Sodium Line Guide Star Laser using Optical Parametric Oscillator

Transcription

1 Sodium Line Guide Star Laser using Optical Parametric Oscillator Khu Tri Vu A thesis submitted for the degree of Bachelor of Science with Honours at The Australian National University November, 2003

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3 Declaration This thesis is an account of research undertaken between February 2003 and November 2003 at the Laser Physics Centre, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, Australia. Except where acknowledged in the customary manner, the material presented in this thesis is, to the best of my knowledge, original and has not been submitted in whole or part for a degree in any university. Khu Tri Vu 14 November, 2003 iii

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5 Acknowledgements There has been so much help from everyone who I know and have met during the course that I would like to thank them deeply. Although I am trying to name some people that I can think of at the moment of writing, there are many others who I may fail to mention. First of all I would like to say thank you to my supervisor, Prof. Barry Luther-Davies. He is the best experimentalist and supervisor I have ever met and worked with. He has given me a great deal of enthusiasm from the first moment he described the project to the last minute. He has been the most encouraging person, who always spent a great amount of time with me during the year. He was always there willing to answer and guide me through any aspect of the project. He was actively thinking ahead of all from the smallest thing to the most important. I would like to thank all the other Honours students for being such a nice and supportive group. My special gratitude to Kristen Feher, Steward Wilson, Trudy Peaty and Tom Argue for being the closest friends. With them I have had great moments of sharing and laughing. I also greatly appreciate the effort made by Craig Savage for being a very supportive coordinator. I would like also express my deep gratitude to all staff and students at the Laser Physics Centre for the welcoming and exciting moments and help they have provided during the year. Life at the Research School of Physical Science and Engineering at Australian National University has been very exciting. I would like to thanks all support staff for making it such an interesting and welcoming place. I also would like to thank Australian National University for supporting me with the Honours Scholarship. I am grateful to the Physics Department, specially Allan Baxters, for letting me demonstrate a first year physics class which has brought not only some financial support, but more importantly a lot of fun. Most important of all, this is a tribute to my parents who are always my love and encouragement even at a very far distance. I am so grateful for every single thing that they have done for me. They are always the motivation for every thing I have done in my life. Finally I would like to thank my girlfriend, Uyen Nguyen, for being so patient and loving. My life is so much happier and more fulfilled with her presence in my heart. v

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7 Abstract The aim of this project is to develop a novel, high average power source of highly coherent radiation at the 589nm sodium D 2 resonance line for use in creating a laser guide star for use with adaptive optics systems such as astronomical telescopes and satellite ranging. Adaptive optics is used to improve the resolution of ground base telescope by compensating the atmospheric turbulence. To achieve this it is necessary to measure and correct the distortions caused by the atmosphere on the beam. A laser guide star provides a beacon which combined with a wavefront sensor allows the effect of the atmosphere to be quantified and mechanical corrections made to the adaptive optics in the telescope to cancel those distortions. Whilst laser guide stars have been demonstrated as effective in a number of astronomical observatories, at present there is no laser technology that dominates this application. This project will demonstrate a novel approach to the creation of high average power optical sources for guide star applications. The aim of the project is to build a prototype of high average power, synchronously pumped optical parametric oscillators pumped by novel high power solid state laser technology. The Magnesium Oxide doped periodically poled Lithium Niobate was used as the nonlinear crystal for the 532nm pumped singly resonant parametric oscillator. The oscillator was pumped by 1.4W of 532nm light which was created from a 3.5W mode-locked Nd:YVO 4 laser by a Lithium Triborate second harmonic generator. With a grating period of 12.05µm, the signal at the wavelength of 594nm was obtained at 150 o C. There was unexpected green absorption of the crystal, which undermined the performance of the oscillator. However, from the experimental results of a Potassium Titanyl Phosphate optical parametric oscillator, it was demonstrated that highly efficient parametric oscillator can be built despite the idler absorption. vii

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9 Contents Declaration Acknowledgements Abstract iii v vii 1 Introduction Motivation Review of Laser Guide Star Optical Parametric Oscillators: An Approach Summary of the Thesis Chapter 2: Nonlinear Optics Introduction to nonlinear optics Wave equation Nonlinear Susceptibilities Properties of the Nonlinear Susceptibility Contracted Notation d-coefficient and effective susceptibility Second Harmonic Generation Wave-Mixing Parametric Amplification Phase Matching Angle Tuning Temperature Tuning Quasi Phase Matching Chapter 3: Optical Parametric Oscillator Parametric Amplification with Gaussian Beam Focused Gaussian Beam (Near-Field) Optimised Confocal Near-Field Gaussian Beam Common Confocal Parameter Gaussian Beams Arbitrary Focused Gaussian Beams Optical Parametric Oscillator Singly Resonant Parametric Oscillator Threshold Efficiency Synchronously Pumped OPOs Parametric Oscillator with Idler Absorption Magnesium Oxide doped Periodically Poled Lithium Niobate Lithium Niobate MgO:Periodically Poled Lithium Niobate PPLN with Ionic-Nonlinearity Enhanced Parametric Gain ix

10 x Contents 4 Chapter 4: Design of Guide Star Optical Parametric Oscillator Overview Threshold Calculation Grating Period of MgO:PPLN for Guide Star OPO Cavity Configuration Mode Matching the Pump Beam Chapter 5: Potassium Titanyl Phosphate Optical Parametric Oscillator Experiment KTP OPO Overview Laser Source Laser Amplifier KTP OPO KTP crystal Cavity Configuration and Pump Focusing Threshold Calculation KTP OPO Measurement and Results Threshold Efficiency Cavity Detuning Signal Bandwidth Implications to Guide Star OPO Chapter 6: Performance of Prototype Guide Star Optical Parametric Oscillator Overview Second harmonic generation MgO:PPLN crystal Cavity Configuration and Pump focusing Threshold Calculation Performance of Guide Star Prototype Conclusion 65 Bibliography 67

11 List of Figures 1.1 A typical set up for adaptive system, the wavefront sensor provide a feedback to the deformation mirror, which corrects the distorted wavefront. A guide star is needed to provide high enough intensity for the wavefront sensor (Source: Gemini adaptive telescope) A comparison between normal telescope image and adaptive optics telescope image. This is the image of a binary star: on the left is without adaptive optics, on the right adaptive optics is used. (Source: Keck Observatory) Second harmonic generation Wave-mixing process in a nonlinear medium Parametric amplification of the signal and idler in plane wave, non-depleted pump approximation The dependence of second harmonic intensity against phase mismatch as a sinc function of kz Dispersion of the refractive indices of a negative uniaxial crystal Angle phase matching for second harmonic generation Poling of crystal in QPM A near-field Gaussian beam. The confocal parameter b is longer than the crystal length L Parametric gain reduction factor as a function of focusing parameter ξ of both pump and signal beams Variation of the SRO function h sm as a function of focusing parameter ξ s = l/b s, with k=1.5 and two values of ξ p = l/b p = 1, Singly resonant parametric oscillator Conversion efficiency as a function of the number of times the pump power exceeds threshold for the SRO with Gaussian beam a) Output signal efficiency as a function of effective detuning time at various peak pump intensity and b)output efficiency as a function of peak pump intensity at various normalised effective detuning times Transmission through a 1-mm x-cut Lithium Niobate crystal at 28 o and 130 o [1] Refractive index of undoped congruently grown Lithium Niobate as a function of wavelength Schematic layout of the guide star optical parametric oscillator Schematic of a cavity of the guide star SROPO An equivalent model of the OPO cavity to calculate the beam parameters at crystal and mirrors Signal waist radius at the crystal as a function of two curved mirrors separation 46 xi

12 xii LIST OF FIGURES 4.5 Signal waist radius at the end mirrors as a function of two curved mirrors separation Schematic of pump beam focusing setup. A convergent lens is used to focus the pump beam to a suitable waist radii. The curved mirror of the cavity acts as a diverging lens Waist of the pump at the crystal using the convergent lens with f 1 =450mm as a function of the distance between the lens and the mirror Distance from the pump waist to the first curved mirror of the cavity d against the distance L between the lens and the curved mirror Layout of the KTP OPO: 1W Nd:YVO 4 SESAM laser, an 3.5 gain doublepass amplifier, and a KTP OPO cavity Top and side view of the amplifying crystal side pumped by two diodes Thermal lensing modified the horizontal profile of the output beam in single pass mode Double pass amplifier; left: side view (vertical plane), right: top view (horizontal plane) Layout of the KTP OPO cavity Pump beam profile at the crystal Measured and calculated threshold of the KTP OPO against the output coupler transmission Pump depletion and output power against pump power Pump depletion and output efficiency against number of times pump exceeded threshold power The signal output power against the cavity length detuning. The effect was not symmetric due to the dispersion of the crystal The signal bandwidth as a function of number of times the pump power exceeded the threshold power Detailed experiment setup of the prototype guide star OPO. There were four main components: Nd:YVO 4 SESAM laser, amplifier, second harmonic generator and parametric oscillator Beam diameter at 13.5% intensity in vertical and horizontal direction at the crystal MgO:PPLN with three poling periods of 2.05µm, 12.15µm, and 12.25µm in z-direction Radii of the signal at the crystal and the end mirrors Temperature tuning of MgO:PPLN Threshold as a function of temperature MgO:PPLN with three poling periods of 2.05µm, 12.15µm, and 12.25µm in z-direction

13 List of Tables 1.1 The estimated efficiencies of the three schemes can be used to produce 589nm Sodium line Phase-matching methods for uniaxial crystals [2] Formulae for the steady-state threshold pump power for singly resonant optical parametric oscillator Optical properties of Lithium Niobate [3], [4] Sellmeier coefficients for congruently grown Lithium Niobate Sellmeier coefficients for congruently grown Lithium Niobate with 5 mol.% MgO Sellmeier coefficients for congruently grown Lithium Niobate for Equation Temperature-dependent Sellmeier coefficients for congruently grown Lithium Niobate with 5 mol.% MgO

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15 Chapter 1 Introduction 1.1 Motivation In 1953 the astronomer Horace Babcock proposed the concept of adaptive optics as a method for improving the resolution of an astronomical telescope by correcting distortions introduced by the atmosphere [5]. The idea was to insert a wavefront corrector between the output of the telescope and the imaging camera to alter the phase of the optical wave to exactly cancel the aberrations caused by the atmosphere. Since the atmospheric distortions are dynamic, the wavefront corrector must be adjustable on the ms time scale (See Figure 1.1). An example of the wavefront corrector is a deformable mirror where actuators can push or pull local regions of a thin glass plate that comprises the mirror surface, locally changing the optical phase [6]. In addition to developing methods of fabricating and adjusting such a mirror, the major challenge for adaptive optics is to determine the phase error on the beam at the positions of the actuators. To achieve this it is necessary to measure the aberrations dynamically and this requires an optical source, a beacon, outside the atmosphere as well as a local phase front sensor. This sensor essentially measures the phase of the incoming light from the beacon relative to a local reference plane wave, and generates an error signal that moves the actuators controlling the deformable mirror to eliminate that error. The beacon itself can of course be a bright star, typically between 5 th and 15 th magnitude, but a significant problem arises because there are not enough stars to correct the atmospheric distortion over more than 1% of the sky. However, atmospheric distortions are very localised and hence even if one monitors the distortions in one direction, they will be different when observing an object in a direction only a few arc seconds away. As a result, an adaptive optics system will only work when the sky is sampled within about one arc-second of the primary observation direction. Artificial laser guide stars are needed to cover any position on the sky. The most useful guide star mechanism involves resonant scattering from sodium atoms in the mesosphere between 85 and 95km above the earth s surface. A high power laser operating at the sodium D 2 resonance line close to 589nm optically pumps the mesospheric sodium layer exciting resonance fluorescence from that layer providing a bright, localised source that forms the artificial beacon. The position of this beacon can be moved to the direction the telescope is pointing to and hence, phase front correction can be made for any observation angle. The main challenge associated with laser guide star technology is to obtain sufficient photon flux from the guide star to provide good signal to noise ratio in the wavefront sensor. In the case of infrared astronomy the required photon flux is quite low at 100photons/cm 2 /s, whereas for visible satellite ranging the required flux is much larger at >1000photons/cm 2 /s [7]. Examples of the laser guide star on Keck Observatory 3

16 4 Introduction Figure 1.1: A typical set up for adaptive system, the wavefront sensor provide a feedback to the deformation mirror, which corrects the distorted wavefront. A guide star is needed to provide high enough intensity for the wavefront sensor (Source: Gemini adaptive telescope) and the improvement in resolution obtained for astronomical observations is shown in Figure 1.2. Ao off AO on Figure 1.2: A comparison between normal telescope image and adaptive optics telescope image. This is the image of a binary star: on the left is without adaptive optics, on the right adaptive optics is used. (Source: Keck Observatory) Laser excitation of mesospheric sodium is a complex matter, but has now been the subject of extensive studies for guide star applications [8]. Since the sodium concentration is rather low ( atoms/cm 3 ) it is easy to saturate the transition and this means that careful attention needs to be paid to the operating mode of the laser to ensure that the highest fluorescence yields. The main D 2 line consists of a manifold of 24 magnetic sub-states spanning a total line-width of 3GHz. The quantum selection rules indicate optical pumping by circularly polarised light is preferable, providing a substantial increase in signal return compared with non-optically pumped populations [9]. The very low saturation intensity (6.3mW/cm 2 ) for narrow band (10MHz wide) excitation strongly suggests

17 1.2 Review of Laser Guide Star 5 pumping over the full 3GHz line-width is desirable to increase the saturation intensity to around 5W/cm 2. The radiative lifetime of the level is 16ns suggesting that CW or long pulse sources are preferred. It is a complex matter to determine the laser power required to achieve particular levels of photon emission since it depends critically on the operating parameters of the laser. Milonni et al. [10] calculated the photon return for the Starfire Optical Ranging and Mauna Kea laser guide star systems with continuous wave, long pulse and short pulse duration. The main parameters here are the laser line-width and power. For astronomical observations, 10W lasers are required; for satellite ranging 100W lasers are required. These laser power requirements are very demanding, especially when the power needs to be delivered in diffraction limited optical beams. 1.2 Review of Laser Guide Star Lasers providing W, frequency stable, diffraction limited beams are very rare. To complicate matters further, the system must be a reliable low maintenance industrial type laser preferably based on solid-state laser technology amenable to pumping by reliable semiconductor diode lasers. Early guide stars used dye lasers that could be readily tuned to emit directly at the 589nm resonance line. These systems could be pumped with high average power copper vapour lasers or frequency doubled solid-state lasers. This technology was developed by Lawrence Livermore Laboratories in the USA to create a 15W average power at 11kHz repetition rate and 100ns pulse source used at the Shane Telescope at the Lick observatory [11]. This was a very complex laser, and difficult to scale to higher average powers. Even if the systems s average power were increased by an order of magnitude, the pulse format was such that strong saturation was encountered, lowering the fluorescence yields from the mesospheric sodium. Hence, on engineering grounds and from the saturation considerations, alternative format sources are desirable. No known solid-laser system can emit directly at the 589nm line and hence, a number of approaches using nonlinear optical processes to convert the emission from the common solid-state lasers have been studied. These include Raman shifting of the 1064nm line from Nd:YAG laser to 1178nm using Calcium Tungstate and then frequency doubling that emission to 589nm as has been demonstrated by Cheung et al.[12]. A second process uses a fluke of nature in that two strong gain lines in Nd:YAG, the 1064nm and 1319nm lines, are nearly perfect for sum frequency generation to produce 589nm emission [13]. This forms the basis of an all solid-state system amenable to diode laser pumping using of of the best developed laser materials. It is, therefore, quite attractive for this application. Since in either of these schemes (and other variations of them which have been studied) nonlinear conversion is essential, it is important to achieve the maximum conversion efficiency in the nonlinear process. In general the higher the peak intensity in the pulse, the better the conversion efficiency. For processes such as sum frequency generation or second harmonic generation using readily available nonlinear crystals, intensities around 1GW/cm 2 will lead to conversion efficiencies of >80%. While it is theoretically possible to obtain such a high intensity using say a 100ns duration pulse, in practice, such long pulses at these intensities will cause optical damage in the nonlinear crystal. Typically for pulses around 100ns in duration, the damage threshold fluences for nonlinear crystal are a few J/cm 2 and this limits the maximum intensity to only 10-30MW/cm 2 at most, strongly limiting the available conversion efficiency. To overcome

18 6 Introduction this problem much shorter pulses should be used. For example, for the pulses around 100ps duration, intensities of 1GW/cm 2 can be achieved at fluences corresponding to only 10-20% of the damage threshold for the nonlinear crystal and this allows the nonlinear process to be very efficient. As a result, most schemes based on nonlinear conversion use pulses of a few hundred picoseconds in duration, which are often delivered as so-called micro-macro pulses-trains of few hundred 100ps duration pulses separated by 10-20ns within a macro-pulse envelope of a few hundred microseconds long. The additional advantage of such a pulse format is that it is very resistant to saturation and returns more signal per watt of transmitted power than obtained using longer 100ns pulses. The shortest possible pulse that can be employed in a guide star is determined by the line-width of the Na D 2 transition, which is 3GHz. For a pulse with a Gaussian temporal profile, the time-bandwidth product is 0.44, which means that the pulse duration should be 146ps. Pulses of this duration are readily produced from mode-locked solid-state lasers. A diode-pumped solid state Nd laser system based on sum frequency generation has been successfully applied to a LGS system at the NOAO Vacuum Telescope Tower at Sac Peak [14]. 12W of emitted power produced a beacon equivalent to a magnitude 9.2 star. Whilst this system is scalable to higher power, there are still considerable challenges to produce well-polarised, diffraction-limited beams from YAG lasers at powers in the W range. Nd:YAG is an isotropic material with no natural birefringence. As a result, at high pump powers, thermal loading in the crystal results in induced birefringence and this degrades the performance of the laser, especially for the nonlinear optics applications (where the pump laser must be well-polarised). In addition, the scheme based on sum frequency generation requires two time synchronised pulsed mode-locked laser oscillators and this poses a number of technical challenges. In order to synchronise the lasers, active mode-locking is required. Actively mode-locked lasers are troublesome because their resonators must be very accurately stabilised in length (to within a few microns) to remain in resonance with the crystal controlled RF source that drives the acousto-optic mode-lockers. This requires the development of the mechanical subsystems which maintain the resonator length stable in the face of changes in the ambient temperature. This had been achieved with relatively limited success in commercial lasers. The gain varies significantly between the 1064nm and 1319nm lines of YAG and this can complicate synchronisation of the envelopes of the pulsed systems. A preferable approach would be to use a single, continuous, passively mode-locked laser where variations in length of the resonator have no effect on the laser performance, thereby, simplifying the resonator design. In addition, to obtain the highest diode pump efficiency and eliminate the problems of thermally induced birefringence, a naturally birefringent gain medium is preferred. For reasons of its birefringence, high gain, and ease of pumping, Neodymium-doped Yttrium Ortho-Vanadate, Nd:YVO 4 has, therefore, become the preferred gain medium for compact high power diode-pumped Nd lasers. Its short upper lasing level lifetime (compared with Nd:YAG) makes it especially suitable for passive mode-locking, making it robust against Q-switched mode-locking which is potentially damaging to nonlinear crystals. It is not possible, however, to use Nd:YVO 4 to create 589nm light through sum frequency generation because its longer wavelength emission occurs at 1342nm rather than 1319nm as for Nd:YAG, placing the sum frequency well away from the D 2 line. As a result, other approaches are required. The aim of this project is to develop a novel 589nm source based on the use of high average power, high brightness Nd:YVO 4 laser to synchronously pump a optical parametric

19 1.3 Optical Parametric Oscillators: An Approach 7 oscillator to produce 589nm emission. 1.3 Optical Parametric Oscillators: An Approach Optical parametric oscillators (OPOs) are the sources of coherent radiation that share some characteristics of laser oscillators. The laser medium is, however, replaced by a second order nonlinear optical crystal and gain due to stimulated emission replaced by parametric amplification. In an optical parametric amplifier (OPA), photons from the pump laser spontaneously decay into pairs of photons whose energy adds up to that of the pump (thus, obeying energy conservation) as in Equation 1.1. By convention the higher frequency photon is known as the signal, and the lower frequency photon is the idler. ω p = ω s + ω i (1.1) Here ω p is the pump frequency, ω s is the signal frequency and ω i is the idler frequency. Once the signal and idler photons are generated by spontaneous decay of the pump by a process known as parametric fluorescence, mixing processes take over where signal (idler) photons mix with the pump via the nonlinearity to create more idler (signal) photons. Energy flows always from the high to the lower frequency waves and thus both signal and idler fields grow at the expense of the pump. For the process to be efficient, the interaction must conserve momentum, as described in Equation 1.2. This relation is known as the phase matching relation. k p = k s + k i (1.2) where k j = n j ω j /c, with n j being the index of refraction of the materials at the angular frequency ω j and c is the speed of light. Equation 1.2 is not normally satisfied because of dispersion (the variation in refractive index of a medium with wavelength). However, in a birefringent crystal it is possible to choose appropriate polarisations for the three interacting waves so that Equation 1.2 can be satisfied. In such conditions, the gain depends exponentially on the distance of propagation (in the absence of pump depletion). If the crystal is placed inside a low loss resonator that reflects signal or idler (or both) waves, the signal and idler photon flux increase until limited by pump depletion. Another way exists to obtain phase matching known as periodic poling or quasi phase matching. This involves treatment of the nonlinear optical crystal to modulate the sign of the optical nonlinearity periodically. In this case the wave vector mismatch, k = k p k s k i is compensated by the grating vector, K g due to the modulation of the nonlinear properties. Thus, in periodically poled materials, birefringence is no longer needed to obtain phase matching and materials can be engineered to use their most favourable nonlinear optical coefficient (which is a tensor property dependent on the orientation of the crystallographic axes relative to the propagation direction of the beams). Amongst the most popular nonlinear crystals, MgO:LiNbO 3 is an excellent candidate due to its high nonlinear coefficient, suitable range of transparency, high photorefractive damage threshold, and commercial availability. Using either of these phase matching approaches, however, perfect phase matching only occurs for a small range of signal and idler frequencies for any particular crystal geometry (grating period or propagation direction). Hence, the parametric oscillator can emit narrow band radiation as is required for guide star application. The tunability of

20 8 Introduction the OPO using quasi phase matching is due to the dependence of grating vector K g on temperature. Therefore, by controlling the temperature of the nonlinear crystal, the signal output can be tailored and stabilised at the desired wavelength or frequency. There are many OPO schemes that can lead to the generation of 589nm light and several of these could be investigated as part of this project. There are three promising possibilities, as following: Scheme 1 Use of a high average power frequency doubled mode-locked continuous wave Nd:YVO 4 laser to pump a periodically poled MgO:LiNbO 3 (MgO:PPLN) OPO with its signal wavelength at 589nm. Scheme 2 The Nd laser is frequency tripled to create a 355nm pump and this used to pump an OPO producing 589nm light. Scheme 3 Nd laser directly pumps a MgO:PPLN OPO generating 1319nm then sum frequency generation with 1064nm to produce 589nm wavelength. Schemes Description Overall signal Efficiency nm SHG 532nm OPO 589nm+ (5500nm) 43% TFG 355nm OPO 589nm + (893nm) 29% nm OPO 1319nm + (5500nm) SFG 589nm 29% Table 1.1: The estimated efficiencies of the three schemes can be used to produce 589nm Sodium line. Out of the three schemes mentioned above, the Scheme 1 is the best in terms of efficiency to the signal. The efficiencies of the schemes depend on the number of nonlinear optical processes involved and the nature of those processes. The maximum possible overall conversion efficiencies of second and third harmonic generations, parametric oscillator and sum frequency generation are 80% and 60% and 60%, respectively. The total energy conversion is in the output of both signal and idler. This aim of the design is to maximise the output energy of the signal. Table 1.1 estimates the overall efficiency to the 589nm signal of the three schemes mentioned above. In Scheme 1, after the OPO the signal carries 90% the converted energy due to the fact that the signal frequency is very close to the pump frequency. Hence, the overall signal efficiency can be up to 43%. However, in the Scheme 2, the signal after OPO carries only 60% of the converted energy; this makes the overall signal efficiency of only 29%. Similarly, the Scheme 3 has only maximum overall signal conversion of 29%. Table 1.1 indicates that scheme 1 is the best option. However, there are disadvantages in the Scheme 1 due to the fact that Lithium Niobate strongly absorbs the 5.4µm idler wave (α = 8cm 1 ). This absorption potentially can lead to a significant increase in the threshold. More importantly, the idler absorption can lead to the thermal effect which may cause distortion of the pump beam and reduce the efficiency of the OPO. Lithium Niobate also has a relatively low photorefractive damage threshold at 532nm. This will lead to some restriction in the design of the practical OPO guide star.

21 1.4 Summary of the Thesis Summary of the Thesis The second chapter is Nonlinear Optics, which briefly describes the principles of nonlinear optics, second harmonic generation, parametric generation and phase matching, especially quasi phase matching. The third chapter, Optical Parametric Oscillator, is devoted to the optical parametric oscillator and optical properties of Lithium Nioates. The principles of synchronously pumped singly resonant optical parametric oscillators are briefly described. The fourth chapter, Design of Guide Star Optical Parametric Oscillator, is the detailed description of a practical guide star base on Magnesium Oxide doped periodically poled Lithium Niobate. The fifth chapter, Potassium Titanyl Phosphate Optical Parametric Oscillator, details the experimental results and the discussion of the KTP OPO. The sixth chapter, Performance of Prototype Guide Star Optical Parametric Oscillator, describes a guide star prototype at low average pump power. The final chapter is the conclusion of the project.

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23 Chapter 2 Chapter 2: Nonlinear Optics In this chapter, the main principles of linear and nonlinear optics are outlined. The phenomena of harmonic generation, wave-mixing, parametric generation and oscillation are examined in detail sufficient for the experimental aims of the project. Finally, different phase matching methods are introduced, and the quasi phase matching method is specifically emphasized. 2.1 Introduction to nonlinear optics Wave equation In a dielectric, free of electric charges and magnetic dipoles, the most general wave equation describing the propagation of the electric field vector is given by [3]: Ẽ + 1 c 2 2 t 2Ẽ = 1 ɛ o c 2 2 t 2 P (2.1) where Ẽ and P are the electric field and polarization vectors (or tensors in more general term), which vary rapidly in time, c = m/s is speed of light in vacuum, and ɛ o = F/m is the electric permittivity of free space. In the linear optics regime, the total polarisation has only a linear component, which is proportional to the applied electric field: P L = ɛ o χ (1) Ẽ (2.2) where χ (1) is the linear dielectric response tensor. In general, the relation between P L and Ẽ is tensorial, and can also be written as P L,i = ɛ o j χ (1) ij Ẽ j (2.3) where the subscript i signifies the i th cartesian coordinate (i = x, y, z) and the sum is over j = x, y, z. The tensor χ (1) thus has nine components. In an isotropic medium there is only one independent nonzero component, and the dielectric response is written as a scalar quality χ. When the intensity of the light is sufficiently high (e.g. from a laser), a small additional polarization will appear in addition to the normal linear polarisation, so that the total polarization can be written as: P = P L + P NL (2.4) 11

24 12 Chapter 2: Nonlinear Optics where P NL is a nonlinear function of the applied field Nonlinear Susceptibilities It can be assumed that the nonlinear polarisation can be written as where P NL = P (2) + P (3) +... (2.5) P (2) = ɛ o χ (2) : ẼẼ (2.6) P (3) = ɛ o χ (3) : ẼẼẼ (2.7) etc. It is important to note that the field in these equations is the total applied field, which can be a superposition of many fields of different frequencies. χ (n) is the n th (n > 1) order dielectric response and is a tensor of rank n+1. When the total field is expanded in terms of its Fourier components, the nonlinear polarization will consist of several terms oscillating at various combination frequencies. For example, if the total field consists of two waves oscillating at frequencies ω 1 and ω 2, the second order nonlinear polarisation will have components oscillating at 2ω 1, 2ω 2, ω 1 + ω 2, ω 1 ω 2 and dc term at zero frequency. Similarly, with three fields oscillating at frequencies ω 1, ω 2, and ω 3, the third order polarisation will oscillate at 3ω 1, 3ω 2, 3ω 3, ω 1 + ω 2 + ω 3, ω 1 + ω 2 ω 3, etc. It is common to write the Fourier components of the nonlinear polarisation in the following way. Consider a second order polarisation oscillating at ω n and ω m, with ω r = ω n + ω m. Then the i th component of the complex polarisation amplitude is expressed as P (2) i (ω r ) = ɛ o jk (nm) χ (2) ijk (ω n + ω m, ω n, ω m )E j (ω n )E k (ω m ) (2.8) Here the indices ijk refer to the cartesian components of the fields. The notation (nm) indicates that, in performing the summation over n and m, the sum ω n + ω m is to be held fixed, although ω n and ω m are each allowed to vary. Following convention, χ (2) is written as a function of three frequency arguments. This is technically unnecessary in that the first argument is always the sum of the other two. To emphasize this fact, the susceptibility χ (2) (ω 3, ω 1, ω 2 ) sometimes written as χ (2) (ω 3 ; ω 1, ω 2 ) as a reminder that the first argument is different from the other two; or it may be written symbolically as χ (2) (ω 3 = ω 1 + ω 2 ) Properties of the Nonlinear Susceptibility Nonlinear susceptibility has been defined as a tensor. In the second order term χ (2) is an tensor with 27 components. For a particular material, it is possible to determine all values of the component. There are a number of restrictions resulting from symmetries that relate the various components of χ (2), hence, fewer than 27 components are usually needed in order to describe the nonlinear coupling. In this section, the second-order χ (0) susceptibility is considered, but the same idea can be extended to higher-order term.

25 2.1 Introduction to nonlinear optics 13 The first symmetry apparent from the form of the Equation 2.8 is due to the lack of difference physically in which order the product of the field amplitudes is given. Thus, the interchange in the order of the product E j (ω 1 )E k (ω 2 ) [i.e. E j (ω 1 )E k (ω 2 ) E k (ω 2 )E j (ω 1 )] will not affect the value or sign of the i th component of the nonlinear polarisation. The nonlinear susceptibility should reflect this symmetry. The symmetry property is expressed as χ (2) ijk (ω 3, ω 1, ω 2 ) = χ (2) ikj (ω 3, ω 2, ω 1 ) (2.9) In other words, if any of the subscripts {jk} are permuted, then the susceptibility will remain unchanged as long as the corresponding set of subscripts {12} is also permuted. Another form of the permutation symmetry can be shown when the nonlinear susceptibility is calculated quantum mechanically. This is usually done using the density matrix method [2], [15]. It can be shown generally, for example in second order, that χ (2) ijk (ω 3, ω 1, ω 2 ) = χ (2) jik (ω 1, ω 2, ω 3 ) = χ (2) kji (ω 2, ω 1, ω 3 ) (2.10) First, it is noted that if the complex conjugate of the susceptibility is taken, it just changes sign of all of the frequencies. This is a consequence of the fact that the rapidly varying nonlinear polarisation is a real mathematical quantity. Then, for any permutation of the Cartesian subscript, the new susceptibility obtained is equal to the original susceptibility if the corresponding frequency subscripts are also permuted. This important result states that the susceptibilities for different physical processes are simply related. The above expressions hold generally when the susceptibilities are complex. This is especially important when any single frequency or any combination frequency is near a natural resonance frequency of the materials. However, it is often the case in many applications that all frequencies and combination frequencies are far from any material resonance. Then, the susceptibilities are be treated as real quantities, and any susceptibility is thus equal to its complex conjugate. In this case, Equation 2.10 may be written as χ (2) ijk (ω 3, ω 1, ω 2 ) = χ (2) jik (ω 1, ω 2, ω 3 ) = χ (2) kij (ω 2, ω 1, ω 3 ) (2.11) Under the condition that the susceptibilities are real (all frequencies are far from any resonance), the susceptibilities are unchanged for the simultaneous permutation of subscripts for the Cartesian set {ijk} and the corresponding subscripts from the frequency set {312}, on the condition that the frequencies carry the proper sign. Another symmetry follows when the susceptibilities exhibit negligible dispersion over the entire frequency range of interest. Thus, in addition to all frequencies being far from any material resonance, this symmetry requires that there is no resonance between any of the frequencies. When dispersion can be ignored, the frequencies can be freely permuted without permuting the corresponding Cartesian subscripts, and vice versa, and the susceptibility will remain unchanged. This is known as Kleinman symmetry. In other words, the second-order susceptibilities can be written as χ (2) ijk (ω 3, ω 1, ω 2 ) = χ (2) jik (ω 3, ω 1, ω 2 ) = χ (2) kij (ω 2, ω 1, ω 3 ) = χ (2) jki (ω 3, ω 1, ω 2 ) = χ (2) kji (ω 3, ω 1, ω 2 ) = χ (2) ikj (ω 3, ω 1, ω 2 ) (2.12) Nonlinear susceptibilities also reflect the structural symmetry of the material. This is

26 14 Chapter 2: Nonlinear Optics important since in many cases this greatly reduces the number of nonzero, independent tensor components needed to describe the medium. One immediate consequence of this is that for all materials that possess a centre of inversion symmetry, all elements of all even-order nonlinear susceptibility tensors are identically equal to zero. Hence, there are no even-order nonlinear processes in these types of materials. The specific form of third order susceptibilities may also be given for each symmetry class. Third order effect are primary in centrosymmetric media. The detailed table of susceptibility tensor for each class of symmetry can be found in Boyd [2] and Shen [15] Contracted Notation d-coefficient and effective susceptibility At this point it is important to note another notation that is used in second order nonlinear optics. Often the susceptibility is represented as the so-called d-coefficient, where d is a tensor given by d ijk = 1 2 χ(2) ijk (2.13) Furthermore, the intrinsic permutation symmetry, as discussed above, is used to contract the last two subscripts and write d ijk d il. The subscripts are then written as numbers instead of letters, using the following rules i : jk : l : x 1 xx 1 y 2 yy 2 z 3 zz 3 yz = zy 4 xz = zx 5 xy = yx 6 (2.14) For example, d xyz = d xzy = d 31 etc. The utility of this notation is that d-coefficients can be expressed as elements of a 3 6 matrix rather than a tensor. To use these coefficients in the nonlinear polarisation of Equation 2.8, just make the substitution χ ijk 2d il i.e. P (2) i (ω r ) = D (2) ɛ o 2d il (ω n + ω m, ω n, ω m )E j (ω n )E k (ω m ) (2.15) jk The nonlinear susceptibility tensor can then be represented as the 3 6 matrix: d il = d 11 d 12 d 13 d 14 d 15 d 16 d 21 d 22 d 23 d 24 d 25 d 26 d 31 d 32 d 33 d 34 d 35 d 36 (2.16) If we now explicitly introduce Kleinman symmetry condition, i.e. that the indices d ijk can be freely permuted, we find that not all of the 18 elements of d il are independent. For instance, it can be shown that and d 12 d 122 = d 212 d 12 d 14 d 123 = d 213 d 25 (2.17)

27 2.2 Second Harmonic Generation 15 By applying this type of argument systematically, it can be found that d il has only 10 independent elements when the Kleinman symmetry condition is valid; the form of d il under these conditions is d il = Effective Value of d (d eff ) d 11 d 12 d 13 d 14 d 15 d 16 d 16 d 22 d 23 d 24 d 14 d 12 d 15 d 24 d 33 d 23 d 13 d 14 (2.18) For a fixed geometry (i.e., for fixed propagation and polarisation directions) it is possible to express the nonlinear polarisation using an effective nonlinear susceptibility d eff. A general prescription for calculating d eff for each of the crystal classes has be well studied (See Boyd[2] and Sutherland[3]). For example, for the crystals of crystal class 3m, the effective value of d is given by the expression d eff = d 31 sinθ d 22 cos θ sin3φ (2.19) under conditions known as type I phase matching, in which the signal and idler waves have the same polarisation; and by d eff = d 22 cos 2 θ sin3φ (2.20) under condition known as type II conditions, in which polarisations of signal and idler are orthogonal. In these equations, θ and φ are the angle between the propagation vector with optical axes of the crystal. It is important to recognise that, in general, d eff is a sum of terms obtained through the contraction process and must be determined for each particular problem based on the polarisations of the interacting optical waves and geometry of the nonlinear crystal. Beyond this point, the nonlinear susceptibility coefficients are always written as d eff or d, because for each of the condition of the angle of propagations of light waves relative to the optical axes of the crystal, it is always possible to calculate this effective coefficient using the standard equations given in the literature. 2.2 Second Harmonic Generation In this section, the mathematical description of the process of second-harmonic generation is considered. It is assumed that the medium is lossless both at the fundamental frequency ω 1 and at the second-harmonic frequency ω 2 = 2ω 1, so that the nonlinear susceptibility obeys the conditions of full permutation symmetry. The first detailed theoretical treatment of second harmonic generation is by Armstrong et al. [16]. The process is shown in Figure 2.1. The total electric field in the nonlinear medium to be given by Ẽ(z, t) = Ẽ1(z, t) + Ẽ2(z, t) (2.21) where each component is expressed in terms of a complex amplitude E j (z) and slowly varying amplitude A j (z) according to Ẽ(z, t) = Ẽ1(z)e iω jt + c.c., (2.22)

28 16 Chapter 2: Nonlinear Optics Pump w w 2w Nonlinear Crystal Figure 2.1: Second harmonic generation where E j (z) = A j (z)e ik jz (2.23) and c.c. stands for complex conjugate. Using the wave equations, the amplitudes of fundamental and second harmonic are found to be related by a pair of coupling equations [2] and da 1 dz = i2ωd eff n ω c A 2A 1e i kz (2.24) da 2 dz = i2ωd eff n 2ω c A2 1e i kz (2.25) where k = 2k 1 k 2 These equations were first solved by Armstrong et al. [16]. In general, both the modulus and phase of the complex field amplitudes are computed. However, to compute the output intensity of the generated wave, only modulus is used. The intensity of a wave at some position z is given by I j = 2ɛ o n j c A j 2. (2.26) The power of a given wave is computed from the area S of the beam and the intensity distribution. P j = S I j ds (2.27) In the undepleted-pump approximation (i.e., A 1 = A ω =constant), Equation 2.25 can be integrated immediately to obtain an expression for the spatial dependence of the second harmonic field amplitude (with initial condition of no second harmonic input at z = 0)i.e. I 2 (z) = 8ɛ oω 2 d 2 ( eff sin n 2ω c 2 A 4 1z kz ) kz (2.28) The intensity I of the pump ω 1 is related to A 1 by Equation Hence the conversion efficiency is given by the formula η 2ω = I 2ω = 8π2 d 2 eff z2 I ω sin 2 ( kz/2) I ω ɛ o n 2 ωn 2ω cλ ω 2 ( kz/2) 2 = 8π2 d 2 eff z2 I ω ɛ o n 2 sinc 2 ( kz/2) (2.29) ωn 2ω cλ ω 2

29 2.3 Wave-Mixing 17 Equation 2.29 indicates that the maximum conversion occurs when k = 0. (2.30) Then it is said that the nonlinear interaction is phase-matched This treatment is the simplest, and applied only to a plane wave that propagates in lossless medium and in the absence of pump depletion. In practice, the pump depletion can reach 80% using a Gaussian pulse. The next approximation is plane wave with pump depletion at perfect phase matching. The efficiency is given by [17] [3] η 2ω = tanh 2 (L/L NL ) L LN = 2.3 Wave-Mixing 1 4πd eff 2ɛo n 2 ωn 2ω λ 2 ω I ω (0) (2.31) Considering the situation shown in Figure 2.2, in which optical waves at frequency ω 1, ω 2 and ω 1 interact in a lossless nonlinear optical medium. It is assumed that the ω 3 is a strong wave (i.e. is undepleted by the nonlinear interaction so that its amplitude is treated as a constant), and for the boundary condition without any ω 2 intensity (A 2 (0) = 0 at the position z = 0). w1 w3 w2 Nonlinear Crystal Figure 2.2: Wave-mixing process in a nonlinear medium The coupling equations for the amplitudes of three waves are [2] da 1 dz da 2 dz da 3 dz = i ω 1d eff n 1 c A 2A 3 e i kz = i ω 2d eff n 2 c A 1A 3 e i kz = i ω 3d eff n 2 c A 1A 2 e +i kz (2.32) where k = k 1 + k 2 k 3 These equations describe the coupling, or mixing of three waves, thus called three wave mixing. Second harmonic generation is a special example. Note that the names of ω 1, ω 2

30 18 Chapter 2: Nonlinear Optics and ω 3 are only relative: one can always swap these symbols as long as the physics process is clearly indicated (i.e. which is the pump, which is the signal). Sum-frequency generation is the phenomenon in which the two pump waves at ω 1 and ω 2 mix to generate a third wave at ω 3 = ω 1 + ω 2. The difference-frequency generation is similar except that the two pump waves combine to produce a wave at different frequency ω 3 = ω 1 ω 2. The latter process is the origin of the parametric generation where the only input to the nonlinear crystal is the pump at ω 3. The production of an initial pair of signal and idler that satisfies the phase matching condition is due to parametric fluorescence, which is a quantum phenomenon. The material acts as a catalyst, generating the new photons through the nonlinear polarisation. As a convention, the higher frequency is called signal and the other is called idler. 2.4 Parametric Amplification Equation 2.32 then can be rewritten as a set of two equations da 1 dz da 2 dz = i ω 1d eff n 1 c A 2A 3 e i kz = i ω 2d eff n 2 c A 1A 3 e i kz (2.33) The solution of these two equations in absence of pump depletion (i.e. A 3 = constant) can be easily found [2],[15],[17],[18] A 1 (z) = A 2 (z) = [ ( A 1 (0) cosh(gz) i k ) 2g sinh(gz) + i κ ] 1 g A 2(0)sinh(gz) e i kz/2 [ ( A 2 (0) cosh(gz) i k ) 2g sinh(gz) + i κ ] 2 g A 1(0)sinh(gz) e i kz/2 (2.34) where the coefficient g is given by g = [ κ 1 κ 2 ( ) ]1 k (2.35) and κ j = ω jd eff n j c A 3 (2.36) For the initial condition of no input for the idler ω 2 Equation 2.34 becomes A 1 (z) = A 2 (z) = [ ( A 1 (0) cosh(gz) i k )] 2g sinh(gz) e i kz/2 [ i κ ] 2 g A 1(0)sinh(gz) e i kz/2 (2.37) For the perfect phase matching situation where k = k 1 + k 2 k 3 = 0, the final solution is simply

31 2.4 Parametric Amplification 19 A 1 (z) = A 1 (0)cosh(gz) A 2 (z) = i κ 2 g A 1(0)sinh(gz) (2.38) These two equations imply that even with no input at ω 2 it is still generated together with the amplification of the signal ω 1. The intensities of the idler and signal with distance z are plotted on Figure 2.3 Amplitude Signal Idler Distance z Figure 2.3: Parametric amplification of the signal and idler in plane wave, non-depleted pump approximation The parametric gain coefficient is defined by G(z) = I 1(z) I 1 (0) 1 = A 1(z) 2 A 1 (0) 2 1 (2.39) For the simplest case as above (i.e. plane wave, no pump depletion, no input at ω 2 and perfect phase matching) the parametric gain coefficient is G(z) = cosh 2 (gz) 1 (2.40) For non-perfect phase matching, the gain coefficient can be calculated from Equation The final result of the gain would be complicated due to the complex number calculation. However, the gain can be simplified by a further assumption that is the gain is normally small, this is generally true for parametric amplification. Milonni and Eberly [19] and Myers et al. [4] shows the approximation that leads to the gain expression for low gain, non-perfect phase matching for plane wave case G(z) = κ 1 κ 2 z 2sin2 1 2 kz ( 1 2 kz)2 (2.41)

32 20 Chapter 2: Nonlinear Optics 2.5 Phase Matching It is clear from the previous section that phase matching is necessary for efficient conversion. Equations 2.29 and 2.41 predict a dramatic decrease in the efficiency of the second harmonic generation when the condition of perfect phase matching, k = 0, is not satisfied due to the sinc function which has the shape as shown on Figure 2.4. Figure 2.4: The dependence of second harmonic intensity against phase mismatch as a sinc function of kz The phase matching condition k = 0 is often difficult to achieve because the refractive index of materials that are lossless in the range from ω 1 to ω 3 shows an effect known as normal dispersion: the refractive index is an increasing function of frequency. As a result, the condition of perfect phase matching with collinear beams, where n 1 ω 1 + n 2 ω 2 = n 3 ω 3 (2.42) ω 1 + ω 2 = ω 3 (2.43) cannot be achieved. For the case of second-harmonic generation, with ω 1 = ω 2, and ω 3 = 2ω 1, these conditions require that n(ω 1 ) = n(2ω 1 ) (2.44) which is clearly not possible when n(ω) increases monotonically with ω. In principle, however, it is possible to achieve the phase-matching condition by making use of anomalous dispersion, that is, the decrease in refractive index with increasing frequency that occurs near an absorption feature. The most common procedure for achieving phase matching is to use birefringence displayed by many anisotropic crystals. Birefringence is the dependence of the refractive index on the direction of polarisation of the optical radiation. Not all crystals display birefringence; for example, crystals with cubic symmetry are optically isotropic. The nonlinear crystals that are phase matchable are from uniaxial (including trigonal, tetragonal, hexagonal) and biaxial (including triclinic, monoclinic, orthorhombic) systems.

33 2.5 Phase Matching 21 Refractive Index no ne w Frequency w 2w Figure 2.5: Dispersion of the refractive indices of a negative uniaxial crystal. In order to achieve phase matching through the use of birefringent crystals, the highestfrequency wave ω 3 = ω 2 +ω 1 is polarised in the direction that gives it the lower of the two possible refractive indices. For the case of a negative uniaxial crystal, as in the example shown in Figure 2.5, this choice corresponds to the extraordinary polarisation. There are two choices for the polarisation of the lower-frequency waves. Type I phase matching is the case where the lower-frequency waves have the same polarisation, and type II is the case where the polarisations are orthogonal. The possibilities are summarised in Table 2.1 Positive uniaxial (n e > n o ) Negative uniaxial (n e < n o ) Type I n o 3 ω 3 = n e 1 ω 1 + n e 2 ω 2 n e 3 ω 3 = n o 1 ω 1 + n o 2 ω 2 Type II n o 3 ω 3 = n o 1 ω 1 + n e 2 ω 2 n e 3 ω 3 = n e 1 ω 1 + n o 2 ω 2 Table 2.1: Phase-matching methods for uniaxial crystals [2] Careful control of the refractive indices at each of the three optical frequencies is required in order to establish the phase-matching condition. In practice, there are two methods to achieve phase matching: angle and temperature tuning [2] Angle Tuning The angle tuning method involves precise angular orientation of the crystal with respect to the propagation direction of the incident light. It is most simply described for the case of a uniaxial crystal. Uniaxial crystals are characterised by a particular direction known as the optical axis (or c axis or z axis). Light polarised perpendicular to the plane containing the propagation vector k and the optical axis is said to have ordinary polarisation. Such light experiences a refractive index n o called the ordinary refractive index. Light polarised in the plane containing k and the optic axis is said to have extraordinary polarisation and experiences a refractive index n e (θ) that depends on the angle θ between the optic axis and k according to the relation 1 n e (θ) 2 = sin2 θ n 2 + cos2 θ e n 2 o (2.45)

34 22 Chapter 2: Nonlinear Optics Here n e is the principal value of the extraordinary refractive index. Note that n e (θ) is equal to the principal value n e for θ = 0 degrees is equal to n o for θ = 0. Phase matching is achieved by adjusting the angle θ to achieve the value of n e (θ) for which the condition k = 0 is satisfied. A special case arises if the phase matching angle turns out to be θ = 90 o. This is possible in some materials at a particular temperature. This situation is referred to as noncritical phase matching. The case of θ 90 o is called critical phase matching. Noncritical phase matching is advantageous for two reasons. First, it is less sensitive to beam divergence, which lowers the conversion efficiency. Second, the walk off angle for this case is zero. c w Ordinary theta k 2w extraordinary Figure 2.6: Angle phase matching for second harmonic generation As an illustration of angle phase matching, the case of type I second-harmonic generation in a negative uniaxial crystal is considered, as shown in Figure 2.6. Since n e is less than n o for a negative uniaxial crystal, the fundamental wave is chosen to be an ordinary and the second harmonic to be an extraordinary wave, in order that the birefringence of the materials can compensate for dispersion. The phase matching condition then becomes or n e (2ω, θ) = n o (ω), (2.46) sin 2 θ n e (2ω) 2 + cos2 θ n o (2ω) 2 = 1 n o (ω) 2 (2.47) In order to simplify this equation, we replace cos 2 θ by 1 sin 2 θ and solve for sin 2 θ to obtain 1 sin 2 n θ = o (ω) 2 1 n o (2ω) 2 (2.48) 1 n e (2ω) 2 1 n o (2ω) 2 This equation shows how the crystal should be oriented in order to achieve the phase matching condition. Then the effective nonlinear coefficient d eff can be calculated based on the information of the particular setting of phase matching condition Temperature Tuning There is one serious drawback to the use of angle tuning. Whenever the angle θ between the propagation direction and the optic axis has a value other than 0 or 90 degrees, the Poynting vector S and the propagation vector k are not parallel for extraordinary rays. As

35 2.5 Phase Matching 23 Figure 2.7: Poling of crystal in QPM. a result, ordinary and extraordinary rays with parallel propagation vectors quickly diverge from one another as they propagate through the crystal. This effect is called walk off. This walk off effect limits the spatial overlap of the two waves and decreases the efficiency of any nonlinear mixing process involving such waves. For some crystals, notably Lithium Niobate, the amount of birefringence is strongly temperature-dependent. As a result, it is possible to phase-match the mixing process by holding θ fixed at 90 degrees and varying the temperature of the crystal Quasi Phase Matching Angle phase matching in birefringent crystals has been the most common form of phase matching for frequency conversion, however, it has some disadvantages. For example, some angles of propagation are not possible. Also, the use of cubic crystals is not possible, even though they have large nonlinear coefficients in some cases. In general, it would be desirable to isolate and use one of the diagonal terms of nonlinear susceptibility since they are generally larger. However, this is not possible in angle phase matching, and the d eff is generally smaller than these diagonal terms. Armstrong et al. [16] were the first to suggest ways to achieve quasi phase matching (QPM). For a frequency conversion process, such as an OPO or a SHG process, the phase mismatch kl = (k 3 k 2 k 1 )L is accumulated by increasing the interaction length (or length of the crystal). After a coherence length, l c = π/ k, the conversion efficiency reduces and energy flows back from the signal or from the doubled wave to the pump. If the nonlinear coefficient is modulated with a period twice the coherence length, in other words, the nonlinear coefficient changes its sign after each coherence length, the accumulated phase mismatch can be offset. Consequently, the conversion efficiency increases monotonously with interaction length. The most common technique is illustrated in Figure 2.7. This structure is called periodically poled. The nonlinear crystal is divided into segments each a coherence length long. Each segment is then rotated relative to its neighbours by 180 o about the axis of propagation. Because of the lack of inversion symmetry, this has the effect of changing the sign of all of the components of the nonlinear susceptibility tensor. Hence, the nonlinear polarisation wave is shifted by π radians each coherence length. Consider second harmonic generation as an example, for n th slab in the stack,

36 24 Chapter 2: Nonlinear Optics d n,eff = ( 1) n 1 d eff (2.49) where d eff is the effective nonlinear coefficient of the first slab for the particular polarisation of the interacting fields and the direction of propagation relative to the crystal principle axes. With further analysis using Fourier series representation [4], the final effective nonlinear coefficient for quasi phase matching is largest for the first-order process d Q = 2 π d eff (2.50) Furthermore, Fourier analysis gives the wave-vector mismatch for QPM interaction as k Q = k 1 + k 2 k 3 + k Q (2.51) where k Q = 2π/Λ is the grating vector of the crystal and Λ is the period of the modulated structure. Hence, the quasi phase-matched condition requires the period Λ to be or k Q = k 3 k 1 k 2 2π Λ = 0 (2.52) n 3 λ 3 n 1 λ 1 n 2 λ 2 1 Λ = 0 (2.53) Using quasi phase matching, a wide range of spectrum can be tuned using grating period tuning and temperature tuning. Many grating Λ can be produced in fine steps on the same crystal. Using different grating periods, the different spectrum region can be tuned to. Then, the temperature tuning allows the fine tuning of the signal wavelength. Summary This chapter has reviewed the principles of nonlinear optics. It has emphasized ideas of second harmonic generation, wave mixing, and parametric amplifications. The angle, temperature and quasi phase matching techniques were outlined.

37 Chapter 3 Chapter 3: Optical Parametric Oscillator The main purpose of this chapter is to introduce the optical parametric oscillators. The parametric amplification process using Gaussian beam is first reviewed, then the threshold of the singly resonant optical parametric threshold formulae are introduced. The effect of idler absorption on threshold power is considered. Finally, the properties of Lithium Niobate are listed. 3.1 Parametric Amplification with Gaussian Beam The treatment of section 2.4 is for the plane wave approximation. To increase the parametric gain, the beams are often focused to a small beam at the crystal. For harmonic generation and other frequency conversion processes, TEM 00 Gaussian beams are generally used. This section describes the parametric process with continuous wave (cw) Gaussian beams Focused Gaussian Beam (Near-Field) This approximation is valid when the crystal thickness L is less than or of the order of the confocal parameters b j = 2nπw 2 j /λ j of the interacting beams (see Figure 3.1). In other words, the focusing parameter, which is defined by ξ = L/b, is less than 1. The pump beam is focused onto the crystal with radius w p at 1 e 2 intensity. The generated signal and idler beams have arbitrary radii of w s and w i, respectively, at 1 e 2 intensity. The parametric gain, which is assumed to be small at perfect phase matching, is given by [3] G = 4π2 d 2 eff L2 ɛ o n 3 pcλ 2 P p (1 δ 2 ) 4M2 p π (3.1) where L is the length of the crystal, λ p, n p and P p are the wavelength, refractive index and power of the pump, respectively; δ is degeneracy factor defined through δ = 2λ p λ s 1 (3.2) and λ s is the signal wavelength in the vacuum; M is called the near field gain reduction parameter, which is defined through the radii of the pump, signal and idler Gaussian waves: 25

38 26 Chapter 3: Optical Parametric Oscillator w s w i w p M = wsw 2 i 2 + w2 swp 2 + wi 2w2 p (3.3) When the interacting beams have a Gaussian radial dependence, they are generally not optimally overlapped. This is because the nonlinear driving polarisation beam is smaller than the beams that generate it. For example, the polarisation source at the signal would be proportional to 2r 2 2r 2 w e p 2 w e i 2 = e 2r2 ( 1 wp w i 2 ) = e 2r 2 w 2 s (3.4) In general w s w s due to the fact that the signal is arbitrarily generated. Since the beams are not optically overlapped, this produces a reduction in the gain coefficient. The key factor is the near field gain reduction parameter M Optimised Confocal Near-Field Gaussian Beam For a fixed value of the signal and idler radii, M 2 is maximised for a pump radius given by 1 w 2 p = 1 w 2 s + 1 w 2 i (3.5) and hence, M 2 Max = 1 4(w 2 s + w 2 i ) (3.6) Although the formula for the gain displays a dependence on L 2, the intensity will only be large over a distance on the order of the confocal parameter. Setting L equal to the common confocal parameter of the signal and idler beams, it follows that ws 2 = λ s L/2πn s and w 2 i = λ il/2πn i. Since w 2 λ and λ 1 p = λ 1 s, if the signal and idler beams are confocally focused, then the pump beam will be confocally focused as well. In this case, the small signal parametric gain becomes +λ 1 i Nonlinear crystal Pump b 2wp Figure 3.1: A near-field Gaussian beam. The confocal parameter b is longer than the crystal length L

39 3.1 Parametric Amplification with Gaussian Beam 27 Note that the gain is now only proportional to L not L 2. G = 2π2 d 2 eff L ɛ o n 2 pcλ 3 P p (1 δ 2 ) (3.7) p Common Confocal Parameter Gaussian Beams Boyd and Kleinman [20] analysed the effect of arbitrarily tightly focused Gaussian beams. They assumed that the signal beam size is determined directly from the pump beam size and that the generated signal has the same confocal parameter as the pump, ξ s = ξ p. Their result for the small signal parametric gain is given by G = 2π2 d 2 eff L ɛ o n 2 pcλ 3 P p (1 δ 2 )h m (B, ξ) (3.8) p where h m (ξ) is a function of confocal parameter ξ of both pump and signal beam, as shown in Figure 3.2. The subscript m denotes that h has been maximised with respect to phase mismatch k. h m (ξ) reaches a maximum of at ξ = Figure 3.2: Parametric gain reduction factor as a function of focusing parameter ξ of both pump and signal beams Arbitrary Focused Gaussian Beams The pump beam size might not have the same confocal parameter ξ as the signal. Hence, the assumption made by Boyd and Kleiman is generally not true in practice. Guha et al. [21] calculated the parametric amplification gain for arbitrary focusing parameters of the pump and signal beams. Their result is given in the following form: G = 32π2 d 2 eff LP ph sm (k, ξ p, ξ s ) ɛ o n s n i λ p λ s λ i (1 + k) (3.9) where k = k p /k s is the ratio of the wave vectors of the pump and signal; h sm (k, ξ p, ξ s ) is a function, which must be evaluated numerically. The plots of h sm (k, ξ p, ξ s ) showing the