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OPTICS LETTERS / Vol. 40, No. 9 / May 1, 2015
Dynamic control of light beams in second harmonic generation Ana Libster-Hershko,1,* Sivan Trajtenberg-Mills,2 and Ady Arie1 1
School of Electrical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel 2 School of Physics, Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel *Corresponding author: [email protected] Received January 29, 2015; revised March 29, 2015; accepted March 30, 2015; posted March 31, 2015 (Doc. ID 233479); published April 20, 2015
In this Letter, we report the dynamic control of the spatial shape of the second harmonic (SH) beam generated in a nonlinear crystal, by controlling the phase of the input fundamental beam before entering the crystal. This method enables 2D beam shaping and does not require any special fabrication beforehand. We have shown in simulation and experiment that this is possible for both short and long crystals: for short crystals, we assume the transverse phase of the SH beam is doubled relative to the input phase of the fundamental beam; for longer crystals, genetic algorithms were used in order to solve the inverse phase problem, which generally does not have an analytical solution. The method we present enables us to dynamically shape a beam in a nonlinear process, using standard crystals and optical equipment, and without the need to use any optical element after the nonlinear crystal. © 2015 Optical Society of America OCIS codes: (190.2620) Harmonic generation and mixing; (100.5070) Phase retrieval. http://dx.doi.org/10.1364/OL.40.001944
Spatial shaping of the second harmonic beam generated in a nonlinear crystal eliminates the need for additional optical elements and provides a compact manner for generating a desired beam shape. It has been explored during the past few years using several different methods: by modulating the quadratic nonlinear susceptibility of the crystal [1–6] or by functionalizing the exit facet of the nonlinear crystal for changing the beam shape at the exit of the crystal [7]. All of these methods require special fabrication procedures, after which they are permanent; thus, they cannot be changed to attain a different beam shape. Also, the first method may sometimes require us to use short crystals [3–5], and, in many cases, the desired shape emerges in multiple diffraction orders, resulting in significant power loss of the SH [4,5,7]. Due to fabrication limitations, these methods are mainly suitable for modulating the beam in only one transverse dimension [1–4,6]. In this Letter, we present a new method of shaping the generated SH beam by controlling the phase of the fundamental beam before it enters the nonlinear crystal, using a spatial light modulator (SLM). This method enables dynamic beam shaping, does not require any special fabrication beforehand, and does not require any use of optical equipment after the crystal. Shaping beams at the fundamental wavelength is highly advantageous in cases in which optical elements or spatial light modulators are unavailable or extremely expensive at SH wavelengths, e.g., in the ultraviolet range. SLM is a rather complex and expensive device, but there are other alternatives such as digital micromirror arrays and deformable mirrors that can be used for generating the desired phase. For sufficiently short crystals, the transverse phase of the fundamental beam is doubled, thereby providing a straightforward method to shape the second harmonic beam, while, for longer crystals, beam shaping can be achieved by optimizing the input beam. We also provide criteria for determining whether the “short crystal” approximation can be used. 0146-9592/15/091944-04$15.00/0
A slowly varying SH complex field envelope A2ω with frequency 2ω obeys the following equation in a quadratic nonlinear media [8]: ∇2T A2ω 2ik2
∂A2ω −κA2ω ei2k1 −k2 z ; ∂z
(1)
where Aω is, in our case, the fundamental Gaussian beam with frequency ω; k1 and k2 are the wave numbers of the fundamental and SH waves, respectively; ∇2T ∂2 ∕∂x2 ∂2 ∕∂y2 is the transverse Laplacian; z is the propagation distance; and κ χ 2 ω22 ∕c2 is the nonlinear coupling coefficient. If the process is (birefringently or quasi-) phase matched, we can ignore the term expi2k1 − k2 z. If the beam diffraction in the crystal is negligible, we may omit the first diffraction term on the left-hand side of Eq. (1) and obtain dA2ω iκ 2 − A : dz 2k2 ω
(2)
For a fundamental beam with a transverse phase φω x; y and complex envelope Aω jAω j expiφω , under the undepleted pump approximation the solution is, simply, A2ω x; y ≈ −
iκ jA x; yj2 ei2φω x;y L; 2k2 ω
(3)
where L is the crystal length. Hence, the SH is a Gaussian beam with an approximated phase of 2φω x; y. For a longer crystal, the variation of the fundamental beam along the crystal has to be considered; therefore, we cannot estimate the output phase in the same simple way. The inverse phase problem has no analytic solution for this type of crystal, but a numerical solution can be found for specific beams, as will be shown below. By using this approximation for short crystals, we can encode a desired phase pattern on the SH beam and, © 2015 Optical Society of America
May 1, 2015 / Vol. 40, No. 9 / OPTICS LETTERS
therefore, create arbitrary 2D beam shapes in the far field. The experimental setup is presented in Fig. 1(a). We used an SLM with 512 × 512 pixels, each pixel of 15 × 15 μm, but, for our applications, we found it easier and sufficient to use only the 256 × 256 centered pixels of the SLM for encoding our desired phase. We used a pulsed 1550 nm signal of our homemade OPO, with a peak power of 180 W, and a 1 mm long periodically poled KTiOPO4 (PPKTP) crystal with a poling period of 24.7 μm held at 35°C. The SLM plane was imaged to the crystal entrance plane with a demagnification of 4.3. The fundamental beam at the entrance to the nonlinear crystal has a Gaussian intensity profile and a phase that is added by the SLM. The generated beams were examined using a CCD camera after passing through a lens for Fourier transform (for far-field images). In order to test the beam-shaping capabilities, we generated several high-order Hermite–Gauss (HG) beams [9] and a 2D Airy beam [10]. An HGlm beam has a complex envelope of the form: Hl
p p 2x 2y Hm A x; y; z × expiZZ; W Z W Z G
(4)
where AG x; y; z is a Gaussian beam with a beam width of W Z, H l u is the Hermite polynomial of order l, and ZZ is a function of z. For HG10 , this complex envelope is proportional to ∝ x × AG x; y; z. We can approximate this form by multiplying a Gaussian beam by a phase step of π on the x axis, effectively creating a beam proportional to ∝ signx × AG x; y; z. When propagated, this beam will take a form very similar to the theoretical HG10 (or HG01 , by using signy) beam. Similar use of simple variations of phase steps for creating HG beams can be further shown for higher orders as well. For generating Hermite–Gauss HG10 (HG01 ) beams at the SH, we used a fundamental Gaussian beam, having a waist size of 250 μm at the crystal entrance plane with a phase step of π∕2 in the X direction (Y direction). According to Eq. (3), the SH beam at the crystal exit plane is, therefore, (approximately) a Gaussian beam with a phase step of π, which evolves to a beam that has a similar shape to the HG01 beam [9]. For generating an HG11 SH beam, we used a fundamental Gaussian beam with phase steps of π∕2 in both X and Y directions. The SH Airy was generated by frequency doubling a Gaussian beam with a cubic phase [9] expibx3 y3 , where (a)
1945
b 1011 m−3 . Here, we used the Fourier transform relation between the cubic phase function and the Airy function. The experimental and simulated results for HG01 , HG10 , HG11 , and Airy beams are presented in Fig. 2. The simulations are based on the split-step Fourier method [11]. Another functionality that can be achieved using this method is beam focusing. In this case, a fundamental beam with a quadratic transverse phase expix2 y2 ∕2λf , where f is the focal length, was sent to the crystal. As predicted by Eq. (3), the SH beam will, therefore, obtain a quadratic transverse phase as well, and the beam will be focused at f after the crystal. Focusing of the SH beam 30 mm after the crystal edge is presented in Fig. 3(a). In order to understand the limits of this method, we evaluated the range of crystal lengths for which our “short crystal” assumption is valid. We simulated a variety of crystal lengths for the phase mismatch conditions of our experiment and calculated the cross correlation of the output SH beams’ phase to the desired phase (using the simplest HG01 phase step for these simulations). Our results clearly show that, if we require a cross correlation of at least 0.95, the region of acceptance is for crystals shorter than ∼4 mm, as seen in Fig. 4(a). We have found that this length is a function of the material’s dispersion (or equivalently the phase mismatch), by repeating the simulation described above for different wavelengths, whereby in each wavelength it is assumed that the crystal is periodically poled to quasi-phase match a collinear interaction. Figure 4(b) shows the results of these
Fig. 2. Simulated (top) and experimental (bottom) results for 1 mm (short) crystal with the following beams: (a) and (b) HG01 ; (c) and (d) HG10 ; (e) and (f) HG11 ; (g) and (i) 2D Airy beam.
SLM fundamental SH phase phase profile profile
(b) SLM
lens
NL crystal
fundamental phase profile lens
SH phase profile NL crystal
Fig. 1. Experimental setup for short (a) and long (b) crystals.
Fig. 3. Experimental measurements of beam focusing of the SH beam for (a) 1 mm and (b) 12 mm crysal, both with a focal lengths of 30 mm. In (b) focusing is only in Y direction.
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OPTICS LETTERS / Vol. 40, No. 9 / May 1, 2015
Fig. 4. Simulation results of the region of validity of the “short crystal” approximation: (a) cross correlation of the output SH phase to the desired phase for the crystal’s phase mismatch at 1550 nm. (b) Threshold crystal length for which the cross correlation shown above is better than 0.95.
simulations for KTiOPO4 , but nearly the same values are obtained for Mg-doped congruent LiNbO3 and for stoichiometric LiTaO3 . This method is also subjected to the input beams’ spatial frequency: the cross correlation of the output beam’s shape with the desired shape is a function of the beam waist. We have shown through numerical simulations that the relation z0 ∕L is constant, where z0 πW 20 ∕λ is the Rayleigh range of the smallest beam that provides a cross-correlation value higher than 95%. For crystals of different lengths ranging between 1 and 18 mm, this relation is 7.56 0.38 for a correlation of 0.95 0.002 to an HG01 beam, indicating that the beam divergence along the crystal length can indeed be neglected. For a long crystal, which provides higher conversion efficiency, the problem is more complex, since we can no longer ignore the variation of the transverse profile of the beams along the interaction length. Therefore, we cannot analytically derive the phase that should be applied to the fundamental beam in order to receive the desired SH beam phase pattern at the output of the crystal. Since the interaction is nonlinear, a numerical solution is also impossible: one cannot propagate the beam in the opposite direction in order to receive the initial phase from the desired known output phase, even though all parameters of the system are deterministic. We can treat this problem as an inverse problem of a beam propagating through a nonlinear material. In order to solve it, an optimization genetic algorithm (GA) was implemented. Optimization algorithms were used in the recent years for focusing [12–14] and imaging [15] beams through turbid media as well as for focusing beams through nonlinear random media [16,17]. Genetic algorithms are optimization search algorithms that imitate the evolution process in nature. Here, we implemented
the algorithm described in [14]. The suggested solution is valid for beams without rapid changes in the amplitude or phase, e.g., low orders of HG beams and lenses. We start with a randomly chosen population of 30 input phase masks, which are then each multiplied by a Gaussian beam to create an input beam with a different transverse phase. Each beam is propagated through the nonlinear media, generating an SH beam. Each phase mask is then ranked according to the cross correlation between the generated SH beam profile and the desired one. In each iteration, 15 offspring phase masks are generated, with random mutations added to them. After each iteration, the lower-ranked phase masks in the population are replaced with a new generation of phase masks, which have higher rank. Higher-ranked phase masks have a better probability of becoming parents. A random binary matrix is used for the breeding process to generate one offspring from “father” and “mother” phase masks. Since the nonlinear crystal is deterministic, all optimization processes can be first simulated and then implemented on the experimental system without further iterations, unlike other optical systems containing a random media [13–15], where the optimization must be done on the experimental setup itself. Convergence occurred after ∼1000 iterations (generations). Typical computation time for a 1D shape running on a single Intel i3 CPU on a Windows 7 computer was around 5 h until convergence. All the computations were performed for transverse phase modulation in a single axis to reduce calculation time but can be implemented for 2D systems as well. As in the case of the short crystal, we used only the centered 256 × 256 pixels of the SLM. We used a 1550 nm diode continues wave (CW) laser with average power of 10 mW and a 12 mm long PPKTP crystal with a poling period of 24.7 μm held at 35°C. The phases, which were obtained using the simulations, were applied to the SLM. The phase on the SLM plane was imaged to the crystal’s entrance plane. This phase was added to a fundamental beam with a Gaussian intensity. The generated SH beams were recorded using a CCD camera (Fig. 5). We measured the beam quality parameter M2 of every beam [18]. For the HG01 , HG10 , HG02 , and HG20 , the theoretical M2 values are 3, 3, 5 and 5. As can be seen in Table 1, the measured M2 is quite close to the theoretical prediction. In addition, the measured cross correlation between the measured beam and the theoretical beam
Fig. 5. Simulated (top) and experimental (bottom) results for 12 mm (long) crystal with the following beams: (a) and (b) HG01 ; (c) and (d) HG10 ; (e) and (f) HG02 ; (g) and (i) HG20 .
May 1, 2015 / Vol. 40, No. 9 / OPTICS LETTERS Table 1. Beam Quality Parameter and Cross Correlation Between Measured and Desired SH Beam in the Long (12 mm) Crystal Desired Beam HG10 HG01 HG20 HG02
M2
Cross Correlation
3.083 3.040 5.288 5.159
0.9887 0.9461 0.9465 0.9696
was above 0.94 in all cases, showing again the high quality of the generated SH beams. As mentioned before, we have confirmed experimentally that the off-line optimization process produces good results with no need for additional iterations on the experimental setup itself. We have also generated an SH beam with a quadratic transverse phase (in 1D), which focuses the beam 30 mm from the edge of the crystal [see Fig. 3(b)]. Since the phase mask was planned only for one dimension, the beam diverges in the orthogonal direction, in which the focusing function is not activated. In conclusion, this Letter presents a new method for shaping the generated SH beam in a nonlinear crystal by shaping the phase of the fundamental beam before it enters the crystal. We separated the solution into two parts: short and long crystals. For a short crystal, we showed that we can approximate the output phase as twice the input fundamental beam phase, therefore providing a straightforward method of shaping. For a long crystal, we used GA to solve the inverse problem. We provided guidelines for determining whether the “short crystal” approximation is valid. This method enables flexible and dynamic shaping of beams, does not require any special fabrication, and does not require any additional optical elements after the crystal. The concepts presented here can be further expanded for temporal shaping of pulses in a nonlinear process [19]. In the future, the methods we presented here can be further explored by using phase-retrieval algorithms for more sophisticated beam shapes [20]. Shaping both phase and amplitude of the fundamental beam before the input of the crystal, as well as combining different beam shaping techniques [1–6] for a
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hybrid approach, can lead to complete dynamic control of the SH beam shape. The authors would like to acknowledge E. Burstein for the technical support. This work was supported by the Israeli Ministry of Science, Technology and Space in the framework of the Israel–Italy bi-national collaboration program. References 1. G. Imeshev, M. Proctor, and M. Fejer, Opt. Lett. 23, 673 (1998). 2. J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. Fejer, IEEE J. Sel. Top. Quantum Electron. 8, 660 (2002). 3. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009). 4. A. Shapira, I. Juwiler, and A. Arie, Opt. Lett. 36, 3015 (2011). 5. A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, Opt. Lett. 37, 2136 (2012). 6. X.-H. Hong, B. Yang, C. Zhang, Y.-Q. Qin, and Y.-Y. Zhu, Phys. Rev. Lett. 113, 163902 (2014). 7. A. Shapira, A. Libster, Y. Lilach, and A. Arie, Opt. Commun. 300, 244 (2013). 8. R. W. Boyd, Nonlinear Optics (Academic, 2008). 9. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2013). 10. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007). 11. G. Agrawal, Nonlinear Fiber Optics (Academic, 2001). 12. I. M. Vellekoop and A. P. Mosk, Opt. Commun. 281, 3071 (2008). 13. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012). 14. D. B. Conkey, A. N. Brown, A. M. Caravaca-Aguirre, and R. Piestun, Opt. Express 20, 4840 (2012). 15. O. Katz, E. Small, and Y. Silberberg, Nat. Photonics 6, 549 (2012). 16. C. Yao, F. J. Rodriguez, and J. Martorell, Opt. Lett. 37, 1676 (2012). 17. C. Yao, F. J. Rodriguez, J. Bravo-Abad, and J. Martorell, Phys. Rev. A 87, 063804 (2013). 18. S. Saghafi and C. J. R. Sheppard, Opt. Commun. 153, 207 (1998). 19. Z. Zheng and A. M. Weiner, Chem. Phys. 267, 161 (2001). 20. R. W. Gerchberg and W. O. Saxton, Optik (Stuttg). 35, 237 (1972).
OPTICS LETTERS / Vol. 40, No. 9 / May 1, 2015
Dynamic control of light beams in second harmonic generation Ana Libster-Hershko,1,* Sivan Trajtenberg-Mills,2 and Ady Arie1 1
School of Electrical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel 2 School of Physics, Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel *Corresponding author: [email protected] Received January 29, 2015; revised March 29, 2015; accepted March 30, 2015; posted March 31, 2015 (Doc. ID 233479); published April 20, 2015
In this Letter, we report the dynamic control of the spatial shape of the second harmonic (SH) beam generated in a nonlinear crystal, by controlling the phase of the input fundamental beam before entering the crystal. This method enables 2D beam shaping and does not require any special fabrication beforehand. We have shown in simulation and experiment that this is possible for both short and long crystals: for short crystals, we assume the transverse phase of the SH beam is doubled relative to the input phase of the fundamental beam; for longer crystals, genetic algorithms were used in order to solve the inverse phase problem, which generally does not have an analytical solution. The method we present enables us to dynamically shape a beam in a nonlinear process, using standard crystals and optical equipment, and without the need to use any optical element after the nonlinear crystal. © 2015 Optical Society of America OCIS codes: (190.2620) Harmonic generation and mixing; (100.5070) Phase retrieval. http://dx.doi.org/10.1364/OL.40.001944
Spatial shaping of the second harmonic beam generated in a nonlinear crystal eliminates the need for additional optical elements and provides a compact manner for generating a desired beam shape. It has been explored during the past few years using several different methods: by modulating the quadratic nonlinear susceptibility of the crystal [1–6] or by functionalizing the exit facet of the nonlinear crystal for changing the beam shape at the exit of the crystal [7]. All of these methods require special fabrication procedures, after which they are permanent; thus, they cannot be changed to attain a different beam shape. Also, the first method may sometimes require us to use short crystals [3–5], and, in many cases, the desired shape emerges in multiple diffraction orders, resulting in significant power loss of the SH [4,5,7]. Due to fabrication limitations, these methods are mainly suitable for modulating the beam in only one transverse dimension [1–4,6]. In this Letter, we present a new method of shaping the generated SH beam by controlling the phase of the fundamental beam before it enters the nonlinear crystal, using a spatial light modulator (SLM). This method enables dynamic beam shaping, does not require any special fabrication beforehand, and does not require any use of optical equipment after the crystal. Shaping beams at the fundamental wavelength is highly advantageous in cases in which optical elements or spatial light modulators are unavailable or extremely expensive at SH wavelengths, e.g., in the ultraviolet range. SLM is a rather complex and expensive device, but there are other alternatives such as digital micromirror arrays and deformable mirrors that can be used for generating the desired phase. For sufficiently short crystals, the transverse phase of the fundamental beam is doubled, thereby providing a straightforward method to shape the second harmonic beam, while, for longer crystals, beam shaping can be achieved by optimizing the input beam. We also provide criteria for determining whether the “short crystal” approximation can be used. 0146-9592/15/091944-04$15.00/0
A slowly varying SH complex field envelope A2ω with frequency 2ω obeys the following equation in a quadratic nonlinear media [8]: ∇2T A2ω 2ik2
∂A2ω −κA2ω ei2k1 −k2 z ; ∂z
(1)
where Aω is, in our case, the fundamental Gaussian beam with frequency ω; k1 and k2 are the wave numbers of the fundamental and SH waves, respectively; ∇2T ∂2 ∕∂x2 ∂2 ∕∂y2 is the transverse Laplacian; z is the propagation distance; and κ χ 2 ω22 ∕c2 is the nonlinear coupling coefficient. If the process is (birefringently or quasi-) phase matched, we can ignore the term expi2k1 − k2 z. If the beam diffraction in the crystal is negligible, we may omit the first diffraction term on the left-hand side of Eq. (1) and obtain dA2ω iκ 2 − A : dz 2k2 ω
(2)
For a fundamental beam with a transverse phase φω x; y and complex envelope Aω jAω j expiφω , under the undepleted pump approximation the solution is, simply, A2ω x; y ≈ −
iκ jA x; yj2 ei2φω x;y L; 2k2 ω
(3)
where L is the crystal length. Hence, the SH is a Gaussian beam with an approximated phase of 2φω x; y. For a longer crystal, the variation of the fundamental beam along the crystal has to be considered; therefore, we cannot estimate the output phase in the same simple way. The inverse phase problem has no analytic solution for this type of crystal, but a numerical solution can be found for specific beams, as will be shown below. By using this approximation for short crystals, we can encode a desired phase pattern on the SH beam and, © 2015 Optical Society of America
May 1, 2015 / Vol. 40, No. 9 / OPTICS LETTERS
therefore, create arbitrary 2D beam shapes in the far field. The experimental setup is presented in Fig. 1(a). We used an SLM with 512 × 512 pixels, each pixel of 15 × 15 μm, but, for our applications, we found it easier and sufficient to use only the 256 × 256 centered pixels of the SLM for encoding our desired phase. We used a pulsed 1550 nm signal of our homemade OPO, with a peak power of 180 W, and a 1 mm long periodically poled KTiOPO4 (PPKTP) crystal with a poling period of 24.7 μm held at 35°C. The SLM plane was imaged to the crystal entrance plane with a demagnification of 4.3. The fundamental beam at the entrance to the nonlinear crystal has a Gaussian intensity profile and a phase that is added by the SLM. The generated beams were examined using a CCD camera after passing through a lens for Fourier transform (for far-field images). In order to test the beam-shaping capabilities, we generated several high-order Hermite–Gauss (HG) beams [9] and a 2D Airy beam [10]. An HGlm beam has a complex envelope of the form: Hl
p p 2x 2y Hm A x; y; z × expiZZ; W Z W Z G
(4)
where AG x; y; z is a Gaussian beam with a beam width of W Z, H l u is the Hermite polynomial of order l, and ZZ is a function of z. For HG10 , this complex envelope is proportional to ∝ x × AG x; y; z. We can approximate this form by multiplying a Gaussian beam by a phase step of π on the x axis, effectively creating a beam proportional to ∝ signx × AG x; y; z. When propagated, this beam will take a form very similar to the theoretical HG10 (or HG01 , by using signy) beam. Similar use of simple variations of phase steps for creating HG beams can be further shown for higher orders as well. For generating Hermite–Gauss HG10 (HG01 ) beams at the SH, we used a fundamental Gaussian beam, having a waist size of 250 μm at the crystal entrance plane with a phase step of π∕2 in the X direction (Y direction). According to Eq. (3), the SH beam at the crystal exit plane is, therefore, (approximately) a Gaussian beam with a phase step of π, which evolves to a beam that has a similar shape to the HG01 beam [9]. For generating an HG11 SH beam, we used a fundamental Gaussian beam with phase steps of π∕2 in both X and Y directions. The SH Airy was generated by frequency doubling a Gaussian beam with a cubic phase [9] expibx3 y3 , where (a)
1945
b 1011 m−3 . Here, we used the Fourier transform relation between the cubic phase function and the Airy function. The experimental and simulated results for HG01 , HG10 , HG11 , and Airy beams are presented in Fig. 2. The simulations are based on the split-step Fourier method [11]. Another functionality that can be achieved using this method is beam focusing. In this case, a fundamental beam with a quadratic transverse phase expix2 y2 ∕2λf , where f is the focal length, was sent to the crystal. As predicted by Eq. (3), the SH beam will, therefore, obtain a quadratic transverse phase as well, and the beam will be focused at f after the crystal. Focusing of the SH beam 30 mm after the crystal edge is presented in Fig. 3(a). In order to understand the limits of this method, we evaluated the range of crystal lengths for which our “short crystal” assumption is valid. We simulated a variety of crystal lengths for the phase mismatch conditions of our experiment and calculated the cross correlation of the output SH beams’ phase to the desired phase (using the simplest HG01 phase step for these simulations). Our results clearly show that, if we require a cross correlation of at least 0.95, the region of acceptance is for crystals shorter than ∼4 mm, as seen in Fig. 4(a). We have found that this length is a function of the material’s dispersion (or equivalently the phase mismatch), by repeating the simulation described above for different wavelengths, whereby in each wavelength it is assumed that the crystal is periodically poled to quasi-phase match a collinear interaction. Figure 4(b) shows the results of these
Fig. 2. Simulated (top) and experimental (bottom) results for 1 mm (short) crystal with the following beams: (a) and (b) HG01 ; (c) and (d) HG10 ; (e) and (f) HG11 ; (g) and (i) 2D Airy beam.
SLM fundamental SH phase phase profile profile
(b) SLM
lens
NL crystal
fundamental phase profile lens
SH phase profile NL crystal
Fig. 1. Experimental setup for short (a) and long (b) crystals.
Fig. 3. Experimental measurements of beam focusing of the SH beam for (a) 1 mm and (b) 12 mm crysal, both with a focal lengths of 30 mm. In (b) focusing is only in Y direction.
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OPTICS LETTERS / Vol. 40, No. 9 / May 1, 2015
Fig. 4. Simulation results of the region of validity of the “short crystal” approximation: (a) cross correlation of the output SH phase to the desired phase for the crystal’s phase mismatch at 1550 nm. (b) Threshold crystal length for which the cross correlation shown above is better than 0.95.
simulations for KTiOPO4 , but nearly the same values are obtained for Mg-doped congruent LiNbO3 and for stoichiometric LiTaO3 . This method is also subjected to the input beams’ spatial frequency: the cross correlation of the output beam’s shape with the desired shape is a function of the beam waist. We have shown through numerical simulations that the relation z0 ∕L is constant, where z0 πW 20 ∕λ is the Rayleigh range of the smallest beam that provides a cross-correlation value higher than 95%. For crystals of different lengths ranging between 1 and 18 mm, this relation is 7.56 0.38 for a correlation of 0.95 0.002 to an HG01 beam, indicating that the beam divergence along the crystal length can indeed be neglected. For a long crystal, which provides higher conversion efficiency, the problem is more complex, since we can no longer ignore the variation of the transverse profile of the beams along the interaction length. Therefore, we cannot analytically derive the phase that should be applied to the fundamental beam in order to receive the desired SH beam phase pattern at the output of the crystal. Since the interaction is nonlinear, a numerical solution is also impossible: one cannot propagate the beam in the opposite direction in order to receive the initial phase from the desired known output phase, even though all parameters of the system are deterministic. We can treat this problem as an inverse problem of a beam propagating through a nonlinear material. In order to solve it, an optimization genetic algorithm (GA) was implemented. Optimization algorithms were used in the recent years for focusing [12–14] and imaging [15] beams through turbid media as well as for focusing beams through nonlinear random media [16,17]. Genetic algorithms are optimization search algorithms that imitate the evolution process in nature. Here, we implemented
the algorithm described in [14]. The suggested solution is valid for beams without rapid changes in the amplitude or phase, e.g., low orders of HG beams and lenses. We start with a randomly chosen population of 30 input phase masks, which are then each multiplied by a Gaussian beam to create an input beam with a different transverse phase. Each beam is propagated through the nonlinear media, generating an SH beam. Each phase mask is then ranked according to the cross correlation between the generated SH beam profile and the desired one. In each iteration, 15 offspring phase masks are generated, with random mutations added to them. After each iteration, the lower-ranked phase masks in the population are replaced with a new generation of phase masks, which have higher rank. Higher-ranked phase masks have a better probability of becoming parents. A random binary matrix is used for the breeding process to generate one offspring from “father” and “mother” phase masks. Since the nonlinear crystal is deterministic, all optimization processes can be first simulated and then implemented on the experimental system without further iterations, unlike other optical systems containing a random media [13–15], where the optimization must be done on the experimental setup itself. Convergence occurred after ∼1000 iterations (generations). Typical computation time for a 1D shape running on a single Intel i3 CPU on a Windows 7 computer was around 5 h until convergence. All the computations were performed for transverse phase modulation in a single axis to reduce calculation time but can be implemented for 2D systems as well. As in the case of the short crystal, we used only the centered 256 × 256 pixels of the SLM. We used a 1550 nm diode continues wave (CW) laser with average power of 10 mW and a 12 mm long PPKTP crystal with a poling period of 24.7 μm held at 35°C. The phases, which were obtained using the simulations, were applied to the SLM. The phase on the SLM plane was imaged to the crystal’s entrance plane. This phase was added to a fundamental beam with a Gaussian intensity. The generated SH beams were recorded using a CCD camera (Fig. 5). We measured the beam quality parameter M2 of every beam [18]. For the HG01 , HG10 , HG02 , and HG20 , the theoretical M2 values are 3, 3, 5 and 5. As can be seen in Table 1, the measured M2 is quite close to the theoretical prediction. In addition, the measured cross correlation between the measured beam and the theoretical beam
Fig. 5. Simulated (top) and experimental (bottom) results for 12 mm (long) crystal with the following beams: (a) and (b) HG01 ; (c) and (d) HG10 ; (e) and (f) HG02 ; (g) and (i) HG20 .
May 1, 2015 / Vol. 40, No. 9 / OPTICS LETTERS Table 1. Beam Quality Parameter and Cross Correlation Between Measured and Desired SH Beam in the Long (12 mm) Crystal Desired Beam HG10 HG01 HG20 HG02
M2
Cross Correlation
3.083 3.040 5.288 5.159
0.9887 0.9461 0.9465 0.9696
was above 0.94 in all cases, showing again the high quality of the generated SH beams. As mentioned before, we have confirmed experimentally that the off-line optimization process produces good results with no need for additional iterations on the experimental setup itself. We have also generated an SH beam with a quadratic transverse phase (in 1D), which focuses the beam 30 mm from the edge of the crystal [see Fig. 3(b)]. Since the phase mask was planned only for one dimension, the beam diverges in the orthogonal direction, in which the focusing function is not activated. In conclusion, this Letter presents a new method for shaping the generated SH beam in a nonlinear crystal by shaping the phase of the fundamental beam before it enters the crystal. We separated the solution into two parts: short and long crystals. For a short crystal, we showed that we can approximate the output phase as twice the input fundamental beam phase, therefore providing a straightforward method of shaping. For a long crystal, we used GA to solve the inverse problem. We provided guidelines for determining whether the “short crystal” approximation is valid. This method enables flexible and dynamic shaping of beams, does not require any special fabrication, and does not require any additional optical elements after the crystal. The concepts presented here can be further expanded for temporal shaping of pulses in a nonlinear process [19]. In the future, the methods we presented here can be further explored by using phase-retrieval algorithms for more sophisticated beam shapes [20]. Shaping both phase and amplitude of the fundamental beam before the input of the crystal, as well as combining different beam shaping techniques [1–6] for a
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hybrid approach, can lead to complete dynamic control of the SH beam shape. The authors would like to acknowledge E. Burstein for the technical support. This work was supported by the Israeli Ministry of Science, Technology and Space in the framework of the Israel–Italy bi-national collaboration program. References 1. G. Imeshev, M. Proctor, and M. Fejer, Opt. Lett. 23, 673 (1998). 2. J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. Fejer, IEEE J. Sel. Top. Quantum Electron. 8, 660 (2002). 3. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009). 4. A. Shapira, I. Juwiler, and A. Arie, Opt. Lett. 36, 3015 (2011). 5. A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, Opt. Lett. 37, 2136 (2012). 6. X.-H. Hong, B. Yang, C. Zhang, Y.-Q. Qin, and Y.-Y. Zhu, Phys. Rev. Lett. 113, 163902 (2014). 7. A. Shapira, A. Libster, Y. Lilach, and A. Arie, Opt. Commun. 300, 244 (2013). 8. R. W. Boyd, Nonlinear Optics (Academic, 2008). 9. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2013). 10. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007). 11. G. Agrawal, Nonlinear Fiber Optics (Academic, 2001). 12. I. M. Vellekoop and A. P. Mosk, Opt. Commun. 281, 3071 (2008). 13. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012). 14. D. B. Conkey, A. N. Brown, A. M. Caravaca-Aguirre, and R. Piestun, Opt. Express 20, 4840 (2012). 15. O. Katz, E. Small, and Y. Silberberg, Nat. Photonics 6, 549 (2012). 16. C. Yao, F. J. Rodriguez, and J. Martorell, Opt. Lett. 37, 1676 (2012). 17. C. Yao, F. J. Rodriguez, J. Bravo-Abad, and J. Martorell, Phys. Rev. A 87, 063804 (2013). 18. S. Saghafi and C. J. R. Sheppard, Opt. Commun. 153, 207 (1998). 19. Z. Zheng and A. M. Weiner, Chem. Phys. 267, 161 (2001). 20. R. W. Gerchberg and W. O. Saxton, Optik (Stuttg). 35, 237 (1972).
