February 15, 2012 / Vol. 37, No. 4 / OPTICS LETTERS
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Fractional second-harmonic Talbot effect Zhenhua Chen,1 Dongmei Liu,1 Yong Zhang,1,4 Jianming Wen,1,2 S. N. Zhu,1 and Min Xiao1,3,5 1
National Laboratory of Solid State Microstructures, School of Physics, School of Engineering and Applied Science, Nanjing University, Nanjing 210093, China 2 Institute for Quantum Information Science, University of Calgary, Calgary, Alberta T2N 1N4, Canada 3
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA 4
e-mail:
[email protected] 5 e-mail:
[email protected]
Received September 30, 2011; revised December 1, 2011; accepted December 15, 2011; posted December 15, 2011 (Doc. ID 155181); published February 14, 2012
We demonstrate the fractional second-harmonic (SH) Talbot effect in a hexagonally poled LiTaO3 crystal. We carefully record the SH Talbot images at 1∕2, 1∕3, and 1∕4 Talbot lengths, which are well matched with the simulated images by using the modified Rayleigh–Sommerfeld diffraction formula. A simplified model with a hexagonal array is adopted in the simulations. Also, we use a modified reciprocal vector theory to analytically explain the evolution of the SH array at fractional Talbot lengths. Our results show that the images are sensitive to the duty circle and the background of the array. © 2012 Optical Society of America OCIS codes: 110.6760, 190.2620, 140.3515.
The Talbot effect, a near-field diffraction phenomenon of periodic object, was first observed by H. F. Talbot in 1836 when he illuminated a diffraction grating with a white light source. [1]. Up to today, the beauty and simplicity of Talbot self-imaging effect still attracts many researchers. Recent progresses on fundamental Talbot effect are made with, for example, single photons and entangled photon pairs [2,3], waveguide arrays [4], metamaterials [5], and electromagnetically induced grating [6]. The effect is more than just an optical curiosity for physicists, and has led to a variety of applications, such as photolithography, optical testing, optical metrology, and array illuminator [1]. The extension to X-ray and terahertz (THz) waves through Talbot interferometers [7,8] is particularly useful because of the lack of efficient optics for these wavelengths. An electron Talbot interferometer has also been demonstrated [9]. Recently, we reported a novel nonlinear Talbot effect in periodically poled LiTaO3 (PPLT) crystals, where the self-imaging is formed by the generated second-harmonic (SH) waves instead of the fundamental input beam [10]. The prerequisite condition to realize the effect is to have a periodic SH pattern, which is fulfilled by the periodic distribution of χ 2 in the PPLT crystals. The observations also offer an optical way to image ferroelectric domain structures in nonlinear crystals without damaging them. Although a theoretical description was developed to interpret the observed effect [11], the focus was mainly on self-images at integer Talbot lengths [10]. Therefore, further studies are necessary for the self-imaging at fractional Talbot lengths. We notice that few theoretical methodologies [12–16] have been proposed for conventional (linear) fractional Talbot effect. Among these methods, the reciprocal vector theory (RVT) [12,13] gives simple analytical solutions for images at any fractional Talbot distances, while the Rayleigh–Sommerfeld (R-S) diffraction formula [17] yields accurate numerical simulations. In this Letter, we choose both methods to interpret the recorded images at the fractional Talbot lengths. 0146-9592/12/040689-03$15.00/0
We find that in order to obtain good agreements between the experiment and theory [12,13,17], the spot size and background of the hexagonal array have to be taken into account in the theory as well as numerical simulations. That is, the geometrical points and complete darkness of the background assumed in original theories would result in discrepancies with our experimental observations. Our work is further motivated by one potential application to detect tiny defects or imperfections in fabricated nonlinear crystals without destroying them. In the experiment, a hexagonally poled LiTaO3 crystal was fabricated through an electric-field poling technique at room temperature [18]. The size of the sample slice is 20 mmx × 20 mmy × 0.5 mmz, and its domain structure with period 9 um and duty cycle ∼30% is shown in Fig. 1. The experimental setup is shown in Fig. 2, where a femtosecond laser operating at a wavelength of 800 nm serves as the pump laser. It is focused by a lens with a focal length of 200 mm and propagates along the z axis of the crystal. The PPLT sample is placed at the focal plane
Fig. 1. The SEM image of the hexagonally poled LiTaO3 crystal. © 2012 Optical Society of America
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of the lens. The laser spot size on the sample surface is about 100 μm in radius. The long-pass filter (filter 1 in Fig. 2) in front of the sample removes visible noise in the fundamental beam. A short-pass filter (filter 2 in Fig. 2) is placed between the crystal and the objective to filter out the near-IR pump field. A 50× objective is used to magnify the SH images. The SH patterns at different imaging planes are imaged by moving the objective along the SH propagation direction. Since the confocal length (20 mm) is much larger than the sample thickness (0.5 mm) in our setup, the input pump is treated as a plane wave in both experiments and simulations. By taking the PPLT crystal as the SH source, the diffraction amplitude at a distance of z from the object can be deduced from the R-S diffraction formula as [13] →
Ur 1
2 iλ
ZZ
→
Σ
→
U i r tr
→
expikjr 01 j → jr 01 j
→ →
× cosn; r 01 ds;
Fig. 3. (Color online) (a) SH pattern at few micrometers away from the sample output surface. (b) Recorded SH self-imaging at the first SH Talbot plane. (c) Modeled object used in our simulations. (d) Simulated mage at the first SH Talbot plane.
→
where λ is the input wavelength, U i r is→the amplitude of the incident light at the object plane, tr is the aperture → → function, and r and r 1 are spatial coordinates at the ob→ is an outward ject and imaging planes, respectively. n → unit→ vector normal in the object plane. r 01 is defined → → → as r 01 r − z − r 1 . After some algebra, the SH Talbot length is zt 3a2 ∕λ, where a is the structure period [10]. We first checked the images at integer SH Talbot planes. In our previous work [10], we chose the patterns exhibiting at the output surface of the crystals as the object. To ease the numerical simulations here, we chose the pattern about a few micrometers from the output surface as the object [Fig. 3(a)]. Figure 3(b) is the self-image at the first SH Talbot plane. The measured SH Talbot length is 300 μm and the calculated zt is 304 μm. The evolution of the SH images from the object plane to the first Talbot plane was recorded in the video (see Media). The images at 1∕N Talbot planes (N 2, 3, 4) are carefully measured and shown in Figs. 4(a), 4(b), and 4(c), respectively. In comparison with the integer case [Fig. 3(b)], remarkable and complicated features, such as period change, lattice rotation, and phase shift of the array, appear in the fractional self-images. To quantitatively →understand these images, we first built a model for tr to simulate the integer case [10] as shown in Fig. 3(c). The parameters used in the model are based on the actual object in Fig. 3(a), where the period is 9 μm and the spot size is 2.6 μm in radius. In current experiments, the hexagonal array in our model consists of more than 750 SH spots. Careful examination on Fig. 3(a) indicates that, behind the periodic object, a notable background exists. One possible origin of this background comes from the unpoled parts of the sample. In
Fig. 2.
(Color online) Experimental setup.
the model, we assume a uniform background, for simplicity, and take I s ∶I b 2.5∶1, where I s and I b are the intensities of the SH spot and the background, respectively. From the simulations, we notice that although the background does not affect the period and rotation of the array, it has important contributions to the details in the SH self-images. The quantitative analysis is based on the modified RVT, and the numerical simulations start with the R-S diffraction formula by assuming uniform phase over the object plane. The integer case is easily calculated and understood with the model described above. For instance, we checked the simulated image at the first Talbot plane [see Fig. 3(d)], which is an exact replica of the pattern in Fig. 3(c). Next, we look at the image at the half Talbot plane, i.e., the SH image at z 152 μm. Quantitative analysis based on the RVT [12,13] states that the image at 1∕2 Talbot length should also be a hexagonal array but with half the input period. However, a discrepancy appears by comparing Fig. 4(a) with Fig. 3(a). That is, the experiment
Fig. 4. (Color online) Recorded fractional self-images at 1∕N SH Talbot lengths in CCD camera (a) N 2, (b) N 3, (c) N 4) and (d)–(f) their corresponding numerical simulations, respectively (Media 1).
February 15, 2012 / Vol. 37, No. 4 / OPTICS LETTERS
further shows that all the bright SH spots in the object [Fig. 3(a)] evolve to the dark holes in the image [Fig. 4(a)], while the dark background in the object becomes bright SH boundary. Alternatively, a phase shift of π occurs at the 1∕2 Talbot length. The apparent contradiction between the theory and experiment can be easily resolved by taking into account the spot size of the array. Recall that the spot size is 2.6 um in radius while the periodicity at the 1∕2 Talbot length is 4.5 um. This implies that the fields from two nearest adjacent spots in the image will partially overlap with each other and consequently, lead to the interference. After noting this correction, the simulated image [Fig. 4(d)] using the modified R-S formula is now consistent with the experimental pattern [Fig. 4(a)]. The fractal structures in the bright boundary in Fig. 4(d) were not experimentally observable due to the limitation of the intensity contrast of the CCD camera. At 1∕3 Talbot plane (z 101 μm), the RVT predicts that: (1) the SH image is a hexagonal array with a reduced period equal to 3 × 31∕2 μm and (2) the basis vectors of the image lattice are rotated 30° with respect to the input. Our experiment confirms these two conclusions, see Figs. 4(b) and 4(e). In this case, because the period 5.2 μm is comparable to the distance between two nearest spots, the fields from the corresponding spots do not overlap and accordingly, no interference is produced to further modify the image. Yet, a careful examination of Fig. 4(b) further reveals that the center SH spot in the marked area is relatively weaker than its six neighbors. This finding does not show in the RVT. Our simulations indicate that the background SH waves from the object are responsible for such a discrepancy. Without considering the background in the object (I b 0), the calculated pattern would simply be a composition of SH spots with same intensities. We also numerically find that as the intensity of the background increases, the brightness of center SH spots decreases and becomes invisible eventually. From the measurements we experimentally estimate I s ∶I b 2.5∶1. This unexpected background is detrimental in realizing meshlike SH patterns. The way to eliminate its effect is to achieve good phase-matching condition and high-quality fabrication. At 1∕4 Talbot length (z 76 μm), the period of the image reduces to 2.25 μm according to the theory. In this case, the interference due to the fields emitted from two neighboring spots becomes a dominant effect, and shall be carefully counted in both theories and simulations. The simulated image given in Fig. 4(f) now coincides well with the recorded one, Fig. 4(c). Note that the background is also considered in the simulation and it leads to a hexagonal array of weak SH spots (which were not clearly observed in the experiment). One possible reason is that the SH background in the object, assumed to be homogeneous in our model, actually consists of small structures beyond the resolution of the used CCD camera.
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In summary, we have investigated the fractional SH Talbot self-imaging effect in a PPLT crystal. In particular, our experiment indicates that the duty cycle of the array in the object has to be taken into account for analyzing the image details at fractional Talbot lengths, so does the background of SH waves. By including these two factors, the modified RVT and R-S diffraction formula agree well with the experiment. Our result also offers a way to shape the fractional images details by engineering phase matching and fabrication, which would be difficult for the conventional case. Further, by making slight changes to the current experimental setup, this work could allow detecting tiny defects in crystals, which will be presented elsewhere. This work was supported by the National Basic Research Program of China (contracts 2012CB921804 and 2011CBA00205), the National Natural Science Foundation of China (NSFC; grants 11004097 and 11021403), the Fundamental Research Funds for the Central Universities (contracts 1095021339 and 1117021306), and the Priority Academic Program Development of Jiangsu Higher Education Institutions. J. Wen was supported by an AI-TF New Faculty Grant and an NSERC Discovery Grant. References 1. K. Patorski, Prog. Opt. 27, 1 (1989). 2. K. H. Luo, J. M. Wen, X. H. Chen, Q. Liu, M. Xiao, and L. A. Wu, Phys. Rev. A 80, 043820 (2009). 3. X. Song, H. Wang, J. Xiong, K. Wang, X. Zhang, K. Luo, and L. Wu, Phys. Rev. Lett. 107, 033902 (2011). 4. R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 95, 053902 (2005). 5. W.S.Zhao, X. Huang,and Z.Lu, Opt. Express19, 15297(2011). 6. J. M. Wen, S. Du, H. Chen, and M. Xiao, Appl. Phys. Lett. 98, 081108 (2011). 7. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Gronzweig, and C. David, Nat. Mater. 7, 134 (2008). 8. P. Peier, S. Pilz, F. Muller, K. A. Nelson, and T. Feurer, J. Opt. Soc. Am. B 25, B70 (2008). 9. B. J. McMorran and A. D. Cronin, New J. Phys. 11, 033021 (2009). 10. Y. Zhang, J. M. Wen, S. N. Zhu, and M. Xiao, Phys. Rev. Lett. 104, 183901 (2010). 11. J. M. Wen, Y. Zhang, S. N. Zhu, and M. Xiao, J. Opt. Soc. Am. B 28, 275 (2011). 12. C.-S. Guo, X. Yin, L.-W. Zhu, and Z.-P. Hong, Opt. Lett. 32, 2079 (2007). 13. L.-W. Zhu, X. Yin, Z.-P. Hong, and C.-S. Guo, J. Opt. Soc. Am. A 25, 203 (2008). 14. J. Leger and G. Swanson, Opt. Lett. 15, 288 (1990). 15. P. Szwaykowski and V. Arrizon, Appl. Opt. 32, 1109 (1993). 16. C. Zhou, S. Stankovic, and T. Tschudi, Appl. Opt. 38, 284 (1999). 17. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996). 18. S. Zhu, Y. Zhu, and N. Ming, Science 278, 843 (1997).
