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OPTICS LETTERS / Vol. 37, No. 10 / May 15, 2012
Controlling the diffused nonlinear light generated in random materials Can Yao,1 Francisco J. Rodriguez,* and Jordi Martorell1,2 1
ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels, Barcelona, Spain 2 Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Terrassa, Spain *Corresponding author:
[email protected] Received February 7, 2012; revised March 16, 2012; accepted March 17, 2012; posted March 19, 2012 (Doc. ID 162628); published May 10, 2012
We describe a method for shaping the wavefront of the second-harmonic light generated in nonlinear materials with a disordered structure using a spatial light modulator on the fundamental beam. We show that, for the case of a transparent strontium-barium niobate crystal with a random distribution of antiparallel domains, the speckle generation can be concentrated into a single spot. © 2012 Optical Society of America OCIS codes: 190.2620, 030.6140, 290.0290.
The light emitted by three-wave-mixing processes, such as second harmonic generation (SHG) in materials with a disordered structure, is generally scattered in many directions. In some cases, this may be partly an effect of multiple scattering due to random variations on the refractive index in the same way as it happens with linear light diffusion [1]. In materials in which disorder happens only in the second-order susceptibility and not in the refractive index, transparent crystals with a random domain structure, the nonlinear light emission is scattered in many different directions [2–6] and a speckle pattern is formed in the spatial intensity distribution [7]. In such transparent materials, the frequency conversion efficiency grows linearly on average with the path length of the fundamental light inside the material, improving the efficiency over a non-phase-matched case [8,9]. In addition, a similar conversion efficiency is obtained over the full transparency range without any angular or temperature adjustment. However, such emission over many different directions restricts the maximum intensity that it is possible to achieve in a specific direction. Here we report a method to concentrate the light generation in one direction or focus it to a point by means of a spatial light modulator (SLM) acting on the fundamental light. Phase control of the incident wavefront by means of phase-only SLMs has recently been used to focus and transmit images through stationary diffusive media [10,11]. Ultrashort pulses can also be time compressed in the same way [12–14]. The method is based on the linearity of the transmission matrix. The complex coefficients of the transmission matrix connect the optical field at points in a surface containing the sources (input modes E i ) with the optical field at points in the detection surface (output modes E o ) through the equation Eo
X i
toi E i .
(1)
With an independent control of the phase of all input modes, one can transform the sum in Eq. (1) in a sum over real numbers in such a way that the output intensity in one mode is maximized. In the case of three-wave mixing, the transmission matrix needs to be replaced by a third rank tensor relating 0146-9592/12/101676-03$15.00/0
the output modes at the resulting frequency with combinations of input modes from the incident beams in the following way: Eo
XX i∈A j∈B
toij E iA EjB :
(2)
For sum-frequency, difference-frequency, or two-beam SHG, we can control the phase of the input modes in one group, A or B, keeping the other group input fixed. Equation (2) can then be reduced to a single sum like Eq. (1) by including the constant E field inside the transmission matrix. On the other hand, if we consider a process like SHG from one fundamental beam, the input modes E iA and E jB belong to the same set (A B) and are, therefore, modified at the same time. In this case, the crossed terms in the double summation may always have different phases for any combination of input phases. Yet, an optimization method could still find a combination of input phases that leads to the concentration of the generated intensity in an output mode. In the optimization methods for linear light transmission, the phase of one or several input modes is modulated from 0 to 2π while the rest of the modes are kept fixed. This produces an intensity modulation of the form I a21 a22 2a1 a2 cosφ − ϕ, where a1 and φ are, respectively, the amplitude and phase of the modulated modes together with their respective transmission matrix elements, and a2 and ϕ correspond to the amplitude and phase of the fixed modes with their matrix elements. After the measurement, the modulated phase is fixed at φ ϕ corresponding to the maximum intensity. This step is repeated multiple times, changing each time the modes that are modulated. When the process is applied to SHG, the intensity follows a different dependence with the modulated phase. In a similar way, as explained above for the linear case, we can reduce the modes in Eq. (2) into two groups: one that includes all terms whose phases are modulated by φ, and the second for the fixed part. Equation (2) is then rewritten as E o ao11 eiϕo11 2φ ao22 eiϕo22 ao12 eiϕo12 φ ; © 2012 Optical Society of America
(3)
May 15, 2012 / Vol. 37, No. 10 / OPTICS LETTERS
where we have also explicitly separated the amplitude and phase for each term in the sum. The intensity of the output mode resulting from it has the form I a2o11 a2o22 a2o12 2ao11 ao22 cos2φ ϕ1 − ϕ2 2ao11 ao12 cosφ ϕ1 2ao22 ao12 cosφ − ϕ2 ; (4) where we have rewritten ϕ1 ϕo11 − ϕo12 and ϕ2 ϕo22 − ϕo12 . As shown in the inset of Fig. 1, this function can have two local maxima in the interval of φ between 0 and 2π. We always choose the highest of the two local maxima during the optimization process. The nonlinear material used in the experiments is a strontium-barium niobate (SBN) crystal with the composition Sr0.61 Ba0.39 Nb2 O3 . To maximize the SHG efficiency, the crystal was poled at room temperature under a 400 V∕mm electric field along the crystallographic c axis with alternation of the field polarization during at least two cycles. The resulting domain structure consists of a random pattern of needle-shaped antiparallel domains with an average width on the order of 3 μm. This is close to the coherent length of the SHG process in the collinear configuration and, when the fundamental beam propagates perpendicular to the c axis, it yields a stronger intensity in directions on the plane perpendicular to the crystal c axis within 20° from the propagation direction of the fundamental beam. The nearly homogeneous refractive index along the structure does not produce any important scattering in the linear-light transmission. On the other hand, the SHG is scattered in the mentioned directions, forming a speckle pattern [7]. The experimental setup is shown in Fig. 1. We used a Q-switched Nd-YAG laser emitting 6 ns pulses at 1064 nm wavelength and 0.7 cm−1 linewidth, with 10 Hz repetition rate. The narrow linewidth allows the observation of the speckle pattern and justifies the use of time-harmonic fields in the presented formalism. The laser beam was expanded before arriving to the SLM (LCoS model X10468-03 from Hamamatsu). The modulated beam
Fig. 1. (Color online) Scheme of the experimental setup. A laser beam is expanded and modulated by a phase only SLM. The image of the SLM is then reduced and projected on the nonlinear crystal by a 4f lens system formed by lenses L1 and L2 . The SHG is imaged at the far field using a CCD sensor on the focal plane of lens L3 . Upper left inset: example of intensity measurement during the phase modulation.
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was then reduced using a 4f system in such a way that the image of the SLM was projected on the nonlinear crystal. The transmitted light at the fundamental wavelength was then rejected using a short pass filter. The detection of the SHG at 532 nm was made at the far field using a CCD sensor at the focal plane of a lens. Several optimization algorithms have been tested by others for focusing light transmitted through diffusive media [15]. In general, they are able to find solutions such that a maximum possible intensity is achieved at the selected point. However, in the case of SHG, it is not possible to be certain that the global maximum would be found following such procedure because the crossed terms in the double sum may provide several local maxima. We have used the partitioning algorithm, which has been shown to provide the fastest growth in the intensity for linear light transmission [15]. The SLM pixels are randomly divided into two equal-sized groups, keeping one group fixed and adding to the pixels on the other group a phase that is modulated continuously from 0 to 2π, while the intensity of the SHG at the desired direction is recorded. We identify the phase at which the maximum intensity occurs, and we fix the phase of the pixels at that value. The process is iterated many times, randomly changing the set of pixels whose phase is altered. We found that this method is able to provide a good enhancement of the intensity at the selected point. One important difference in the results from the optimization algorithm between the linear and the SHG case arises from the quadratic dependence of the SHG intensity with the fundamental intensity. In general, the intensity at any detection point can be increased just by focusing the fundamental light in the crystal. If this is not taken into account, the optimization algorithm can configure the SLM as a sort of kinoform lens. The effect is an overall increase of the intensity and a growth in the speckle size because of the smaller illuminated area. One way to avoid that is to place the crystal at the image plane of the SLM. At that plane, there will be no variations in the intensity for any phase pattern of the SLM (Fig. 2). Another possibility, not considered here, is to take for optimization the intensity ratio between the desired point and a wide reference area around it in such a way that the intensity will grow only in one point and not everywhere. The reference measurement has the additional advantage of reducing the noise introduced by pulse to pulse intensity fluctuations. With a setup like the one presented in Fig. 1 with f 1 500 nm, f 2 500 mm, and f 3 150 mm, and
Fig. 2. (Color online) (a) Fundamental beam intensity distribution at the crystal obtained by focusing the image of the crystal on the CCD without the short-pass filter. (b) SHG pattern at the far field when the SLM is configured with a constant phase, obtained by placing the CCD at the focal plane of L3 .
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Fig. 3. (Color online) Results of the optimization process. (a) Initial CCD image of the SHG pattern at the far field. The picture brightness is increased ten times for clarity. The image is also magnified compared with Fig. 2(b). (b) CCD image after the optimization process. (c) Intensity growth at the chosen point during the process normalized to the initial average. (d) Intensity profile normalized to the background average.
dividing the SLM in a 30 × 40 matrix, we obtained the results displayed in Fig. 3. The original SLM dimensions of 12 mm × 16 mm were reduced by the 4f system to 1.2 mm × 1.6 mm at the central position of the crystal. The crystal length was 5 mm, which makes the intensity homogeneity at the crystal to be only approximate. The optimization process took about 50 min to reach a ratio between the intensity at the selected point and the average intensity at the areas around it of nearly 700. At each step, 20 measurements with different phases were taken and then fitted with Eq. (4), as shown in the inset in Fig. 1. The phase φ was set at the value corresponding to the maximum intensity. Around 1500 iterations were necessary to reach the saturation value of the intensity. Similar results were obtained by focusing the light directly on the CCD without lens L3 . In all cases, the optimized points were close to the propagation direction of the fundamental beam. Points within a few degrees, with respect to that direction, behave in the same manner. The results are close to the ones obtained for linear light diffusion [15]. The enhancement is of the same order as the number of modulated pixels in the SLM, and it takes about the same number of iterations to reach it. It should be possible to improve the enhancement in processes like sum- or difference-frequency generation or two-beam SHG in which each input beam is separately controlled by a SLM. It is possible to enhance the intensity of several points at the same time by optimizing their sum. However, with the used algorithm, it is difficult to get an even intensity in all of them, as already noted in [15]. The use of different algorithms, or the measurement of the transmission tensor [11], could bring the possibility of forming images in the emitted light. The stability of the crystal structure allows maintaining the optimized emission during long periods of time without changing the phase pattern in the SLM. The enhanced spot was still observed without appreciable loss in
intensity when the stored optimized wave front was displayed again in the SLM six days after the optimization process. The broadband SHG characteristic of the random structure allows the simultaneous frequency doubling or mixing of different wavelengths in the same sample. The optimization of different processes would probably require a phase-pattern change in the SLM. In any case, the possibility of red, green, and blue emission through frequency doubling or mixing of infrared lasers in the same crystal [16], together with beam shape controlled by the wavefront modulation as demonstrated here, could find some applications, for example, in laser display technology. In summary, we have presented a method to control the intensity distribution of the SHG from materials with a random structure by controlling the wavefront of the fundamental light. We expect this method, demonstrated here for a transparent crystal, to be directly generalizable to diffusive materials with changes in the refractive index. The transmission tensor for such multiple scattering media would still be similar, although if shorter pulses are used the pulse length control would have to be taken into account. We acknowledge financial support from Ministerio de Ciencia e Innovacion under grants MAT2011-28665 and CONSOLIDER NANOLIGHT CSD2007-00046. F. J. Rodriguez acknowledges also grant JCI-2009-04860. References 1. S. K. Kurtz and T. T. Perry, J. Appl. Phys. 39, 3798 (1968). 2. S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, Appl. Phys. Lett. 73, 768 (1998). 3. A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003). 4. R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, Appl. Phys. Lett. 89, 191105 (2006). 5. J. Bravo-Abad, X. Vidal, J. L. Domínguez-Juárez, and J. Martorell, Opt. Express 18, 14202 (2010). 6. W. Wang, K. Kalinowski, V. Roppo, Y. Sheng, K. Koynov, Y. Kong, C. Cojocaru, J. Trull, R. Vilaseca, and W. Krolikowski, J. Phys. B At. Mol. Opt. Phys. 43, 215404 (2010). 7. F. J. Rodríguez, C. Yao, J. L. Domínguez-Juarez, J. Bravo-Abad, and J. Martorell, Opt. Lett. 36, 1347 (2011). 8. M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, Nature 432, 374 (2004). 9. X. Vidal and J. Martorell, Phys. Rev. Lett. 97, 013902 (2006). 10. I. M. Vellekoop and A. P. Mosk, Opt. Lett. 32, 2309 (2007). 11. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010). 12. J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011). 13. D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011). 14. O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photon. 5, 372 (2011). 15. I. M. Vellekoop and A. P. Mosk, Opt. Commun. 281, 3071 (2008). 16. J. J. Romero, D. Jaque, J. García Solé, and A. A. Kaminskii, Appl. Phys. Lett. 81, 4106 (2002).
