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Abstract
Twist phase is a nontrivial second-order phase that only exists in a partially coherent field, providing a new degree of freedom for manipulating statistical properties of random light. However, in the existing experimental methods either it is difficult to control the twist phase or requires a large number of pseudo/random modes to approximately synthesize the twisted beams. In this work, we demonstrate a simple and efficient approach, based on the superposition of mutually orthogonal Laguerre-Gaussian modes with appropriate mode weights, to generate the twisted beams with a controllable twist phase. We show that a smaller number of modes are required to synthesize the twisted beams, compared to the pseudo-mode superposition, because the orthogonal coherent modes are used in the present method. We experimentally generate the twisted Gaussian Schell-model beams with controllable strength of twist phase and measure their degree of coherence and average intensity behavior during propagation. The experimental results agree well with the theoretical predictions. Our approach will promote the application of the twisted partially coherent beams in optical imaging and free-space optical communications.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In 1993, Simon and Mukunda introduced the concept of twist phase into Gaussian Schell-model (GSM) beams, which has opened a door for partially coherent light fields manipulation [1]. Unlike usual quadratic phases, the twist phase depends on two non-separable spatial position vectors, i.e., it is expressed as $\textrm{exp}[{ - ik\mu ({{x_1}{y_2} - {x_2}{y_1}} )} ]$, where k is the wavenumber, $\mu $ is the twist factor, and $({x_1},{y_1})$ and $({x_2},{y_2})$ are two transverse spatial positions. The twist factor is bounded by $\mu \le 1/({k{\delta^2}} )$, where $\delta $ represents the transverse coherence width of the partially coherent beam. Hence, the twist phase only exists in the partially coherent beams and vanishes in their coherent counterparts, i.e., when $\delta \to \infty $, $\mu \to 0$. Up to now, a great deal of work has been devoted to studying the propagation characteristics of twisted beams passing through complex optical systems, random media, and optical cavities [2–16]. The most remarkable feature of the twist phase is that it will induce the beam carrying orbital angular momentum (OAM) [17–19] and is responsible for the beam spot rotation during propagation [1,20]. Owing to its intriguing feature, the twisted beams could increase both transverse and longitude ranges in trapping a dielectric Rayleigh particle [21]. When they are served as illuminations in optical imaging system, they could overcome Rayleigh diffraction limit [22]. In addition, the twisted beams also can reduce the turbulence-induced negative effects and enhance the visibility in ghost imaging system [23–26], which have potential applications in free-space optical communications and high-order correlation imaging. Recently, by interacting the twist phase with the vortex phase, it was found the OAM of light can be greatly enhanced [27].
Despite abundant theoretical research on twisted beams, only a few reports were devoted to their experimental generation [28–31]. The earliest method reported by Friberg and coauthors was the use of six cylindrical lenses (CLs) to transform an anisotropic GSM beam into a twisted GSM (TGSM) beam [28]. The main drawbacks of the proposed optical system are somewhat complicated and lack of flexibility. The generated twist phase depends not only on the parameters of the anisotropic GSM beam, but also on the focal lengths of the CLs. Although recently such optical system has been simplified by reducing the number of the CLs from six to three, it is still difficult to generate the twisted beams with adjustable twist phase [29]. Another alternative method is based on the optical modes superposition, i.e., the TGSM beam is considered as an incoherent superposition of a set of optical modes. The practical experimental system usually involves a spatial light modulator or a digital mirror device to generate these optical modes in time series, and the twisted beam is obtained by time averaging over the modes. Two different superposition approaches have been proposed, up to now. In the first approach [30], the optical modes are approximately obtained by the discretion of Gori’s integrals [31–33], known as pseudo-mode superposition. In the second approach [34], the random modes obtained by the complex random screen method [35] are used. Both approaches enable one to synthesize the TGSM beams with controllable twist phase by governing the spatial distribution and the modal weight of each mode.
Nevertheless, a large number of modes are required in the experiment to represent the TGSM beam accurately, which is time-consuming in the process of synthesis. This is because the orthogonal condition [36–38] for the optical coherent modes is released in the above two pseudo-mode superpositions. In this work, we experimentally demonstrate an efficient way for the generation of TGSM beams of controllable twist phase with the incoherent superposition of mutually orthogonal optical fully coherent modes. Compared to the above pseudo-mode superpositions, we show in our experiment that a smaller number of modes are required to accurately represent the TGSM beam, which improves the efficiency of twisted beam synthesis. Our findings open an avenue for the fast generation of the partially coherent sources with novel complex coherence structures and may find applications in coherence based optical imaging [39] and optical encryption [40].
2. Theory
In space-frequency domain, the second-order statistical properties of a TGSM source (at $z\; = \; 0$), propagating along $z$-axis, are characterized by the cross-spectral density (CSD) function specified at ${{\textbf r}_1}$ and ${{\textbf r}_2}$ as [1]
In general, one can represent the CSD function of a statistically stationary partially coherent source of any state of coherence in the form [36]
It is not an easy task to obtain the analytical eigenfunctions from Eq. (3), and therefore only a very limited partially coherent sources find their eigenmode superposition. Thanks to the pioneer work of Simon and Gori [1,41], the TGSM source can be represented as the incoherent superposition of mutually orthogonal Laguerre-Gaussian (LG) modes with appropriate eigenvalues, given by
To assess the accuracy of the LG modes superposition, we use the normalized eigenvalues, defined as ${{{{\bar{\lambda }}_{nm}} = {\lambda _{nm}}} / {{\lambda _{00}}}}$, to measure the convergence of the mode representation. The beam parameters are chosen as $\sigma \; $= 0.5 mm, $\delta \; $ = 0.28 mm, and $ \mu $ = 0.001 mm−1, which are the same as those used in the pseudo-mode superposition in [30]. Figure 1 shows the variation of the normalized eigenvalue with the mode indices m and n. As expected, ${\bar{\lambda }_{nm}}$ decrease with the increase of the indices n and |m|, implying that the modes with large mode indices have less contribution. In the experiment, the number of modes can be truncated by a setting cutoff value ${\bar{\lambda }_{cut}}$, i.e., the modes whose eigenvalues are smaller than ${\bar{\lambda }_{cut}}$ are omitted. To show the accuracy of the truncated mode series, the average intensity distributions $I({\boldsymbol {\rho} }) = W({\boldsymbol {\rho} },{\boldsymbol {\rho} })$ with three different cutoff values are shown in Fig. 2. For a better comparison, the average intensity distributions (black solid curves) calculated from the theoretical model shown in Eq. (1) are also given. It can be seen that when the cutoff value equals to 0.015, the average intensity distributions between the LG modes superposition and theoretical model match perfectly. Under this circumstance, the involved number of LG modes is 35 ($n\; = \; 0,\; m\; = \; - 1$ to 23 and $n\; = \; 1,\; m\; = \; 0$ to 9). It is worth noting that the number of LG modes depends on the coherence width and the twist factor. The smaller the coherence width or twist factor is, the larger the number of LG modes should be involved. In the following analysis, the cutoff value is fixed at 0.015.
Fig. 1. The distributions of the normalized eigenvalues of $\lambda _{nm} / \lambda _{00}$ of LG modes.
Fig. 2. Average intensity distributions (red dashed curves) of the TGSM beams using LG modes superposition with three different cutoff values. The black solid curves are the corresponding distributions calculated from theoretical models.
3. Generation of a TGSM beam via LG mode superposition: experiment
We now pay our attention to the optical system for the generation of a TGSM beam based on the LG modes superposition. A schematic for the experimental setup is shown in Fig. 3. A linear polarized Gaussian beam ($\lambda \; $= 632 nm) emitted from a He-Ne laser is expanded by a beam expander (BE) and reflected by a mirror, then goes towards a SLM (Holoeye LC2002) to modulate its amplitude and phase. The SLM has the number of $1024 \times 768$ pixels with each pixel size 18 μm ×18 μm and operates at transmission mode. As shown in Eq. (3), generation of high-quality LG modes and accuracy control of their eigenvalues are the key factors for synthesizing the TGSM beams. In our experiment, the holograms used to generate LG modes are first synthesized in advance and then saved to computer memory. Figure 4 shows the typical computer generated holograms (CGHs) which are used to generate different LG modes. Here, the 1D grating in CGHs is associated to separate the desired LG modes from the zero-order diffraction and other background noise. After the SLM, the diffraction light passes through a 4f optical system composed of lens L1 and L2. A circular aperture placed in the rear focal plane of L1 is used to block other unwanted diffraction orders (only allows the first order passes). In the rear focal plane of L2, the LG modes are generated. Figure 5 presents the experimental results for the intensity distributions of the LG modes with different mode orders. The LG mode with $m\; = \; n\; = \; 0$ reduces to the fundamental Gaussian mode [see in Fig. 5(a)]. The index m corresponds the number of intensity rings and the azimuthal index n is related to the radius of the rings [see in Figs. 5(b)–5(h)]. It can be seen that the intensity on the ring becomes more and more nonuniform for the large n. This is because the varying of the vortex phase with high topological charge n becomes fast in the azimuthal direction. The accuracy of the phase modulation decreases owing the finite pixel size. Nevertheless, this nonuniform intensity distributions of the LG modes has little effect on the subsequent generation of the TGSM beam.
Fig. 3. Experimental setup for generating TGSM beams. BE, beam expander; RM, reflecting mirror; SLM, spatial light modulator; L1, L2, and L3, thin lenses; PH, pinhole; CL, thin cylindrical lens; CCD, charge-couple device; PC1 and PC2, personal computers.
Fig. 5. Experimental results of the normalized intensity distributions of LG modes with different mode indices.
The synthesis of a TGSM beam is divided into three steps in experiment. First, the number of LG modes are determined according to the criterion (${\bar{\lambda }_{cut}} = 0.015$) discussed in Sec. 2.1. The holograms for the generation of the LG modes are then prepared. Second, the normalized eigenvalue of each mode is regarded as the appearance probability of its corresponding LG mode, and therefore, holograms play in sequence on the SLM screen according to their appearance probabilities. The SLM operates in such a way that at each time step the chronologically earliest hologram is removed from the SLM’s screen and replaced by a new hologram. At last, a CCD records the series of intensity distributions of the LG modes with the integration time being 15 ms. The average intensity distribution is obtained by averaging over the series of recorded intensity distributions. In the experiment, the beam parameters of the TGSM beam are set as $\sigma \; $= 0.5 mm, $\delta \; $ = 0.28 mm, and $ \mu \; = $ 0.001 mm−1. Under this circumstance, the number of involved LG modes is 35 when the cutoff value ${\bar{\lambda }_{cut}}$ is 0.015. And 350 holograms (10 times of the involved LG modes) based on their appearance probabilities play in cycle on the SLM. The refresh rate of the SLM is about 30 Hz, and the CCD takes about 15 s to capture 450 intensity distributions with the frames per second (fps) being 30.
Figure 6(a) shows the experimental result of the average intensity distribution of the generated TGSM beam in the source plane. The corresponding crossline ($y\; $= $ 0$, circular dots) for the average intensity is shown in Fig. 6(b). For the comparison, the calculated result (solid curve) is also plotted in Fig. 6(b). We find the beam profile at source is nearly of Gaussian distribution, which is consistent with that calculated from theoretical model. The experimental results of the modulus of the degree of coherence (DOC) and the corresponding crossline at $y $ = $ 0$ are shown in Figs. 6(c) and 6(d), respectively. As expected, the DOC pattern is also of Gaussian profile. In order to verify the generated source carrying the twist phase, a cylindrical lens (CL) is placed in the rear focal plane of L2, and the CCD detects the average intensity distributions of the beam at several propagation distances after the CL. It is known that the twist phase is responsible for the rotation of beam spot upon propagation if the circular symmetry of the beam profile is broken [29,34]. Figure 7 presents the experimental (second row) and the corresponding theoretical (first row) results for the intensity distributions at several propagation distances. We find the beam spots display clockwise rotation direction upon propagation. The experimental results agree well with the theoretical predictions.
Fig. 6. Theoretical calculations and experimental results of the normalized intensity distributions and the degree of coherence of the TGSM beam with $\sigma = 0.5 mm , \delta = 0.28 mm , \mathrm{\mu} = 0.001 m{m^{ - 1}}.$
Fig. 7. Theoretical (first row) and experimental (second row) results for the average intensity distributions of the TGSM beam passing through a CL and L3 at several propagation distances. The top left corner of each picture displays the rotation angle formed by the dashed red line and the vertical axis.
The above experimental results have shown that the LG modes superposition is an efficient method for the generation of the TGSM beams with high accuracy. One of the advantages of the method is that it is convenient to adjust the beam parameters including transverse coherence width and the twist factor without changing the experimental layout. One only need to prepare the LG modes’ CGHs and calculate the eigenvalue for each mode according to the TGSM beam’s parameters. To verify the flexibility, we generate two kinds of TGSM beams with transverse coherence width $\delta $ being 0.20 mm and 0.24 mm. The experimental results of the modulus of DOC are shown in Figs. 8(a)–8(d). The other parameters are chosen $\sigma \; $ = 0.5 mm and $ \mu \; = $ 0.001 mm−1. From the fitting of the experimental data [solid curves in Figs. 8(b) and 8(d)], it is found that the measured transverse coherence widths in Figs. 8(b) and 8(d) are 0.2019 mm and 0.2443 mm, respectively, which are very close to the prescribed values.
Fig. 8. Experiment results of the modulus of the degree of coherence. (a), (b) the results for the transverse coherence width are $\delta \; $ = 0.20 mm. (c), (d) the results for $\delta \; $ = 0.24 mm.
The twist phase is difficult to measure and visualize in the experiment, but it is closely related to the rotation angle of beam spot during propagation after passing through a CL. Hence, we measure the evolution of the beam spot during propagation to observe the different twist factors. The experimental results are shown in Fig. 9. The values of the twist factor in the first, second, and third rows are -0.0006 mm−1, 0.0006 mm−1, and 0.0002 mm−1, respectively. The other beam parameters are the same as those used in Fig. 6. The top left corners of subpicture show the measured rotation angles formed by the dashed red line and vertical axis, and the corresponding theoretical results are shown in the right top corners. One can see that when the twist factor is negative, the beam spot rotates in the anti-clockwise direction upon propagation, which is opposite to the rotation direction for the positive twist factor (see in the second row of Fig. 9). The measured rotation angles are reasonable consistent with the theoretical calculations. In the third row, it is found that when the twist factor becomes larger, the rotation speed of beam spot is faster than that for a smaller twist factor within the first $z\; $= $100$ mm propagation distance, whereas this situation is inversed after $z\; $ = 100 mm.
Fig. 9. Variation of beam spots of the TGSM beams passing through a cylindrical lens with the propagation distances. The twist factors in the first, second and third rows are −0.0006 mm−1, 0.0006 mm−1 and 0.0002 mm−1, respectively. The angle formed by the long axis of beam spot (red dashed line) and the vertical axis are shown in the top left corner (experimental results) and in the top right corner (theoretical results) in each subpicture.
4. Comparison with pseudo-mode superposition
In this section, we compare our LG-mode superposition with the pseudo-mode superposition reported in [30]. The two methods are similar, i.e., the representation of the CSD function as the incoherent superposition of spatially coherent modes. In pseudo-mode superposition, the CSD function of the TGSM beam represents as follows
The relation of the parameter a above with the beam’s spatial coherence width and twist factor is ${\delta ^{ - 2}} = a + {k^2}{\mu ^2}/({4a} )$. The modal weight $\gamma $ decays as the value of $|{{{\textbf u}_{mn}}} |$ increases. In the experiment, ${{\textbf u}_{mn}}$ is discretized by the following manner: ${u_{xm({yn} )}} = \left[ {\frac{{2({m(n )- 1} )}}{{N - 1}} - 1} \right]\sqrt {2({4{\sigma^2} + 1/a} )} $, when $\gamma $ is truncated by the value exp(-4).
To show the advantage of the LG-mode superposition under the same condition, we show in Fig. 10 the experimental results of the average intensity distributions with different mode number N for two methods. It is found that when the number of modes is larger than 200, the results of the two methods are almost the same. However, when the mode number $ N\; = \; 100$, the LG-mode superposition is well converged, and the beam profile is close to the theoretical result [see in Figs. 10(c1) and 10(c2)], while the intensity profile for the pseudo-mode superposition obviously deviates from the Gaussian profile. The results imply that the convergence speed of the coherent-mode superposition is much faster than that of the pseudo-mode superposition. The LG-mode superposition shows the superior to the pseudo-mode superposition in the situation when the high speed is required.
Fig. 10. Experimental results for the normalized average intensity distributions of the TGSM beams generated by coherent-mode superposition (left two panels) and pseudo-mode superposition (right two panels) with different mode numbers (a1)-(a4) N = 300, (b1)-(b4) N = 200, and (c1)-(c4) N = 100. The red circular dots are the experimental data at the crossline $ x = 0$ and the black curves are the corresponding theoretical results.
5. Summary
In conclusion, we have experimentally demonstrated a convenient and efficient method to generate high quality TGSM beams with the use of temporally randomized, properly truncated sequence of spatially coherent LG modes. The LG modes involved in the incoherent supposition are realized by the CGHs loaded on the SLM and their corresponding eigenvalues depend on the appearance probabilities of the CGHs. Our experimental results agree well with the theoretical results. In contrast to the conventional optical system using cylindrical lenses [28,29], the LG mode superposition enables us to generate the TGSM beams with dynamic controllable beam parameters (e.g., spatial coherence width and twist factor) without changing the experimental layout. In addition, because the mutually orthogonal modes are used, the LG mode superposition involves less modes to accurately represent the TGSM beam, compared to other pseudo-mode superposition method [30,34], which could save time in the process of the beam synthesis. Our method for the fast synthesis of twisted beam may find uses in the applications where the fast coherence modulation speed is needed.
Funding
National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11774251, 11874046, 11904247, 11974218, 62075149, 91750201); Local Science and Technology Development Project of the Central Government (YDZX20203700001766); Innovation Group of Jinan (2018GXRC010); Priority Academic Program Development of Jiangsu Higher Education Institutions; Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX21_2935); Natural Science Foundation of Jiangsu Province (BK20201406).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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