1. Introduction

Microlens arrays (MLAs), a 2D array of close-packed lenslets, are optical elements designed to split the wavefront into multiple beamlets of equal energies. MLAs have found a great deal of attention for their wide range of applications in wavefront sensing [1], digital holography [2], and also as a tool for fundamental studies in supercontinuum generation [3]. Having a series of periodic lenslets, they are also being used as array illuminators and in ultrafast laser filamentation experiments [4]. In 1836, H. F. Talbot observed a unique self-imaging effect while working on periodic structures [5]. When a plane wave is incident on a periodic grating, the image of the grating field is replicated at fixed positions in the propagation direction, known as the Talbot planes appearing at the distances, $z=N \Lambda ^{2}/ \lambda$. Here, $\Lambda$ is the period of the grating object, $\lambda$ is the wavelength of the input radiation, and N is a positive integer. For odd values of N, the Talbot images are shifted laterally by an amount of $\Lambda /2$ with respect to the grating position. Therefore, the distance between two consecutive non-shifted image planes is $z_{T}=2\Lambda ^{2}/\lambda$, referred to as Talbot length [6]. The Talbot effect, being a near-field effect, is observed as a consequence of the diffraction of the input field from the periodic object and the subsequent interference between the diffracted fields. Near-field diffraction is enriched with several important properties. For example, if one explores the strong imaging regime [7], where frequencies are higher than $1/\lambda$, superoscillation of the state can be used for subwavelength imaging with grating waves, can access certain advantages over imaging with evanescent waves [8]. Imaging exploiting the superoscillation has been demonstrated a few years ago [9]. The Tablot effects have been observed in various research areas. For example, in atomic physics, the manipulation of Talbot phenomena in cold atomic systems represents a promising frontier in optical imaging and manipulation at the quantum level [10]. As such, efforts have been made with theoretical and experimental studies of first- and second-order Talbot effects using electromagnetically induced grating [11], and the spontaneous parametric four-wave mixing by modulating the third order nonlinear optical coefficient [12] through a strong standing wave in cold atomic medium and molecules. Additionally, the experimental observations on dynamic behaviors of wave packets in non-Hermitian photonic graphene, 2D honeycomb optical lattice structures realized in three-level $\Lambda$-type atomic vapor configurations [13] shows the possibility of the development of all-optical-controllable Talbot–Lau interferometers, and also improvement in the performance of cold atom imaging and lithography.

On the other hand, MLAs, being 2D periodic objects, are also expected to produce self-imaging effects. It has been verified that MLAs also exhibit the Talbot effect with almost similar properties [14,15], however, the additional phases in the case of lenslets that are not closely packed resulted in quasi-Talbot effects. This self-imaging property of MLAs has been found useful in a variety of applications in science and technology. Employing the Talbot effect in X-ray Talbot interferometers, the phase structures of 3D samples can be identified from the Moiré fringe patterns [16]. Similarly, using fractional Talbot images, high-resolution fluorescence images are obtained in fluorescence Talbot microscopy [17]. Although Talbot imaging with MLAs has been studied in the past, no significant efforts have been made on single-pass image upconversion of periodic objects through the second harmonic generation (SHG) and retaining Talbot effect [10] in the new wavelength where the high-efficiency image sensors are abundant. In 2010, the SH Talbot effect was observed for the first time in periodically poled lithium tantalate (PPLT) crystals commonly used for highly efficient nonlinear frequency conversion processes. However, in this case, the poling periodicity of the crystal was used as the periodic structure for the study of the Talbot effect [18] by illuminating the crystal in the direction orthogonal to the poling direction. As a result, the direct benefit of periodic poling of the nonlinear crystal after accessing the highest nonlinear gain and long crystal length was unexplored, resulting in low nonlinear frequency conversion efficiency. Recently, SHG of Talbot images with pump beam modified by spatial light modulator (SLM) has been reported [19]. Again, the lower damage threshold of SLM has restricted such studies to lower power levels. As a result, the control over the Talbot planes in both pump and SHG with high nonlinear upconversion efficiency over long distances has not been addressed so far. Given that the Talbot length is proportional to the square of the period of the object for a fixed wavelength [5], one can vary the Talbot length by using MLAs of different periods in different array architecture as observed [13] in case of atomic coherence gratings, instantaneously tunable complex honeycomb potential in atomic and atomic-like ensemble. However, it is essential to access tunable Talbot length while keeping all physical parameters of the grating constant at fixed wavelength for high resolution imaging and lithographic application.

Here, we report, for the first time to our knowledge, high-power, single-pass frequency-upconversion of the periodic object producing Talbot planes with controllable lengths. Using the MLA as the 2D periodic object and 1.2 mm long BiBO crystal for frequency upconversion, we have demonstrated Talbot effect in both pump and upconversion wavelength with Talbot lengths tunable across $z_{T}=26$ cm to $z_{T}=62.4$ cm at the pump and $z_{T}=12.4$ cm to $z_{T}=30.8$ cm at the green wavelength.

2. Experimental configuration

The schematic of the experimental setup is shown in Fig. 1. A 5 W Yb-fiber laser with a spectral bandwidth of 15 nm centered at 1064 nm providing femtosecond pulses of $\sim 260$ fs duration at a repetition rate of 78 MHz is used as the pump radiation [20]. The input power to the setup is varied using a half-wave plate $(\lambda /2)$ and a polarizing beam-splitter (PBS) cube. The input beam is expanded and collimated using a telescopic lens combination comprising two lenses, L1 and L2, of focal length, $f_{1} = 50$ mm, and $f_{2} = 100$ mm, respectively, to collimate the laser beam into a near plane wavefront. The second $\lambda /2$ plate is used to control the polarization of the laser beam depending upon the orientation of the nonlinear crystal for optimum phase-matching in the experiment. A lens, L3, of focal length, $f_{3} = 150$ mm, is used to Fourier transform the phase profile of the microlens array (MLA) (Thorlabs MLA 300-14AR) placed at a distance, $d$, from the back focal plane of the lens. The MLA consisting of 391 lenslets of focal length, $f_{MLA} = 18.6$ mm, with a lenslet period of $\Lambda = 300$ $\mu$m [21], is used as the periodic object to generate the Talbot images. A 1.2-mm-long bismuth triborate (BiBO) crystal [22] with an aperture of $4 \times 8$ $mm^{2}$ is placed at the back focal plane of the lens, L3. The crystal is cut for type-I ($e + e \to o$) frequency-doubling in the optical yz-plane ($\Phi$ = 90$^o$) with an internal angle of $\theta$ = 168.5$^o$ to the normal incidence and is used for second harmonic generation (SHG) of the Talbot images at 1064 nm into the green at 532 nm. The frequency-doubled green beam is extracted from the undepleted pump using a wavelength separator, S, and subsequently imaged using a CCD camera after the lens, L4, of focal length, $f_{4} = 150$ mm. The lens, L4, is placed at a distance, $f_{4}$, from the crystal to inverse Fourier transform the array of the spatial structure of the input beam (here Gaussian beam) generated due to the Fourier transform of the MLA at the back focal plane of the lens, L3. To record the Talbot images of both pump and SHG beams along the propagation distance, the CCD camera is placed on a translation stage. The insets of Fig. 1 shows the line profiles of the self-images of both the pump and the SHG beams recorded along with propagation distance at an interval of 1 mm.

 

Fig. 1. Schematic of the experimental setup to study the nonlinear interaction of the Talbot effect. $\lambda /2$, half-wave plate; PBS, polarizing beam splitter cube; L1-4, plano-convex lenses; MLA, microlens array; BiBO, nonlinear crystal; S, wavelength separator. (Inset) Images of light carpet at pump and upconversion wavelength.

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3. Results and discussion

To verify the Talbot effect of the MLA at the pump wavelength, we illuminated the MLA with the collimated pump beam by removing the lenses, L3 and L4, and recorded the images right after the MLA at an interval of 1 mm along the propagation direction. The results are shown in Fig. 2. To avoid the mechanical constraint of the experiment, the images were recorded starting from at a distance of about $\sim 5$ cm from the MLA, up to a distance of 45 cm, covering a range of 40 cm. However, such a delayed start of the image recording does not affect the overall experimental results, as the Talbot planes are replicated at every Talbot length irrespective of the initial position of occurrence. To improve the visibility and exact detection of the Talbot planes, we plot the line profiles of the CCD images along propagation distance by extracting the intensity profile of the CCD images from a fixed pixel line (here, pixels in the vertical line) of width 4.4 $\mu$m (corresponding to the size of a single pixel) and concatenate the line intensity images with propagation distance forming a continuous image (light carpet). The results are shown in Fig. 2(a). As evident from Fig. 2(a), the image locations identified with dashed white lines are almost identical, indicating the positions of Talbot planes of the pump beam. The intensity profiles, indicated by the first and third dashed lines, are almost identical, indicating the positions, $z$ = 1 mm and 171 mm respectively. Therefore, the experimental Talbot length of the MLA with a pitch of $\Lambda$ = 300 $\mu$m is $z_{T_{p}}$ = 170 mm. As reported previously [23], the microlens array can be approximated as a 2D sinusoidal phase grating. Therefore, the Talbot images recorded in the present study form a 3D light carpet comprising both self and fractional Talbot images and complicated fractal intensity distribution appearing as a function of distance across the image [24], confirming the successful generation of light carpet for MLA. For the lenslet pitch of $\Lambda =300$ $\mu$m, the theoretical Talbot length for the pump wavelength of $\lambda _{p}=1064$ nm can be calculated to be, $z_{T_{p}} = 2\Lambda^{2}/\lambda{p} =169.1$ mm, in close agreement to the experimental value, $z_{T_{p}}$ = 170 mm.

 

Fig. 2. Light carpet along propagation distance for the periodic object (microlens array) at (a) pump, and corresponding (b) upconversion wavelengths confirming Talbot effect. The white dashed lines identify different Talbot planes.

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To observe the upconversion of the Talbot effect, we placed the nonlinear crystal at the focal plane, $f_{MLA}=18.6$ mm, of MLA, and recorded the intensity profile of the SHG beam along the propagation direction, with an increment of 1 mm, after extraction from the undepleted pump using the wavelength separator, S. However, due to the mechanical constraints in positioning the wavelength separator, the nearest attainable limit of the CCD camera position was about 15 mm from the crystal plane. Using the same technique of extracting the line profile of fixed pixel line from the CCD images along propagation and concatenating the images, we observed the variation of the SHG beam along propagation, with the result shown in Fig. 2(b).

As evident from Fig. 2(b), the images of the SHG beam are almost identical, as marked by the dashed lines along the propagation axis, indicating the positions of Talbot planes. However, from Fig. 2(a) and Fig. 2(b), it is evident that the SHG images are reproduced at a propagation distance twice that of the pump images. Theoretically, the Talbot length of the SHG beam, $z_{T_{SH}} = 2\Lambda ^{2}/\lambda _{SH} = 2z_{T_{p}}= 338$ mm, is twice that the Talbot length of the pump beam, $z_{T_{p}}=169.1$ mm, since $\lambda _{SH}=\lambda _{p}/2$. Using the results of Fig. 2(b), we measured the Talbot length of the single-pass frequency-doubled Talbot effect to be $z_{T_{SH}} = 330$ mm, close to the theoretically predicted value of $z_{T_{SH}}= 338$ mm. By carefully analyzing the line profiles, one can clearly see the change in the spatial frequencies of the SHG beam in fractional Talbot planes along the propagation distance. The decrease in the intensities with propagation distance can be attributed to the divergence effect of the lenslets of MLA. It is evident from the experimental results of Fig. 2 that the Talbot lengths of the pump and the SHG beams strictly depend on the input parameters of the MLA. To overcome this stringent dependence and to obtain long-range self-images with tunable Talbot lengths, we controlled the scale of the Fourier transform of the MLA by placing it after the lens L3 at a distance $d$ from the back focal plane of the lens. According to the Fourier transformation theory [25], adjusting the separation distance, $f_{3}-d$, between the MLA and lens L3, we can control the scale of the Fourier transform at the focal plane of the lens L3 due to the change in the illuminated region of the MLA. As reported previously [21], the Fourier transform of the MLA will produce the array of spots. Each spot of the array has an intensity profile same as the input beam (here Gaussian). The pitch of the Gaussian arrays can be represented as [26],

Here, $\Lambda$ is the pitch of the MLA. Now, the inverse Fourier transform of the Gaussian array using the lens L4, of focal length, $f_{4}$, placed at a distance, $f_{4}$, from the Fourier plane of the lens L3, produces scaled Talbot effect due to the scaling of the spatial distribution of the MLA. The effective lenslet period of the MLA at the back focal plane of the lens L4 can be written as,

Consequently, the modified Talbot length of the pump beam can be expressed as,

As evident from Eq. (3), the Talbot length, $z_{T_{p}}$, is modulated with a factor, $(f_{4}/d)^{2}$, the ratio of the focal length of the inverse Fourier transforming lens, L4, and the position of the MLA away from the back focal plane of the Fourier transforming lens, L3. Therefore, for a fixed focal length of the lens, L4, the Talbot length can be controlled by simply adjusting the position of the MLA. For the SH Talbot effect, the crystal is placed at the Fourier plane of lens L3, keeping the rest of the experimental setup fixed. Similar to Eq. (2), the effective lenslet period of the MLA in the SHG process, can be obtained as,

It is interesting to note that the variable Talbot length in the upconversion process, as indicated by Eq. (5), is precisely one-half of the variable Talbot length of the pump beam. While our experimental results in Fig. 2 demonstrate the doubling of the Talbot length in the upconversion process, the frequency-doubling of the array and the inverse Fourier transform at the upconverted wavelength effectively reduce the pitch of the microlens array (MLA) to one-half. Consequently, the Talbot length of the upconverted beam is halved compared to the Talbot length at the pump wavelength.

To gain further insight of controlled Talbot effect, we measured the period of the MLA (MLA300-14AR-M) used in the experiment and validated the performance of the imaging setup with the object positioned between lenses L3 and L4, of focal length $f_{3}=f_{4}=150$ mm, as depicted in Fig. 1. In doing so, we removed the BiBO crystal and the wavelength separator, S, and placed the CCD at the back focal plane of the lens, L4, to record the intensity pattern of the MLA at the pump wavelength. The results are shown in Fig. 3. The pictorial design of the MLA, as reproduced in Fig. 3(a) from the vendor (Thorlabs), has a two-dimensional array of convex lenses at a pitch of 300 $\mu$m. However, using a microscope, we imaged a section of the MLA, as shown in Fig. 3, and found the pitch to be 300 $\mu$m. A closer look at the image shows concentric circular rings confirming the convex shape of the lens. Now, placing the MLA at two distances, $d$ before the back focal plane of the lens $L3$, we recorded the image of the MLA, with the results shown in Fig. 3(c) and (d). As evident from the images of Fig. 3(c) and (d) corresponding to the $d$ value of 132 mm and 120 mm, respectively, we clearly see the change in the pitch of the MLA. Counting the number of pixels and multiplying the pixel size, we experimentally measured the change of MLA pitch from 340 $\mu$m to 375 $\mu$m due to the change of $d$ from 132 mm to 120 mm. Dividing the measured pitches with the magnification factor, $(f_4/d)$, we can find the actual pitch of the MLA to be 300 $\mu$m, the same as the value specified by the vendor and also measured using the microscope. Additionally, a closer look at the images of Fig. 3(c) and (d) shows the concentric rings, as observed in the microscopic image (see Fig. 3(b)) due to the convex shape of the microlenses in the array. The high-quality imaging with the object in between the imaging lens pair, L3 and L4, coupled with simple control of the magnification factor, confirms the possibility of measurement and control of the Talbot length of the periodic object. This novel experimental scheme facilitates the measurement of extremely small pitches of the periodic object by circumventing the limitations imposed by mechanical constraints in the experiment.

 

Fig. 3. (a) Schematic design of microlens array (MLA) as provided by the vendor (Thorlabs). (b) Microscopic image of the MLA showing the circular diffraction rings due to each microlens. Images recorded at the back focal plane of the lens, L4 for the MLA position, (c) $d=132$ mm and (d) $d=120$ mm, respectively.

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Confirming the performance of the experimental scheme in terms of simple control of the Talbot length, we recorded the images of the MLA at different fractional Talbot planes and the Talbot planes for both pump and upconverted wavelengths. We removed the BiBO crystal and the wavelength separator to identify the Talbot planes for the pump. But for the upconverted Talbot effect, we placed the BiBO crystal at the back focal plane of the lens ($L_3$) and the wavelength separator after the crystal to remove the undepleted pump. Keeping the MLA at a distance, $d$ = 120 mm and 132 mm, from the back focal plane of the lens, $L_3$, we recorded the Talbot images, with the results shown in Fig. 4. As evident from the first row of Fig. 4, the image of the MLA placed at $d$ = 120 mm illuminated by the pump wavelength changes with propagation distance, $z$. Interestingly, the images at $z$ = 0 and $z$ = $z_T$ are identical, confirming the Talbot effect. The Talbot length of the MLA for the pump wavelength for $d$ = 120 mm is measured as $z_T = z_{T_{p}}=268$ mm. As expected, for $z$ $\not$= $z_T$, we observe the image recorded at $z$ = 0.5$z_T$ has a transverse shift with respect to the image recorded at $z$ = 0. A similar shift is observed between the images recorded at $z$ = 0.75$z_T$ and $z$ = 1.25$z_T$ due to the $\pi$ phase shift, as commonly observed in Talbot effect [5,25]. This observation confirms the presence of the Talbot effect. It is to be noted that the circular diffraction pattern due to the convex surface of microlenses, as observed at $z$ = 0 plane, is not visible at other planes along propagation distance. It is evident from Eq. (5) that the Talbot length of the periodic object during the upconversion process is half of the Talbot length observed for the object at the pump wavelength. In fact, the images of the upconverted Talbot effect, as shown in the second row of Fig. 4, show a 2D array of one-half of MLA’s pitch due to the frequency doubling process. As expected, the upconverted Talbot images are identical at $z$ = 0 and $z$ = $z_T$, and for propagation distance, $z$ $\not$= $z_T$, the images vary. Like the Talbot images for MLA at the pump wavelength, in the upconversion process, we also observe the expected transverse shift among the images recorded at $z$ = 0.5$z_T$ and $z$ = 0, and $z$ = 0.75$z_T$ and $z$ = 1.25$z_T$, due to the $\pi$ phase shift. The Talbot length of the upconverted MLA with its position at $d$ = 120 mm is measured to be $z_T = z_{T_{SH}}$ = 124 mm, close to half of the Talbot length at pump wavelength. Having the MLA at $d$ = 132 mm, we observed the Talbot effect for pump and upconverted wavelengths, as shown in the third and fourth rows of Fig. 4, respectively. However, due to the increase in the $d$ value while other parameters remain constant, we expect a decrease in the Talbot length due to the decrease in the effective pitch of the MLA, as predicted by Eq. (2) and Eq. (3). Experimentally, we measured the Talbot length to be $z_T$ = 220 mm and 104 mm for the pump and upconversion wavelength, respectively. The slight variation in the upconverted Talbot length compared to half of the pump Talbot length can be attributed to the precise positioning of the crystal center concerning the focal plane of lens $L_4$, considering the focal length of the MLA lenslets at 18.6 mm and the associated beam shift induced by the crystal. Other contributing factors to the error may include human error in accurately identifying the Talbot plane at SH wavelength, which is more pronounced at lower values of $d$ due to higher longitudinal magnification of the Talbot plane range. It is noteworthy that the upconverted Talbot images lack the circular diffraction pattern typically observed for the pump wavelength. In addition, the boundaries of the arrays appear blurred. This phenomenon can be attributed to the spatial filtering effect inherent in the nonlinear frequency conversion process. It is well known that for any grating structure, the intensity of higher-order spatial frequencies diminishes inversely with the order of the spatial frequency. On the other hand, in the nonlinear parametric process, the efficiency depends on the intensities of the pump beam at the nonlinear crystal plane (in this case, the Fourier plane of lens $L_3$). Therefore, the intensity of higher-order spatial frequencies of the periodic object is significantly lower in the upconversion process, as if the higher-order spatial frequencies of the object undergo natural spatial filtering during the nonlinear process, leading to the blurring of array boundaries observed in the upconverted Talbot images. This study also demonstrates the potential to control the Talbot length at both the fundamental and upconversion wavelengths by adjusting the position of the periodic object.

 

Fig. 4. Transverse intensity distributions along propagation distance at pump (first and third rows) and SHG (second and fourth rows) wavelengths for two different MLA positions, $d=120$ mm, and $d=132$ mm. Self-image planes at $0$, $z_{T}/2$, $z_{T}$ and fractional Talbot images at $0.75z_{T}$, and $1.25z_{T}$ are shown along different columns, at both pump and SHG beam

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To gain a deeper understanding and verify the predicted simple control of the Talbot length according to Eq. (3) and Eq. (5), we conducted measurements of the Talbot length at both pump and upconversion wavelengths while varying the MLA distance. The results, depicted in Fig. 5, corresponds to the MLA distances, $d$, of 80 mm, 90 mm, 100 mm, 120 mm, and 125 mm for both pump and upconversion wavelengths. It is evident from Fig. 5 that the pump Talbot length (represented by black dots), $L_{T}$, decreases from 62.4 cm to 26 cm as the MLA separation, $d$, increases from 80 mm to 125 mm. The theoretical (experimental) magnification factor of the pitch of the periodic object, ranging from 1.87 (1.91) to 1.2 (1.23) due to the current experimental geometry, results in a magnification of Talbot length from 3.51 (3.67) to 1.44 (1.52) at the pump wavelength. Similarly, the upconverted Talbot length (indicated by red dots) decreases from 30.8 cm to 12.4 cm for the same MLA separation. Both the pump and upconverted Talbot lengths exhibit an inverse square law dependence on the MLA separation, $d$, closely aligning with the theoretical predictions given by Eq. (3) (black line) and Eq. (5) (red line). This observation convincingly confirms the potential of the current technique for the simple control of Talbot lengths without altering the parameters of the periodic object. While our study focused on the MLA position, $d$, at a distance of 80 mm, further reduction in the $d$ value can increase the magnification factor of the pitch of the periodic object. It is essential to note that there is a lower limit for the $d$ value to ensure the Talbot effect by illuminating the object with a beam size much larger than the pitch of the periodic object. Since the increase in the Talbot length signifies the amplification of the periodicity of the object, this technique offers the potential to measure periodic objects with a small pitch.

 

Fig. 5. Variation of Talbot length at pump and upconversion wavelength as a function of distance, $d$ of MLA away from the back focal plane of the lens, $L3$. The black and red lines are the theoretical results derived using Eq. (3) and Eq. (5), respectively.

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Furthermore, it was reported previously [22] that the upconversion (here, SHG) process with very high efficiency is feasible by controlling the input pump intensity and employing a nonlinear crystal with a high figure-of-merit. While periodically poled crystals offer higher figure-of-merit, their low aperture poses a limitation. In our setup, we opted for a BiBO crystal, which, although possessing a moderately high figure-of-merit, features a larger clear aperture. To boost the pump intensity, we replaced lenses L3 and L4 with shorter focal length lenses, setting $f_{3}=f_{4}=25$ mm. With an average pump power of 3.47 W, our setup produced upconverted Talbot images with an average power of 0.351 W, achieving a single-pass nonlinear frequency upconversion efficiency as high as 2.91${\% } W^{-1}$. This efficiency represents a substantial improvement, exceeding the efficiency reported in the prior study [19] by a factor of $10^{6}$.

4. Conclusion

In summary, our study has successfully demonstrated single-pass second harmonic generation of the Talbot effect with notable conversion efficiency. Employing the microlens array as a two-dimensional periodic object illuminated with pump radiation at a wavelength of 1064 nm, we investigated the single-pass upconversion of the periodic object in a 1.2-mm-long BiBO crystal, confirming the doubling of the Talbot length in the upconversion (SHG) process. Through an experimental approach based on the Fourier transformation technique, we introduced a straightforward method to control and vary the Talbot lengths in both pump and upconverted wavelengths while accessing significantly enhanced nonlinear conversion efficiency. This adaptable technique enabled precise adjustments of the Talbot length for specific desired planes, allowing measurements of periodic objects with low pitch without the need for sophisticated microscopes. The highly efficient nonlinear upconversion of Talbot imaging demonstrated in this study holds promise for applications in Talbot lithography and other high-energy scenarios involving the Talbot effect.