The perpendicular (blue) and parallel polarised (red) states are with reference to the output orientation of the laser Degree of polarisation for the fast and the slow axes of the fibre An illustration of pulse diffraction by a low numerical aperture (NA) lens. (a) shows how the path length and the NA affect the pulse distribution as the temporal envelope passes through the focus. (b) shows the observation frames of the intensity profile in the focus The temporal effects of a focused hyperbolic secant ultrashort pulse propagating through the focus of a low NA (.1) objective. (a) On axis diffraction centred at the focal point (the full temporal evolution of the hyperbolic secant on the axis is described in Appendix C). (b) On axis diffraction centred at u = 5π. (c) Radial and axial diffraction pattern centred at the focal point (the full temporal evolution of the hyperbolic in the radial and axial direction is described in Appendix C). (d) The intensity matrix used to obtain the temporal and axial intensity information for the stationary and nonstationary observation frames xiii

20 4.3.2 The coherence time of a focused hyperbolic secant ultrashort pulse for the stationary and the non-stationary cases. (a) Axial and radial distribution of the coherence time for the.1 NA lens for the stationary case; (b) Axial and radial distribution of the coherence time for the.1 NA lens for the non-stationary case; (c) Effect of NA on the coherence time on the axis for the stationary case; and (d) Effect of NA on the coherence time on the axis for the non-stationary case The coherence time illustrating the effect of the variation in temporal phase through the addition of chirp through the chirp parameter C(ps 2 ) for the stationary (a) and nonstationary (b) observation frames. τ is the initial coherence time before the objective Mean frequency distribution of a focused hyperbolic secant ultrashort pulse in the axial and radial plane of a.1 NA lens for stationary (a) and non-stationary (b) cases The effects of RS on the coherence time for a focused electromagnetic wave by a lens of NA =.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a power of 1 W (relating to the field in Fig ). (a) stationary observation frame and (b) a nonstationary observation frame The effects of TOD on the coherence time for a focused electromagnetic wave by a lens of NA =.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a power of 1 W (relating to the field in Fig ). (a) stationary observation frame and (b) a nonstationary observation frame The effects of RS on the coherence time for a focused electromagnetic wave by a lens of NA =.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a fibre length of.3 m (relating to the field in Fig ). (a) stationary observation frame and (b) a nonstationary observation frame xiv

21 4.4.4 The effects of TOD on the coherence time for a focused electromagnetic wave by a lens of NA =.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a fibre length of.3 m (relating to the field in Fig ). (a) stationary observation frame and (b) a nonstationary observation frame The temporal effects of a SC propagating through the focus of a low NA (.1) objective. (a) On axis diffraction centred at the focal point (the full temporal evolution of the SC on the axis is described in Appendix C). (b) On axis diffraction centred at u = 5π. (c) Radial and axial diffraction pattern centred at the focal point (the full temporal evolution of the SC in the radial and axial direction is described in Appendix C). (d) Complete axial and temporal diffraction field The coherence time within a focused SC for the stationary and the non-stationary cases. (a) the axial and radial distribution of the coherence time for the.1 NA lens for the stationary case; (b) the axial and radial distribution of the coherence time for the.1 NA lens for the non-stationary case; (c) the effect of NA on the coherence time on the axis for the stationary case; (d) the effect of NA on the coherence time on the axis for the non-stationary case Propagation of an ultrashort hyperbolic secant pulse through a nonlinear PCF. (a) field propagation as a function of fibre length; (b) coherence time for the stationary observation frame in the focal region of a.1 NA lens for different length fibre and (c) coherence time for the nonstationary observation frame in the focal region of a.1 NA lens for different length fibre. The peak input power to the photonic crystal fibre is 25 W with a pulse duration of 1 fs.(1) represents the cross section used for Fig c (blue) and (2) represents the cross section used for Fig d (blue) xv

22 4.5.4 Propagation of an ultrashort hyperbolic secant pulse through a nonlinear PCF. (a) variation of output temporal envelope by varying the input power. (b) the coherence time of the stationary observation frame of the focal region of a.1 NA lens for different for the field obtained from different input powers. (c) the coherence time of the nonstationary observation frame of the focal region of a.1 NA lens for different for the field obtained from different input powers Mean frequency distribution of the focused SC in the axial and radial plane of a.1 NA lens for stationary (a) and non-stationary (b) cases A comparison between the coherence times for a lens of NA = 1 and.1 with hyperbolic secant ultrashort pulse with a width of.1 ps The coherence time of the diffraction by a lens of varying numerical aperture along the x (a), y (b) and z (c) axes. These coherence times are calculated for the autocorrelation of the electric field in the direction of the E i (E x ). (d) the coherence times for the diffraction by a lens of NA = 1 along the x axis, which contains the autocorrelation and cross-correlation coherence times with respect to the E x and E z fields The mean coherence time of a SC as a function of NA for the x, y and z axes The coherence time of the autocorrelation of the diffraction by a lens of NA = 1 the electric field E x with variation in the fibre length The power dependence of coherence time in the focus of a NA = 1 lens for input fields generated by the nonlinear PCF of varying input power. The coherence time is for a linear polarised field orientated along the x direction The PCF output field for an incident polarisation state at 45.(a) the horizontal (x) polarisation state, (b) the vertical (y) polarisation state, (c) the horizontal (x) polarisation state as a function of fibre length, and (c) the vertical (y) polarisation state as a function of fibre length xvi

23 5.3.6 The coherence time for the autocorrelations and cross correlations calculated for the diffraction by a lens of NA = 1 along the optical axis for the SC field generated in Fig The coherence time for the autocorrelations and cross correlations calculated for the diffraction by a lens of NA = 1 as a function of fibre length along the optical axis. (a) coherence time produced by the autocorrelation of E x ; (b) coherence time produced by the cross correlation E x and E y ; and (c) coherence time produced by the autocorrelation of E y The power dependence of coherence time in the focus of a NA = 1 lens for input fields generated by the nonlinear PCF of varying input power. The coherence matrix is for a linear polarised field orientated at 45 to the x direction. (a) coherence time produced by the autocorrelation of E x ; (b) coherence time produced by the cross correlation E x and E y ; and (c) coherence time produced by the autocorrelation of E y xvii

24 List of Tables Parameters for the intermediate broadening model Dispersion data for the polarised mode of the nonlinear fibre Contributions to the field E for the x y and z axes xviii

25 Chapter 1 Introductory Literature Review At the forefront in the development of optical devices is the ability to generate, propagate and process radiation, which is highlighted in the result of last years Nobel prize with Kao awarded half the prize for his work in fibre optic communications. Kao showed that by producing a high glass purity, fibre losses could be minimised and allow the propagation of light to approach 1 km s. 1,2 These general principles of generation, propagation and processing in applying radiation can be seen in all photonic applications. The advancement made by Kao could be regarded as the foundation for the motivation of current fibre optic technology. The more recent technological advancement in optical fibre technology has led to the use of a photonic crystal structure to enhance the fibre guidance properties which leads to a modification of the optical properties such as dispersion. The idea in the use of the photonic crystal structure in the cladding of the fibre, known as a photonic crystal fibre (PCF) 3 has led to many changes in optical properties and in essence is similar to Kao in providing the ability to enhance propagation. The PCF has led to the improvement and generation of many optical applications and none more fascinating than the supercontinuum (SC). SC generation is a remarkable source of radiation formed by the construction of broadband light from the severity of nonlinear optical processes, which has extraordinary temporal and spectral features. SC generation is an ideal source 1

26 Chapter 1 for many applications and it provides two key photonic capabilities; optical imaging using many microscopy techniques simultaneously for biophotonic and biomedical applications; optical processing information on a tremendous scale since its bandwidth could provide multiple channels to carry information, leading to the realisation of all-optical computing. The realisation of the applications described involve the propagation and control of light on the microscale and a method for the control is the use of diffraction. For SC applications predominantly involving diffraction and interference, it is imperative that a fundamental treatment should be constructed in these optical phenomena. This chapter is divided into two major sections: an introductory literature review of SC generation and an introductory literature review of diffraction by a lens. The review on SC generation highlights the optical properties, which generates its spectral and temporal characteristics and makes SC generation a desirable form of radiation. The review is then turned toward diffraction to present the reasons why diffraction has been a heavily investigated optical phenomenon. Finally, we highlight the necessity for a fundamental study into the diffraction of complex light such as a SC. 1.1 Introduction to Supercontinuum Generation SC generation in microstructured optical fibre, also known as PCF 4 6 has become an imperative instrument for many applications SC is generated in PCF by the interplay between nonlinearity induced by the Kerr effect originating from the tight modal confinement and tailored modal dispersion. SC generation caused by this engineered dispersion and nonlinearity can be easily demonstrated with high power continuous wave lasers or pulsed laser systems. This section provides an overview of the design of PCFs and its particular optical properties which generate the SC. The discussion involves the dispersion properties which are important for polarisation maintaining PCFs, and the nonlinear properties such as self phase modulation, cross phase modulation, self steepening and Raman 2

27 Chapter 1 Λ d φ Figure Illustration depicting the structure of a PCF. Λ is the pitch or periodicity, d is the hole diameter and φ is the core diameter. scattering Nonlinear photonic crystal fibre PCFs are a type of optical fibre which have a periodic micro-structure of air holes within the cladding to reduce the refractive index. The design freedom of PCFs allow the tuning of the refractive index in the cladding, which alters the effective refractive index of the guided mode changing its dispersion profile. The basic geometry of a PCF is shown in Fig A defect is placed at the centre of the crystal to form a waveguide which guides under the conditions of total internal reflection. To determine the optical characteristics of the PCF structure, the mode profile can be solved using algorithms such as the finite difference method 13, plane wave theory or the multipole method By solving for the mode, the solution gives information about the effective refractive index and the effective modal area, which is used to determine the dispersion and nonlinear characteristics, respectively. Three structural properties of PCFs allow the modification of dispersion and 3

28 Chapter 1 nonlinearity; the air-fill fraction, a one-dimensional asymmetry and the core size. Dispersion engineering by the manipulation of the air hole structure of a PCF allows the guidance of the optical wave to be influenced by different dispersion effects. The development of PCFs has become widely used due to the pioneering work by Russell and co-workers in The results showed a single mode core optical fibre could be achieved for a wavelength region of nm. In 1997 Birks et al. extended their work to create the endlessly single mode PCF. 21 Furthermore Mogilevtsev et al. 24 showed that by tuning the PC structure the group velocity could be altered to shift the zero dispersion wavelength (ZDW) to below 1.27 µm. 24 These developments have led to an extensive range of PCFs with different guidance properties. The SC in PCFs was first demonstrated in 2 by Ranka et al. 5 Experimentally it was presented that the modification of the photonic crystal lattice can shift the dispersion profile into the visible wavelength region, creating the ability to couple an ultrashort laser pulse within the anomalous dispersion region (the increase in refractive index with wavelength). By doing so, this coupling enhances the nonlinear effects which were relatively weak in the standard silica fibres. This research has led to extensive studies on how the PCF structure influences the generation of a SC both experimentally and theoretically Dispersion The dispersion properties of a PCF are due to two components; the material and the modal properties. The effect of material dispersion comes from the material s frequency dependent refractive index and is well approximated using the Sellmier equation, 25 which is given by 3 n 2 (ω) = 1 + ω 2 j=1 j B j ωj 2 (1.1.1) ω2, where ω is the frequency, ω j and B j are the frequency and the oscillator strength of the j th material resonance. 4

29 Chapter n eff d/λ =.5 d/λ =.83 Fused Silica Wavelength (µm) Figure A comparison between the effective refractive index of two PCF s with different d to Λ ratios and the material dispersion of fused silica. The parameters in the calculation were d =.6 µm and Λ = 1.2 µm (dashed line), d = 1 µm and Λ = 1.2 µm (dot dashed line). The effective refractive index produced by the waveguide must be added to the material contribution to obtain the dispersion properties. For standard optical fibres the contribution from the waveguide effective refractive index is small. However for a PCF the contribution from the waveguide is increased by the addition of air holes in the cladding which reduces the effective refractive index. It is this property of PCFs that allows the tailoring of the dispersion profile by modification of the air hole diameter (d) and the pitch (Λ) of the photonic crystal. Figure shows the effect of changing the d to Λ ratio on the effective refractive index of a PCF and a comparison to the material refractive index of silica. The dispersion by an optical waveguide is determined by the mode propagation constant β(ω) which is calculated by β (ω) = n (ω) ω c, (1.1.2) where c is the speed of light. The two most dominant dispersion properties in the pulse propagation are the 2 nd and the 3 rd order dispersion effects which can be determined by expanding Eq. (1.1.2) into a high order Taylor series about the 5

30 Chapter 1 1 β 2 (ps 2 km 1 ) 1 d/λ =.5 d/λ =.83 Fused Silica Wavelength (µm) Figure A comparison between the dispersion of two PCF s with different d to Λ ratios and the material dispersion of fused silica. The parameters in the calculation were d =.6 µm and Λ = 1.2 µm (dashed line), d = 1 µm and Λ = 1.2 µm (dot dashed line). carrier frequency ω, which is given by 25 β j (ω) = β j + β 1j (ω ω ) + β 2j 2! (ω ω ) 2 + β 3j 3! (ω ω ) 3, (1.1.3) where j refers to the particular mode. An important parameter for the generation of a SC is the second order dispersion term (β 2 ) as it determines the regime the pulse propagates within. For the case of β 2 > the dispersion is in the normal dispersion region where the red frequencies travel faster then the blue frequencies, and for the regime where β 2 < the dispersion is in the anomalous dispersion region. The dispersion profile, in particular the slope of β 2, can be tailored by changing the PCF structure. Figure shows a comparison between the β 2 profiles for two PCFs and the material dispersion of fused silica. The balance of nonlinearity and dispersion, in particular the second order term determines the pulse evolution and ultimately the effects which govern SC generation. The β 2 term acts to broaden or compress an optical pulse depending which regime the pulse is propagating within. Figure shows the broadening of a.1 ps ultrashort pulse when affected by varying amounts of β 2. 6

31 Chapter 1 Intensity (W) β 2 = (ps 2 /m) β 2 =.5 (ps 2 /m) β 2 =.1 (ps 2 /m) Time (ps) Figure The effect of β 2 dispersion on an optical pulse after propagation for 1 m. The third order dispersion term β 3 is an important parameter as it provides the dominant dispersive component when considering the optical pulses in the femtosecond regime and pumping near the ZDW. Figure shows the effects of β 3 in the absence and presence of β 2. This term places an asymmetric phase shift on the pulse and becomes increasingly dominant with fibre length Birefringence All optical fibres, whether they are single-mode or multi-mode support two orthogonal linearly polarised (LP) modes for the same spatial modal distribution, which is called birefringence. For standard single mode fibre the difference between these LP modes is small and they are said to exhibit weak birefringence. A pivotal development in SC generation was the use of highly birefringent PCF. When an asymmetry is incorporated into an optical fibre, there is a difference between the modal effective refractive indices for the two LP modes. The difference in the modes can be exploited to maintain the incident polarisation direction. In 25, Zhu and Brown presented experimental and theoretical studies 26,27 on the polarisation properties of SC generation in a birefringent PCF, or also known as polarisation maintaining PCF. It was shown that the polarisation state could be maintained for polarised pulsed coupling aligned with a birefrigent axis of the fibre and was 7

32 Chapter 1 Intensity (W) β 2 =,β 3 = β 2 = β 2 =.2 (ps 2 /m) Time (ps) Figure The effect of β 3 on an ultrashort pulse propagating along a 3 m PCF. Except for where specified the coefficient β 3 = ps 3 /m. otherwise elliptically polarised. The birefringence of an optical fibre can be calculated by where x and y represent the two LP modes. B = n x eff n y eff = β j β k k, (1.1.4) The difference between these two polarised modes can be enhanced by the inclusion of an asymmetry in the photonic crystal structure. This is achieved by modifying two holes either side of the core region or by making all the holes of the photonic crystal slightly elliptical which is illustrated in Fig Mathematically, the birefringence of the optical wave guide creates a mismatch between the two polarised fundamental modes propagation constants, which is given by β (ω) = β + β 1 (ω ω ) + β 2 2! (ω ω ) 2 + β 3 3! (ω ω ) 3, (1.1.5) where the and 1 st order coefficients represent the phase mismatch and group velocity mismatch, respectively. 8

33 Chapter 1 Λ Λ d d φ φ Figure Illustration depicting the introduction of birefringence into a PCF structure Nonlinearity Along with the guidance properties of modal dispersion, nonlinearity plays an important role in the generation of SC spectra. In 1999, Broderick et al. reported that the effective modal area enhances the effective nonlinearity of an optical fibre. 28 This enhancement leads to a range of nonlinear processes which combine with dispersion to form the complex temporal spectral features that are seen in a SC today. The nonlinearity in a PCF is due to the Kerr effect, which is the refractive indices dependence on intensity and is determined by the following equation n (ω) = n (ω) + n 2 (ω)i, (1.1.6) where n (ω) and n 2 (ω) are the linear and nonlinear refractive indices, respectively and I is the intensity. To induce a nonlinear response from bulk silica, high peak intensity laser sources are required. There exists a temporal and spatial requirement for the confinement of the propagating light to be able to reach such high intensities. In PCF, this can be achieved by reducing the core size of the structure. The nonlinear response is quantified through the nonlinear coefficient γ which is given by 28,29 γ = n 2 (ω )ω ca eff (ω ), (1.1.7) 9

34 Chapter 1 where A eff (ω ) is the effective modal area of the PCF. From a numerical simulation perspective, the nonlinear coefficient controls the strength of nonlinearity achievable within the PCF. Optical nonlinearity is composed of an expanse of effects, with the most relevant of these processes being self phase modulation (SPM), cross phase modulation (XPM), self steeping and stimulated Raman scattering Self phase modulation (SPM) An increase in the intensity modulates the refractive index, which changes the phase on the field and is known as SPM. The first report of SPM in optical fibres was by Stolen in ; he showed that with a single-mode silica fibre the pulse from a mode-locked argon laser could produce defined frequency shifts, which were in good agreement with theoretical predictions of SPM. SPM is an important process in SC generation as it balances the chirp from β 2 which assists in the formation of non dispersive waves known as optical solitons. The formation and optical properties of solitons have been studied extensively in optical communications as they provide a means of sending information unperturbed by dispersion Cross phase modulation (XPM) As described earlier (Section 1.1.3) there exists two fundamental LP modes in a PCF. When light is coupled into a PCF in such a way that a portion of the power is propagating in both LP modes, power from one mode can transfer to the other LP mode which modulates the effective refractive index of that mode. This effect is called XPM which is an important process as it can be used as a means of introducing gain to a co-propagating wave and is not necessarily restricted to the two fundamental LP modes in the case of multimode propagation. However, XPM in the situation of low birefringent PCF can degrade the degree of polarisation of a mode. The first report of XPM in silica fibre was shown by Chraplyvy and Stone in and a theoretical description was described by Agrawal. G.P. in

35 Chapter Self steepening In the femtosecond regime third order nonlinear effects must be taken into account. The two major third order nonlinear effects that induce phase shifts onto an ultrashort pulse are self steepening and stimulated Raman scattering. Self steepening is an asymmetric nonlinear phase shift which acts upon the effects of SPM and XPM and was first described by De Martini et al. in Since the speed of the optical pulse varies across its envelope due to the intensity dependent refractive index, the trailing edge of the pulse steepens as it catches up with the peak component of the pulse. The effects seen from self steepening is a gradual asymmetric temporal shift across the pulse caused by the temporal gradient of the intensity of the pulse Raman scattering Raman scattering is the process where photons of a given energy are scattered by a molecule and form a photon of lower energy. The process of Raman scattering is an important optical property as it can transfer energy from a pump beam to a Stokes beam. For silica the molecular vibration levels span a wide frequency range and therefore the gain in the stokes beam occurs over a wide frequency range. The Raman gain for fused silica has been extensively studied with the most significant contributions to the field produced by Stolen et al. in 1989 on the Raman response function in silica fibres 33 and a theoretical description in optical fibres by Blow and Wood in When simulating pulse propagation in silica optical fibres either the experimentally measured response function or the theoretical approximate response functions are used. The theoretical approximation used in this thesis was developed by Hollenbeck and Cantrell 35 which uses a multiple-vibration-mode model and is repeated here for completeness. The Raman response function using the intermediate broadening model for silica optical fibres is given by 11

36 Chapter 1 a Amplitude (A.U.) b Amplitude (A.U.) S(ω) R(ω) Time (ps) Frequency shift (THz) Figure Raman response function modelled using Eqs. (1.1.8), (1.1.9) and (1.1.1). (a) Raman transfer function, (b) The real (R(ω)) and imaginary (S(ω)) components of the Raman response function in the frequency domain. h R (t) = 13 i=1 A i ω v,i e iγ it e Γ2 i t2 /4 sin (ω v,i t)θ(t), (1.1.8) where the parameters related to the intermediate broadening model are shown in Table and Θ (t) is a unit step function. The Raman response function is shown in Fig The Fourier transform of the Raman response function determines the real (R(ω)) and imaginary (S(ω)) frequency components, where the imaginary component is related to the Raman gain bandwidth. These two components are shown in Fig b and are described by the following equations 35 S (ω) = R (ω) = 13 l=1 13 l=1 A l 2 A l 2 (cos [(ω v,l ω)t] cos [(ω v,l + ω)t])e iγ lt e Γ2 l t2 /4 dt, (1.1.9) (sin [(ω v,l ω)t] + sin [(ω v,l + ω)t]) e iγ lt e Γ2 l t2 /4 dt. (1.1.1) The nonlinear Schrödinger equation The theoretical description of pulse propagation has been understood for decades and begins with the formulation of Maxwell s equations to form what is called the nonlinear Schrödinger equation (NLSE). Pulse propagation in fibre waveguides owes 12

37 Chapter 1 Table Parameters for the intermediate broadening model 35 Mode A i ω v,l γ i Γ i # [TRads 1 ] [TRads 1 ] [TRads 1 ] A i is the peak intensity of the mode, ω v,l is the frequency of the component position, Γ i is the Gaussian full width at half maximum of the mode and γ i is the Lorentzian full width at half maximum of the mode 13

38 Chapter 1 its development to Agrawal as his texts 25 have laid the foundation to the modern interpretation of such formulation. The initial use of the NLSE for SC generation was investigated by Husakou and Herrmann in the develop of an understanding of soliton fission dynamics leading to SC generation The formulation used in these studies neglects Raman scattering, which as shown later, is a dominant nonlinear process which influences the pulse propagation leading to SC generation. The most significant contribution to the theoretical description of the NLSE was developed by Kodama and Hasegawa. 39 This derivation incorporated higher order dispersion effects and nonlinar effects such as Raman scattering. A significant contribution by Blow and Wood in 1989 was the description of a wave equation for the modelling of transient stimulated Raman scattering. 34 The model also incorporated a method of numerical integration for the nonlinear response which has become commonplace. Although the mathematical treatment as described in the nonlinear integration formulation has been misprinted, this description has formed a basis for the theoretical treatment of pulse propagation in optical fibres (a correction to this formulation is provided by Cristiani et al. in 24 4 ). A more accurate form of the wave equation incorporating stimulated Raman scattering was provided later by Mamyshev and Chernikov in The validity of the NLSE is limited since the theory involves a slow varying envelope approximation (Section 2.2.2). The limit of this theory occurs as the pulse width approaches the oscillation period of the carrier wave (sub-cycle regime). Extensions to the slow varying envelope equation have been derived to extend the NLSE into sub-cycle regime. 42,43 The theoretical description in this thesis is based on the derivation provided by Agrawal 25 in Chapter 2 with the description of stimulated Raman scattering as provided by Hollenbeck and Cantrell in Section Discussion As described in the preceding sections, SC generation is the formation of broadband light from the involvement of all the discussed optical processes. In the anomalous 14

39 Chapter 1 dispersion regime, initially in the evolution of the ultrashort pulse along the PCF, the phase shift incurred is a balance between the chirp from the influence of β 2 and the chirp associated with SPM. The pulse compresses and forms a high order soliton. Throughout the initial evolution, the soliton accumulates additional phase and expands spectrally. The ability to control the propagation of an ultrashort pulse was first investigated in PCFs by Reeves et al. 23 It was shown that by coupling a pulse into a PCF within the anomalous dispersion regime that the effects from β 2 could be balanced by the chirp associated with self phase modulation (SPM). The key parameter which determines the wavelength of the coupled laser pulse is the ZDW. The extent of SC generation is enhanced in the anomalous dispersion regime, where the carrier frequency of the coupled light is close to the ZDW. In this region four wave mixing is the strongest causing strong Stokes and Anti-Stokes frequencies. 44 In the femtosecond regime the influence of third order dispersion is the dominant broadening mechanism and a high degree of curvature in dispersion is required to enhance the third order term. 45 Since the geometry of fibres with two ZDWs inherent these characteristics, they are of particular interest in generating extensive SC spectra. At a particular point along the fibre, the phase accumulation makes the high order soliton unstable and begins fissions into stable fundamental solitons. Throughout this process the soliton sheds energy into dispersive waves which form because the soliton is phase matched to the wave vector of the dispersive waves. Since the intensity is now spread over several fundamental solitons which are temporally independent, the effect of nonlinearity becomes less profound and higher order dispersion effects begin to dominate. What forms is the red shifted radiation from Raman solitons and a blue shifted dispersive wave 46. Highly birefringent PCF has become an important area of investigation in SC generation as it provides a key insight into a number of dispersive and nonlinear processes. In considering birefringence the scalar description of the NLSE is no longer appropriate as vectorial effects caused by modal dispersion differences affect pulse propagation 47. The soliton formation and fission processes have been shown to 15

40 Chapter 1 be highly polarisation sensitive 25. By coupling an input pulse at polarisation angles other than the primary axes of the fibre, polarisation sensitive nonlinear effects can be enhanced which could give more insight into the dynamics of SC generation. The incorporation of PCFs with two ZDWs can enhance the spectral extent of the SC. The change in slope of the second order dispersion term effectively changes the influence of the third order dispersion, which has been shown by Gaeta to produce red shifted dispersive waves, 48 enhancing the SC in the near infra-red region. Experimentally, the effects of the two ZDWs have been investigated 44,49 51 and confirm the results presented by Gaeta. 1.2 Introduction to Diffraction Theory The origin of diffraction theory dates back centuries and its theoretical description is still debated. Diffraction theory can owe its development to the pioneering works of Huygens and Fresnel, who have constructed the basis for the optical wave theory. The diffraction of a wave occurs when light propagating through an aperture or around an obstruction changes its propagation direction. The effect seen at a particular distance from the aperture is the superposition of the incident electromagnetic waves causing interference. The theoretical description of diffraction is summarised into two forms, Fresnel diffraction and Fraunhofer diffraction also known as near-field and far-field diffraction, respectively. The diffraction of the electromagnetic field due to a lens is caused by refraction where the curvature and the refractive index of the lens converge or diverge the field. For a lens the diffraction is determined by Fresnel diffraction of a circular aperture. This theory works well for lens focusing under low NA conditions (lens systems with a long focal length), however for high NA (short focal length) there exists a transfer of energy from the incident polarisation state to the orthogonal transverse field and the longitudinal field component, which is known as depolarisation. This section serves as an introduction to Fresnel diffraction of a lens and the 16

41 Chapter 1 r (µm) z (µm) S/S (norm. 1log 1 ) Figure Diffraction by a lens of NA =.1 for an incident wavefront with λ =.78 µm. S/S is the normalised intensity, z and r are the axial and radial dimensions, respectively. extension for the high NA condition known as vectorial diffraction theory. Presented is a literature review of the research which has led to the currently accepted diffraction theory Fresnel Diffraction The diffraction of a polychromatic wave such as a SC wave is described by what is known as the Huygens-Fresnel principle 52 where the incoming wave produces a set of secondary wavelets which superimpose and mutually interfere to form an Airy pattern. Figure shows the diffraction of an electromagnetic wave and what can be seen is the diffraction shows zero intensity locations, known as singular points. The mathematical expression for the Fresnel diffraction by a lens, under the paraxial approximation is given by 52,53 E (r,z,ω) = iωna2 e ik z c a b E i (ω)j (k rnaρ)e 1 2 ik zna 2 ρ 2 ρdρ, (1.2.1) where E i is the incident electromagnetic field upon the lens, k is the free space wave number, z and r are the axial and radial dimensions, respectively. Using complex analysis it is evident that the equation has singular points at discrete positions in both the radial and axial directions. These points have been rigorously studied in 17

42 Chapter 1 the case of a generalised polychromatic wave and create spectral anomalies in the focal region 54,55. Gbur et al. showed that these singular points cause the spectrum of a polychromatic wave to red or blue shift. This occurs because the singular point exists in a discrete position but also changes with wavelength (frequency) implying that the focal region has localised regions of wavelength dependent singularities. However, these points are not limited to the focal region of a lens. They also occur in many other optical systems involving diffraction and interference 56,57. The effects of phase singularities have been rigorously studied by Berry 58 and provide the key initial knowledge of the effects of singularities (or Caustics). Experimentally the effect of the singularities on the focal region have been verified by Popescu and Dogariu in 22 59, using a Michelson interferometer deviced from a 2 2 fibre coupler. A comparison is made between the light in the reference arm and the test arm. The test arm comprises the focusing lens and a reflecting object, which in this case is a spherical mirror. This experiment is pivotal to the understanding of Fresnel diffraction under polychromatic wave illumination as it provides a direct means of verifying the anomalous behaviour previously theoretically described Vectorial Diffraction Vectorial diffraction was developed by Richards and Wolf in 1959 to describe the changes in the distribution of the focal region of a high NA lens. 6,61 The polarisation of a field in general can be described as containing three polarised vector field components relating to the spatial dimensions of the system, for example the diffraction system shown in Fig The change in the focal distribution occurs due to the phenomena known as depolarisation which is when an incident polarisation state containing a single vector component (e.g. the electric field vector E x ) undergoes a vector rotation which transfers energy from this state to an orthogonal transverse field (E y ) and a longitudinal field (E z ). The validity of this approach was further investigated by Wolf and Li in 1981 to verify such conditions of high NA. 62 Figure shows the geometry of vectorial diffraction under conditions of a 18

43 Chapter 1 a y(j) E (φ) E (ρ) E (i) x(i) b E (ρ) ρ E (φ) θ O α z(k) Figure The geometric illistration of vectorial diffraction 53 of a incident electric field (E i x) in the x direction. E i r and E i φ are the polar components of Ei x

44 Chapter 1 high NA. Depolarisation occurs because at the extremities of the lens there exists an increase in refraction which causes the radial component of the incident vector field to rotate. The transfer of energy due to depolarisation changes the effects of singularities on the focal distribution which can be seen in Fig The singularities that occurr in Fresnel diffraction by a low NA lens only exist in certain directions in the focal region of the lens. Ganic et al. showed that the spectral splitting emphasised by Gbur et al. for the low NA system 54 do extend to vectorial diffraction, but only for the particular directions and no longer existing along the optical axis 63. In considering the diffraction by a lens, what has previously been investigated is the effect of temporal phase associated with an ultrashort pulse. 64,65 However, such effects become more complicated when the temporal, spectral and phase complexity of a SC is considered. 1.3 Applications The temporal structure and the extensive bandwidth of the SC field have changed the experimental design of applications such as optical microscopy. SC provides multiple microscopy techniques in the one compact source and is a simple extension to a conventional ultrafast laser and microscope. In 24 Shi et al. 12 presented experimental evidence that SC generation could be applied to the conventional confocal microscope and then in 25 Isobe et al. 11 was able to use the structured temporal envelope of the SC to perform two photon microscopy. The intuitive extension of SC generation is the application of coherent anti-stokes Raman scattering microscopy, where the SC field is used to pump the molecular energy states of a chemical or biological sample. In 23 Paulsen et al. 8 presented a study on the use of a SC and ultrashort pulse as a co-propagating pump and Stokes beams to excite molecular vibrations of a chemical sample. In optical coherence tomography, low coherence interferometry is used to section biological tissue by both in vivo and in situ. The resolution of this imaging system 2

45 Chapter 1 a y (µm) b x (µm) x (µm) c z (µm) y (µm) z (µm) S/S (1log ) S/S (1log ) S/S (1log ) Figure Diffraction by a lens of N A = 1 for an incident wavefront with λ =.78 µm. (a) the xy plane; (b) the xz plane and (c) the yz plane. 21

46 Chapter 1 is limited by the bandwidth, since the resolution is inversely proportional to the bandwidth. In 21 Hartl et al. 7 showed that by using SC generation predominantly spanning the near infra-red region they could achieve an axial resolution of 2.5 µm. The singular points within the diffraction plane of a lens are an important optical property as they can be used as a signature to phase unwrap information. Typically, the field of optical vortex metrology regards these points as an obstacle and the main focus is their removal by adding a phase vortex map to the beam which contains phase singularities of opposite phase. However, in 25 by Wang et al. it was shown that a random distribution of phase singularities could be used as method for displacement measurement. 7 Another application which would benefit from SC generation is optical data storage where the trend is to encode data into spectral and polarisation properties of an optical material. An example of this is the research completed by Zijstra et al., where five-dimensional recording was achieved by surface plasmon mediation in gold nanorods. 71 The gold nanorods in this study photo-thermally melt under single pulse laser illumination causing them to reshape, which changes their absorption, fluorescence and polarisation characteristics. SC generation is ideal for this application as it provides pulsed and spectral features capable of encoding these modalities simultaneously. It is quite clear that these applications involve diffraction and interference. However, what is not understood is how the complexity of a broadband source such as a SC is affected by diffraction and interference, and what is absent from the literature is the behaviour of a SC field within the focal region. 1.4 This thesis The major theme of the research completed in this thesis is the characterisation of SC generation and its focal distribution. The use of SC generation has become an important laser source because its optical features, whether it be temporal or spectral, can be tailored to the experiment through the manipulation of dispersion 22

47 Chapter 1 and nonlinearity. However, the manipulation is not limited to just the fibre characteristics and would be strongly dependent on the application where diffraction and interference would implicate changes in spatial and temporal properties. Although the characteristics of diffraction in an optical lens system have been studied for a polychromatic wave such as an ultrashort pulse, no such investigation has been completed for the complex field of a SC. It is evident that this study would have a far outreaching influence as it has consequences that relate to photonics applications such as microscopy. The objectives of this thesis are to provide three main scientific contributions: the development of a theoretical and experimental comparison into highly birefringent PCF, an understanding of how a SC field is affected by Fresnel diffraction, and an understanding of how a SC field is affected by vectorial diffraction. Chapter 2 provides a detailed theoretical background to both SC generation and diffraction by a lens. The development of the theory behind SC generation begins with Maxwell s equations to derive the wave equation. Two key equations are derived to solve the modal properties and the propagation equation, where the modal properties characterise the dispersion and the nonlinearity of the optical fibre. These parameters are then used within the propagation equation to derive the nonlinear Schrödinger equation (NLSE). The treatment continues to make a modification to incorporate the two orthogonal LP modes of the PCF which leads to the coupled mode nonlinear Schrödinger equation (CMNLSE). The scientific contribution to SC generation is to provide a detailed description of pulse propagation in highly birefringent PCF constructed with two ZDWs. This is different to other previous studies in the way that the birefringence in the selected PCF is much higher than previously experimentally and theoretically investigated. Also, the contribution is also novel by the use of polarisation maintaining PCF with two ZDWs. Chapter 3 involves a theoretical and experimental study of pulse propagation in highly birefringent PCF with two ZDWs. The CMNLSE derived in Section is applied to a commercial developed highly birefringent PCF (NLPCF-75, Crystal- 23

48 Chapter 1 Fibre 72 ) which has two ZDWs. Experimentally an investigation was completed which was verified by the theoretical observations. Soliton dynamics provides the key characteristics which determine the temporal and spectral difference between the linear polarised modes. The diffraction by a lens of a field such as a SC field, is a pivotal step in the scientific contribution, as it lays the theoretical optical foundation for the investigations into the interaction between a SC and imaging samples. What is required is an in depth study into the characteristics of the focal region of a focused SC. These effects need to be quantified through scientifically appreciated characteristics such as optical coherence. The effects of diffraction need to be considered in both the classical optics viewpoint of Fresnel diffraction and the more modern understanding of vectorial diffraction theory. The next topic carried out in Chapter 4 investigates the modification of the SC field through the Fresnel diffraction by a lens. Using the mathematical framework described in Section the focal distribution of the lens is characterised. It is demonstrated that the spatial modification through phase modification couples with the temporal phase of the SC field which modifies the radiation. These effects are characterised through temporal and spatial correlations known as the degree of coherence, which are quantified through the parameters of coherence time and mean frequency. The coherence of the field can be observed from two viewpoints: a stationary and a nonstationary observation frame. The coherence times are influenced by the phase associated with the destructive interference around points of singularity and become complicated for a nonstationary reference frame, which is influenced by the path differences of the rays extending over the aperture. Chapter 5 investigates the influence of diffraction by a high NA lens through the mathematical description of vectorial diffraction developed in Section The theoretical model not only provides key insight into the effects produced by spatial and temporal coupling between the lens and the incident field, but also investigates the effects on the polarisation state. Since the high NA focusing produces depolarisation, this should correlate to changes in the temporal profile of the SC field. These effects are characterised though correlations which investigate 24

49 Chapter 1 polarisation coherence and are quantified through the coherency matrix. The influence of depolarisation can be observed through the transverse and axial directions of the focal region, where the optical axis shows the most significant change, which is due to the superposition no longer forming points of destructive interference. Chapter 6 provides a summary of the conclusions drawn from the investigation. The chapter highlights the key aspects of pulse propagation through highly birefringent PCFs and the role of the two ZDWs and the modifications of the SC field caused by diffraction of a lens. The implications of this study and the future work that needs to be investigated are provided in Section

50 Chapter 1 26

51 Chapter 2 Theory 2.1 Introduction This chapter provides a basis for the theory behind nonlinear pulse propagation in optical fibre. The theory used in this chapter was extensively presented by Agrawal in The aim is to provide a concise description of how this theory has been developed to lead to polarised pulse propagation. The outline here starts from Maxwell s equations to derive the wave equation for an electromagnetic field in a dielectric waveguide. The transverse and longitudinal properties of the field are separated to derive the formulae for the dispersion and the nonlinear parameters which leads to what is known as the nonlinear Schrödinger equation. An extension is then made to incorporate modal birefringence and how this relates to variations in dispersion and nonlinearity, to derive the coupled-mode nonlinear Schrödinger equation. 2.2 Nonlinear pulse propagation Maxwell s equations The derivation starts from Maxwell s equation which are given by 27

52 Chapter 2 E = B t, (2.2.1) H = J + D t, (2.2.2) D = ρ, (2.2.3) B =, (2.2.4) where E and D are the electric field intensity and density, respectively. H and B are the magnetic intensity and density, respectively. For a dielectric material such as a silica waveguide there are no free charges so the current density J = and charge density ρ =. D and E are related to E and H through the following equations D = ǫ E + P, (2.2.5) B = µ H + M, (2.2.6) where ǫ and µ are the free space permittivity and permeability respectively. P and M are the electric and magnetic induced polarisations. For a dielectric material M =. From this point the wave equation for the electric field can be derived and is much simpler using the Fourier transform relationship of Maxwell s equations, which leads to E = iωb, (2.2.7) H = J + iωd, (2.2.8) D = ρ, (2.2.9) B =. (2.2.1) By taking curl of Eq. (2.2.7) and using Eq. (2.2.8), the wave equation in the frequency domain can be shown to be given by 28

53 Chapter 2 E = ω2 c 2 E µ ω 2 P. (2.2.11) Eq. (2.2.11) can be simplified by using the vector identity E = ( E) 2 E. (2.2.12) Since E =, Eq. (2.2.11) now becomes 2 E = ω2 c 2 E + µ ω 2 P. (2.2.13) P can be divided into two components, the linear contribution P L and a nonlinear contribution P NL such that P = P L + P NL. (2.2.14) The linear and nonlinear components are related to the material s susceptibilities χ which are given by P L (r,t) = ǫ χ 1 (t t )E(r,t)e iω (t ) dt, (2.2.15) P NL (r,t) = ǫ χ 3 E (r,t) t R (t ) E (r,t t ) 2 dt, (2.2.16) where R (t t ) is the Raman response of fused silica which includes instantaneous and delayed components defined by R (t) = (1 f R )δ (t) + f R h R (t). (2.2.17) Here f R is the Raman contribution and for fused silica is equal to.18. The nonlinear response to the induced polarisation is treated as a small perturbation 29

54 Chapter 2 with the relative permittivity determined by ǫ (ω) = 1 + χ 1 + ǫ NL, (2.2.18) where ǫ NL is the nonlinear permittivity. Eq. (2.2.13) can be simplified and is now given by 2 E + ǫ (ω) ω2 E =. (2.2.19) c Slow varying envelope equation The electric field and the induced electric polarisation terms contain slow amplitude and rapidly varying components and for this derivation are separated from the slow varying field, which are given by E (r,t) = 1 2ˆx [ E (r,t) e iω t + ], (2.2.2) P L (r,t) = 1 2ˆx [ P L (r,t) e iω t + ], (2.2.21) P NL (r,t) = 1 2ˆx [ P NL (r,t) e iω t + ]. (2.2.22) Here, ˆx is the transverse spatial dimension of the propagating mode. The solution to Eq. (2.2.11) depends on the transverse and the longitudinal components of the field, and how they vary spatially and temporally. The transverse modal properties of the field are treated to be invariant in the propagation direction (z) and are separated from the longitudinal field components. By assuming a solution to be of the form the solution to Eq. (2.2.19) becomes E (r,ω ω ) = E (x,y)e(z,ω ω )e iβ z, (2.2.23) 3

55 Chapter 2 2 E (x,y) x 2 2iβ E (z,ω ω ) z + 2 E (x,y) + [ǫ (ω)k 2 y β ] 2 E (x,y) =, (2.2.24) 2 + ( β2 β 2 where β is the wave number of the fibre mode. ) E (z,ω ω ) =, (2.2.25) Equation (2.2.24) is used to determine the modal properties of the optical fibre such as dispersion coefficients and the nonlinear cross section. By using the coefficients obtained from Eq. (2.2.24) and using Eq. (2.2.25) the mathematical formulae for nonlinear pulse propagation can be determined Optical properties of photonic crystal fibre The dispersion and nonlinear properties of a photonic crystal fibre (PCF) are determined by solving Eq. (2.2.24) to determine the wave number β. Solving Eq. (2.2.24) involves an iterative procedure to determine β (ω) and the field distribution E (x,y), which can be used to determine the effective refractive index n eff (ω). A numerical solution to Eq. (2.2.24) has been achieved by a wide range of numerical models including the finite element method 73,74, the multipole method 17 2,the plane wave expansion method and the finite difference method. 13 A point to add here is that the perturbations from the nonlinear component discussed earlier have no effect on the modal distribution and are only considered for the longitudinal propagating field. The method that is used in the investigation in this thesis uses the plane wave expansion method as this numerical method has been shown to be a fast and an accurate method. The software package used to achieve this is RSoft Photonics CAD Suite

56 Chapter Nonlinear Schrödinger equation The wave number β (ω) can now be used to include the perturbations from the nonlinear polarisation for which β (ω) is now given by β (ω) = β (ω) + δβ (ω). (2.2.26) by 25 The change in the wave number δβ (ω) is related to the modal field and is given δβ (ω) = k δn E (x,y) 2 dxdy E (x,y) 2 dxdy. (2.2.27) The change in the effective refractive index is related to the nonlinear index by δn (ω) = n 2 R (ω) E (z,ω) 2 + iα 2k, (2.2.28) where we have replaced χ 3 with n 2 and therefore δβ (ω) now becomes δβ (ω) = k γ (ω)r(ω) E (z,ω) 2 + iα 2. (2.2.29) The term γ (ω) is the nonlinear coefficient described by γ (ω) = n 2ω ca eff and α (ω) is the loss coefficient. Similarly to the expansion of the wave number in Section 1.1.2, γ (ω) and α (ω) can also be expanded. The effective modal area A eff is determined by A eff = ( ) 2 E (x,y) 2 dxdy E (x,y) 4 dxdy. (2.2.3) Equation (2.2.25) now can be solved where the approximation for β 2 β 2 = ) 2β ( β β is used and therefore Eq. (2.2.25) becomes 32

57 Chapter 2 E (z,ω) z = i [β (ω) + δβ β ]E(z,ω). (2.2.31) Substituting Eqs. (1.1.3) and (2.2.29) into Eq. (2.2.31) and taking the Fourier transform leads to the generalised NLSE, which is given by E (z,t) z i m+1 m! β m E (z,t) m t m m 2 ( ) = iγ 1 + iτ E (z,t) t ( ) R (t ) E (z,t t ) 2 dt. (2.2.32) The equation has been shifted in time to create a moving observation frame, also known as a retard time given by the relation t = τ β 1 z. The important process in supercontinuum (SC) generation is the formation of high order optical solitons which can be demonstrated with the nonlinear Schrödinger equation (NLSE). Figure 2.2.1a shows the formation of the third order soliton. The input optical pulse used in this model was an ultrashort hyperbolic secant pulse ( t =.5 ps) and a peak power of 2 W. The input ultrashort pulse undergoes a transformation through phase caused by the balanced chirp contributions from β 2 and self phase modulation (SPM), which rapidly expands the field spectrally. The incorporation of β 3 introduces a phase shift on the soliton which becomes increasingly dominant with fibre length. The influence of β 3 is to perturb the high order soliton which introduces the nonlinear and dispersive waves resulting in effects such as four wave mixing. Figure 2.2.1b shows the effects of β 3 in the propagation of a third order soliton. Dispersive effects are not the only form of phase perturbations on the ultrashort pulse. Intra-pulse Raman scattering is also a dominant effect which red shifts the solitary waves. Figure 2.2.1c shows the effects of stimulated Raman scattering on the formation of a soliton. As the soliton propagates it sheds energy into phase matched dispersive waves and in shedding the energy forms a fundamental soliton. 33

58 Chapter 2 a Time (ps) S/S (1log 1 ) Frequency (THz) S/S (1log 1 ) Fibre length (m) Fibre length (m) 4 b Time (ps) S/S (1log 1 ) Frequency (THz) S/S (1log 1 ) Fibre length (m) Fibre length (m) 4 c Time (ps) S/S (1log 1 ) Frequency (THz) S/S (1log 1 ) Fibre length (m) Fibre length (m) 4 Figure The formation of the third order soliton. The parameters in the simulation were (a) β 2 =.5 ps 2 /m and γ =.95 W/m; (b) β 2 =.5 ps 2 /m, β 3 = ps 3 /m and γ =.95 W/m; and (c) β 2 =.5 ps 2 /m, γ =.95 W/m and R (t) determined by Eq. (1.1.7). All other terms were neglected. 34

59 Chapter Coupled mode nonlinear Schrödinger equation Similar to the procedure presented in Section 2.2.4, the coupled mode nonlinear Schrödinger equation can be derived and as described earlier when considering birefringent PCF there needs to be consideration for the two linearly polarised (LP) modes. The major difference that exists between the modes is the effect of P NL. For fused silica there are three major contributions to χ 3 (since χ 3 is a 4 th Rank tensor) and they are of similar strength. It can be shown that the contribution P NL forms the following equation t P NL j (r,t) = ǫ χ 3 E j (r,t) + ǫ χ 3 E k (r,t t ) where j,k = x or y. t R (t ) [ E j (r,t) ] E k (r,t t ) 2 dt 1 3 R (t )E j (r,t t )E k (r,t t ) dt, (2.2.33) It can be shown that for a linear birefringent PCF the propagation equation now becomes the coupled mode nonlinear Schrödinger equation (CMNLSE) and is given by = iγ E j (t) z ( 1 + iτ t ( E j (t) β + β 1 t ) ( E j (t) (1 f R ) ) i m+1 m! β m E j (t) mj t m m 2 [ E j (t) ] ) E k (t) 2 + f R R j (z,t), (2.2.34) R j (z,t) = t h R (t t ) ( E j (t) 2 + E k (t) 2) dt, (2.2.35) where E j and E k are the field components with j and k = x or y (x y), z is a propagation coordinate, the time coordinate moving in a reference frame is given by t = τ (β 1j + β 1k )z/2, β m is the m th order propagation coefficient, β = (β j β k ) is the phase mismatch, β 1 = (β 1j β 1k ) is the group velocity mismatch, γ and τ are the nonlinearity and optical shock coefficients respectively. 35

60 Chapter 2 The method for solving Eqs. (2.2.32) and (2.2.34) can be achieved by the split step Fourier method. A description of the numerical implementation of this method is presented in Appendix A. For polarised pulse propagation the term governing the degree of polarisation of the output is the group velocity mismatch β 1 which relates to the walk off length between the two LP modes. If the group velocity mismatch was neglected, co-propagating pulses in adjacent modes would interact for the duration of the fibre length causing coupled polarisation effects. At this point, a method for describing the spectral and temporal characteristics is introduced, which is through a spectrogram. Experimentally the spectrogram can be obtained by using a frequency resolved optical gating (FROG) system. The mathematical description of the spectrogram is determined by S (ω,τ) = E (t)g (t τ)e iωt dt 2, (2.2.36) where g (t τ) is a delayed gate pulse at a delay of τ. The spectrogram is used to compare the temporal and spectral characteristics between the coupled fibre modes. Figure shows the spectrograms for the two LP modes of a PCF of 1 metre length. The dispersion and nonlinear properties for the two modes are the same and as β 1 = ps/m the two modes have the same spectrogram. However, when the group velocity mismatch between the modes is increased to β 1 = 2 ps/m (Fig ) the spectrograms for the two modes are different since the group velocity mismatch introduces a walk off length for which the two modes can interact within and hence have a small time frame for which XPM effects can influence the pulses. This situation arises in highly birefringent PCF s as the modal mismatch is enhanced. 36

61 Chapter 2 a 4 1 b 4 1 Frequency (THz) S (norm. 1log 1 ) Frequency (THz) S (norm. 1log 1 ) Time (ps) Time (ps) 2 Figure Ultrashort ( t =.5 ps) pulse propagation using the CMNLSE. (a) y polarised mode and (b) the x polarised mode. The parameters used in the simulation were β j2 =.5 ps 2 /m, β k2 =.5 ps 2 /m, γ =.95 W/m and a β 1 = ps/m. a 4 b 4 Frequency (THz) S (norm. 1log 1 ) Frequency (THz) S (norm. 1log 1 ) Time (ps) Time (ps) 2 Figure Ultrashort ( t =.5 ps) pulse propagation using the CMNLSE. (a) y polarised mode and (b) the x polarised mode. The parameters used in the simulation were β j2 =.5 ps 2 /m, β k2 =.5 ps 2 /m, γ =.95 W/m and a β 1 = 2 ps/m. 37

62 Chapter Diffraction theory: low numerical aperture Introduction The basic construction of any optical imaging system is the lens or the microscope objective because it delivers the capability to optically image with magnification. The knowledge that has developed the current understanding of how a lens performs, known as diffraction theory was constructed by Huygens and Fresnel which has led to the Huygens-Fresnel principle. 53 This chapter serves as a theoretical background to the understanding of the Huygens-Fresnel principle which intuitively leads to the diffraction integral for the low numerical aperture (NA). The theoretical background then extends to Debye theory leading to the vectorial diffraction by a high NA lens Huygen-Fresnel principle As described in Section the Huygens-Fresnel principle considers the primary wave front being diffracted by an aperture as a source of secondary spherical wave fronts. The diffraction at a point after the aperture the field is the superposition of the primary wave front and the secondary spherical wave fronts which interfere. Mathematically the Huygens-Fresnel principle can be described as E (r 2,z 2 ) = C A e ikr r E (r 1 )da, (2.3.1) where E (r 1 ) is the primary wave, e ikr /r describes a secondary spherical wavelet, C is a constant and A is aperture as shown in Fig Fresnel approximation The development of the theoretical treatment of diffraction has had many important contributions which have led to the current formulation. Contributions from researchers such as Rayleigh, Fraunhofer, Somerfield and Kirchhoff have been 38

63 Chapter 2 E i Figure Illustration of mutual interference caused by the superposition of the primary wavefront and secondary spherical waves. 53 instrumental in the development of this theory. 52 The mathematical treatment in this section will not show the historical development of diffraction but provides the most significant formulation which leads to the final form of the diffraction integral. The diffraction of an electromagnetic wave can be described by E 2 (x 2,y 2 ) = iω E i (x 1,y 1 ) e ik r cos (n,r) dx 1 dy 1. (2.3.2) 2πc r The integral of Eq. (2.3.2) describes the superposition of the incident wave front (E i (x 1,y 1 )) and a set of secondary spherical wave fronts. The cos (n,r) factor is the vector component of r in the direction of n. Equation (2.3.2) can be simplified by using a few assumptions. Firstly, the directional cosine can be assumed to be unity since for the case of a low NA lens the majority of r is in the direction n. Secondly, the r in the denominator is replaced by z. Finally, the formulation makes an assumption for the vector r, which is as follows 39

64 Chapter 2 r 2 = z 2 + (x 2 x 1 ) 2 + (y 2 y 1 ) 2 [ ] = z (x 2 x 1 ) 2 + (y 2 y 1 ) z r z [ 1 + (x 2 x 1 ) 2 + (y 2 y 1 ) 2 2z 2 ],, (2.3.3) where the approximation of the form 1 + x 1 + x 2 is used. Therefore the formula for the Fresnel diffraction of an electromagnetic wave is given by E (x 2,y 2 ) = ie ikz λz E (x 1,y 1 )e ik 2z[(x 2 x 1 ) 2 +(y 2 y 1 ) 2 ] dx1 dy 1. (2.3.4) The Fresnel diffraction of a circular aperture can be obtained from Eq. (2.3.4) and is simplified by using a cylindrical coordinate system which is given by E (r 2 ) = iω c e ikz e ikr 2z where a is the aperture radius. 2 2 a ( ) E (r 1 )e ikr 2 1 kr1 r 2 2z J r 1 dr 1, (2.3.5) z Fresnel diffraction by a circular lens The diffraction formula (Eq. (2.3.5)) can be modified to determine the diffraction by a circular lens. The focal length f of the lens is assumed to be approximately equal to the distance z. The scalar diffraction theory for a circular lens is given by 52,53 E (u,v,ω) = iωna2 c a e iu/na2 E i (ω)j (vρ)e 1 2 iuρ2 ρdρ. (2.3.6) b 4

65 Chapter 2 The dimensionless parameters u and v are given by u = ω c (NA)2 z and v = ω (NA)r respectively, where r and z are the radial and axial coordinated of the c lens image space. The parameters a and b are the aperture radius and the integral lower bound for the lens, NA is the numerical aperture, J is a zero order Bessel function of the first kind, ω is the angular frequency and c is the speed of light. 2.4 Diffraction theory: high numerical aperture The Debye integral The Debye approximations are used when a high NA objective is considered. The assumptions in Debye theory is the field in the focal region is a superposition of plane waves with their associated propagation vectors originating from within the aperture. The Debye integral can be expressed as E (x 2,y 2,z 2 ) = i λ E i e is R dω, (2.4.1) where E i is the incident electric field with s and R related to the geometric representation shown in Fig For a circular coordinate system the dot product of the unit vector s and the vector R is given by Ω s R = r 2 sin θ cos (φ ψ) + z 2 cos θ. (2.4.2) The integral over the solid angle can be replaced by dω = sin θdθdφ. (2.4.3) The incident electromagnetic field at the lens aperture is assumed to fully 41

66 Chapter 2 y p 2 r p 1 fs R x θ z E i f Figure Illustration of the geometry of vectorial diffraction. 53 spatially coherent and enters the aperture as a plane wave. The diffraction by a circular lens under the Debye approximations 53 is calculated by E (r,ψ,z,ω) = i λ E i (θ,φ,ω) Ω e ikr sin θ cos(φ ψ) ikz cos θ sin θdθdφ. (2.4.4) Evaluation of the vectorial diffraction formula The treatment of the diffraction formula for a high NA lens begins with the formulation described by Wolf and Richards. 6,61 Consider an incident electromagnetic wave at the back of a high NA lens described by E i (ω) = E i x E i y E i z e iω t, (2.4.5) where the incident field is represented by its polarisation components. 42

67 Chapter 2 However, the incident field undergoes refraction as it propagates through the lens which causes a vectorial rotation of the polarisation (Fig ). The unit vector related to the radial component of the field (E(a ρ )) is transformed into the angular component θ (E(a θ )). The incident electric field as it propagates past the lens can now be described by E i (θ,φ) = P(θ) cos φa θ, (2.4.6) sin φa φ where cos θ cos φi a θ = cos θ sin φj,a sin φi φ =. (2.4.7) cos φj sin θk Using Eqs. (2.4.6) and (2.4.7), the vectorial form of the refraction of the incident field through the lens can be obtained. The incident field can be separated into its spatial dimensions (θ, φ) and frequency components(ω) where the spatial component is given by (cos θ + sin 2 φ(1 cos θ))i E i (θ,φ) = P(θ) cos φ sin φ(cosθ 1)j. (2.4.8) cos φ sin θk Here P(θ) = cos θ, which is the apodisation function of the lens determined by the sine condition. The vectorial form of the diffraction by a lens for a spherical coordinate system can now be obtained by substituting Eq. (2.4.8) into Eq. (2.4.4) which can be shown to be given by 43

68 Chapter 2 E (r,ψ,z,ω) = i λ Ω (cos θ + sin 2 φ(1 cos θ))i E i (ω)p(θ) cos φ sin φ(cosθ 1)j cos φ sin θk e ikr sin θ cos(φ ψ) ikz cos θ sin θdθdφ. (2.4.9) For a circular symmetric incident field the diffraction of the field can be simplified to being only dependent on θ. The diffraction of the incident field with a horizontal polarisation direction can be shown to be given by E x (v,u,ψ,ω) [I E h (v,u,ψ,ω) = E y (v,u,ψ,ω) = iω + cos (2ψ)I 2 ]i sin (2ψ)I 2c 2 j, (2.4.1) E z (v,u,ψ,ω) 2i cos (ψ)i 1 k where I I 1 I 2 = α (1 + cosθ)j (v sin θ/ sin α) E i (ω) cos 1/2 θ sin θ (sin θ)j 1 (v sin θ/ sin α) (1 cos θ)j 2 (v sin θ/ sin α) e iu cos θ/ sin2 α dθ, (2.4.11) where u and v represent the normalised axial and radial dimensionless parameters of the imaging system given by u = kz 2 sin 2 α and v = kr 2 sin α. Essentially what occurs is the incident polarisation rotates slightly to increase the strength of the orthogonal transverse and longitudinal field polarisation states. For an incident polarisation state in the vertical direction the field components are given by 44

69 Chapter 2 E x (v,u,ψ,ω) sin (2ψ)I E v (v,u,ψ,ω) = E y (v,u,ψ,ω) = iω 2 i [I 2c (v,u) cos (2ψ)I 2 ]j. (2.4.12) E z (v,u,ψ,ω) 2i sin (ψ)i 1 k The diffraction formula is now composed of the horizontal and vertical polarisation components determined by E = ae h + be v, (2.4.13) where a and b are the polarisation coefficients. When the NA of the lens is reduced, the vectorial diffraction theory under the paraxial approximation for the horizontally polarisation state is given by E(v,u,ω) = E x (v,u,ω) = iω 2c I (v,u)i, (2.4.14) which converges to scalar diffraction theory described in Section Coherence The modification of the temporal and spatial behavior of a focused wave can be characterised through the degree of coherence (g 1 (u,v,τ)) which is generalised through the correlation between two points and is calculated by 76 g 1 E (z 1,t 1 )E(z 2,t 2 ) (z 1,t 1 : z 2,t 2 ) =, (2.5.1) [ E (z 1,t 1 ) 2 E (z 2,t 2 ) 2 ] 1 2 where z and t are the axial and temporal coordinates. For a stationary beam (e.g. in a Mach Zehnder interferometer) within the 45

70 Chapter 2 diffraction field, the spatial parameters u and v remain constant; hence Eq. (2.5.1) becomes an autocorrelation technique determined by g 1 (u,v,τ) = E (u,v,t),e(u,v,t + τ), (2.5.2) [ E (u,v,t) 2 E (u,v,t) 2 ] 1 2 where g 1 (u,v,τ) depends on the position (u,v ) of the detector. However, when considering the wave packet (in a nonstationary observational frame) the calculation becomes spatiotemporal and is given by 76 g 1 (u,v,τ) = E 1 (u,v,t)e 1 (u + u,v,t + τ), (2.5.3) [ E 1 (u,v,t) 2 E 1 (u,v,t) 2 ] 1 2 where the variables u and τ are related by c = u /τ. This nonstationary frame of reference has been investigated for nonstationary polychromatic waves Using Eqs. (2.5.2) and (2.5.3) the coherence time for the field can be calculated through 76 τ c (u,v ) = g 1 (u,v,τ) 2 dτ. (2.5.4) 46

71 Chapter 3 Pulse Propagation in Nonlinear Photonic Crystal Fibre 3.1 Introduction The ability to control supercontinuum (SC) radiation is reliant on the optical properties of the photonic crystal fibre (PCF) waveguide and the precise knowledge of its modal characteristics of nonlinearity and dispersion. The dispersion characteristics determine the regime which the ultrashort pulse propagates within and the ability to maintain the polarisation state of the incident beam. The motivation behind this chapter is to provide an understanding of pulse propagation in a highly birefringent PCF. The aim is to present that under the condition of high birefringence ( 1 3 ) the dispersion profiles of the two linear polarised modes must be treated separately. The interesting consequence in using highly birefringent PCF is that the structure enforces the incorporation of two zero dispersion wavelengths (ZDWs), which could be important in generating extensive spectra. 47

72 Chapter y/λ 1.6 ε x/λ Figure The geometry as defined in the simulation using a refractive index profile resolution for the PCF of pixels and a supercell size 1 1 unit cells. 3.2 Photonic crystal fibre characteristics The numerical study presented in this chapter uses the coupled mode nonlinear Schrödinger equation (CMNLSE) as described in Section and Eq. (2.2.34). The effective refractive index for the two fundamental propagating modes were calculated using the plane wave expansion method as described in Section The dispersion coefficients and phase mismatch coefficients were determined by the Taylor series expansion as described by Eqs. (1.1.3) and (1.1.5). The field distributions can be used to determine γ, using the methods described by Hainberger and Watanabe 84, and similarly with the optical shock coefficient τ described by Blow and Wood and Karasawa et al.. 34,85 The changes in optical nonlinearity and optical shock are insignificant between modes and for this study are.95 W/m and.57 f s, respectively. The phase mismatch and the group velocity mismatch are ± /m and ± ps/m, respectively and are determined by Eq. (1.1.5). The PCF geometry is shown in Fig The birefringent axis is in the y direction with the two birefringent holes being.2 µm larger than the rest of PCF air holes. Using R-Soft Photonics CAD Suite 8.1 a theoretical model was developed to model the PCF structure. The pitch of the fibre was 1.2 µm with an air hole size of.7 µm. The supercell size, which relates to a collection of unit 48

73 Chapter 3 a β 1 (ps/m) [1x1 3 ] 5.2 β 1y 5.1 β 1x β 2y.3 β 2x β 2 (ps 2 /m) Wavelength (µm) b β(1/m) Wavelength (µm) β 1 (ps/m) Figure The dispersion coefficients related to the mode propagation constant β. (a) shows the first- and second-order dispersion coefficients for the two fundamental modes. (b) shows the phase mismatch ( β ) and the group velocity mismatch ( β 1 ) between these modes. cells should be within the range of 6 6 to 1 1 unit cells. The most important parameter used is the eigenvalue tolerance and for these calculations was set at 1 16 which means a precision of the order of 1 8. Figure 3.2.2a shows β 1 and β 2 dispersion terms. The difference between the two polarised modes is significant and conveys the importance of the group velocity mismatch, which is shown in Fig b. The β 1 dispersion term is related to the group velocity term by the relation β 1 = 1/v g, which shows that the y-polarised mode travels faster than the x-polarised mode and are called the fast axis and slow axes, respectively. The dispersion properties used in this study are presented in Table Table Dispersion data for the polarised mode of the nonlinear fibre. β m mode y (ps m /m) mode x (ps m /m) β β β β β β β β The high order dispersion properties are important as they lead to the generation 49

74 Chapter 3 a Fibre length (m) S/S (1log 1 ) b Fibre length (m) S/S (1log 1 ) Time (ps) Frequency (THz) 4 Figure The effects of TOD originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a peak power of 1 W. (a) the time domain and (b) the frequency domain. of dispersive waves and ultimately determine the dispersive waves phase matched wave vectors. 3.3 Nonlinear and dispersion effects The theoretical analysis throughout this thesis involves the investigation into SC generation. The dominant processes that form the SC need to be isolated in order to understand the effects such a field has in an optical system, for example a lens. During the propagation of an ultrashort pulse through a PCF the nonlinear and dispersion effects become dominant at different stages. The two terms which have the most influence in shifting radiation are intra-pulse Raman scattering (RS) (see Section ) and the third order dispersion (TOD) effect (see Section 1.1.2). SC generation in the anomalous dispersion regime is dominated by soliton dynamics where the solitary wave forms due to second order dispersion and self phase modulation. To develop an understanding of the processes which occur in a highly birefringent PCF with two ZDWs, the dominate phase terms influencing the propagation of the ultrashort pulse are investigated. Figure shows the propagation of an ultra-short pulse affected by the TOD effect through the PCF for an input polarisation state orientated along the fast axis (y direction) of the PCF. The high order dispersion and third order nonlinear terms have been neglected. The input peak power of 1 W was chosen to illustrate the 5

75 Chapter 3 effects of soliton fission from a third order soliton. The soliton fission dynamics are modified by dispersive waves generated at phase matched frequencies associated with the propagation constant of the soliton and nonlinear phase. 86,87 It can be seen in Fig that energy from the initial soliton is transferred to a resonant wave at a lower frequency. This is an important process as it is the key component in formation of visible wavelengths within the spectrum of a SC field. These dispersive waves do not shift with the fibre length after their initial ejection and therefore the low wavelength component of the SC is fixed by initial soliton dynamics. RS is also a dominant effect in SC generation. After the formation of a high order soliton, the soliton begins fissions into fundamental solitons which shift with the Raman spectrum of the fibre, Fig. (3.3.2). The dynamics of these process are determined by the nonlinear and second order dispersion characteristics which form the soliton and subsequent pulse widths of the ejected fundamental solitons. 88,89 The temporal envelope shows the formation of these solitons and the eventual separation from the remaining dispersive temporal features. The rate at which the soliton pulses shifts has been formulated by Gordon in and is given by ν R z β 2. (3.3.1) Under conditions of strong birefringence the walk off length for energy transfer T 4 a.3 1 b.3 3 Fibre length (m) S/S (1log 1 ) Fibre length (m) S/S (1log 1 ) Time (ps) Frequency (THz) 4 Figure The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a power of 1 W. (a) the time domain and (b) the frequency domain. 51

76 Chapter 3 a Fibre length (m) c Fibre length (m) Time (ps) Time (ps) S/S (1log 1 ) S/S (1log 1 ) b d Fibre length (m) Fibre length (m) Frequency (THz) Frequency (THz) S/S (1log 1 ) S/S (1log 1 ) Figure The effects of TOD originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and in each mode of power of 75 W. The coupled polarisation state was 45. (a) the time domain of the y polarised mode (b) the frequency domain of y polarised mode (c) the time domain of x polarised mode and (d) the frequency domain of x polarised mode. between the linear polarised modes, occurs within a very short spatial window ( mm) and does not have a strong influence on propagation. As can be seen with the previous investigation dealing with TOD and RS effects, the temporal envelope is slanted due to the difference between the effective group velocities of the two modes. Again, dispersive and nonlinear effects can be isolated for each mode. The power injected into each mode is the same (1 W); however the second order dispersion coefficients are different (Table 3.2.1). Figure shows the TOD effect and what can be seen is the difference in the effect of second order dispersion in both the formation of solitary waves and the generation of dispersive waves. Since second order dispersion is stronger for the x mode, the pulse compresses more than the y mode and red shifts the ejected solitons to greater extent. This leads to the dispersive waves at higher resonant frequencies (shorter wavelengths), which can be seen in Fig The Raman shifting of the co-propagating waves generated by the equal coupling 52

77 Chapter 3 a Fibre length (m) c Fibre length (m) Time (ps) Time (ps) S/S (1log 1 ) S/S (1log 1 ) b d Fibre length (m) Fibre length (m) Frequency (THz) Frequency (THz) S/S (1log 1 ) S/S (1log 1 ) Figure The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and in each mode of power of 75 W. The coupled polarisation state was 45. (a) the time domain of the y polarised mode (b) the frequency domain of y polarised mode (c) the time domain of x polarised mode and (d) the frequency domain of x polarised mode. 53

78 Chapter 3 a Input power (W) Time (ps) S/S (1log 1 ) b Input power (W) Frequency (THz) S/S (1log 1 ) Figure The effects of TOD originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a fibre length of.3 m. (a) the time domain and (b) the frequency domain. a Input power (W) Time (ps) S/S (1log 1 ) b Input power (W) Frequency (THz) S/S (1log 1 ) Figure The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a fibre length of.3 m. (a) the time domain and (b) the frequency domain. of the fundamental modes can be seen in Fig The frequency shift associated with the soliton formed in the x mode shifts further than the soliton in the y mode, which reiterates the influence of the stronger compression from the larger second order dispersion term. The difference between the dispersion coefficients for the two fundamental modes is therefore important as it sets the conditions for the red shifting of the fundamental solitons and the transfer of energy to dispersive waves. The effects seen for ultra-short pulse propagation as a function of fibre length can also be verified through an investigation into a variation in input power. Figure shows the dependence of the temporal and spectral properties on the input pulse pulse peak power and how it is affected by TOD. The coupled peak power determines the initial soliton order N, which will then fission into the N fundamental solitons. Fig confirms that the TOD effect initiates the growth of blue shifted radiation 54

79 Chapter 3 a 75 b 75 2 Input power (W) S/S (1log 1 ) Input power (W) S/S (1log 1 ) c Time (ps) 75 d Frequency (THz) 75 2 Input power (W) S/S (1log 1 ) Input power (W) S/S (1log 1 ) Time (ps) Frequency (THz) Figure The effects of TOD originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a fibre length of.3 m. The coupled polarisation state was 45. (a) the time domain of the y polarised mode (b) the frequency domain of y polarised mode (c) the time domain of x polarised mode and (d) the frequency domain of x polarised mode. for which after a particular power the spectral expansion begins to slow down. Similarly to the investigation of the TOD effect, the effects of RS can also be understood and is presented in Fig The figure shows the clear formation and fission of solitons (Fig a) and the extensive red shifted radiation. In the absence of TOD there are no blue shifted dispersive waves and the spectra is dominated by near infra-red radiation from soliton self frequency shift. The power dependence of co-progating modes confirms the observations discussed for the dependence of fibre length and are shown in Figs and The difference between the dispersion terms for the two creates a difference in the soliton and dispersion dynamics which leads to a difference in temporal and spectral behaviour. The figures depicting the power dependence show that the formation and self frequency shift of the solitons occurs from a balance between the power within the pulse and its pulse width. As the pulse propagates along the fibre the pulse duration and peak power are adjusted to maintain solitary shape. The initial 55

80 Chapter 3 a Input power (W) S/S (1log 1 ) b Input power (W) S/S (1log 1 ) c Input power (W) Time (ps) Time (ps) S/S (1log 1 ) d Input power (W) Frequency (THz) Frequency (THz) S/S (1log 1 ) Figure The effects of RS originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a fibre length of.3 m. The coupled polarisation state was 45. (a) the time domain of the y polarised mode (b) the frequency domain of y polarised mode (c) the time domain of x polarised mode and (d) the frequency domain of x polarised mode. 56

81 Chapter 3 Input Power (mw) Input Power (mw) (a) (b) (c) θ = θ = 45 θ = 9 3 (d) (e) (f) θ = θ = 45 θ = Frequency (THz) Ouput Power (dbm) Figure Theoretically obtained spectra of propagation within a 13 mm NL-PCF with a 87 fs pulse. Figures (a), (b) and (c) are the spectra for the y polarised output field with (d), (e) and (f) for the x polarised output field. θ is the input polarisation angle with respect to the y axis. peak power is important because it sets the strength of the nonlinearity with in the fibre which along with second order dispersion determines the temporal duration of the soliton and hence the ability of the pulse to shift in frequency. 3.4 Supercontinuum generation Equation (2.2.34) was used to calculate 87 f s pulses propagating with different input polarisation orientations. A pulse width of 87 fs and a fibre length of 13 mm were chosen to coincide with experimental conditions. Figure shows the output spectra obtained from a nonlinear PCF pumped with 87 fs pulses at input polarisation orientations of (y axis), 45 and 9 (x axis) degrees. 91 The simulation shows that the polarisation state is maintained for light coupled into either the x-polarised or y-polarised axis of the fibre. The x-polarised mode has the most extensive spectra and this is because the pump wavelength undergoes a stronger initial compression caused by the stronger second-order dispersion term. 57

82 Chapter y mode x mode γ MI Frequency (THz) Figure Modulation instability gain for the y and the x modes. For an input polarisation orientation of 45 degrees, the degree of polarisation is not maintained due to an equal coupling between modes and the spectra are different due to the relative strength of the second order dispersion. The difference in the dispersion curves creates a change in the wave numbers for the generation of dispersive waves which are determined by the phase matching condition for each individual mode. This occurs because of the modulation instability which gives rise to gain and amplifying the nonlinear processes. Modulation instability is determined by 25 ( ) γ MI = Im Q (Q + 2γP ), (3.4.1) where Q is the propagation constant containing only the even order coefficients and P is the peak power of the input pulse. Figure shows the modulation instability gain for the two polarised modes and quantifies that the differences in the dispersion of the two mode can change the output spectra. When the pulse enters the nonlinear fibre, it undergoes a transformation to form a high order soliton. The high order soliton then breaks up in a fission process which converts the soliton into fundamental solitons. 46 The order of the initial soliton and the length at which soliton fission occurs is given by 92 58

83 Chapter 3 ( ) γp T 2 1/2 N =, (3.4.2) β 2 L f = T 2 1 β 2 N, (3.4.3) where P is the peak power and T is the full width at half maximum. The soliton order is higher along the y axis in comparison to the x axis, which is due to the smaller second order dispersion term. However, the soliton fission length is.465 m and.35 m for the y axis and x axis, respectively, which could be contributing to the more extensive spectra. The self frequency shift of the soliton is inversely proportional to the fourth power of the temporal duration of the soliton. The initially stronger compression by the second order dispersion in the x mode leads to further spectral expansion in comparison with the y mode. Although there are some differences between the spectral components between the input polarisation orientations, the spectra are similar and hence both axes could be used for polarised broadband applications. Figure confirms that the output spectra are different between the two fundamental modes. The output spectra for the y polarised mode travels at a higher speed compared with the x polarised mode which is due to the group velocity mismatch. Figures (a) and (f) show the difference in the pulse structure which is due to the different dispersion properties and fission processes of the coupled axes. The effects observed would be important to consider in time-resolved polarised illumination applications since there is a delay between spectral features Experimental study The experimental setup is shown in Fig A pulsed light beam from a Ti:Sapphire laser was coupled into a nonlinear PCF (Crystal-fibre). Two Glan Thomson polarisers were used to vary the input power and a half wave plate was used to alter the input polarisation orientation. The output pulse was analysed with a Glan Thomson polariser. Spectra were observed and recorded using an 59

84 Chapter Frequency (THz) Frequency (THz) (a) θ = (b) θ = 45 (c) θ = Ouput Power (dbm) 25 (d) θ = (e) θ = Time (ps) (f) θ = Figure Theoretically obtained spectral and temporal profile of 87 f s pulsed propagation within a 13 mm nonlinear PCF. Figures (a), (b) and (c) are the spectra for the y polarised output field with (d), (e) and (f) for the x polarised output field. Ando spectrometer and Princeton Instruments CCD (pixis 1). The pulsed propagation spectra were obtained for different input polarisation orientations. The characteristics of the fibre used in this experimental study are shown in Fig and Table Figure shows the spectra for the two output modes of the PCF coupled with 78 nm, 87 fs pulses and 15 mw average power. The spectra show the high degree of polarisation for the input pulse orientations of and 9 degrees. In addition, a large degree of the red-shifted radiation attributed to stimulated Raman scattering is present. The and 9 degree spectra are different and confirm the theoretically obtained results. For degrees it is apparent that there is a blue-shift of radiation and is attributed to the dispersive wave generation and four-wave mixing. In addition, a small amount of radiation is coupled into the orthogonal mode which is attributed to the depolarisation by the high numerical aperture input coupling. The degree of polarisation (DOP), defined as DOP = ( ) ( ) I I / I + I, is shown in Fig The experimental curves show a strong degree of polarisation 6

85 Chapter 3 SA GT GT 1 WP 2 GT Ti:Sa Fibre Spec Figure Optical arrangement used in this study. GT - Glan Tomson, WP - Wave Plate, Spec - Spectrograph and SA - Spectrum Anaylser for all wavelengths except for the pump bandwidth. This confirms the high degree of polarisation measured in Fig The theoretical degree of polarisation shows the effects of depolarisation are not attributed to cross coupling and must be introduced by the input coupling. 3.5 Conclusion A highly birefringent PCF with two ZDWs is beneficial to produce extensive highlypolarised optical spectra due to the nonlinear and dispersive properties inherent from its geometry. A methodology for generating broadband pulsed light from a two ZDWs PCF is presented. The theoretical and experimental observation shows that the spectra maintain their linear polarisation state and that the extent of the spectra is stronger at either of the fundamental mode axes. Qualitatively, the dispersion properties of the the two fundamental polarised modes of the PCF produce different SC spectral features and is due to the strength of the second order dispersion term. This difference creates different order solitons in each mode when coupled with the same power, which spectrally shift at different rates. The dispersion polynomials for the two modes set the conditions for the radiation of dispersion waves, which determines the extent of blue shifted radiation. These characteristics are achieved because of the highly birefringent PCF. 61

86 Chapter 3 a 2 25 θ in = o Intensity (db) b Frequency (THz) θ in = 45 o Intensity (db) c Frequency (THz) θ in = 9 o Intensity (db) Frequency (THz) Figure Spectral properties of the polarised modes of the nonlinear PCF. The perpendicular (blue) and parallel polarised (red) states are with reference to the output orientation of the laser. 62

87 Chapter 3 Degree of polarisation Parallel Perpendicular Fast axis (theory) Frequency (THz) Figure Degree of polarisation for the fast and the slow axes of the fibre. 63

88 Chapter 3 64

89 Chapter 4 Fresnel Diffraction 4.1 Introduction The phase associated with an electromagnetic wave can affect the way by which the field correlates in different optical phenomena. Within the last decade the fundamental description of diffraction in optical systems such as a lens was heavily investigated because of the influence of spatial phases which forms points of destructive interference. This effect has been extensively studied and in particular from two points of view: firstly from a interference through an interferometer such as a Michelson Mach Zehnder interferometer, and secondly from diffraction by a lens. The study in thesis will investigate the later case. As stated earlier in Chapter 2 there has been recent interest in the quantitative description and formulation of the diffraction of a polychromatic wave. Around the point of destructive interference, the focused wave shows the behaviour of red shifted and blue shifted radiation. Physically, the addition of temporal phase onto an electromagnetic wave can change the superposition condition of the focal distribution. Since there exists a temporal and spatial phase coupling through the wavefront propagation in a lens, it is intuitive that a temporal phase variation of supercontinuum (SC) field would affect the diffraction by the lens. Presented in this chapter is an understanding of SC generation under conditions 65

90 Chapter 4 of Fresnel diffraction, which is considered because it is a simplified and accurate description of the diffraction by a lens. 52 Also presented in this chapter is the coupling relationship between the temporal and spatial phase within a focused SC field, and how the diffraction modification through the steep phase gradient associated with the points of destructive interference enhances the degree of coherence of a SC field in the focal region. 4.2 Numerical methodology The diffraction of a polychromatic wave such as a SC wave is calculated using the scalar diffraction theory, which can be given, under the paraxial approximation, by Eq. (2.3.6). 53 If b =, the diffraction is for the complete aperture and for a non-zero b is a diaphragm. E (ω) is the Fourier transform of the SC wave using the dispersion parameters, nonlinear parameters and method described in Chapter 3, and the Fourier transform of E 1 (u,v,ω) is used to obtain the temporal profile E 1 (u,v,t). An analytic solution (v = ) for the diffraction field can be obtained by the following equation ( ) E 1 (u,v,ω) = ωna2 e iu/na2 E (ω) e 1 2 ib2u e 1 2 ia2 u. (4.2.1) uc From Eq. (4.2.1) it can be seen that when a 2 u/2 = ±2nπ and if b =, the equation is equal to which is a singularity. Since u is dependent on both ω and z u there exists a region of singularity. As the electromagnetic field propagates through a lens it is diffracted and modified through the spatial phase of the wavefront. The superposition of the wavefront from the outer aperture to the inner aperture has an inherent path difference (Fig ). The destructive interference at certain frequency components of the SC wave produces a singularity or a null in intensity. These points occur at discrete positions in both the axial and radial directions and can be determined 66

91 Chapter 4 through the parameter u = ω c (NA)2 z and occur at u = ±4nπ (where n is an integer, for the radial direction this point is the zero of a zero order Bessel function of the first kind). The parameters u and v (Fig a) are defined as the normalised axial and radial coordinates of the optical system and are given by u = 2π λ (NA) 2 z and v = 2π λ (NA)r, where NA is the numerical aperture, z and r are the axial and radial dimensions (in µm), and λ is the centre wavelength of the original pulse coupled to the nonlinear photonic crystal fibre (PCF) (the input pulse is a hyperbolic secant pulse which is used to represent a mode-locked laser pulse). The analysis used to understand how a field propagates through the focal region depends on the method of its detection or observation (Fig b). In a conventional optical system with a single point detector, the intensity that is collected depends on the diffraction for an axial (and radial) position (stationary observer, S (t,z ) ) and evolves with time. The intuitive observation, however, would be to view the focal plane from the side, where the intensity is both temporally a U (t) f v (t,r) r u (t,z) O (t=,z=) b v (ω ) u (ω ) S (t,z ) Stationary observer S (t = t z /c,z) Nonstationary observer Figure An illustration of pulse diffraction by a low numerical aperture (NA) lens. (a) shows how the path length and the NA affect the pulse distribution as the temporal envelope passes through the focus. (b) shows the observation frames of the intensity profile in the focus. 67

92 Chapter 4 a 1.8 b.4.3 S/S.6 S/S c 6π 3π 3π 6π u (axial) d π 2π 4π 6π u (axial) 1 3π 5 v (radial) 3π 2 3π S/S (1log 1 ) Time (ps) S/S (1log 1 ) 3π 35 6π 3π 3π 6π u (axial) 4 1 6π 3π 3π 6π u (axial) Figure The temporal effects of a focused hyperbolic secant ultrashort pulse propagating through the focus of a low NA (.1) objective. (a) On axis diffraction centred at the focal point (the full temporal evolution of the hyperbolic secant on the axis is described in Appendix C). (b) On axis diffraction centred at u = 5π. (c) Radial and axial diffraction pattern centred at the focal point (the full temporal evolution of the hyperbolic in the radial and axial direction is described in Appendix C). (d) The intensity matrix used to obtain the temporal and axial intensity information for the stationary and nonstationary observation frames. and axially dependent since the leading intensities of the pulse are modified by the diffraction for an axial position and differs from the trailing intensities (S (t, z)), which is referred to as a nonstationary observer. E 1 (ω) is the intensity distribution for the stationary observation frame where the intensity for the nonstationary observation frame is obtained by taking the diagonal of the the matrix E 1 (u,t) for different v. 4.3 Ultrashort hyperbolic secant pulse For an ultrashort pulse its phase is linear, a phase modification is expected from the diffraction by the lens (Fig a) and when the pulse encounters a point 68

93 Chapter 4 a b v (radial) 3π 3π τ c (ps) v (radial) 3π 3π τ c (ps) 3π 2 3π π.1 3π c.17 6π 3π 3π 6π u (axial) d.17 6π 3π 3π 6π u (axial) τ c (ps).13 τ c (ps) NA =.1 NA =.14 NA =.2 6π 3π 3π 6π u (axial) NA =.1 NA =.14 NA =.2 6π 3π 3π 6π u (axial) Figure The coherence time of a focused hyperbolic secant ultrashort pulse for the stationary and the non-stationary cases. (a) Axial and radial distribution of the coherence time for the.1 NA lens for the stationary case; (b) Axial and radial distribution of the coherence time for the.1 NA lens for the non-stationary case; (c) Effect of NA on the coherence time on the axis for the stationary case; and (d) Effect of NA on the coherence time on the axis for the non-stationary case. of destructive interference its temporal profile changes significantly (Fig b) in which the pulse envelope and spectrum are split. The phase modification also extends to the radial direction (Fig c) caused by the zero intensity location due to the zero order Bessel function. For a stationary observer the temporal intensity information is obtained for a constant axial position (data contained in the columns of Fig d). The nonstationary observation frame is complicated since the intensity information is linked in both the temporal and axial coordinates and is obtained through the diagonal of the matrix S(u,t) (diagonal of Fig d). Both observation frames may vary in the radial direction to obtain the three dimension intensity information. The temporal characteristics of the field are quantified through the coherence time, which is mathematically described by Eqs. (2.5.1)-(2.5.3). For an ultrashort 69

94 Chapter 4 pulse the coherence time is 6 % larger than its initial coherence time (τ =.164 ps) before the lens (Figs a and 4.3.2b). The variation in the coherence time for the stationary case is expected to be narrow for an ultrashort pulse due the narrow bandwidth of the field. The observation frames are different because of the geometric path difference (Figs a and b), which is illustrated by changing the NA in the non-stationary observation frame (Fig d). The coherence time is not enhanced for a hyperbolic secant because there is no temporal phase contribution. An increase in chirp is effectively increasing the amount of phase which creates an enhancement in the coherence time (Fig ). The spatial phase incurred because of diffraction has a change of negative phase to positive phase across the focal plane and has a greater path difference on the inside of the focal plane (the rays from the extremity of the lens approach the focal length the further away from the lens), which explains the asymmetric distribution in the coherence time with an increase in the chirp parameter. The coherence time is related to the bandwidth ν by τ c = 1/ ν. The mean frequency µ can validate the effects observed in the temporal correlation, which is illustrated in Fig The variation in the mean frequency caused by the change in phase associated with the frequency dependent point of destructive interference is similar to the previous literature presented by Gbur Nonlinear and dispersive phase To determine how the phase associated with electromagnetic field from a PCF behaves in the focal region of a lens, the dominant phase terms can be isolated. Third order dispersion (TOD) and stimulated Raman scattering (RS) are dominant processes which act to perturb an optical soliton. To isolate these effects the coupled mode nonlinear Schrödinger equation is reduced to the dependence of β 2, SPM and the coupled mode phase mismatch terms β and β 1. The incorporation of RS is shown in Fig In the initial stages of the propagation, the temporal phase affecting the coherence time is the influence of soliton formation. At a length corresponding to approximately the soliton fission 7

95 Chapter 4 a.4.2 (τ c τ )/τ.2.4 C = C =.1 C = 1 b.6.1 6π 3π 3π 6π u (axial) (τ c τ )/τ C = C =.1 C = 1.4 6π 3π 3π 6π u (axial) Figure The coherence time illustrating the effect of the variation in temporal phase through the addition of chirp through the chirp parameter C ( ps 2) for the stationary (a) and nonstationary (b) observation frames. τ is the initial coherence time before the objective. 71

96 Chapter 4 a 3π x v (radial) 3 2 π 3 2 π 1 1 (µ µ )/µ b v (radial) 3π 3π 3 2 π 3 2 π 3π 6π 3π 3π 6π u (axial) 6π 3π 3π 6π u (axial) 2 x (µ µ )/µ Figure Mean frequency distribution of a focused hyperbolic secant ultrashort pulse in the axial and radial plane of a.1 NA lens for stationary (a) and non-stationary (b) cases. a Fibre length (m) (τ c τ )/τ b Fibre length (m) (τ c τ )/τ 2 1 8π 4π 4π 8π u (axial).5 8π 4π 4π 8π u (axial) Figure The effects of RS on the coherence time for a focused electromagnetic wave by a lens of NA =.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a power of 1 W (relating to the field in Fig ). (a) stationary observation frame and (b) a nonstationary observation frame. 72

97 Chapter 4 a Fibre length (m) π 4π 4π 8π u (axial) (τ c τ )/τ b Fibre length (m) π 4π 4π 8π u (axial) 4 3 (τ c τ )/τ 2 1 Figure The effects of TOD on the coherence time for a focused electromagnetic wave by a lens of NA =.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a power of 1 W (relating to the field in Fig ). (a) stationary observation frame and (b) a nonstationary observation frame. length the coherence (Fig ) properties change significantly. The coherence time becomes complicated by the radiative and shifting processes of soliton dynamics. The steep phase gradient of the point of destructive interference caused by the diffraction by the lens, contributes to a modification of the field through the removal of frequency components which modifies the temporal coherence. The point of destructive interference associated with the carrier frequency of the input pulse to the PCF occurs at u = 4π and, as expected, the coherence time and bandwidth shift around this point. Since RS shifts the soliton toward infra-red wavelengths and the point of destructive interference is frequency dependent, the temporal change in the field by these points should move axially outward, which would correspond to an increase in coherence. As described earlier in Section 3.3 the TOD effect is a significant contribution in SC generation. The perturbation that the TOD effect places on the field is a dominant phase contribution and should strongly affect the coherence of its focal field. Figure shows how the PCF field perturbed by third order dispersion, relates to modifications in the focal plane of a lens of.1 NA. As expected the stationary observation frame for both the field effected by RS and the TOD effect shows strong spectral broadening which is caused by the rapid spectral expansion of the field associated with soliton formation. The nonstationary reference frame is different as it is affected by the path difference associated with the lens. 73

98 Chapter 4 a Input power (W) b (τ τ )/τ c Input power (W) (τ c τ )/τ 2 1 8π 4π 4π 8π u (axial).5 8π 4π 4π 8π u (axial) Figure The effects of RS on the coherence time for a focused electromagnetic wave by a lens of NA =.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a fibre length of.3 m (relating to the field in Fig ). (a) stationary observation frame and (b) a nonstationary observation frame. a 1.6 b 1 4 Input power (W) (τ c τ )/τ Input power (W) (τ c τ )/τ 1 8π 4π 4π 8π u (axial).6 8π 4π 4π 8π u (axial) Figure The effects of TOD on the coherence time for a focused electromagnetic wave by a lens of NA =.1, originating from a PCF pumped with an ultra-short pulse with a pulse duration of.1 ps and a fibre length of.3 m (relating to the field in Fig ). (a) stationary observation frame and (b) a nonstationary observation frame. The coherence times for the cases of RS and the TOD effect should also convey the previous effects when considering a variation in input peak power and is important as it determines the soliton order, which then relates to the structure temporal and spectral properties. Figures and shows the power dependence of both the stationary and nonstationary observation frames for the RS and the TOD effect. For the stationary case, RS and the TOD effect are dominant on the outside and inside (with respect to the origin) of the points of destructive interference, respectively. The spectral shifting associated with the spatial phase around the point of destructive interference causes an increase in coherence which is amplified by the spectral shifting properties associated with RS and the TOD effect. 74

99 Chapter 4 a Diffraction field Input field b.5.4 S/S S/S c 3π 6π 3π 3π 6π u (axial) d 2π 4π 6π u (axial) 1 v (radial) 3π 2 3π S/S (1log 1 ) Time (ps) S/S (1log 1 ) 3π 6π 3π 3π 6π u (axial) 4 1 6π 3π 3π 6π u (axial) 4 Figure The temporal effects of a SC propagating through the focus of a low NA (.1) objective. (a) On axis diffraction centred at the focal point (the full temporal evolution of the SC on the axis is described in Appendix C). (b) On axis diffraction centred at u = 5π. (c) Radial and axial diffraction pattern centred at the focal point (the full temporal evolution of the SC in the radial and axial direction is described in Appendix C). (d) Complete axial and temporal diffraction field. 4.5 Supercontinuum generation The influence of the lens results in a superposition of amplitude and phase which determines the diffracted focal distribution (Fig a). Since the SC field contains structured temporal components, as it encounters singularities, it is expected that the temporal modification would be more significant. 93 If the frequency distribution of a temporal field coincides with the frequency dependence of the points of destructive interference, a pulsed feature would be removed (Fig b). Since there is a path difference incurred across the aperture and an increased temporal and spectral extent of the SC, the diffraction in the focal plane is more dramatic (Fig c). The focal distribution on the axis (Fig d) shows the complexity of the points of destructive interference of the lens diffraction and how they manipulate the SC field temporal structure. 75

100 Chapter 4 The correlation function of a field point essentially provides a measure of the frequency component variation. The degree of coherence within a focused SC field is expected to vary dramatically due to the removal of frequencies within the temporal profile caused by the points of destructive interference and is quantified through the coherence time τ c (Figs a-d). For the stationary observation frame, the coherence time changes around the region of the points of destructive interference (Fig a) and in fact an enhancement of the coherence time occurs because of the spectral redistribution that modifies the bandwidth. Compared with the coherence time, τ =.5 ps, of the SC field before it is focused, τ c at the points of destructive interference is enhanced by a factor of 2. The coherence time in this situation is symmetric with respect to the focal plane, which is physically expected since it is contributed by a single axial position. In this case, the spatial phase contribution from the lens diffraction is unchanged during the correlation measurement, since the diffraction equation is symmetric with respect to the focal plane. This symmetry holds for larger NA lenses and thus the coherence time shows little variation with NA (Fig c). However, depending on the observational view the calculated coherence time is different. For the nonstationary observation frame (e.g. in a time resolved experiment), the coherence time shows a remarkable difference (Fig b) and is caused by the path difference incurred by the rays which pass the extremities of the lens compared to rays on the optical axis. Further, the path difference is not symmetric with respect to the focal plane. This effect can be confirmed by changing the NA (Fig d) where the coherence time changes dramatically. Such an effect leads to the enhancement of the coherence time by a factor of 3 near the point of destructive interference before the focal plane. This effect occurs because of the variation of the path difference through the focal plane and the change in sign of the spatial phase on either side of the focus. Though both observational frames are valid in a laboratory measurement, the nonstationary observation frame has greater consequences. The coherence time is strongly dependent on the temporal variance of the input field as well as the spatial phase contribution from the lens diffraction. This effect would have a strong impact on time resolved (or frequency resolved) measurements and would rely on the characteristics of the SC field and the NA of 76

101 Chapter 4 a b 3π.11 3π.18 v (radial) 3π 2 3π τ c (ps) v (radial) 3π 2 3π τ c (ps) c 3π.11 6π 3π 3π 6π u (axial) NA =.1 NA =.14 NA =.2.5 d 3π.2 6π 3π 3π 6π u (axial) NA =.1 NA =.14 NA = τ c (ps).7 τ c (ps).1.5 6π 3π 3π 6π u (axial).5 6π 3π 3π 6π u (axial) Figure The coherence time within a focused SC for the stationary and the non-stationary cases. (a) the axial and radial distribution of the coherence time for the.1 NA lens for the stationary case; (b) the axial and radial distribution of the coherence time for the.1 NA lens for the non-stationary case; (c) the effect of NA on the coherence time on the axis for the stationary case; (d) the effect of NA on the coherence time on the axis for the non-stationary case. 77

102 Chapter 4 the lens. The phase on the SC temporal profile is dependent on the physical origin of nonlinear and dispersive effects that occur because of the nonlinear PCF. The dominant effect in the initial pulse propagation through a PCF is the balance between self phase modulation and second order dispersion, as the pulse proceeds further into the fibre higher order dispersive effects become more dominant (Fig a). The ultrashort pulse initially forms a higher order soliton and at a particular point in the propagation, fissions into many fundamental solitons. The phase contribution caused by these effects can be isolated by observing the coherence time in the diffraction of a lens using an input field generated by a PCF with varying length (Fig b). Initially, the temporal coherence behaves similar to a chirped hyperbolic secant pulse shape (partially coherent source) with a predictable structure, but at a particular length corresponding to the fission length of the higher order soliton the coherence time dramatically changes. This observation confirms that an increase in phase complexity added to the original ultrashort pulse (with linear phase 25 ) is coupled with the spatial phase from the lens diffraction to modify the correlation of the electromagnetic field. The formation of high order solitons is power dependent due to SPM and it would be expected that the temporal coherence in the focal region would change dramatically. Fig shows the temporal coherence of focused electromagnetic field produced by a PCF coupled with different input power pulses. It can be seen that the stationary observation frame has a more predictable structure. As the power of the input pulse increases the spectral and temporal features of the output change (Fig a). In the focal region these spectral features coincide with frequency shifting property of the phase singularities which changes the temporal coherence. This is evident in Fig b as it can be seen that the strength of the coherence time variation changes from side to side around the phase singularity. The coherence time variation is much stronger on the inside of the phase singularity which coincides with red shifted radiation (Gbur et al. 54 ) which is understandable since this would correspond to the removal blue shifted dispersive waves from the SC spectra and hence the diffraction superposition of red shifted soliton. Similar to 78

103 Chapter 4 a b Fibre length (m) Fibre length (m) Time (ps) (1) S/S (τ c τ )/τ c π 3π 3π 6π u (axial).5 3 Fibre length (m) (2) 6π 3π 3π 6π u (axial) (τ c τ )/τ Figure Propagation of an ultrashort hyperbolic secant pulse through a nonlinear PCF. (a) field propagation as a function of fibre length; (b) coherence time for the stationary observation frame in the focal region of a.1 NA lens for different length fibre and (c) coherence time for the nonstationary observation frame in the focal region of a.1 NA lens for different length fibre. The peak input power to the photonic crystal fibre is 25 W with a pulse duration of 1 fs.(1) represents the cross section used for Fig c (blue) and (2) represents the cross section used for Fig d (blue) 79

104 Chapter 4 a 25 1 Input power (W) S (norm.) b Input power (W) Time (ps) (τ c τ )/τ c 25 6π 3π 3π 6π u (axial).5 Input power (W) (τ c τ )/τ 6π 3π 3π 6π u (axial) Figure Propagation of an ultrashort hyperbolic secant pulse through a nonlinear PCF. (a) variation of output temporal envelope by varying the input power. (b) the coherence time of the stationary observation frame of the focal region of a.1 NA lens for different for the field obtained from different input powers. (c) the coherence time of the nonstationary observation frame of the focal region of a.1 NA lens for different for the field obtained from different input powers. 8

105 Chapter 4 numerical simulation for the variation in fibre length, the nonstationary observation frame becomes complicated due to the spatial and temporal coupling effect (Fig c). Statistically, SC generation varies from pulse to pulse due to fluctuations created by noise such as spontaneous Raman scattering. The correlation and therefore the coherence time in the focus would also vary at the single pulse level. However, since the majority of applications involving a SC field involve the ensemble measurement, these fluctuations would average out and should result in minimal fluctuations in the coherence time. Physically, the temporal coherence of a field is related to the bandwidth ν by τ c = 1/ ν. Though the definition of the bandwidth is not straightforward in the case of the point of destructive interference, the mean frequency µ, introduced previously for the description of focusing a polychromatic wave 54, can be used to confirm the temporal correlation and relative frequency shifting of the focused SC wave. From both observational frames, the mean frequency would be related to the inverse of its temporal coherence (Fig ). For a SC it is expected that the frequency shifting would be much broader due to the increase in bandwidth. Since the increase in bandwidth results in a wider region of singular points, it is also expected that the spatial location of spectral shifting would be broader. However, Fig a shows a behaviour which is different from what is seen in previous literature. 54 The frequency shifting in the radial plane from the focal point moving radially outward becomes less profound. The superposition of the diffraction field and the input SC in this region makes the bandwidth narrower, causing a reduction in the magnitude of the mean frequency. Specifically, for the stationary observation frame (Fig a) the mean frequency is symmetric about the focal point which is due to the symmetric nature of the diffraction process and the spatial phase contribution remains constant. However, for the nonstationary observation frame (Fig b) the result becomes asymmetric because of the observed phenomenon in Fig b. 81

106 Chapter 4 a 3π.1 v (radial) 3π 2 3π (µ µ )/µ b 3π 6π 3π 3π 6π u (axial).1 v (radial) 3π 3π 2 3π 2 3π.5.5 (µ µ )/µ 6π 3π 3π 6π u (axial) Figure Mean frequency distribution of the focused SC in the axial and radial plane of a.1 NA lens for stationary (a) and non-stationary (b) cases. 82

107 Chapter Conclusion To summarise, it has been demonstrated that there exists a coupling between the temporal and spatial phases that arise from the diffraction of a SC field by a lens. The contribution of temporal phase from the input source superimposes with the diffraction field of a low NA lens to modify the bandwidth of the input which alters its correlation. At a particular point where the pulse incurs a phase in the evolution through the PCF, which is attributed to soliton formation and fission, changes the correlation from a simple predictable structure to a complex structure. These effects can be observed from two different observation frames which gives rise to significantly different coherence times. Consequently, for a nonstationary observer with the addition of complex temporal input phase can enhance the coherence time by a factor of 3. The alteration of bandwidth is extremely important and would change the excitation frequency range that can be applied in microscopy applications involving multiple wavelength excitation. The interesting effects on the bandwidth and temporal correlation at points of destructive interference could provide interesting dynamics for applications such as optical vortex metrology where these singular regions provide signatures for phase unwrapping. In all these applications, the SC source provides the capability to tailor the temporal coherence and bandwidth within the focal region for a particular application. 83

108 Chapter 4 84

109 Chapter 5 Vectorial Diffraction 5.1 Introduction When an electromagnetic field is focused by a high numerical aperture (NA) lens, energy is transferred from the incident polarisation state to the transverse orthogonal and the longitudinal field components, which is called depolarisation. The transfer of energy due to depolarisation is related to a change in coherence, which physically can be quantified through the coherence time of each vectorial component and the coherence time of the cross correlation between vectorial components. The theoretical treatment of the coherence effects of a vector field have been previously investigated by Wolf in and by Dennis in However, these studies only investigate the frequency dependence of an incident polychromatic wave. The extension that is made in this thesis is to investigate how these correlations influence the temporal aspect of a propagating wave such as supercontinuum (SC) field in the focal region of a high NA lens. Physically, the spatial and temporal phase coupling which was presented in Chapter 4 is not restricted to scalar fields and would manifest as a cross phase coupling between vectorial field components. Birefringence is an important property of a photonic crystal fibre (PCF) because it allows the capability of maintaining the polarisation state by creating both strong modal guidance and spectrally dependent vectorial field components. The complicated temporal phase associated 85

110 Chapter 5 with the birefringent modes of the SC field couples with the spatial phase from the diffraction by the lens 93, which would produce interesting correlations under vectorial diffraction conditions. The aim of this chapter is to provide a detailed theoretical description of the degree of coherence of a SC field under vectorial diffraction conditions. We present the coherence relationship between the field components produced by depolarisation under high NA diffraction and the relationship between the SC fields produced by a highly birefringent PCF when diffracted by a lens of high NA. 5.2 Three-dimesional coherence matrix The characterisation of the degree of coherence for a vectorial field E (V,ω) begins with the correlation equation given by Eq. (2.5.2), which extends to a coherence matrix and is calculated by gmn 1 E m (V,t),E n (V,t + τ) (V,τ) = [ E m (V,t) 2 E n (V,t) 2 ] 1/2 g xx(v,τ) 1 gxy(v,τ) 1 gxz(v,τ) 1 = g yx(v,τ) 1 gyy(v,τ) 1 gyz(v,τ) 1, (5.2.1) gzx(v,τ) 1 gzy(v,τ) 1 gzz(v,τ) 1 where m and n are the polarisation states in the spatial directions x,y,z and V represents the collective dimensions of the diffraction volume. For m and n = x, g 1 represents the autocorrelation of the electric field component with a polarisation orientation in the x direction. For m n, g 1 represents the cross correlation of the vector components of the field. Physically, this matrix quantifies the transfer of energy between field components and provides the ability to analyse the polarisation properties of the degree of coherence for the focal region. The components of the field for a linear polarisation state with an arbitrary polarisation angle under vectorial diffraction can be determined by Eqs. (2.4.1), 86

111 Chapter 5 (2.4.12) and (2.4.13). When combined these equations form the following set of equations E x (V,t) = iω 2c (ai (V,t) + (a cos (2ψ) + b sin (2ψ))I 2 (V,t)), (5.2.2) E y (V,t) = iω 2c (bi (V,t) + (a sin (2ψ) + b cos (2ψ))I 2 (V,t)), (5.2.3) E z (V,t) = iω 2c (2i (a cos (ψ) + b sin (ψ))i 1 (V,t)). (5.2.4) 87

112 Chapter 5 Table Contributions to the field E for the x y and z axes. axis ψ E h ( iω 2c ) Ev ( iω 2c ) E( iω 2c ) E h x = I (z,t) + I 2 (z,t) E v x = E x = ai (z,t) + ai 2 (z,t) z E h y = E v y = I (z,t) I 2 (z,t) E y = bi (z,t) bi 2 (z,t) E h z = E v z = E z = 88 E h x = I (y,t) I 2 (y,t) E v x = E x = ai (y,t) ai 2 (y,t) y 9 or 27 E h y = E v y = I (y,t) + I 2 (y,t) E y = bi (y,t) + bi 2 (y,t) E h z = E v z = 2iI 1 (y,t) E z = b2ii 1 (y,t) E h x = I (x,t) + I 2 (x,t) E v x = E x = ai (x,t) + ai 2 (x,t) x or 18 E h y = E v y = I (x,t) I 2 (x,t) E y = bi (x,t) bi 2 (x,t) E h z = 2iI 1 (x,t) E v z = E z = a2ii 1 (x,t)

113 Chapter 5 The vectorial field components which contribute to the degree of coherence for the x, y and z axes are shown in Table The degree of coherence for a linear polarisation state with an arbitrary polarisation angle propagating in the directions x, y and z can be determined in terms of the field components I, I 1 and I 2, which are given by gxx 1 (x,τ) = ai (x,t) + ai2 (x,t),ai (x,t + τ) + ai 2 (x,t + τ) [ ai (x,t) + ai 2 (x,t) 2 ai (x,t) + ai 2 (x,t) 2 ] 1/2, (5.2.5) gxy 1 (x,τ) = ai (x,t) + ai2 (x,t),bi (x,t + τ) bi 2 (x,t + τ) [ ai (x,t) + ai 2 (x,t) 2 bi (x,t) bi 2 (x,t) 2 ] 1/2, (5.2.6) g 1 xz (x,τ) = ai (x,t) + ai2 (x,t),a2ii 1 (x,t + τ) [ ai (x,t) + ai 2 (x,t) 2 a2ii 1 (x,t) 2 ] 1/2, (5.2.7) gyx 1 (x,τ) = bi (x,t) bi2 (x,t),ai (x,t + τ) + ai 2 (x,t + τ) [ bi (x,t) bi 2 (x,t) 2 ai (x,t) + ai 2 (x,t) 2 ] 1/2, (5.2.8) gyy 1 (x,τ) = bi (x,t) bi2 (x,t),bi (x,t + τ) bi 2 (x,t + τ) [ bi (x,t) bi 2 (x,t) 2 bi (x,t) bi 2 (x,t) 2 ] 1/2, (5.2.9) g 1 yz (x,τ) = bi (x,t) bi2 (x,t),a2ii 1 (x,t + τ) [ bi (x,t) bi 2 (x,t) 2 a2ii 1 (x,t) 2 ] 1/2, (5.2.1) gzx 1 (x,τ) = a2ii 1 (x,t),ai (x,t + τ) + ai 2 (x,t + τ) [ a2ii 1 (x,t) 2 ai (x,t) + ai 2 (x,t) 2 ] 1/2, (5.2.11) gzy 1 (x,τ) = a2ii 1 (x,t),bi (x,t + τ) bi 2 (x,t + τ) [ a2ii 1 (x,t) 2 bi (x,t) bi 2 (x,t) 2 ] 1/2, (5.2.12) g 1 zz (x,τ) = a2ii1 (x,t),a2ii 1 (x,t + τ) [ a2ii 1 (x,t) 2 a2ii 1 (x,t) 2 ] 1/2, (5.2.13) 89

114 Chapter 5 gxx 1 (y,τ) = ai (y,t) ai2 (y,t),ai (y,t + τ) ai 2 (y,t + τ) [ ai (y,t) ai 2 (y,t) 2 ai (y,t) ai 2 (y,t) 2 ] 1/2, (5.2.14) gxy 1 (y,τ) = ai (y,t) ai2 (y,t),bi (y,t + τ) + bi 2 (y,t + τ) [ ai (y,t) ai 2 (y,t) 2 bi (y,t) + bi 2 (y,t) 2 ] 1/2, (5.2.15) g 1 xz (y,τ) = ai (y,t) ai2 (y,t),b2ii 1 (y,t + τ) [ ai (y,t) ai 2 (y,t) 2 b2ii 1 (y,t) 2 ] 1/2, (5.2.16) gyx 1 (y,τ) = bi (y,t) + bi2 (y,t),ai (y,t + τ) ai 2 (y,t + τ) [ bi (y,t) + bi 2 (y,t) 2 ai (y,t) ai 2 (y,t) 2 ] 1/2, (5.2.17) gyy 1 (y,τ) = bi (y,t) + bi2 (y,t),bi (y,t + τ) + bi 2 (y,t + τ) [ bi (y,t) + bi 2 (y,t) 2 bi (y,t) + bi 2 (y,t) 2 ] 1/2, (5.2.18) g 1 yz (y,τ) = bi (y,t) + bi2 (y,t),b2ii 1 (y,t + τ) [ bi (y,t) + bi 2 (y,t) 2 b2ii 1 (y,t) 2 ] 1/2, (5.2.19) gzx 1 (y,τ) = b2ii 1 (y,t),ai (y,t + τ) ai 2 (y,t + τ) [ b2ii 1 (y,t) 2 ai (y,t) ai 2 (y,t) 2 ] 1/2, (5.2.2) gzy 1 (y,τ) = b2ii 1 (y,t),bi (y,t + τ) + bi 2 (y,t + τ) [ b2ii 1 (y,t) 2 bi (y,t) + bi 2 (y,t) 2 ] 1/2, (5.2.21) g 1 zz (y,τ) = b2ii1 (x,t),b2ii 1 (x,t + τ) [ b2ii 1 (x,t) 2 b2ii 1 (x,t) 2 ] 1/2, (5.2.22) gxx 1 (z,τ) = ai (z,t) + ai2 (z,t),ai (z,t + τ) + ai 2 (z,t + τ) [ ai (z,t) + ai 2 (z,t) 2 ai (z,t) + ai 2 (z,t) 2 ] 1/2, (5.2.23) gxy 1 (z,τ) = ai (z,t) + ai2 (z,t),bi (z,t + τ) bi 2 (z,t + τ) [ ai (z,t) + ai 2 (z,t) 2 bi (z,t) bi 2 (z,t) 2 ] 1/2, (5.2.24) gyx 1 (z,τ) = bi (z,t) bi2 (z,t),ai (z,t + τ) + ai 2 (z,t + τ) [ bi (z,t) bi 2 (z,t) 2 ai (z,t) + ai 2 (z,t) 2 ] 1/2, (5.2.25) gyy 1 (z,τ) = bi (z,t) bi2 (z,t),bi (z,t + τ) bi 2 (z,t + τ) [ bi (z,t) bi 2 (z,t) 2 bi (z,t) bi 2 (z,t) 2 ] 1/2, (5.2.26) g 1 xz (z,τ) = g 1 zx (z,τ) = g 1 yz (z,τ) = g 1 zy (z,τ) = g 1 zz (z,τ) =, (5.2.27) respectively. The set of coherence functions can be use to determine the coherence 9

115 Chapter 5 times of the focus under conditions of vectorial diffraction, which are given by τmn(v c ) = gmn 1 (V,τ) 2 dτ τ xx(v c ) τxy(v c ) τxz(v c ) = τ yx(v c ) τyy(v c ) τyz(v c ). (5.2.28) τzx(v c ) τzy(v c ) τzz(v c ) For a horizontal polarisation state (a = 1,b = ), the coherence times for the x, y and z axes are given by τxx c (x) = τxz c (x) = τzx c (x) = τzz c (x) = τxx c (y) = τxx c (z) = g 1 xx (x,τ) 2 dτ, (5.2.29) g 1 xz (x,τ) 2 dτ, (5.2.3) g 1 zx (x,τ) 2 dτ, (5.2.31) g 1 zz (x,τ) 2 dτ, (5.2.32) g 1 xx (y,τ) 2 dτ, (5.2.33) g 1 xx (z,τ) 2 dτ. (5.2.34) The coherence times that are not defined in Eqs. (5.2.29) - (5.2.34) are equal to zero, which are caused by the polarisation coefficient b =. When the NA is below.7 the effects of depolarisation can be neglected and the terms I 1 and I 2 =. Under these conditions the field E reduces to a scalar field determined by I, where the degree of coherence and the coherence time are given by 91

116 Chapter 5.5 NA = 1 NA =.1.25 (τ c τ )/τ π 3π 3π 6π u (z) Figure A comparison between the coherence times for a lens of NA = 1 and.1 with hyperbolic secant ultrashort pulse with a width of.1 ps. g 1 mn (V,τ) = g 1 xx (V,τ) = τ c xx (V ) = I (V,t),I (V,t + τ) [ I (V,t) 2 I (V,t) 2 ] 1/2, (5.2.35) g 1 xx (V,τ) 2 dτ, (5.2.36) respectively. Equations (5.2.29) - (5.2.36) are used in Section to characterise the focus of a SC field under vectorial diffraction conditions. Consider the general case of a hyperbolic secant with a pulse duration of.1 ps. The coherence time for a NA =.1 under vectorial diffraction conditions is shown in Fig , which gives an identical result to the coherence time produced by Fresnel diffraction. Under high NA vectorial diffraction conditions the coherence of the field is no longer influenced by the point of destructive interference, which is due to depolarisation. The final mathematical analysis involves an incident field with a linear polarisation orientation at 45. The incident field is given by E i 45 = E hi + E vi = Ex hi Ey hi E hi z Ex vi Ey vi E vi z. (5.2.37)

117 Chapter 5 When diffracted by a high NA the degree of coherence for the field E 45 becomes complicated. For this investigation the degree of coherence is calculated for only the optical axis where the coherence matrix is determined by Eqs. (5.2.23) - (5.2.26). The coherence times generated by the degree of coherence for E 45 is given by τmn c (z) = τ xx c (z) τxy c (z). (5.2.38) τyx c (z) τyy c (z) The observation is along the optical axis, where the vectorial diffraction contribution from the E z component is zero and is why the coherence time has only contributions from E x and E y. Equations (5.2.23) - (5.2.26) and (5.2.38) are used in Section to understand the influence of cross coupling in the degree of coherence for a SC field in the focal region. 5.3 Vectorial diffraction of a supercontinuum Linear Polarisation For a linear polarisation state the degree of coherence and the coherence time are determined by the theoretical derivations in Section 5.2. It is expected that the coherence times for the electric field in the direction of the incident polarisation state E x would be influenced by the points of destructive interference. The coherence time for the SC diffraction by a lens is shown in Fig for the x, y and z axes. The input polarisation state to the PCF is in the x direction and the analysis is for the autocorrelation of the field component E x determined by Eqs. (5.2.29), (5.2.33) and (5.2.34). Figure shows three key effects: the influence of spatial phase through the points of destructive interefence on the field; the reduction of the coherence time with increased NA; and a lateral (x and y axes) and a longitudinal (z axis) shift in the coherence time. The gradual shift inward (y) and outward (x) is due to the change in superposition of the wave as it passes through the lens. The modification of the field by the spatial phase associated with the lens changes the field E x to becoming slightly asymmetric (over the xy plane) which is only seen 93

118 Chapter 5 a NA =.1 NA =.3 NA =.7 NA = 1 b NA =.1 NA =.3 NA =.7 NA = 1 (τ c xx τc )/τc.8.4 (τ c xx τc )/τc.8.4 c (τ c xx τc )/τc.4 4π 2π 2π 4π v (x) NA =.1 NA =.3 NA =.7 NA = 1.4 8π 4π 4π 8π u (z) d (τ c mn τc )/τ.4 4π 2π 2π 4π v (y) mn = xx mn = xz = zx mn = zz.4 4π 2π 2π 4π v (x) Figure The coherence time of the diffraction by a lens of varying numerical aperture along the x (a), y (b) and z (c) axes. These coherence times are calculated for the autocorrelation of the electric field in the direction of the E i (E x ). (d) the coherence times for the diffraction by a lens of NA = 1 along the x axis, which contains the autocorrelation and cross-correlation coherence times with respect to the E x and E z fields. under higher NA conditions. Along the optical axis a more significant change occurs and is due to the superposition condition no longer forming points of destructive interference under higher NA conditions. The coherence time for a NA =.1 is identical to the coherence time obtained for the stationary observation frame shown in Fig c. For the transverse axis x the field component E z leading to a coherence matrix containing cross coupling correlation terms between E x and E z. Figure 5.3.1d shows the coherence times simulated using Eqs. (5.2.29) - (5.2.32). The interesting observation is that the coherence time generated by the cross coupling of the field components (mn = xz) is not a simple superposition of the coherence times generated by the autocorrelations (mn = xx and mn = zz). This effect is understandable because the correlation is dependent on the phase structure of 94

119 Chapter 5.4 Mean of τ c xx x y z NA Figure The mean coherence time of a SC as a function of NA for the x, y and z axes. each component of the incident field and the transfer energy due to depolarisation. Effectively, the coherence time formed by the cross coupling between the polarisation states is a measure of the longevity of the elliptical polarisation produced by depolarisation. The influence of the NA can be quantified by calculating the mean of the coherence time as a function of NA and is shown in Fig By increasing the NA, there exists a redistribution of energy within the E x component from the y axis to the x axis, which alters the degree of coherence of the field. Along the optical axis the change in the degree of coherence is different as there no longer exists points of destructive interference, leading to a change in the mean coherence time by an order of magnitude from low NA (.4) to the high NA (1). The diffraction by a lens has been shown in Chapter 4 to be influenced by phase. Since the SC field is generated by an accumulation of phase associated with nonlinearity and dispersion, it becomes important to assess how the degree of coherence changes with phase under vectorial diffraction conditions. A method for analysing this change is by calculating the diffraction of the SC produced by a variation in fibre length (Fig ) along the optical axis of the focal region of a lens of NA = 1 and is shown in Fig As the initial ultrashort pulse travels through the optical fibre it accumulates phase which changes the spectral and temporal components. There exists a point in the evolution where the temporal coherence dramatically changes which is due to the formation and annihilation of a 95

120 Chapter 5 Fibre length (m) π 4π 6π u (z) (τ c xx τc )/τc Figure The coherence time of the autocorrelation of the diffraction by a lens of NA = 1 the electric field E x with variation in the fibre length. Input power (W) π 4π 6π u (z) (τ c xx τc )/τc Figure The power dependence of coherence time in the focus of a NA = 1 lens for input fields generated by the nonlinear PCF of varying input power. The coherence time is for a linear polarised field orientated along the x direction. high order soliton and is similar to behavior discussed in Chapter 4. However, under vectorial diffraction conditions the fluctuation in coherence is less profound, which is attributed to depolarisation and the reduced influence of the points of destructive interference. Similar to Chapter 4, the degree of coherence is dependent on the input power to the PCF and is shown in Fig For low input powers ( 5 W) the variation in coherence time is small, which occurs because the phase on the pulse is dominated by dispersion effects and has little influence from soliton dynamics since the soliton order is small (1 5). With increased power the coherence time is expected to change due to the increased dominance of nonlinearity. The higher input power increases the initial order of the soliton which then after fission changes 96

121 Chapter 5 the degree of coherence, which enhances the coherence time Coupled mode propagation So far the analysis has been restricted to a linear incident polarisation state, which emphasises the depolarisation inherent from a high NA lens. The coupled mode nonlinear Schrödinger equation allows the ability to simulate a SC field with a polarisation orientation at 45 which can occur in highly birefringent PCF. The output spectrograms of the SC field emerging from a highly birefringent PCF is shown in Fig The field was generated using the dispersion and nonlinear parameters discussed in Chapter 3 with a pulse duration of 1 fs and a peak power of 25 W. Also shown is the propagation of the ultrashort pulse along the fibre with a length of.15 m. a 55 b 55 Frequency (THz) S (norm. 1log 1 ) Frequency (THz) S (norm. 1log 1 ) c Fibre length (m) Time (ps) Time (ps) S/S d Fibre length (m) Time (ps) Time (ps) S/S Figure The PCF output field for an incident polarisation state at 45.(a) the horizontal (x) polarisation state, (b) the vertical (y) polarisation state, (c) the horizontal (x) polarisation state as a function of fibre length, and (c) the vertical (y) polarisation state as a function of fibre length. In the focal plane of the lens, these coupled modes should affect the coherence matrix and the coherence times. Fig shows the coherence times produced 97

122 Chapter 5.15 (τ c mn τc )/τc mn = xx mn = xy = yx mn = yy.15 6π 3π 3π 6π u (z) Figure The coherence time for the autocorrelations and cross correlations calculated for the diffraction by a lens of NA = 1 along the optical axis for the SC field generated in Fig from Eq. (5.2.38). It is evident that the coherence time for the cross correlation between the modes is no longer the superposition between the autocorrelated fields and occurs because of their non-constant relative phase. The phase due to nonlinearity and dispersion can be isolated along the fibre length to understand the influence of polarisation on the degree of coherence. Figure shows the polarisation coherence times occurring due to the coupled modes of the PCF along the optical axis as a function of the fibre length. The autocorrelations behave in the same manner as depicted in Fig and Fig , the degree of coherence changes with input phase. The cross correlated degree of coherence in Fig b shows modulations which occur due to the differences in phase between the fibres modes. As discussed in Chapter 3 and 4, the soliton fisson dynamics in SC generation contains spectral expansion and contractions processes. The differences between the soliton fluctuations of the fibre modes could be attributed to the modulations shown in the cross correlated degree of coherence and the coherence time in Fig b. In both cases of a linear polarised SC field and a 45 polarised SC field, the coherence time has a greater variance after the formation of the soliton and the rapid spectral expansion of the SC field. After this point, the coherence within the focal region becomes dominated by interference between dispersive waves and the fundamental solitons. The expansive spectral features of a SC can only be obtained by coupling an 98

123 Chapter 5 a.15.1 Fibre length (m) (τ c xx τc )/τc b Fibre length (m) c π 4π 6π u (z) 2π 4π 6π u (z) (τ c xy τc )/τc Fibre length (m) (τ c yy τc )/τc 2π 4π 6π u (z) Figure The coherence time for the autocorrelations and cross correlations calculated for the diffraction by a lens of NA = 1 as a function of fibre length along the optical axis. (a) coherence time produced by the autocorrelation of E x ; (b) coherence time produced by the cross correlation E x and E y ; and (c) coherence time produced by the autocorrelation of E y. 99

124 Chapter 5 ultrashort pulse of sufficient power to instigate the formation of solitary waves. The effects of temporal phase on the focal region can also be analysed by observing the change in coherence time as function of input power to the PCF (Fig ). At low input powers, the output spectra of the PCF is dominated by the dispersion of the fundamental solitons and under these conditions the phase accumulated through propagation is relatively simple. The cross coupling between the focused coupled modes in this case is shown as a small change in coherence time. With the increase in input power the SC is formed by the amalgamation of nonlinear and dispersive processes, and as expected rapidly expands the bandwidth. The cross coupling between the focused coupled modes becomes complicated because of the superposition of their differing phase, which results in subtle changes in the coherence time (Fig b). 5.4 Conclusions In summary, the optical field components occurring because of depolarisation by the diffraction of a high NA lens reduces the coherence time along the optical axis which is attributed to the superposition of the wavefront no longer forming points of destructive interference. Under conditions of vectorial diffraction, the mean coherence time will change by an order of magnitude when the NA changes from a low NA ( -.4) to a high NA of 1. For the transverse axes the mean coherence time increases and decreases in the x direction and y directions, respectively, which is also due to depolarisation. When considering the case of a vector field, the components of the field create interesting cross coupling characteristics, which are determined by a coherence matrix. When the SC modes of a highly birefringent PCF are focused by a high NA objective, the coherence times produced by their autocorrelations are different due to the phase differences between the modes. In addition, the coherence time for the degree of coherence between these two modes (cross coupled coherence time) is significantly different. In these cases of auto and cross correlation the temporal phase is significantly contributing to the degree of coherence in the focal region, to 1

125 Chapter 5 a 25 Input power (W) (τ c xx τc )/τc 2π 4π 6π u (z) b 25 Input power (W) (τ c xy τc )/τc c 2π 4π 6π u (z) 25.6 Input power (W) π 4π 6π u (z) (τ c yy τc )/τc Figure The power dependence of coherence time in the focus of a NA = 1 lens for input fields generated by the nonlinear PCF of varying input power. The coherence matrix is for a linear polarised field orientated at 45 to the x direction. (a) coherence time produced by the autocorrelation of E x ; (b) coherence time produced by the cross correlation E x and E y ; and (c) coherence time produced by the autocorrelation of E y. 11