Talbot effect for aberrated optical waves A. Goloborodko,∗ D. Podanchuk, M. Kotov Optical Processing Laboratory, Faculty of Radiophysics, Electronics and Computer Systems, Taras Shevchenko National University of Kyiv. 64/13, Volodymyrska str., Kyiv, 01601, Ukraine ∗
[email protected]
Abstract: Theoretic and experimental studies of the Talbot effect for aberrated optical wave is performed. The diffractions of periodic and quasi-periodic square aperture arrays in Fresnel fields are analyzed according to the scalar diffraction theory. Talbot images of the quasi-periodic mask restored by aberrated optical wave are predicted to appear, and scale factors are obtained. The quasi-periodic square aperture arrays are produced with the aid of a spatial light modulator, and the self-images of the quasi-periodic gratings are measured successfully in the experiment. OCIS codes: 050.1950, 070.6760, 260.1960.
Talbot effect (self-imaging phenomenon of the periodic structure in a Fresnel diffraction zone) has been applied in laser array illumination, the interference measurement, wavefront sensing, and even enhancement of the nonlinear effect [1, 2]. The Talbot effect of periodic masks has been attracting lots of research interest [2, 3] and, as one knows, the quasi-periodic structure exhibits a long-range order. Here, we concentrate on discussing the diffraction of the aberrated wave on the two-dimensional (2D) quasi-periodic mask (Talbot effect for aberrated wave). To our knowledge, the Talbot effect for aberrated wave has not been studied. In Fourier optics, the complex transmittance of the 2D quasi-periodic mask with the two orthogonal periods D1 and D2 is expressed in the Fourier series form: t(x, y) = ∑ ∑ An,m e j2mπ x/D1 e j2nπ y/D2 , where m and n are integers, {An,m } are the Fourier coefficients associated with the basic element of the 2D mask. When it is illuminated by plane wave with the wavelength λ , the diffraction intensity at a point with the propagation distance z in the Fresnel diffraction zone can be written as a double integral [4]. If the mask aperture is larger enough, the upper and lower limits of the integral tend toward infinity, and the amplitude can be expressed in a simple form [5]: ∞
− j 2λπ z
u(x, y|z) = u0 e
∞
∑ ∑
An,m e
j2π
(
ny mx D1 + D2
)
e
( ) 2 2 − j π m2 + n 2 λ z D D 1
2
.
(1)
n=−∞ m=−∞
While the periods D1 and D2 are equal (D1 = D2 ≡ D) and the propagation distance z takes 2ND2 /λ (N is an integer), 2 2 2 2 two exponential terms in the above equation (e− jπ m /D and e− jπ n /D ) are equal to 1. One can see that (1) has the same structure as the 2D mask transmittance t(x, y). This is the Talbot image of the mask and the corresponding distance is the Talbot distance zT = 2D2 /λ . So the amplitude distribution at the Talbot distance contains all the information of the mask, naturally, it seems like the 2D mask structure. To obtain the self-image of quasi-periodic mask (D1 ̸= D2 ), the distance needs to satisfy ZT = NzT 1 = MzT 2 (N and M are integer), which is just the common Talbot distance of two gratings. Since in the amplitude distribution at Talbot distance, all spatial frequency information of the two masks appears. Generally, if the quasi-periodic mask is the superimposition of multiple masks with different periods, Talbot image of the such mask can appear at multiple certain distances as long as the propagation distance is the joint Talbot distance of all these gratings. Obviously, the limits of diffraction integrals are not infinite, and the aperture effects may cause a lack of diffraction orders and makes the Talbot image of the grating blurry [6]. Therefore, we can predict that the amplitude distribution even at the common Talbot distance of two gratings ZT will deviate from the Talbot image of this quasi-periodic grating. Image of quasi-periodic mask (D1 ̸= D2 ) could be also obtained if the mask illuminated by aberrated optical wave with the orthogonal specific wavefront curvature radii (R1 and R2 ). Amplitude of the aberrated optical wave in the Fresnel diffraction zone can be expressed as [7]: − j 2λπ z
u(x, y|z) = u0 e
e
( ) 2 2 jπ λxR + λyR 1 2
∞
πR λ z
∞
∑ ∑
n=−∞ m=−∞
An,m e
πR λ z
2 2 1 2 πR m πR n j (z+R1 )D 2x j (z+R 2)D 2y − j (z+R )D2 m − j (z+R )D2 n 1 1 2 2 1 1 2 2 e e e .
(2)
Now consider the plane zT 1 and the radii R1 → ∞, 2D21 D22 R2 = . λ (D22 − D21 )
(3) (4)
Substituting (3) and (4) into (2) results on the arguments of the two last exp function in (2) and it becomes a multiple of 2π : ( ) ∞ ∞ R2 ny mx j2π R2 y j2π D (z +R )D 1e T1 2 2 = t u(x, y|zT 1 ) ∼ ∑ ∑ An,m e x, . (5) z T 1 + R2 n=−∞ m=−∞ So Talbot image is the distorted image of the mask with the scale factor sy = (zT + R2 )/R2 for y axis. The scaling factor depends on the curvature of the incident wavefront and Talbot distance zT . Formation of the image was considered in assumption that the grating was illuminated with a divergent orthogonal spherical waves. In our notation such a wave has positive radiuses of curvature R1 and/or R2 . Transition to the convergent orthogonal waves are performed choosing R1 < 0 and/or R2 < 0. Experimental setup was used to measure the diffraction intensity distributions of periodic and quasi-periodic masks at the different propagation distances to verify above theoretic results (Fig.1). The light beam emitted from a He-Ne
Fig. 1. Experimental setup in the Fresnel diffraction of aberrated wave on a 2D mask. laser (λ = 0.63µ m) goes through a collimator (C) and then illuminates lenses L1–L3, which forms a test wavefront. The wavefront with an spherical (sph. +2.0 for L1 and L2) or astigmatic (sph. −2.5, cyl. +4.0 for L3) shape can be generated by L1–L3. Aberrated wave illuminates 2D periodic or quasi-periodic mask, which is made by the help of two polaroids (P and A) and spatial light modulator (SLM) – high-resolution VGA TFT display with size of the pixel is 42µ m. In the experiment the mask period is changed from 10 to 1 pixel. CCD camera (12 bit high-resolution digital camera with 1344×1024 pixels with the size of 6.45µ m×6.45µ m) receives the diffraction intensity distribution in the Fresnel field. To adjust conveniently, the CCD camera is placed on a movable platform. In the next experiments, the dependence of the curvature of the test spherical wave (1/R) on the Talbot image blurring was measured (Fig.2). Obtained images show the increasing wavefront curvature changes Talbot image of the
(a) zT /R = 0
(b) zT /R = −0.01
(c) zT /R = −0.02
(d) zT /R = −0.03
Fig. 2. Fragments of the diffraction images in the Talbot plane (D1 = D2 = 336µ m (8 pixels), zT = 356.8mm) for the wavefronts with different curvature (1/R). mask, but it is partially reproduced until zT /R < −0.025. According to [5] image in Talbot plane is partially reproduced
if the the phase 2π zT /R within the interval [−π /4, π /4] for all signicant terms of the series t(x, y). One should note the number of significant terms of the series t(x, y) depends on duty factor of the mask (D/d, where d is a pit length). Fig. 3 shows the diffraction intensity distribution of the astigmatic wavefront on the masks with different periods. As
(a) D = 168µ m
(b) D = 210µ m
(c) D1 = 168µ m; D2 = 210µ m
Fig. 3. Diffraction images in the Talbot plane for the astigmatic wavefront (ZT 1 = 356.8mm). one can see partially reproduced images could be obtained for the masks with different periods on the same distance Z. However self-imaging phenomenon obtained by quasi-periodic mask with different orthogonal periods (D1 ̸= D2 ). It should be noted that aberrated wave distorts Talbot image with the scale factors of: sx = 0.315 and sy = −0.194. In summary, theoretical and experimental studies about the Talbot effect for aberrated optical wave are first performed in this paper. The theoretic analysis about the Fresnel diffraction of 2D quasi-periodic mask, which can be taken as the superimposition of two orthogonal one-dimensional masks, is carried out, and the self-imaging phenomenon of this quasi-periodic mask by the aberrated wave is predicted. Experimentally, the periodic and quasi-periodic masks are obtained with the aid of the an SLM, and their diffraction intensity distribution in the Fresnel field is recorded by a CCD. The experimental results show that the Talbot image of the quasi-periodic mask is restored by aberrated wave with scale factors, which are determined by Talbot distance and specific curvature radius s = (zT + R)/R. The experimental results are consistent with the theoretic ones. We believe study of the Talbot effect of the quasi-periodic mask may extend its applications and will be helpful for the diffraction of more complex quasi-periodic masks. References 1. N.H. Salama, D. Patrignani, L.De Pasquale, E.E. Sicre, ”Wavefront sensor using the Talbot effect,” Optics & Laser Technology 31, 269-272 (1999). 2. Jianming Wen, Yong Zhang, Min Xiao, ”The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Advances in Optics and Photonics 5, 83–130 (2013). 3. P. Latimer, R. F. Crouse, ”Talbot effect reinterpreted,” Applied Optics 31, 80-89 (1992). 4. D. V. Podanchuk, A. A. Goloborodko, M. M. Kotov, ”Features of the wavefront sensor based on the Talbot effect,” in Proceedings of International Conference on Advanced Optoelectronics & Lasers CAOL 2013, O.V.Shulika, I.A.Sukhoivanov, eds. (IEEE, 2013), pp.337–339. 5. D. Podanchuk, A. Kovalenko, V. Kurashov, M. Kotov, A. Goloborodko, V. Danko, ”Bottlenecks of the wavefront sensor based on the Talbot effect,” Applied Optics 53, B223–B230 (2014). 6. D. Podanchuk, V. Kurashov, A. Goloborodko, V. Dan’ko, M. Kotov, N. Goloborodko, ”Wavefront sensor based on the Talbot effect with the precorrected holographic grating,” Applied Optics 51, C125–C132 (2012). 7. D. V. Podanchuk, A. A. Goloborodko, M. M. Kotov, D. A. Petriv, ”Talbot sensor with diffraction grating adaptation to wavefront aberrations,” Ukrainian Journal of Physics 60, 10–14 (2015).
