TEL AVIV UNIVERSITY. Time to frequency mapping of optical pulses using accelerating quasi-phase-matching

Transcription

1 TEL AVIV UNIVERSITY The Iby and Aladar Fleischman Faculty of Engineering The Zandman-Slaner School of Graduate Studies Time to frequency mapping of optical pulses using accelerating quasi-phase-matching A thesis submitted toward a degree of Master of Science in Electrical and Electronic Engineering by Mor Konsens April 2015

2 TEL AVIV UNIVERSITY The Iby and Aladar Fleischman Faculty of Engineering The Zandman-Slaner School of Graduate Studies Time to frequency mapping of optical pulses using accelerating quasi-phase-matching A thesis submitted toward a degree of Master of Science in Electrical and Electronic Engineering by Mor Konsens This research was carried out in The School of Electrical Engineering The Department of Electrical Engineering- Physical Electronics This work was carried out under the supervision of Dr. Alon Bahabad April 2015

3 Abstract Pulse characterization is an important problem in ultrafast optics to which there are various solutions based on nonlinear optics [1]. The most famous and commercially available are Frequency Resolved Optical Gating (FROG) [2] and Spectral Phase Interferometry for Direct Electric field Reconstruction (SPIDER) [3]. Methods for direct time to frequency mapping of the amplitude of pulses are also known. These are based on the time-frequency duality between first-order dispersion and quadratic phase modulation and allow to estimate the intensity profile of pulses using spectral measurement after modulation of the original pulse [4, 5, 6, 7, 8, 9, 10, 11]. In this work we show that direct time to frequency mapping can be achieved by frequency converting an optical pulse to another frequency band following an interaction with a nonlinear spatiotemporal photonic crystal [12]. Efficient generation of new frequency components during nonlinear optical processes requires compensation for phase-mismatch caused by dispersion. Spatial Quasi-Phase-Matching (QPM) modulation can be used to compensate for momentum-mismatch and accomplish phase-matching in energy-conserving nonlinear optical processes [13, 14]. More generally, temporal or spatiotemporal QPM can accommodate a mismatch in energy while momentum is conserved, or can recompense for both momentum and energy mismatch [12]. Such spatiotemporal QPM was realized using an all-optical modulation to enhance the efficiency of high order harmonic generation [15]. When the QPM modulation accelerates, light passing through the nonlinear material at different times would be subjected to different phase-matching conditions, allowing to manipulate nonlinear frequency-conversion processes and shape the temporal and spectral profiles of the generated fields [16]. This basic idea is used here to show that an i

4 accelerating QPM modulation can facilitate time-to-frequency mapping of optical pulses. We show that a judicious selection of the QPM geometry yields a spectrum which approximates the temporal intensity profile of the input pulse, while prior knowledge of the dispersion properties of the medium allows to extract the exact profile. While we develop this concept for second-harmonic generation it should be applicable to any frequency conversion process. ii

5 Table of Contents List of Figures 1 1: Introduction Nonlinear optics Second-harmonic-generation Phase-matching condition Quasi-phase-matching Time-to-frequency conversion : Theory Spatiotemporal quasi-phase-matching Accelerating quasi-phase-matching Quasi-phase-matching geometry Solution of the wave equation for the second-harmonic-generation process in the undepleted-pump approximation Influence of dispersion on the time-to-frequency conversion : Numerical simulations Development of the numerical model Numerical results

6 4: Conclusions 50 References 52 iv

7 List of Figures 1.1 Energy-band scheme describing second-harmonic-generation process. The dashed lines represent the virtual energy levels, and the solid line represent the ground state of the atom The dependence of the efficiency of the second-harmonic-generation process in the momentum mismatch Illustration of a nonlinear material in the form of (a) homogeneous and (b) periodically poled crystal Wave-vector representation of second-harmonic-generation process using quasi-phase-matching Spatial variation of the high-harmonic field amplitude generated in a nonlinear optical process for three different phase-matching conditions. Curve (c) shows the oscillations of the generated field amplitude for the case of nonzero momentum mismatch. Curve (a) depicts the high-harmonic field for perfect phase-matching, so the field amplitude increases linearly with distance. Curve (b) assumes quasi-phase-matching interaction, where the modulation period is twice the coherence length. The quasi-phase-matching modulation recompense for the momentum mismatch so the field amplitude is continuously growing with distance, however not as fast as in the perfect phase-matching condition Refractive index approximation error in percents for BBO

8 2.2 Space-time diagram of the real part of an example accelerating spatiotemporal QPM modulation. l c = k π 0 is the coherence length when energy is conserved, and t c = π ω 0 is the coherence time when momentum is conserved Frequency dependence of r(z, ω) at three different propagation distances: z/l D = 0.1 (blue), z/l D = 1 (green) and z/l d = 2.5 (red) Space-frequency diagram of the function r(z, ω) in a dispersive medium Space-frequency diagram of the function r(z, ω) in a non-dispersive medium FH field temporal envelope FH field temporal envelope squared SH spectrum for bτ0 2 = 40 at z/l D = 0.1 as given by the numerical solution (solid blue line) and by the analytical expression (dashed red line) SH spectrum for bτ0 2 = 40 at z/l D = 1 as given by the numerical solution (solid blue line) and by the analytical expression (dashed red line) SH spectrum for bτ0 2 = 40 at z/l D = 2.5 as given by the numerical solution (solid blue line) and by the analytical expression (dashed red line) SH spectrum normalized by r(z,ω) for bτ0 2 = 40 as given by the numerical solution (solid blue line) and by the analytical expression (dashed red line) SH spectrum for bτ0 2 = 4 at z/l D = 0.1 as given by the numerical solution (solid blue line) and by the analytical expression (dashed red line) SH spectrum for bτ0 2 = 4 at z/l D = 1 as given by the numerical solution (solid blue line) and by the analytical expression (dashed red line)

9 3.9 SH spectrum for bτ0 2 = 4 at z/l D = 2.5 as given by the numerical solution (solid blue line) and by the analytical expression (dashed red line) SH spectrum normalized by r(z,ω) for bτ0 2 = 4 as given by the numerical solution (solid blue line) and by the analytical expression (dashed red line) Numerical and analytical results for the time to frequency mapping. (a) The temporal envelope of the FH pulse (left) and its square (right), (b) SH spectrum at three different propagation distances: z/l D = 0.1, z/l D = 1 and z/l d = 2.5 (top, middle and bottom row respectively) and (c) the SH spectrum normalized by r(z,ω) at propagation distances of z/l D = 2.5, where bτ0 2 = 4 (left column) and bτ0 2 = 40 (right column) as given by the numerical solution (solid blue line) and by the analytical expression (dashed red line) The SH spectrum as given by the numerical solution (solid blue line) for chirped FH pulse with chirp rate γ/b = 0.01 where bτ0 2 = 40 at three different propagation distances: z/l D = 0.1, z/l D = 1 and z/l d = 2.5 (top, middle and bottom row respectively). The dashed red line illustrates the analytic solution when the chirp present in the FH field is negligible The SH spectrum as given by the numerical solution (solid blue line) for chirped FH pulse with chirp rate γ/b = 0.1 where bτ0 2 = 40 at three different propagation distances: z/l D = 0.1, z/l D = 1 and z/l d = 2.5 (top, middle and bottom row respectively). The dashed red line illustrates the analytic solution when the chirp present in the FH field is negligible The SH spectrum as given by the numerical solution (solid blue line) for chirped FH pulse with chirp rate γ/b = 1 where bτ0 2 = 40 at three different propagation distances: z/l D = 0.1, z/l D = 1 and z/l d = 2.5 (top, middle and bottom row respectively). The dashed red line illustrates the analytic solution when the chirp present in the FH field is negligible

10 1 Introduction 1.1 Nonlinear optics Presence of intense light can modify the optical properties of a material system. Nonlinear optics is the study of phenomena that transpire as a consequence of this modification. The discovery of second-harmonic-generation by Franken et al. (1961) [17] constitute the first stage of the field of nonlinear optics. Interaction of electric field with a dielectric medium applies force on the electrons and nucleuses of the material and create a separation of positive and negative electrical charges. The dielectric is then polarized, and the polarization P can be expressed as a power series in the applied field strength E: P i = ε 0 [ χ (1) j i j E j + j,k ] χ (2) i jk E je k + χ (3) i jkl E je k E l +... j,k,l (1.1) where ε 0 is the vacuum permittivity, the subscripts i, j,k refer to the Cartesian components of the electric field and the coefficients χ (n) are the n-th order susceptibilities, which are frequency-dependent tensors of rank n+1. In general, χ (n) refers to n th order nonlinear processes, where n photons of the the fundamental field are destroyed, and a new photon of frequency ω q is created. The various frequency components of n th order nonlinear processes are given by: ω q = n i=1 S i ω i (1.2) where S i can be equal either to 1 or 1. From Eq.(1.1) we can deduce that nonlinear atomic response can cause the polarization to produce new frequency components not present in the incident electric field. Each atom oscillates at those 4

11 new frequency components, and the oscillating atoms radiate and generate new frequency components in the electric field. For mathematical description of the nonlinear optical interactions, we start with Maxwell s equation in a non-magmetic mediun, so that: B = µ 0 H (1.3) that contain no free currents or free charges: D = 0 (1.4) H = 0 (1.5) E = µ 0 H t H = D t (1.6) (1.7) where µ 0 is the vacuum permeability and D is the electric displacement field corresponds to: D = ε 0 E + P (1.8) we take the curl of Eq.(1.6), interchange the order of space and time derivatives and substitute Eqs.(1.7) and (1.8) we get: E E c 2 t 2 = µ 2 P 0 t 2 (1.9) where c = 1 µ0 ε 0 is the speed of light. By using the identity: E = ( E) 2 E (1.10) and neglecting the term ( E), Eq.(1.9) becomes: 2 E 1 2 E c 2 t 2 = µ 2 P 0 t 2 (1.11) We can express the polarization and the electric displacement field as a sum of 5

12 linear and nonlinear components: P = P (1) + P NL (1.12) D = D (1) + P NL (1.13) D (1) = ε 0 E + P (1) = ε 0 ε (1) E (1.14) where ε (1) is a frequency-dependent 2-rank tensor. Eqs.(1.12),(1.13) and (1.14) can be introduced into the wave equation (1.11) to get: 2 E ε(1) c 2 2 E t 2 = µ 2 P NL 0 t 2 (1.15) This equation is the wave equation for the electric field where the nonlinear polarization acts as a source function Second-harmonic-generation Second-harmonic-generation is a second-order nonlinear process in which two photons of frequency ω 0 interacting with a nonlinear material are destroyed, and a new photon of frequency 2ω 0 is generated, as illustrated in Figure 1.1. Secondorder nonlinear optical interactions can occur only in noncentrosymmetric crystals, that is, in crystals that do not display inversion symmetry. Let us consider a nonlinear crystal for which the second-order susceptibility χ (2) is non-zero. When the nonlinear crystal is illuminated by a laser beam whose electric field strength is: E(t) = E 0 e iωt + c.c. (1.16) where: E 0 = A 1 (z)e ik 1z (1.17) the second-order nonlinear polarization that is created corresponds to: P (2) (t) = ε 0 χ (2) E 2 (t) = 2ε 0 χ (2) E 0 E 0 + ε 0 χ (2)( E 2 0e 2iωt + c.c. ) (1.18) The first term in the polarization expression, 2ε 0 χ (2) E 0 E0, is a DC component (zero frequency) which leads to a static electric field across the material, in a pro- 6

13 Figure 1.1: Energy-band scheme describing second-harmonicgeneration process. The dashed lines represent the virtual energy levels, and the solid line represent the ground state of the atom. cess known as optical rectification. This term is time invariant, so its second time derivative is zero and it does not generate electric field radiation. The second term, ε 0 χ (2)( E 2 e 2iωt + c.c. ) oscillates at frequency 2ω and leads to the generation of electric field radiation at the second-harmonic frequency. We will now derive the coupled-wave equation for second-harmonic-generation in a loss-less medium which maintains the condition of full permutation symmetry. When the nonlinear polarization is not too large, the second-harmonic electric field can be written as: E 2ω (z,t) = A 2 (z)e i(k 2z 2ωt) + c.c. (1.19) where k 2 = 2ω n(2ω) c, n(2ω) = ε (1) (2ω) and A 2 (z) is the slowly varying amplitude of the generated field. Substituting the polarization given by Eq.(1.18) and the second-harmonic electric field given by Eq.(1.19) into the wave equation (1.15) results in: ( 2 A 2 z 2 + 2ik A 2 2 z k2 2A 2 + (2ω)2 n 2 ) (2ω) c 2 A 2 e i(k2z 2ωt) + c.c. = 4ε 0 µ 0 χ (2) ω 2 A 2 1e i(2k 1z 2ωt) + c.c. (1.20) 7

14 The third and fourth expressions in the left-hand side of the equation cancel and we can drop the complex conjugate terms to obtain: ( 2 ) A 2 z 2 + 2ik A 2 2 = 4ε 0 µ 0 χ (2) ω 2 A 2 z 1e i(2k 1 k 2 )z (1.21) In the slowly varying envelope approximation we can neglect the first term on the left-hand side of the last equation and get the coupled-wave equation for the second-harmonic field amplitude: A 2 z = 2iχ(2) ω 2 k 2 c 2 A 2 1e i kz (1.22) where: k = 2k 1 k 2 (1.23) is the momentum mismatch. In analogous way we can derive the equation for the fundamental field envelope: A 1 z = 2iχ(2) ω 2 k 1 c 2 A 2 A 1e i kz (1.24) When the conversion of the fundamental electric field to the second-harmonic field is not too large, the amplitude A 1 can be taken as a constant. This assumption is known as the undepleted pump approximation. In this case, Eq.(1.22) can be integrated from z = 0 to z = L to obtain the second-harmonic field amplitude at the end of the interaction region: A 2 (z = L) = 2iχ(2) ω 2 L k 2 c 2 A 2 1 e i kz dz = 2iχ(2) ω 2 [ e 0 k 2 c 2 A 2 i kl ] 1 1 i k and the intensity is given by: (1.25) I 2 = χ(2)2 ω 2 L 2 ( ) kl 2n 2 n 2 1 ε 0c 3 I2 1sinc 2 2 (1.26) We see from the last equation that as k L increases, the efficiency of the secondharmonic-generation process decreases, with some oscillations. Figure 1.2 shows the dependence of the SH field intensity in the momentum mismatch. The con- 8

15 dition for efficiency k = 0 is known as the phase matching condition and is discussed in the following section. Figure 1.2: The dependence of the efficiency of the second-harmonicgeneration process in the momentum mismatch Phase-matching condition Many of the nonlinear processes are phase-sensitive, and require phase-matching to be efficient. This means that to get amplitude contributions from different locations to the product wave, a proper phase relationship between the interacting waves is required. Consider the second-harmonic-generation process described in the previous section. The fundamental field propagates in the nonlinear medium, accumulates phase according to e ik1z and develops a polarization wave corresponds to e i2k1z that continuously generates second-harmonic radiation that accumulates phase according to e ik2z. Because of dispersion in the material 2k 1 k 2, so second-harmonic radiation emitted from two different positions at a distance 9

16 l apart in the medium would accumulate a phase distance of (2k 1 k 2 )l. When l increase, energy can flow from the second-harmonic field back into the firstharmonic field, and so forth, so the second-harmonic field amplitude oscillates with propagation distance. The coherence length is the characteristic distance over which the fundamental and the second-harmonic fields exchange energy and is related to the momentum mismatch according to: l c = π k (1.27) When the condition of perfect phase matching is fulfilled, k = 0, there is a fixed phase relationship between the polarization wave and the second harmonic wave, and energy is continuously flowing from the first-harmonic field to the second-harmonic field. In this case, the radiation emitted by all the dipoles constructively interfere. From Eq.(1.25) and (1.26) we can see that the secondharmonic amplitude A 2 increase linearly with z, and the intensity increase quadratically with z. The most common technique for achieving phase matching in nonlinear crystals is birefringent phase matching. Birefringent is the optical property of a material in which the index of refraction depends on the light polarization and propagation direction. Most of the materials possess normal dispersion, in which the refractive index is an increasing function of frequency. Phase-matching can be achieved through the use of birefringent material by determining the polarization of the highest-frequency wave in the process to be in the direction that gives the lower index of refraction Quasi-phase-matching Quasi-phase-matching is a technique which allows a positive net flow of energy from the fundamental fields to the field generated in a nonlinear optical process by creating a periodic structure in the nonlinear medium. In the previous section we have seen that when the condition of perfect phase-matching is not satisfied the second-harmonic field amplitude oscillates with a period of 4l c. To avoid the oscillations and achieve continuously growth of the second-harmonic field amplitude, the sign of the nonlinear susceptibility χ (2) is inverted periodically as a 10

17 function of the position within the nonlinear material. Figure 1.3 illustrates homogeneous nonlinear crystal (part (a)) and a periodically poled nonlinear crystal (part (b)). The idea is that every time the high-harmonic amplitude is about to start to Figure 1.3: Illustration of a nonlinear material in the form of (a) homogeneous and (b) periodically poled crystal. diminish because of phase-mismatch, the sign of χ (2) is inverted so the field amplitude continues to grow. To allow that, the sign of the nonlinear susceptibility should change in multiples of the coherence length of the nonlinear interaction, so the period of the poled material is equal to twice the coherence length Λ = 2l c. Another way to understand this process is by examining the wave-vector relationship shown in Figure 1.4, where the wave-vector of the periodic modulated structure can compensate for the wave-vector mismatch of the interacting waves. The wave-vector is related to the modulation period according to: k qpm = 2π Λ, and by determining k qpm = k we obtain: Λ = 2π k = 2l c. The propagation-distance dependence of the field amplitude of the high-harmonic field generated in a nonlinear process is demonstrated in Figure 1.5 for three different phase-matching conditions- perfect phase-matching (a), quasi-phase-matching, where the modulation period has been set equal to twice the coherence length (b) and nonzero phase mismatch (c). Mathematical analysis of quasi-phase-matching can be formulated by determining the nonlinear susceptibility χ (2) to be spatially varying quantities. First, we define: d = χ(2) 2 (1.28) 11

18 Figure 1.4: Wave-vector representation of second-harmonic-generation process using quasi-phase-matching. Figure 1.5: Spatial variation of the high-harmonic field amplitude generated in a nonlinear optical process for three different phase-matching conditions. Curve (c) shows the oscillations of the generated field amplitude for the case of nonzero momentum mismatch. Curve (a) depicts the high-harmonic field for perfect phase-matching, so the field amplitude increases linearly with distance. Curve (b) assumes quasi-phasematching interaction, where the modulation period is twice the coherence length. The quasi-phase-matching modulation recompense for the momentum mismatch so the field amplitude is continuously growing with distance, however not as fast as in the perfect phase-matching condition. 12

19 The most trivial case for achieving quasi-phase-matching is for the square-wave modulation: [ ( )] 2πz d(z) = d e f f sign cos Λ (1.29) where d e f f is the nonlinear coefficient of the homogeneous material. d(z) is a periodic function and can be expressed as a Fourier series: where k n = 2πn Λ d(z) = d e f f D n e ik nz n= and D n are the Fourier coefficients given by: D n = 2 ( ) nπ nπ sin 2 (1.30) (1.31) By substituting the spatially varying nonlinear susceptibility given by Eq.(1.30) into the coupled-wave-equation for the second-harmonic field amplitude given by Eq.(1.22) we get: A 2 z = 4iω2 d e f f k 2 c 2 A 2 1 n= D n e i( k+k n)z (1.32) Quasi-phase-matching can be achieved through one of the Fourier component of d(z) by determining k + k m = 0. The nonlinear coupling coefficient D n tends to decrease with increasing n, so the most efficient quasi-phase-matching coupling is through the first-order component of the modulation: as we have achieved before. 2k 1 k 2 2π Λ = 0 Λ = 2π 2k 1 k 2 = 2l c (1.33) 1.2 Time-to-frequency conversion In this work we show that an accelerating quasi-phase-matching modulation can be used to measure temporal shapes of optical pulses through time-to-frequency conversion. Other methods for direct time to frequency mapping of the amplitude of pulses are also known. These are based on the time-frequency duality 13

20 between first-order dispersion and quadratic phase modulation and allow to estimate the intensity profile of pulses using spectral measurement after modulation of the original pulse [4, 5, 6, 7, 8, 9, 10, 11]. Kauffman et al. (1994) [5] demonstrated time-to-frequency conversion technique for measuring the waveform of optical pulses that is based on the space-time dualism, from which derives the analogy between the operation of temporal optical system on light pulses and the operation of spatial optical system on light beams. The technique uses a temporal-optical system consist of a dispersive delay line applied by a grating pair and a quadratic phase modulator that constitute a time lens, implemented by electro-optic phase modulator. According to Fourier optics, the field distribution at the output plane of a spatial lens is related to the field distribution at the front focal plane of the lens by a spatial Fourier transform. In accordance with space-time analogy, they showed that the same relation holds for a temporal optical system, so by passing the input pulse through a dispersive medium for one focal time and then through a time lens, the resulting pulse at the output plane of the time lens is the temporal Fourier transform of the input pulse. The output spectrum is the Fourier transform of the output field, so is related to the input pulse by two Fourier transforms, and thus posses it s functional distribution. Therefore, measuring the power of the output spectrum gives the temporal profile of the input pulse. A similar technique was described by Mouradian, et al. (2000) [6]. The proposed system for characterization of femtosecond pulses consists of a dispersive delay line and a nonlinear single-mode optical fiber. In the dispersive line, the input pulse expands in time and experience a negative chirp. Afterward, the chirped pulse is subjected into the nonlinear fiber, in which the chirp is canceled through the use of cross-phase-modulation process (time lens). The resulting pulse spectrum is an image in the spectral domain of the input pulse temporal waveform. For mathematical description, consider the input field amplitude: E In (t) = A 0 (t)e iω0t. The spectrum at the dispersive line output can be described as: H 1 (ω) = H 0 (ω)e iφ(ω), where H 1 (ω) and H 0 (ω) are the dispersive line output and input spectra respectively and e iφ(ω) is the dispersive line transfer function, where the phase can be written as: φ(ω) = φ 0 + φ (ω ω 0 ) + φ (ω ω 0 ) 2. The temporal 14

21 envelope at the dispersive line output is given by the convolution integral: A 1 (t) e i t2 2φ [ ] A(t )e i t 2 2φ tt i e φ dt = φ e i t2 2φ FT 1 A( φ ω)e i φ ω 2 2 where the last form is achieved by the exchange ω = t φ. Because of normal dispersion, the group-delay-dispersion is negative, so φ < 0 and thus the field amplitude A 1 (t) experience negative quadratic phase modulation, described by the term: e i t2 2φ. Then, the nonlinear optical fiber is used to cancel this quadratic phase modulation via cross-phase modulation, where an additional intense pulse is used to modify the desired phase modulation in the nonlinear fiber. This is equivalent to the propagation of the pulse through the dispersive medium for one focal time before entering the lens. The accuracy of this technique relies on this exact cancellation. The nonlinear fiber compensate for the chirp induced by the [ dispersive line, ] so the output field amplitude is given by: A 2 (t) φ FT 1 A( φ ω)e i φ ω 2 2 and the measured spectrum of the output signal is: H 2 (ω) A 0 ( φ ω)e i φ ω 2 2. Finally, the resulting intensity profile obeys: H 2 (ω) 2 A 0 ( φ ω) 2 I 0 ( φ ω) so the output spectrum is a scaled image of the temporal envelope of the input pulse. Azana, J. (2003) [18] implemented time-to-frequency mapping using a single time lens. In this work it was shown that there exist a time-frequency dualism between first-order dispersion and quadratic phase modulation (time lens), so that the action of first-order dispersion in the frequency domain (time domain) is analogous to the action of time lens in the time domain (frequency domain). To demonstrate the potential of this dualism, the dual regime of temporal Fraunhofer, which they called spectral Fraunhofer, was described. Optical pulse passing through a dispersive medium would experience temporal expansion due to first-order dispersion. If the dispersion is sufficiently strong, according to temporal Fraunhofer, the temporal profile of the dispersed pulse would be proportional to the Fourier transform of the temporal envelope of the fundamental pulse. So the spectral information of the fundamental pulse is converted from the frequency domain to 15

22 the time domain. According to the time-frequency dualism between first-order dispersion and time lens, it was demonstrated that a time lens can be used to map the input pulse from the time domain to the frequency domain when the temporal phase modulation is sufficiently strong. In this case, the achieved spectrum at the output of the time-lens is proportional to the temporal envelope of the pulse at the time-lens input, so ultrafast temporal pulse shapes can be measured with spectrometer. Li, B. and Azana, J. (2014) [19] performed temporal imaging of incoherent light waveforms via a time-domain equivalent apparatus of a spatial pinhole camera that performs time-to-frequency conversion process followed by frequency-totime mapping. Illumination from broadband, temporally incoherent optical source is subjected to an intensity modulator that implements the input waveform. The modulated light then propagates through a dispersive line, referred to as the input dispersion line, contributes a linear group delay with slope Φ In. The dispersed light is then transmitted through a temporal intensity modulator with a short pulse waveform, constituting the temporal pinhole. This is followed by another dispersive line, referred to as the output dispersion line, applies a linear group delay with slope Φ Out. The resulting averaged optical intensity is then a temporally scaled image of the input intensity profile. The input pulse experience time-to-frequency conversion process, implemented by the input dispersion line and the temporal pinhole, followed by frequency-to time conversion process, produced by the output dispersive line. When the optical source is a white noise, the system can be fully described by its impulse response, so temporal impulse at t 0 is considered. The input dispersion line induces linear chirp of slope Φ 1 to the input pulse, so In the input pulse is mapped from the spectral domain along the time domain. Then, the time-domain pinhole performs temporal filtering of the chirped pulse at a reference time, so the filtered light is a very short pulse and it s spectrum is much narrower the fundamental pulse spectrum and is centered at ω = t 0 Φ In. The resulting spectrum shows that time-to-frequency conversion is implemented. Than, the output dispersion line induces linear chirp of slope 1 Φ Out, so a frequency component ω i is mapped to the time t i = Φ Out ω i and for our case we get: t = Φ Out Φ In t 0, and the frequency-to-time process is performed. The entire system maps temporal impulse located at t 0 to Temporal impulse located at Φ Out Φ In t 0, where Φ Out Φ In is the 16

23 temporal scaling factor of the imaging process. Arons, E. et al. (1997) [20] exploited time-to-frequency conversion of chirped optical pulses to achieve high resolution range gating. The presented technique is based on ranging method used in microwave radar. A radiation of a chirped pulse is reflected from different object points, so each reflection is delayed by a time t i = 2r i c, with r i being the the range of the corresponding object point. The total reflected field thus consist of a sequence of identical chirped pulses, each delayed by a different amount of time. The returned field is then mixed with a reference field of the same form and opposite chirp through a nonlinear sumfrequency-generation process, so every replica of the chirped pulse is converted to a different constant frequency component in the mixed field spectrum according to: f i = 2 f 0 + α(t r t i ), where f 0 is the central carrier frequency of the chirped signals, t r and t i are respectively the temporal center of the reference pulse and the transmitted pulse corresponds to r i and α is the chirp rate. In this manner, the return signal is converted from the time domain to the frequency domain and there is a one-to-one matching between range and frequency, allowing to achieve range gating through the use of frequency filter. 17

24 2 Theory In this work we show theoretically that the temporal profile of an optical pulse can be mapped to the spectral profile of its up-converted harmonic when an accelerating quasi-phase-matching modulation is used. Thus the problem of temporal measurement at a given frequency band is converted to the problem of spectral measurement at another frequency band. The basic idea is that when an accelerating QPM modulation is applied, light passing through the nonlinear material at different times would be subjected to different phase-matching conditions, allowing to manipulate nonlinear frequency-conversion processes and shape the temporal and spectral profiles of the generated fields. We develop this concept for second-harmonic generation, although it can be applicable to any frequency conversion process. 2.1 Spatiotemporal quasi-phase-matching In the previous section we described nonlinear quasi-phase-matched process which involves energy conservation and momentum-mismatch. A complementary process can be envisioned that conserve momentum exactly at the expanse of energy mismatch, or where a mismatch is allowed in both energy and momentum. The requirement for momentum conversation is a consequence of continuous spatial translation symmetry, and spatial quasi-phase-matching is used to break this symmetry. Analogously, the requirement for energy conversation is a consequence of continuous temporal translation symmetry, so in this case a temporal quasiphase-matching is necessary. In the general case of both energy and momentum mismatch a spatiotemporal quasi-phase-matching is required. As a representative case, consider second-harmonic-generation process. As 18

25 described in the past sections, in traditional second-harmonic-generation two photons of a fundamental frequency ω are up-converted to a frequency w SH = 2ω, so energy is conserved: E = h(2ω ω SH ) = 0 (2.1) and there is a non-zero momentum mismatch given by: p = h 2ω c [ n(ω) n(2ω) ] (2.2) To compensate for the momentum mismatch, a spatial quasi-phase-matching modulation is required. The simplest spatial quasi-phase-matching scheme uses a periodic modulation with period Λ = 2π k = 2l c, allowing quasi momentum conservation and quasi-phase-matching is achieved. Now, let us consider up-conversion process from the fundamental frequency ω to the frequency ω SH = 2ω ω, where ω SH is chosen such that momentum is conserved: p = h c [ ω n(ω) ωsh n(ω SH ) ] = h k = 0 (2.3) Since ω SH 2ω, energy is not conserved in this process, and the energy mismatch is given by: E = h(2ω ω SH ) = h ω (2.4) To compensate for the energy mismatch, a temporal quasi-phase-matching is required. Equivalently to the coherence length, the coherence time of this process is given by t c = π ω (2.5) The simplest quasi-phase-matching scheme in this process uses a periodic temporal modulation with period T = 2t c. The more general case allows both energy and momentum mismatch. The generalized phase-mismatch relation that exist between the energy mismatch and the momentum mismatch is given by [12]: k = n(ω SH) ω c + qω 0 c [n(ω 0) n(ω SH )] (2.6) where q is the number of photons that participate in the nonlinear process (in 19

26 the case of second-harmonic-generation q = 2). The generalized phase-mismatch relation can also be derived for other nonlinear processes that involve more that two interacting waves. Quasi-phase-matching can be achieved by using periodic spatiotemporal modulation with periods Λ = 2l c, T = 2t c. One important advantage of the use of spatiotemporal quasi-phase-matching is that the generated field frequency ω SH can be continuously tuned. The processes described above using periodic quasi-phase-matching modulation, so every temporal slice of the fundamental pulse, or different pulses passing through the nonlinear crystal at different times would experience the same quasiphase-matching modulation and therefore be up-converted to the same frequency ω SH. The modulation in this case can be generated using a light pattern, such as a pulse train, propagating with a constant velocity [15]. 2.2 Accelerating quasi-phase-matching We now describe a complementary and more general process that consider the case in which the phase-matching condition between the interacting fields is time dependent. In that case, every temporal slice of the fundamental pulse would be subjected to a different quasi-phase-matching modulation and thus be up-converted to a different frequency component. In this process the modulation can be created using an accelerating light pattern. The energy mismatch and momentum mismatch become spatial-temporal varying quantities, and to achieve phase-matching the nonlinear modulation g(z,t) = e iφ(z,t) has to be such that the instantaneous spatial and temporal frequencies of the phase function φ(z,t) must obey: Φ(z,t) z Φ(z,t) t = k(z,t) (2.7) = ω(z,t) (2.8) Accelerating quasi-phase-matching modulation enables us to manipulate the spectral and temporal profiles of the field generated in the nonlinear process. We now consider an important example of accelerating quasi-phase-matching in which a specific quasi-phase-matching condition is satisfied along a coordinate 20

27 τ = t v z in a frame propagating with a constant velocity v. Depending on the desired application, the frame velocity can represent, for example, the group velocity or the phase velocity of the fundamental field. Such modulation allows an efficient frequency up-conversion process of a broad and continuous spectrum. If the frame velocity is set equal to the group velocity of the fundamental field, each time-interval t i within the fundamental pulse would continuously experience phase-matching condition corresponding to up-conversion process to a frequency ω i, so that every frequency component of the up- converted field would consecutively experience, along the entire interaction region, an appropriate quasiphase-matching condition required for efficient generation of electric field at that frequency component. 2.3 Quasi-phase-matching geometry Consider a non-linear optical frequency up- conversion process from a fundamental frequency ω 0 to ω SH = 2ω 0 ω. To describe this process, we start with the one-dimensional wave equation in the frequency domain for the second-harmonic (SH) electric field Ẽ 2ω0 (z,ω) in a non-magnetic medium under the non-depletion approximation: 2 Ẽ 2ω0 (z,ω) z 2 + β 2 (ω)ẽ 2ω0 (z,ω) = µ 0 ω 2 P NL (z,ω) (2.9) where β(ω) = n(ω)ω c, n(ω) is the index of refraction and c = 1 µ0 ε 0 is the speed of light with µ 0 and ε 0 being the vacuum permeability and permittivity respectively. P NL (z,ω) is the Fourier transform of the material 2nd order non-linear polarization: P NL (z,t) = ε 0 χ (2) g(z,t)e ω0 (z,t) 2 (2.10) where χ (2) is the second-order electric susceptibility, E ω0 (z,t) is the fundamentalharmonic (FH) electric field and g(z,t) = e iφ(z,t) which describes the QPM geometry is the spatiotemporal modulation of the non-linear electric susceptibility facilitating the spatiotemporal nonlinear photonic crystal. The spatial (temporal) frequency of Φ(z,t) can be used to phase match a momentum (energy) mismatch of the nonlinear process. 21

28 To convert the temporal information of the FH field to spectral profile of the SH field, we want to transform every time interval of the fundamental pulse t i to a different frequency component ω SHi. Thus the conditions for phase matching correspond to momentum mismatch k i and energy mismatch ω i which are satisfied along a coordinate τ in the frame moving at the FH pulse group velocity v g1 : τ = t z v g1, ξ = z. Linear time-to-frequency mapping is therefore achieved through a judicious selection of the QPM geometry that implements a linear dependence of the energy mismatch in the reduced time τ: ω(τ) = a + bτ, with a [sec 1 ] and b [sec 2 ] being the modulation parameters determines the displacement in the central frequency of the SH spectrum and the chirp rate of the timeto-frequency mapping respectively. To find the phase function Φ(z,t) obeying this condition, we use the generalized phase-mismatch condition that exist between the energy mismatch ω, the momentum mismatch k and the material dispersion: k(τ) = ω(τ) n(ω SH) c + 2ω 0 c [n(ω 0) n(ω SH )] (2.11) To achieve phase matching, the instantaneous spatial and temporal frequencies of the phase function must obey: Φ(z,t) z = k(z,t) In the τ, ξ frame : Φ(z,t) t = ω(z,t) Φ(ξ,τ) ξ = k(ξ,τ) 1 v g1 ω(ξ,τ) (2.12) Φ(ξ,τ) τ = ω(ξ,τ) (2.13) Since the energy mismatch obeys: ω = ω(τ), one can conclude from Eq.(2.11) that the momentum mismatch is also a function of τ alone. Eq.(2.12) can be inte- 22

29 grated to give: [ Φ(ξ,τ) = ξ k(τ) 1 ] ω(τ) + f (τ) (2.14) v g1 where f (τ) is a general function of τ. Substituting Eq.(2.14) into Eq.(2.13) yields: [ d k(τ) ξ 1 ] d ω(τ) + d f (τ) = ω(τ) dτ v g1 dτ dτ To fulfill the last equation, the term d k(τ) dτ v 1 d ω(τ) g1 dτ must be equal to zero: d k(τ) dτ 1 d ω(τ) = 0 v g1 dτ We now check what this condition entails on the index of refraction. Using Eq. (2.11) we get: 1 c n SH(τ) d ω(τ) + 1 dτ c ω(τ)dn SH(τ) 2ω 0 dn SH (τ) 1 d ω(τ) = 0 dτ c dτ v g1 dτ [ nsh ( ω) c + ω(τ) dn SH ( ω) 2ω 0 dn SH ( ω) 1 ] d ω(τ) = 0 c d ω c d ω v g1 dτ 0 and the last equation is equiv- The energy mismatch depends on τ, so d ω(τ) dτ alent to: n SH ( ω) + ω(τ) dn SH( ω) d ω 2ω 0 dn SH ( ω) d ω The solution of this differential equation is given by: n SH ( ω) = c v g1 ω + c 1 ω 2ω 0 c v g1 = 0 (2.15) where c 1 is the integration constant chosen to be: c 1 = 2ω 0 n(2ω 0 ) so n SH ( ω = 0) = n(2ω 0 ) and we get: n SH ( ω) = c v g1 ω 2ω 0 n(2ω 0 ) ω 2ω 0 (2.16) The meaning is that if the refractive index depends on the energy mismatch in this 23

30 way than we can find a matching modulation for any frequency mapping ω(τ) we choose. This approximation for the refractive index can also be achieved when group velocity dispersion can be neglected such that v g1 v g2 in the Taylor expansion of n. Figure 2.1 plots the approximation error of the refractive index for BBO in percents and we can see that there is a good match. Figure 2.1: Refractive index approximation error in percents for BBO When the refractive index corresponds to Eq.(2.16), the expression k(τ) 1 v g1 ω(τ) is a constant: k(τ) 1 ω(τ) = 2ω 0 [ n(2ω0 ) n(ω 0 ) ] k 0 (2.17) v g1 c and k 0 = π l c, where l c is the coherence length in an energy-conserving SHG process. Therefore, Eq.(2.14) becomes: Φ(ξ,τ) = k 0 ξ + f (τ) (2.18) By introducing Eq.(2.18) into Eq.(2.13) and determining the energy mismatch to be: ω(τ) = a + bτ (2.19) 24

31 we obtain: d f (τ) d(τ) = a bτ = f (τ) = aτ 1 2 bτ2 + A 0 where A 0 is an arbitrary constant. Finally, the phase function is: Φ(ξ,τ) = k 0 ξ aτ 1 2 bτ2 + A 0 (2.20) and the modulation is given by: [ ] g(ξ,τ) = e j Φ(ξ,τ) (2.21) Figure 2.2 depicts the space-time diagram of the real part of the accelerating spatiotemporal QPM modulation g(z,t) with the phase function given by Eq.(2.20) for b = π, where t tc 2 c = π ω 0 is the coherence time in a momentum-conserving SHG process corresponds to energy mismatch of ω 0 = 2ω 0 [ n(2ω0 ω 0 ) n(ω 0 ) ] (2.22) n(2ω 0 ω 0 ) For this figure we set A 0 = 0 and a = bt 0 with t 0 being at the middle of the FH pulse to achieve the central frequency of the SH field at 2ω Solution of the wave equation for the second-harmonic-generation process in the undepleted-pump approximation In the undepleted-pump approximation the fundamental electric field is: E ω0 (z,t) = A(t z v g1 )e i(ω 0t k 0 z) (2.23) where k 0 = n(ω 0)ω 0 c and A(t v z g1 ) is the pulse envelope we are interested in imprinting upon the SH spectrum. We now represent the envelope function with a Fourier series multiplied by a Gaussian window function which isolates one in- 25

32 Figure 2.2: Space-time diagram of the real part of an example accelerating spatiotemporal QPM modulation. l c = π k 0 is the coherence length when energy is conserved, and t c = π ω 0 is the coherence time when momentum is conserved. stance of the periodic arrangement: A(t) = e t2 T 2 f n e i ω nt n= (2.24) where ω n = 2πn T with an arbitrarily large time period T and f n are the Fourier series coefficients for this periodic expansion of the envelope. For such a representation we require that: T T τ 0 with τ 0 being the temporal width of the FH. Under these conditions we can write the envelope as: A(t z ) v = e t2 T 2 g1 f n e i ω n(t z ) vg 1 n= (2.25) Notice that T can be set as large as required so the Gaussian function need not be moved together with the Fourier components. Substituting the modulation phase function given by Eq.(2.20) and the fundamental electric field given by Eqs. (2.23) 26

33 and (2.25) into the nonlinear polarization expression given by Eq.(2.10) yields: P NL (z,t) = ε 0 χ (2) g(z,t)e ω0 (z,t) 2 ( ) P NL (z,t) = ε 0 χ (2) e i k 0 z a(t v z ) 1 g1 2 b(t v z ) 2 g1 e i(2ω 0t 2k 0 z) t2 2 e T 2 n= m= f n f m e i( ω n+ ω m )(t z v g1 ) P NL (z,t) = ε 0 χ (2) e i k0z e i(a b v z )t g1 e i 1 2 bt2 e ia v z g1 e i 1 2 b z2 v 2 g1 e i(2ω 0t 2k 0 z) t2 2 e T 2 n= m= f n f m e i( ω n+ ω m )(t z v g1 ) P NL (z,t) = n= m= ( ) ε 0 χ (2) f n f m e i k 0 +2k 0 + a+ ω n+ ωm v z i 1 g1 2 b z2 e v 2 g1 ( ( ) e i 2ω 0 +a+ ω n + ω m b v )t z 2 g1 e T 2 i b 2 t 2 (2.26) Applying a Fourier transform in the time domain results in: P NL (z,ω) = n= where we defined: m= ( ) ε 0 χ (2) f n f m e i k 0 +2k 0 + a+ ω n+ ωm v z i 1 g1 2 b z2 e v 2 g1 e 4 ib T 2 (Ω+b v z ) 2 g1 8 T 2 2ib (2.27) Ω = ω 2ω 0 a ω n ω m (2.28) By selecting T such that: 8 T 2 << 2b, the right-most term in the last equation becomes: e (Ω+b v z ) 2 g1 8 T 2 2ib = e 8 Ω2 2Ωb z b 2 z2 v g1 v 2 T 2 2ib 8 g1 e T 2 2ib 8 e T 2 2ib e 8 Ω2 T 2 2ib e iω v z g1 e i 1 2 b z2 v 2 g1 27

34 so from Eq.(2.27) we get: P NL (z,ω) = Now we define: n= m= ( ) ε 0 χ (2) f n f m e i k 0 +2k 0 + a+ ω n+ ωm+ω v z g1 e ib K(ω) = k 0 + 2k 0 + a + ω n + ω m + Ω v g1 = k 0 + 2k 0 + ω 2ω 0 v g1 so the non-linear polarization can be written as: P NL (z,ω) = ε 0χ (2) ib e ik(ω)z n= m= f n f m e 8 Ω2 T 2 2ib (2.29) 8 Ω2 T 2 2ib (2.30) By substituting the non-linear polarization given by Eq.(2.30) into the wave equation (2.9) we get a second-order differential equation for the spectrum of the SH electric field: 2 Ẽ 2ω0 z 2 + β 2 (ω)ẽ 2ω0 = µ 0 ω 2 ε 0χ (2) ib e ik(ω)z n= m= Let um examine the sum expression in the last equation: n= m= f n f m e 8 Ω2 T 2 2ib = n= m= f n f m e f n f m e (ω 2ω 0 a ω n ωm) 2 8 T 2 2ib = 8 Ω2 T 2 2ib (2.31) = n= m= applying 8 T 2 << 2b : f n f m e (ω 2ω 0 a)2 8 T 2 2ib e (ω 2ω 0 a)( ωn+ ωm) 4 ( ω n+ ωm) 2 T 2 ib 8 e T 2 2ib e (ω 2ω 0 a)2 8 T 2 2ib n= m= f n f m e i (ω 2ω 0 a) b ( ω n + ω m ) e i ( ω n+ ωm) 2 2b As the ω n frequencies span the FH envelope their largest absolute values would be bounded by 1/τ 0 (this means that the Fourier coefficients for larger ω n would be negligible). On the other hand 2(ω 2ω 0 a) is bounded by bτ 0. This 28

35 means that for bτ0 2 1 we can neglect the expression ( ω n+ ωm) 2 e i 2b P NL (z,ω) = ε 0χ (2) [ e i (ω 2ω 0 a)2 2b e ik(ω)z A ( ω 2ω 0 a) ] 2 ib b and get: (2.32) Where A(t) is the temporal envelope of the input pulse. Substituting to the wave equation given by Eq.( 2.9) leads to 2 Ẽ 2ω0 z 2 + β 2 (ω)ẽ 2ω0 = µ 0ε 0 χ (2) ω 2 ib [ e i (ω 2ω 0 a)2 2b A ( ω 2ω 0 a b ) ] 2e ik(ω)z (2.33) By solving Eq.(2.33) with the initial conditions: Ẽ 2ω0 (z = 0,ω) = zẽ2ω 0 (z = 0,ω) = 0 we obtain the spectrum of the SH electric field: where: [ Ẽ 2ω0 (z,ω) = r(z,ω) A ( ω 2ω 0 a) ] 2 b (2.34) r(z,ω) = µ 0ε 0 χ (2) ib ω 2 (ω 2ω 0 a) 2 K(ω) 2 β(ω) 2 e i 2b ( e ik(ω)z + cos ( β(ω)z ) i K(ω) β(ω) sin( β(ω)z ) ) (2.35) These last two equations constitute the major result of this work. The term r(z,ω) represents the distortions in the time to frequency mapping due to dispersion of the FH during the nonlinear interaction. Still, knowledge of the medium s dispersion allows to extract the original FH amplitude regardless of the interaction length. In this case, normalizing the spectrum of the SH electric field by r(z,ω) would give us the square of the FH pulse temporal envelope with the exchange: t ω 2ω 0 a b. Figure 2.3 plots the absolute value of r(z,ω) as a function of frequency at three different propagation distances: z/l D = 0.1, z/l D = 1 and z/l d = 2.5. In most cases K(ω)/β(ω) 1 for the relevant frequency range, in which case we can approximate the amplitude of Eq.2.35 as: r = µ 0ε 0 χ (2) b [ (K(ω) β(ω))z sinc 2 ] zω 2 K(ω) + β(ω) (2.36) 29

36 Figure 2.3: Frequency dependence of r(z, ω) at three different propagation distances: z/l D = 0.1 (blue), z/l D = 1 (green) and z/l d = 2.5 (red). From this expression we can estimate the distance at which the distortions get significant as min ω {2π/(K(ω) β(ω))}. This value depends on the bandwidth of the generated SH pulse, however, assuming this bandwidth would be wider than the FH bandwidth we can express the limiting case as the following dispersion length: l D = τ0 2/ β 2 where β 2 = max{β2 FH,β2 SH } is the maximum Group Velocity Dispersion (GVD) over the GVD of the SH and of the FH pulses. For interaction lengths which are smaller than the dispersion length the SH spectrum would constitute a good approximation to the envelope function (as long as we also keep bτ0 2 1). An important point to consider is that due to the phase matched conditions under which this interaction takes place, the SH field is being built up efficiently and so a long interaction length increases the generated SH signal strength. Note that in our derivation we made an implicit assumption that any chirp present in the envelope A(t) is much smaller than the chirp rate b of the QPM modulation and so can be neglected. Significant chirp in the envelope would smear the time to frequency mapping. Figure 2.4 depicts the space-frequency diagram of the function r(z,ω) in a dispersive medium. Non-dispersive case 30

37 Figure 2.4: Space-frequency diagram of the function r(z, ω) in a dispersive medium. When there is no dispersion: n(ω) = n = const, so k 0 = 0, v g1 = n c K(ω). In this case, the spectrum of the SH electric field becomes: [ Ẽ 2ω0 (z,ω) = r(z,ω) A ( ω 2ω 0 a) ] 2 b and β(ω) = (2.37) where: r(z,ω) = µ 0ε 0 χ (2) ω 2 (ω 2ω0 a)2 e i ( ) 2b 4 ibβ(ω) 2 e 2iβ(ω)z 2iβ(ω)z 1 e iβ(ω)z (2.38) Figure 2.5 depicts the space-frequency diagram of the function r(z,ω) in a nondispersive medium. r(z, ω) is a slowly varying linear function of frequency, so when the SH spectral width is not too large it can be approximated as a constant and the input pulse temporal shape can be given by scaling the frequency axis of the non-normalized SH spectrum to a time axis. 31

38 Figure 2.5: Space-frequency diagram of the function r(z, ω) in a nondispersive medium. 2.5 Influence of dispersion on the time-to-frequency conversion Now we evaluate with more details the range of interaction lengths in which distortion takes place and normalization of the SH spectrum is required for good accuracy of the deduced FH waveform. Different frequency components of the SH 1 field travel through the medium at different velocities according to: vω = β(ω) ω, where β(ω) = n(ω) ω c. Thus every frequency component is delayed by different amount of time and the pulse is temporally expanded. If the spectrum of the SH pulse is in the range of [Ω 1,Ω 2 ], than the maximum temporal delay between the different components of the SH field is: T max = L ( 1 v 2 1 v 1 ) where v i is the velocity of the frequency component Ω i. To avoid distortions, the maximum temporal delay have to be small relative to the unstreched SH pulse 32