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Abstract
This work explains diffraction of Laguerre-Gaussian (LG) beams having non-zero radial indices from one dimensional (1D) periodic structures and their transformation into Hermite-Gaussian (HG) modes, theoretically, verifies using simulations and demonstrates the phenomenon experimentally. We first report a general theoretical formulation for such diffraction schemes, and then use it to investigate the near-field diffraction patterns from a binary grating having a small opening ratio (OR) by providing numerous examples. Results show that for OR$\lesssim$ 0.1, at the Talbot planes, mainly at the first Talbot image, the images of individual lines of the grating obtain HG modes’ intensity patterns. Therefore, the topological charge (TC) of the incident beam and its radial index can be determined from the observed HG mode. In this study, the effects of the OR of the grating and the number of Talbot plane on the quality of the generated 1D array of HG modes are also investigated. The optimum beam radius for a given grating is also determined. The theoretical predictions, are well confirmed by a number of simulations based on the free space transfer function and fast Fourier transform approach, and by experiments. The observed phenomenon, the transformation of LG beams into 1D array of HG modes under the Talbot effect, in addition of providing a way for characterization of LG beams with non-zero radial indices, itself is interesting and may be used in other fields of wave physics, especially for long-wavelength waves.
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1. Introduction
Structural beams, often carrying topological charge (TC), and periodic structures, commonly known in optics as gratings, are often used to generate or characterize each other. For instance, with the interference of a plane wave and an optical vortex beam, the structure of a fork linear grating can be created, and the diffraction of a plane wave from a fork linear grating creates a vortex beam. In addition, the far-field diffraction of vortex beams such as Laguerre-Gaussian (LG) beams from gratings can be used for changing their TC values, and the near-field diffraction of a vortex beam from a two dimensional (2D) grating can be used for multiplication of the impinging beam at the Talbot planes [1–5].
The diffraction of an LG beam having zero radial index from a one dimensional (1D) grating was successfully used for the characterization of impinging beam including determining its topological charge (TC) and winding direction [6–8]. It was shown that the image of an individual line of the grating at the self-imaging or Talbot planes includes inclined dark stripes. The number and orientation of these dark stripes reveal TC of the optical vortex. It has also been revealed that for a 1D grating having a small opening ratio (OR), the contrast of the inclined intensity stripes formed at the image location of individual lines of the grating increases significantly [9].
Now in this work we consider the diffraction of LG beams having non-zero radial indices from 1D periodic structures. A comprehensive theory is presented in general which is applicable for the diffraction of vortex beams from a given periodic structure, and for a binary grating having small values of opening ratio (OR) a detailed study is investigated. Investigations reveal a physical surprising, which is the formation of intensity patterns of Hermite-Gaussian (HG) modes at the image locations of each of the individual lines of the grating. The transformation of LG beams into 1D array of HG modes under the Talbot effect is the first important and interesting result of the current work. The conversion between the two modes as well as the rotating characteristic due to cross phase were also discussed in [10,11], and it is shown that the HG beams carrying a cross phase can evolve into the LG modes, and vice versa, an LG mode with the cross phase can also transform to the HG mode. However, the methods used for such conversions were different.
The second important result of the current work is determining the TC and its sign based on this type of diffraction from a 1D array. It is worth noting that this type of measurements have been reported in many works, but they have often been limited to the characterization of LG beams with zero radial index [12–23].
It is worth mentioning that, for the first time, based on our knowledge, the Talbot images of a pair of linear gratings in a moiré deflectometry setup was used for the characterization of vortex beams in [24]. The same arrangement was also used for detection of TC’s sign changing under reflection in [25]. Both of these works used highly spaced moiré fringes and benefiting their advantages, such as the moiré effect magnification, therefore the moiré deflectometry approach is still a more convenient method for the wavefront sensing of the vortex beams. It is also worth mentioning that in the two mentioned works, only the measurement of the wavefront of the LG beams with a zero radial index has been reported, therefore, the study of the wavefront of LG beams having non-zero radial indices with the aid of moiré deflectometry method can be also an interesting study.
Another important issue which is necessary to note here is that, apart from the surprising nature of the observed transformation of LG beams into a 1D array of HG modes over the image plane of the individual lines of the grating, still the ease of the TC measurement in the observed phenomenon is not comparable with a method which is recently introduced in [26,27]. In that method, LG beams having non-zero radial indices can be characterized via far-field diffraction from an amplitude parabolic-line linear grating printed on a transparent plastic sheet.
2. Diffraction of optical beams from 1D periodic structures: a general formulation
Here we present a general formulation for the diffraction of optical beams from periodic structures. The transmittance of a 1D periodic structure having a fundamental period $\Lambda$ can be expressed as a Fourier expansion in the following form [28,29]:
where $f=\frac {1}{\Lambda }$ is the fundamental frequency of the structure and $t_m$ indicate $m$th Fourier coefficient. Taking a 2D Fourier transform, the spatial spectrum of $t(x)$ can be expressed as follows [28]: where $(\xi,\eta )$ denotes the coordinates in the spectral domain and $\delta$ is the impulse symbol. On the other hand, the complex amplitude of an optical wave field in a given plane $z$, $\psi (x,y,z)$, is related to the complex amplitude at $z=0$ plane, $\psi (x,y,0)$, by [30]: where $\otimes$ denotes the convolution operation, ${FR_z} \left [ \quad \right ]$ is known as Fresnel propagator or Fresnel transform, and $h(x,y,z)$ indicates the impulse response of the free space given byAs is apparent, $\psi (x,y,0)$ is product of two 2D functions. The near-field propagation for such a light field distribution is presented in Refs. [5,31,32]. Accordingly, the light field at the propagation distance $z$ can be expressed as
It should be mentioned that $\frac {{m\lambda z}}{\Lambda }$ indicates the coordinate of $m$th diffraction order so that Eq. (9) can be considered as a coordinates transformation to the center of $m$th diffraction order. It is also worth mentioning that Eq. (8) presents a general formulation for the diffraction of optical beams from 1D periodic structures based on the interference of the diffraction orders instead of the direct solving of the Fresnel integral. As a typical example, we consider diffraction of an LG beam from 1D periodic structures.
3. Diffraction of an LG beam from a 1D periodic structure
The complex amplitude of an LG beam at $z=0$ can be expressed as follows:
Now suppose that a 1D periodic structure, with a transmittance indicated by Eq. (1), is illuminated by an LG beam at its beam waist shown by Eq. (10). Considering Eqs. (8) and (12), it is easy to show that, the complex amplitude of diffracted Gaussian beam at distance $z$ from a 1D periodic structure, can expressed as follows:
4. Diffraction of an LG beam from a 1D binary grating
Here, we consider the diffraction of an LG beam, expressed by Eq. (10), from a 1D binary grating schematically depicted in Fig. 1, second row, first and second columns. The transmittance of this grating can be expressed as follows:
Fig. 1. First column, the intensity profile of an LG beam having $l$ = 3 and $p$ = 2, transmission function of a binary grating with $\Lambda$ = 0.2 mm and a = 0.02 mm, and intensity distribution immediately after the grating illuminated by the LG beam. Second column, the corresponding insets show the details. Third column, the near-field diffraction pattern under propagation. Last column, insets show images of three individual lines of the grating (see also Visualization 1).
4.1 Theoretical predictions
Figure 1 shows the results of analytical prediction of the diffraction of an LG beam having a non-zero radial index from a 1D binary grating with a sufficiently small value of OR. The first column, first to third rows, show the intensity profile of an LG beam having $l$ = 3 and $p$ = 2, transmission function of a binary grating with $\Lambda$ = 0.2 mm and $a$ = 0.02 mm, and intensity distribution immediately after the grating illuminated by the LG beam, respectively. In the second column insets show the details of the intensity and grating structures. Third column shows the near-field diffraction pattern under propagation, and in the fourth column, insets show the image patterns of three individual lines of the grating. In the figure, $z_{T}$ indicates Talbot distance of the grating and its value is obtained by $\frac {2\Lambda ^2}{\lambda }$. As seen in the last column, at the first Talbot plane, the images of individual lines of the grating obtain HG modes’ intensity patterns. Since the Talbot image of each of the lines of the grating contains an $(l + p + 1)$ by $(p + 1)$ two-dimensional array of intensity spots, the TC of the incident beam and its radial index can be determined from the observed HG mode. In the background Visualization 1, the width of the illustrated diffraction pattern formed at the first Talbot plane gradually increases when an LG beam with $l$ = 3 and $p$ = 2 illuminates an amplitude binary grating with $\mu$ = 0.1.
Apart from the ability of this diffraction arrangement for characterizing LG beams, the generation of HG modes intensity patterns at the image locations of very narrow individual lines of the grating is very interesting. Such transformation of LG beams into a 1D array of HG modes over the Talbot image plane of the individual lines of the grating is an amazing physical phenomenon and it can attract the attention of physicists, especially in optical sciences.
In the following we investigate the effects of the OR of the grating and the number of Talbot plane on the quality of generated HG modes.
Figure 2 shows the first Talbot images of a 1D binary grating with OR = 0.1 under different LG beams illumination. Insets, in the even columns from the left, present the images of an individual line of the grating with an enlarged scale along horizontal direction.
Fig. 2. The odd columns from the left show a small part of the first Talbot images of a 1D binary grating with OR = 0.1 under different LG beams illumination. Insets show horizontally enlarged images of an individual line of the grating.
Figure 3 considers the effect of OR of the binary grating on the analytically calculated first Talbot images of the individual lines of the grating illuminated by different LG beams. The corresponding phase profiles are also presented in the even rows. It may be seen that as the OR of the grating decreases, the intensity profiles become more similar to the intensity profiles of the HG modes and the phase jumps between the adjacent intensity spots reach $\pi$ values, as it is desired for an ideal HG beam.
Fig. 3. Calculated intensity and phase profiles of the first Talbot image of an individual line of a binary grating with different ORs under different LG beams illumination.
Figure 4 illustrates the TC sign effect on the individual line image of a binary grating with OR = 0.1 formed at the first Talbot image under different LG beams illumination. As seen the direction of the 2D array of spots changes by changing the sign of the TC.
Fig. 4. Illustrating the TC sign effect on the individual line image formed at the first Talbot image.
Figure 5 shows the calculated intensity profiles of an individual line image of a binary grating with OR = 0.1 for different propagation distances under different LG beams illumination. As can be seen, the HG modes produced in the first Talbot images are more similar to the intensity profiles of ideal HG beams.
Fig. 5. Illustrating the effect of the propagation distance on the formation of HG modes over the individual line image.
Figure 6 presents the results of the investigation of the effects of the values of three parameters including the beam waist, and the period and OR of the grating on the resulted diffraction patterns. As seen in the first row, when $d$ = 0.2 mm and OR = 0.1, an optimum beam waist is about $w$ = $5d$. Patterns presented in the second row show that smaller values of OR provides better diffraction patterns. The third row shows that for the given values of the beam waist and grating OR there is an optimum value for the grating period.
Fig. 6. Effects of the beam waist, and the period and OR of the grating on the resulted diffraction patterns.
In Fig. 7, for compassion, the intensity and phase profiles of an individual line first Talbot image of a binary grating having an OR = 0.1 under different LG beams illumination, and the intensity and phase profiles of an ideal HG beam with the same parameters are presented. Here again the similarity between the intensity and phase profiles of the individual diffraction patterns and the ideal HG beams increases as the grating OR decreases.
Fig. 7. The even columns form the left, intensity and phase profiles of an individual line first Talbot image of a binary grating having an OR = 0.1 under different LG beams illumination. The odd columns form the left, intensity and phase profiles of an ideal HG beam with the same parameters.
To gain an intuitive insight into the obtained results we have to present a concise review of the previous related works. It has been shown that in the near-field diffraction of a paraxial beam from a 2D orthogonal grating, with a sufficiently small value of OR, a 2D orthogonal lattice of the 2D Fourier transform of the incident beam is generated on the Talbot planes [5,33]. In analogy, when a paraxial beam illuminates a 1D grating with a sufficiently small value of OR a 1D array of 1D Fourier transform of the incident beam is generated on the Talbot planes, see [32]. The key point is that an LG beam is invariant under the 2D Fourier transform while 1D Fourier transform of an LG beam leads to HG modes. Therefore, illumination of a 2D orthogonal grating, having an appropriate value of OR, by an LG beam leads to multiplication of the incident beam on the Talbot planes. While by illuminating a 1D grating, with an appropriate value of OR, a 1D array of HG modes is obtained. Any generated pattern is similar to the pattern produced in the focal plane of a cylindrical lens.
5. Simulation and experiments
Here we confirm the theoretical predictions with the aid of a number of simulations of the light field propagation and experimental works. The simulation is realized by the free-space light field propagation from the grating plane to the observation plane with the aid of free space transfer function and fast Fourier transform approach.
Figure 8 shows the experimental setup used for the generation and characterization of LG beams having different $l$ and $p$ parameters. Different LG beams were generated by a conventional Spatial Light Modulator (SLM) extracted from a video projector (3M X50, resolution:1024$\times$768, display 0.7 in, polysilicon LCD) under propagation of a Gaussian beam thought it. A typical pattern implemented on the SLM, in transmission mode, is illustrated in the inset of the figure. Generated LG beams were also characterized under diffraction from a curved-line linear grating for a proper verification of the proposed method (for more details such as the pattern shape applied in SLM see [27]). A generated LG beam passes thought a binary linear amplitude grating having a sufficiently small OR and at the first Talbot plane the diffraction provides a 1D array of HG modes.
Fig. 8. Experimental setup used for characterizing LG beams with different non-zero $l$ and $p$ parameters via transformation of the impinging LG beam into 1D array of HG modes at the Talbot plane. S.F. and $z_T$ stand for spatial filter and Talbot distance, respectively.
To do the experiments, the plane wave of the second harmonic of a neodymium-doped yttrium aluminum garnet (ND:YAG) diode-pumped laser beam having a wavelength of $\lambda$= 532 nm and a Gaussian profile was passed through a spatial light modulator. The produced LG beam with given $l$ and $p$ indices passed through a 1D amplitude linear grating having a period 0.15 mm and an OR = 0.1. The near-field diffracted pattern was directly recorded on the sensitive area of a camera (Nikon D7200). To obtain a high-resolution and aberration-less image of the near-field diffraction pattern, we recorded the diffracted pattern directly on the camera sensor by removing the camera imaging lens. Since the first Talbot plane locates at a distance $z_{T}=\frac {2\Lambda ^2}{\lambda }$ = 8.5 cm from the grating, in practice direct recording of the first Talbot image on the camera sensor is possible.
Figure 9 shows the enlarged first Talbot image of an individual line of an amplitude binary linear grating having an OR = 0.1 under LG beams with $l$ = 2 and $p$ = 0, 1, 2. We see that, each individual diffraction pattern consists of $(l + p + 1)$ by $(p + 1)$ 2D array of intensity spots, and is similar to the intensity pattern of HG mode with $l$ and $p$ indices.
Fig. 9. Theoretical (red) and experimentally recorded (green) diffraction patterns formed in front of an individual line of an amplitude binary linear grating having an OR = 0.1 at the first Talbot plane when the grating is illuminated by LG beams with $l$ = 2 and $p$ = 0, 1, 2. For all patterns the grating period and the beam wavelength were $\Lambda$ = 0.15 mm and $\lambda$ = 532 nm, respectively. For the theoretical patterns $w\,=\,5\Lambda$, and the value of $w$ for the experimental works was $1\pm 0.1$ mm.
A comparison of theoretical predicted patterns (red), simulations of the free-space propagation (gray), and experimentally recorded (green) results are presented in Figs. 10 and 11. As seen, there is a good agreement between all results.
Fig. 10. Theoretical (red), simulated (gray), and experimentally recorded (green) diffraction patterns at the first Talbot plane when an LG beam with different $l$ and $p$ indices passed through a 1D amplitude linear grating having an OR = 0.1.
Finally, it is important to identify the factors limiting the operation range of the method for characterization of LG beams. Since the HG mode is formed in an area with a width equal to the grating period and a length equal to the size of the incident beam, by increasing the grating period and the beam size, and by using a high resolution imaging sensor the operating range for the method can be increased, and an LG beam with higher radial and azimuthal indices can be characterized.
6. Conclusion
In this work, a theory for the diffraction of LG beams having non-zero radial indices from 1D periodic structures was presented, and detailed features of the diffraction patterns at the near-field regime when the structure is a binary grating with a sufficiently small OR were presented. It was shown that mainly at the first Talbot plane, the image of each of the individual lines of the grating has an intensity pattern that is similar to HG modes, where TC of the incident beam and its radial index can be estimated from the observed HG mode pattern. Observing intensity patterns of HG modes at the image location of an individual line of the grating may not be of critical use in the visible wavelengths, but the transformation of the LG beam into HG modes at the image location of a very narrow line of the grating has many physical surprises which can attract the attention of physics enthusiasts. Furthermore, the observed phenomenon, the transformation of LG beams into 1D array of HG modes under the Talbot effect might find applications in other area of physics, especially at large wavelengths.
Funding
Institute for Advanced Studies in Basic Sciences (G2023IASBS12632); Iran National Science Foundation (4020609).
Acknowledgment
The author Saifollah Rasouli would like to acknowledge the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy, for the Senior Associate Fellowship.
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
References
1. H. Gao, Y. Li, L. Chen, J. Jin, M. Pu, X. Li, P. Gao, C. Wang, X. Luo, and M. Hong, “Quasi-talbot effect of orbital angular momentum beams for generation of optical vortex arrays by multiplexing metasurface design,” Nanoscale 10(2), 666–671 (2018). [CrossRef]
2. S. Rasouli and D. Hebri, “Theory of diffraction of vortex beams from 2d orthogonal periodic structures and talbot self-healing under vortex beam illumination,” J. Opt. Soc. Am. A 36(5), 800–808 (2019). [CrossRef]
3. B. Knyazev, O. Kameshkov, N. Vinokurov, V. Cherkassky, Y. Choporova, and V. Pavelyev, “Quasi-talbot effect with vortex beams and formation of vortex beamlet arrays,” Opt. Express 26(11), 14174–14185 (2018). [CrossRef]
4. D. A. Ikonnikov, S. A. Myslivets, M. N. Volochaev, V. G. Arkhipkin, and A. M. Vyunishev, “Two-dimensional talbot effect of the optical vortices and their spatial evolution,” Sci. Rep. 10(1), 20315 (2020). [CrossRef]
5. D. Hebri and S. Rasouli, “Theoretical study on the diffraction-based generation of a 2d orthogonal lattice of optical beams: physical bases and application for a vortex beam multiplication,” J. Opt. Soc. Am. A 39(9), 1694–1711 (2022). [CrossRef]
6. P. Panthong, S. Srisuphaphon, A. Pattanaporkratana, S. Chiangga, and S. Deachapunya, “A study of optical vortices with the talbot effect,” J. Opt. 18(3), 035602 (2016). [CrossRef]
7. P. Panthong, S. Srisuphaphon, S. Chiangga, and S. Deachapunya, “High-contrast optical vortex detection using the talbot effect,” Appl. Opt. 57(7), 1657–1661 (2018). [CrossRef]
8. S. Deachapunya, S. Srisuphaphon, and S. Buathong, “Production of orbital angular momentum states of optical vortex beams using a vortex half-wave retarder with double-pass configuration,” Sci. Rep. 12(1), 6061 (2022). [CrossRef]
9. S. Buathong, S. Srisuphaphon, and S. Deachapunya, “Probing vortex beams based on talbot effect with two overlapping gratings,” J. Opt. 24(2), 025602 (2022). [CrossRef]
10. G. Liang and Q. Wang, “Controllable conversion between hermite gaussian and laguerre gaussian modes due to cross phase,” Opt. Express 27(8), 10684–10691 (2019). [CrossRef]
11. G. Liang and Q. Wang, “Power steered modes conversions and rotations in nonlocal nonlinear media,” J. Opt. 21(12), 125504 (2019). [CrossRef]
12. V. Y. Bazhenov, M. Vasnetsov, and M. Soskin, “Laser beams with screw dislocations in their wavefronts,” Jetp Lett 52, 429–431 (1990).
13. N. Heckenberg, R. McDuff, C. Smith, H. Rubinsztein-Dunlop, and M. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24(9), S951–S962 (1992). [CrossRef]
14. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002). [CrossRef]
15. H. Sztul and R. Alfano, “Double-slit interference with laguerre-gaussian beams,” Opt. Lett. 31(7), 999–1001 (2006). [CrossRef]
16. Q. S. Ferreira, A. J. Jesus-Silva, E. J. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011). [CrossRef]
17. J. Hickmann, E. Fonseca, W. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010). [CrossRef]
18. M. W. Beijersbergen, L. Allen, H. Van der Veen, and J. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]
19. R. D. Niederriter, M. E. Siemens, and J. T. Gopinath, “Simultaneous control of orbital angular momentum and beam profile in two-mode polarization-maintaining fiber,” Opt. Lett. 41(24), 5736–5739 (2016). [CrossRef]
20. B. Khajavi and E. J. Galvez, “Determining topological charge of an lg beam using a wedged optical flat,” in Frontiers in Optics, (Optica Publishing Group, 2017), pp. FM3B–4.
21. D. Hebri, S. Rasouli, and M. Yeganeh, “Intensity-based measuring of the topological charge alteration by the diffraction of vortex beams from amplitude sinusoidal radial gratings,” J. Opt. Soc. Am. B 35(4), 724–730 (2018). [CrossRef]
22. O.-V. Grigore and A. Craciun, “Method for exploring the topological charge and shape of an optical vortex generated by a spiral phase plate,” Opt. Laser Technol. 141, 107098 (2021). [CrossRef]
23. Y. Shang, W. Wang, Z. Mi, B. Wang, L. Zhang, K. Han, C. Lei, Z. Man, and X. Ge, “Determining the topological charge of optical vortex by intensity distribution of a quasi-airy vortex beam,” Opt. Commun. 529, 129075 (2023). [CrossRef]
24. M. Yeganeh, S. Rasouli, M. Dashti, S. Slussarenko, E. Santamato, and E. Karimi, “Reconstructing the poynting vector skew angle and wavefront of optical vortex beams via two-channel moiré deflectometery,” Opt. Lett. 38(6), 887–889 (2013). [CrossRef]
25. S. Rasouli and M. Yeganeh, “Use of a two-channel moiré wavefront sensor for measuring topological charge sign of the vortex beam and investigation of its change due to an odd number of reflections,” International Journal of Optics and Photonics 7, 77–83 (2013).
26. P. Amiri, A. M. Dezfouli, and S. Rasouli, “Efficient characterization of optical vortices via diffraction from parabolic-line linear gratings,” J. Opt. Soc. Am. B 37(9), 2668–2677 (2020). [CrossRef]
27. S. Rasouli, S. Fathollazade, and P. Amiri, “Simple, efficient and reliable characterization of laguerre-gaussian beams with non-zero radial indices in diffraction from an amplitude parabolic-line linear grating,” Opt. Express 29(19), 29661–29675 (2021). [CrossRef]
28. S. Rasouli, D. Hebri, and A. M. Khazaei, “Investigation of various behaviors of near-and far-field diffractions from multiplicatively separable structures in the x and y directions, and a detailed study of the near-field diffraction patterns of 2d multiplicatively separable periodic structures using the contrast variation method,” J. Opt. 19(9), 095601 (2017). [CrossRef]
29. S. Rasouli and D. Hebri, “Contrast enhanced quarter-talbot images,” J. Opt. Soc. Am. A 34(12), 2145–2156 (2017). [CrossRef]
30. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).
31. J. T. Winthrop and C. Worthington, “Convolution formulation of fresnel diffraction,” J. Opt. Soc. Am. 56(5), 588–591 (1966). [CrossRef]
32. V. Arrizón and G. Rojo-Velázquez, “Fractional talbot field of finite gratings: compact analytical formulation,” J. Opt. Soc. Am. A 18(6), 1252–1256 (2001). [CrossRef]
33. C. Schnébelin and H. G. de Chatellus, “Spectral interpretation of talbot self-healing effect and application to optical arbitrary waveform generation,” Opt. Lett. 43(7), 1467–1470 (2018). [CrossRef]
