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Abstract
We propose a generalized spiral transformation scheme that is versatile to incorporate various types of spirals such as the Archimedean spiral and the Fermat spiral. Taking advantage of the equidistant feature, we choose the Archimedean spiral mapping and demonstrate its application in high-resolution optical orbital angular momentum (OAM) mode sorting. Experimental results show 90% efficiency and cross-talk of -8.78 dB that is sufficient to separate adjacent OAM modes. This generalized transformation scheme may also find various applications in optical transformation and can be easily extended to other fields related to conformal mapping.
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1. Introduction
Vortex is a natural feature existing in a variety of physical phenomena spanning from spin of the Galaxy to rotation of molecules. It has been widely known that light can carry a spin angular momentum(SAM) of ${\pm} \hbar $ per photon [1,2]. In 1992, Allen et al. pointed out that a vortex beam with a spiral phase structure in the form of ${e^{il\mathrm{\varphi }}}$, can carry an orbital angular momentum(OAM) of $l\hbar $ per photon, where $\varphi $ is the azimuthal angle in the cross section and the integer l is the topological charge [3]. Being theoretically infinite in the state space and mutually orthogonal, these OAM states of light provides a new dimension and boosts the capacity for optical communication in both classical [4–6] and quantum [7,8] fields.
A number of methods have been demonstrated to generate light with different OAM states for optical communication, such as spiral phase plates [9,10], forked grating holograms [11], q-plates [12], mode conversion [13,14], and spin-orbit coupling [15]. On the receiver side, OAM mode sorter plays a critical role in OAM-based optical communication. One straightforward method is using a forked grating hologram [16]. As one hologram can measure only one specific state at a time, the efficiency of this method is restricted to 1/N, where N is number of the OAM states. The Mach-Zehnder (M-Z) interferometer OAM sorter is capable of sorting OAM modes at the level of a single photon. However, N different states requires a complex system of cascading N-1 interferometers. An OAM sorter based on the log-polar coordinate transformation circumvents such complexity gracefully and measures all OAM states simultaneously [17]. Unfortunately, the overlapping between adjacent states barely meets the Rayleigh criterion, deteriorating the signal-to-noise ratio (SNR) of neighboring channels. Incorporating an additional fan-out element can greatly decrease the overlapping and substantially increase the SNR [18]. The scheme can be integrated to a compact form for practical use [19]. Another solution to tackle the crosstalk problem is to apply the logarithmic spiral mapping technique [20]. The number of turns of the logarithmic spiral determines the multiples by which the resolution can be increased.
The logarithmic spiral is an equiangular spiral, the outer spirals are increasingly and significantly wider than the inner spirals. Given an annular intensity distribution and a minimum spiral width, the number of turns of logarithmic spirals are thus quite limited as shown in Fig. 1(a). On the contrary, the Archimedes spiral, of equidistant feature, has equal width for every spiral turn. Consequently, given the same annular intensity distribution and the same minimum spiral width, the number of turns can be increased noticeably as shown in Fig. 1(b). Figure 2 depicts a vortex beam transformed by the Archimedes spiral mapping.
Fig. 1. Vortex beam decomposed by (a) logarithmic spiral and (b) Archimedes spiral. The number represents the turns of the spiral.
In this work, inspired by the previous work on spiral mapping [20,21], we propose and demonstrate a generalized optical transformation scheme that can utilize various types of spiral mapping for sorting OAM states including the Archimedes spiral, the Fermat spiral, etc. Taking advantage of the equidistant feature, we present detailed theoretical analysis, numerical simulation, and experimental demonstration of OAM sorting based on the Archimedes spiral mapping. Experimental results show 90% efficiency and cross-talk of -8.78 dB that is sufficient to separate adjacent OAM modes. The efficiency and cross-talk can be further improved using high-resolution and large-area phase elements. The generalized transformation scheme can also be extended to other research fields based on conformal mapping.
2. Theory
The optical mapping between two planes under paraxial approximation is expressed in terms of the phase gradient by the following equations [22]:
where $(x,y)$ is the Cartesian coordinates of the input plane, $(u,v)$ is Cartesian coordinates of the output plane, $Q(x,y)$ is the phase distribution in the input plane, ${Q_x}$ and ${Q_y}$ are the partial derivatives with respect to the corresponding variables, k is the free-space wave number, and d is distance between the put plane and the output plane. New coordinates $(s,\theta )$ are introduced to replace the Cartesian coordinates $(x,y)$ in the input plane: where $(r,\varphi )$ are the polar coordinates of the input plane and m is an integer indicating the number of spiral turns. The input plane is decomposed to a set of spirals of infinite length and finite width. A point located at $(x,y)$ can be represented by a parameter s that indicates a particular spiral line that goes through the point and a parameter θ that indicates the unwrapped azimuthal angle along the spiral line. Since a spiral is of infinite length, the upper bound of the parameter θ goes to infinity. In other words, the range of angle extends from $[{0,2\pi } )$ to $[{0, + \infty } )$. The type of spirals is governed by the equation $r = r(s,\theta )$, e.g., an Archimedean spiral is expressed as $r = s + {a_{_1}}\theta $, where ${a_1} > 0$.The underlying principle for OAM sorting based on spiral mapping is to convert the spiral phase of OAM states into a linear phase that can be later spatially separated by a simple lens. That is to say, the points with the same azimuthal angle $\varphi $ in the input plane must be mapped to the points with the same coordinate u in the output plane:
which can also be expressed by the spiral coordinatesThe coordinate v depends both on the parameter s and $\theta $.
To obtain one possible solution of the transformation phase function $Q(x,y)$, we assume that the phase function has a continuous second-order partial derivative, i.e., ${Q_{xy}} = {Q_{yx}}$. From Eq. (1), it is obtained that ${u_y} = {v_x}$. Combining with Eq. (4), the following equation can be derived,
Using the relationship between the Cartesian coordinates and the spiral coordinates (Eq. (2)), Eq. (5) can be rewritten as follows,
Equation (6) is a condition that must be satisfied throughout the entire input plane. For simplicity, we set the expression in the parentheses equal to zero:
From Eq. (4), it is seen that the coordinate u depends only on the angle $\theta $, which means that $\frac{{du(\theta )}}{{d\theta }} = f(\theta )$. Therefore, it is convenient to assume that $u^{\prime}(\theta ) = a$, i.e., $u = a\theta $, where a is a constant that controls the size of the transformed beam. Using the above assumptions, the following relationship can be obtained,
For a spiral expressed in the form of $r = r(s,\theta )$, Eq. (8) can be integrated to yield
where $\theta $ is the angle along the spiral line, r is the distance between the point on the spiral line and the origin, and b is the parameter that controls the longitudinal displacement of the transformed beam. Thus, the mapping condition that applies to various types of spirals is derived. As an anti-holomorphicity conformal mapping, this mapping must follow the Cauchy-Riemann conditions [23]Through introducing the complex variables $Z = x + iy$ and $W = v + iu$ [20], the conformal mapping between the input plane (Z) and output plane (W) can be expressed based on Eq. (9),
where m is the number of spiral turns that is related to a specific spiral type. If m is set to 0, the spiral coordinate will degenerate into an ordinary Cartesian coordinate, and the mapping will reduce to the traditional log-polar mapping.The phase function can be obtained based on Eq. (1),
In the output plane, another phase function should be applied to recollimate the beam. This phase function, generally known as the corrector phase, is expressed as [22],
where $(u,v)$ are the Cartesian coordinates in the output plane. A lens is placed after the phase corrector to spatially separate light beams with different linear phase. The position of the focal spot depends on the gradient of the linear phase, or the topological charge l of the original spiral phase, where f is the focal length and $\lambda $ is the wavelength.3. Numerical simulation and experimental results
A typical example of the transformation phase function and corrector phase function for the Archimedean spiral transformation are plotted in Fig. 3(a) and Fig. 3(b), respectively. It is obvious that there exists discontinuity in the transformation phase, which is caused by the introduction of the spiral coordinates. Consequently, there will be a minimum spiral width requirement in order to properly apply the optical transformation. The optical setup is shown in Fig. 3(c). A He-Ne laser at 633 nm is used as the light source. A polarizer is utilized to align the direction of linear polarization of the light with the horizontal direction because the Spatial Light Modulator (SLM) is responsive only to the horizontal direction. A spatial filter, composed of two lenses (L1 and L2) and a pinhole, filters out stray light from the laser. In this experiment, two SLMs are used. The first SLM is divided into two halves, the left half is used for generating the superposition of OAM states and the right half is used to display the transformation phase $Q(x,y)$. A small piece of black paper is placed between BS1 and BS2 to block the light from transmitting through. The second SLM is used to project the correction phase $P(u,v)$ . Finally, the transformed beams with different linear phase gradient are focused by a lens to different horizontal positions on the CCD camera.
Fig. 3. Profile of (a) the transformation phase and (b) the corrector phase. (c) Schematic overview of the setup. P1, Polarizer; L1, L2 and L3, Fourier lens; BS1, BS2 and BS3, Non-polarizing beam splitter; RAP, Right-angle prism; SLM1 and SLM2, Spatial light modulator. (a.1) and (b.1) are the details of the transformation phase and corrector phase respectively. The colorbar indicates the phase range in Fig. 3(a) and Fig. 3(b).
In this experiment, L1 and L2 are convex lenses with a focal length of 100 mm. L3 is a virtual lens of a focal length of 111.1 mm implemented by SLM2. The SLMs are Holoeye PLUTO-NIR-011 with a resolution of 1920(H)×1080(V) and pixel size of 8 µm×8 µm.
In this work, the minimum spiral width is setting equal to the radius of the first turn, that is ${r_0}$. The parameters being used are $d = 202.5\textrm{mm}$, $2\pi a = 2\textrm{mm}$, ${r_0} = 0.183\textrm{mm}$, ${a_1} = \frac{{0.183}}{{2\pi }}\textrm{mm}$ for the Archimedean spiral mapping. Figure 4 shows the numerical and experimental results of the separated OAM modes with different topological charges ($- 2 \le l \le 2$) using the Archimedean spiral transformation [Figs.4(a) and 4(b)]. The area of output plane has been divided into parallelogram regions of interest. The width of these regions is determined by Eq. (16) and given by $\Delta u = \frac{{\lambda f}}{{2\pi a}}$. The height of these regions is defined by the full width at half maxima (FWHM) of the intensity in the u direction. Furthermore, the slope of the parallelogram region is calculated by Eq. (9) and given by,
Fig. 4. Separated OAM modes with different topological charges ($- 2 \le l \le 2$). (a) numerical simulations and (b) experiments for the Archimedean spiral transformation. The colorbar indicates the intensity of different OAM modes.
Figure 5 depicts the efficiency map for vortex beams of different topological charge. The efficiency is defined by the total energy in the corresponding region divided by the total energy in all regions of the camera. Both numerical simulation and experimental results exhibit a good diagonal response, with an average efficiency of approximate 90%. The vortex beam used for numerical simulation is a perfect vortex beam with a fixed ring diameter [24].
Fig. 5. Efficiency map for vortex beams with different topological charges ($- 2 \le l \le 2$). (a) numerical simulations and (b) experiments for the Archimedean spiral transformation. The colorbar indicates the efficiency of different vortex beams.
To quantify the cross-talk of relative modal power, we use the definition introduced in [25]. The cross-talk XT of the channel corresponding to a selected value $l = {l^ \ast }$ is given by
4. Discussions and conclusions
The experiment results demonstrate the high resolution of the proposed method in this paper. The comparison of the Archimedes spiral transformation with the logarithmic spiral transformation is shown in Figs. 6 and 7. The parameters used for the logarithmic spiral transformation are $d = 202.5\textrm{mm}$, $2\pi a = 2\textrm{mm}$, $2\pi a = 2\textrm{mm}$ ${r_0} = 0.183\textrm{mm}$, $2\pi {a_1} = \log (2)$. These parameters ensure that the logarithmic spiral has the same minimum spiral width as the Archimedes spiral.
Fig. 6. Separated OAM modes with different topological charges ($- 2 \le l \le 2$). Experiment results for (a) logarithmic spiral transformation and (b) Archimedean spiral transformation. The colorbar indicates the intensity of different OAM modes.
Fig. 7. Efficiency map for input OAM beams with different topological charges ($- 2 \le l \le 2$). Experiment results for (a) logarithmic spiral transformation and (b) Archimedean spiral transformation. The colorbar indicates the efficiency of different Vortex beams.
The average efficiency of logarithmic spiral transformation is about 90%, which is similar to the Archimedean spiral transformation. And the average cross-talk of logarithmic spiral transformation is -8.29 dB, which is slightly higher than the Archimedean spiral transformation. The current performance of the Archimedean spiral transformation is limited by the pixel size of the SLM being used. An Archimedean spiral with a minimum spiral width of 0.183 mm has only 16 pixels for each turn in the radial direction. Increasing the number of turns will require a SLM of higher pixel density or larger active area.
The optical finesse is defined as the separation of adjacent OAM states divided by the FWHM a specific OAM state along a reference line [19]. Setting the minimum spiral width to $\alpha {r_0}$, the width of the circular incident light to $\beta {r_0}$, the Archimedean spiral mapping will have a finesse of $\frac{{\frac{\beta }{\alpha } + 1}}{{0.8859}}$ and the logarithmic spiral mapping $\frac{{{{\log }_{1 + \alpha }}(1 + \beta ) + 1}}{{0.8859}}$. Figure 8 shows the optical finesse as a $\alpha $ function of the minimum spiral width . The Archimedean spiral mapping has a finesse $\frac{{\frac{\beta }{\alpha } + 1}}{{{{\log }_{1 + \alpha }}(1 + \beta ) + 1}}$ times better than that of the logarithmic spiral mapping. The minimum spiral width is limited by the resolution of phase elements and the beam size.
Fig. 8. Optical finesse as a function of the minimum spiral width α. The width of the circular incident light $\beta {r_0}$ is 18${r_0}$.
It is recognized that the beam splitters used in the system cause a significant energy loss. This energy loss can be alleviated by replacing the SLMs with diffractive optical elements (DOEs) and changing from reflective to transmissive system. Through integrating the two phase elements into one component, the intensity distortion caused by misalignment can be reduced and the system can be made more compact and integratable [26,27].
Except for the Archimedean spiral, the new transformation scheme is flexible in exploiting different types of spirals, such as the Fermat spiral, which is determined by the formula $r = s + {a_1}\sqrt \theta$, and the parameter m of this spiral can be expressed as
The method presented in this paper can also be utilized to sort high-order OAM beams with non-null radial number, for instance, high-order Bessel beams with different radial component ${k_r}$[28]. The Fourier transformation of a Bessel beam is a perfect vortex. The radius of a perfect vortex is related to its radial component ${k_r}$. Based on the mapping in this paper, beams with different radius will be mapped to different positions in the v direction, resulting in different positions of focused spot in the vertical direction in CCD. Therefore, high-order Bessel beams with same azimuthal mode index l and different radial component ${k_r}$ will be distinguished in the vertical direction in CCD. However, we must consider the limitations imposed by the minimum resolvable feature in CCD and the aperture of optical elements, which constrain the range of resolvable radial components ${k_r}$.
In conclusion, we have proposed and demonstrated a generalized optical transformation scheme that is suitable for various types of spiral mapping. In particular, we choose the Archimedean spiral mapping due to its equidistant feature and demonstrate its application in high-resolution OAM mode sorting. The soring resolution can be further increased using high-resolution and large-area phase elements. The generalized transformation method proposed in this paper lays a backbone for applications in various disciplines related to conformal mapping.
Funding
National Natural Science Foundation of China (61875245, 92050202); Wuhan Science and Technology Bureau (2020010601012169).
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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