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Abstract
We report the generation of prescribed optical orbital angular momentum (OAM) spectrum with spiral complex field modulation. Both symmetric and asymmetric OAM spectrums are generated with a vector beam generator and measured with a hybrid conformal mapper. Three methods for OAM spectrum generation ranging from the pure spiral phase modulation, the spiral amplitude and phase modulation, to the spiral phase and polarization modulation are demonstrated. The OAM spectrum generation with spiral complex field modulation may find applications in optical trapping and high-speed data communication.
© 2017 Optical Society of America
1. Introduction
Light with helical phase fronts e(ilφ) carries orbital angular momentum (OAM) of where l can take any integer value [1,2]. The intrinsic nature of dynamic rotation and unbounded state space has empowered the twisted light to find application in diverse disciplines of physics and engineering ranging from optical tweezers [3], edge-enhanced microscopy [4], and high-bandwidth data transmission [5], to quantum optics [6,7].
The generation of a prescribed OAM spectrum (simultaneous generation of multiple OAM states) is useful for constructing three-dimensional optical trapped structures [8], building size selective optical tweezers [9], and multiplexing beams for high-speed optical communication [10]. Conventional methods for the generation of OAM spectrum encompass collinear superposition of two or more helical beams [9], intentional misalignment of a single-OAM-state beam [11], and phase modulation including alternating phase jumps [8] and azimuthal phase modulation in a more general way [12].
In the current paper, we propose and experimentally demonstrate the generation of a prescribed OAM spectrum with spiral complex field modulation. The generated OAM spectrum comprises multiple OAM states of various intensities, phases and even polarization states. The simultaneous generation of arbitrary OAM states may spur numerous applications that harness the intriguing properties of the twisted light.
2. Principles
The OAM state space consists of orthogonal and unbounded number of OAM states. Consequently, an arbitrary complex optical field can be decomposed to the summation of multiple or an infinite number of OAM states (OAM spectrum). The decomposition can be evaluated by the overlap integral of the arbitrary complex optical field with a range of OAM states [13,14]:
where Ea is the arbitrary complex optical field, El is the OAM state of index l, A is the size of the beam and * denotes the complex conjugate operation. A bar chart of all ηl renders the OAM spectrum that can be generated by the spiral complex field Ea.2.1 Pure spiral phase modulation
The spiral phase modulation is the most facile kind of spiral complex field modulation. It is efficient and easily implemented especially when the number of OAM states is small. The spiral phase modulation is capable of obtaining an OAM spectrum with selective intensity distributions. Several algorithms have been reported to derive the required phase to generate the desired OAM states [12,15–17]. Here we design a quickly converged algorithm shown in Fig. 1 to find the phase modulation pattern. The initial optical field E0 is uniform in amplitude and random in phase. The overlap integral of the initial field with selected OAM states are calculated [18–20].
This complex coefficient is different from the ηl in Eq. (1) that represents the fraction of power that resides in OAM order l. The initial optical field is then expressed as the sum of the selected OAM states with individual complex coefficients. The amplitude of the coefficients al is replaced with the desired values a’l and the phase of the coefficients bl is retained.The amplitude of the summation is expressed as A(x,y) and dropped out. The phase of the summation is expressed as ϕ(x,y) and utilized as the phase of the new optical field exp(iϕ(x,y)). After iterations, the phase ϕ(x,y) is stabilized and the OAM spectrum generated by the optical field exp(iϕ(x,y)) approaches closely to the desired OAM spectrum.
Fig. 1 The algorithm to find the pure spiral phase modulation pattern for the OAM spectrum generation.
We derive two phase modulation patterns to generate the OAM spectrum with coefficients (η-1:η0:η1 = 1:1:1) and the OAM spectrum with coefficients (η-1:η0:η1 = 2:1.5:1), respectively. Figures 2(a) and 2(b) show the phase modulation patterns to generate the OAM spectrum shown in Fig. 2(c). The power ratio of OAM orders η-1:η0:η1 = 1:1:1. Besides the three desired orders (−1, 0, 1), a small fraction of power is diverted to orders (−3, −2, 2, 3). Figures 2(d) and 2(e) show the phase modulation patterns to generate the OAM spectrum shown in Fig. 2(f). The power ratio of OAM orders η-1:η0:η1 = 2:1.5:1. Again, it is unavoidable to generate undesired OAM orders with pure spiral phase modulation. The pure spiral phase modulation can be seen as a counterpart of a fan-out pure phase grating. Those phase gratings have been rigorously demonstrated to have an upper limit in efficiency. In other words, a fraction of power will inevitably be distributed to undesired orders.
Fig. 2 (a)(b) The spiral phase modulation patterns to generate the OAM spectrum (η-1:η0:η1 = 1:1:1) shown in (c); (d)(e) The spiral phase modulation patterns to generate the OAM spectrum (η-1:η0:η1 = 2:1.5:1) shown in (f).
2.2 Spiral amplitude and phase modulation
The pure spiral phase modulation has its own limitations. With an increasing number of desired orders, it becomes more difficult to obtain a converged result and the actual OAM spectrum deviates further from the desired OAM spectrum. The power ratio of desired orders becomes inaccurate and the undesired orders reduce the efficiency and the signal-to-noise ratio.
The spiral complex field modulation is not limited to pure phase modulation but can be extended to include the spatial modulation of the optical field in amplitude, phase and polarization state. We will first consider the case with the spiral amplitude and phase modulation. Suppose the amplitude and relative phase of a desired OAM order l are denoted as a’l and bl respectively, the complex optical field to generate such an OAM spectrum can be expressed by Eq. (3). A(x,y) is the amplitude of the optical field and ϕ(x,y) is the phase of the optical field. No optimization algorithm is required to calculate the field distribution because this is a time reversal problem. Unlike the pure spiral phase modulation where A(x,y) must approach unity after iterations, the spiral amplitude and phase modulation allows A(x,y) to be space-variant. The amplitude modulation A(x,y) is easily obtained from the summation in Eq. (3). Figure 3(a) and 3(b) show the amplitude modulation patterns and Fig. 3(c) and 3(d) show the phase modulation patterns to generate the OAM spectrum (η-1:η0:η1 = 2:1.5:1) shown in Fig. 3(e). Figure 4(a) and 4(b) display the amplitude modulation patterns and Fig. 4(c) and 4(d) display the phase modulation patterns to generate the OAM spectrum (η-2:η-1:η0:η1:η2 = 1:1.5:2.5:3:1.5) as shown in Fig. 4(e).
Fig. 3 (a)(b) The spiral amplitude modulation patterns and (c)(d) the spiral phase modulation patterns to generate the OAM spectrum (η-1:η0:η1 = 2:1.5:1) shown in (e).
Fig. 4 (a)(b) The spiral amplitude modulation patterns and (c)(d) the spiral phase modulation patterns to generate the OAM spectrum (η-2:η-1:η0:η1:η2 = 1:1.5:2.5:3:1.5) shown in (e).
The spiral amplitude and phase modulation has several advantages compared to the pure spiral phase modulation. It does not need iterative optimization process and has no optimization convergence problem. It is capable of controlling not only the power ratio of individual OAM orders but also their relative phase. The generated OAM spectrum has no undesired orders and the intensity and phase of the desired OAM orders are accurate. The drawback of a lowered efficiency becomes less significant when the number of desired OAM orders in the spectrum is large.
2.3 Spiral phase and polarization modulation
A more advanced way to generate an OAM spectrum with high efficiency is to utilize the spiral phase and polarization modulation. The modulation can be decomposed into two individual modulations on the x- and y-polarization components respectively. The equivalent amplitude modulation is achieved by exchanging power between the two orthogonal polarizations and thus there is no power loss from the modulation. The x-polarization component of the complex field modulation pattern is expressed as and it is the summation of the x-polarization components of the (2M + 1) desired OAM orders (Eq. (4); the y-polarization component of the complex field modulation pattern is expressed as and it is the summation of the y-polarization components of the (2M + 1) desired OAM orders (Eq. (5).
The parameters θl regulate the power distribution between the x and y components of the OAM order l. The parameters θl, blx, and bly are used as the optimization parameters to obtain the overall amplitude and phase distributions Ax(x,y), Ay(x,y), ϕx(x,y), and ϕy(x,y). The target of the optimization is:
The requirement expressed by Eq. (6) ensures that the spiral complex field modulation include only the spiral phase and polarization modulation.The simulated annealing algorithm is exploited to derive the complex optical field for tailoring an OAM spectrum [21–23]. Figure 5(a)-5(c) characterize the spiral complex optical field to generate the OAM spectrum shown in Fig. 5(f). The common phase ϕx(φ), the amplitude ratio Ay(φ)/Ax(φ), and the retardation ϕy(φ)- ϕx(φ) as a function of the azimuth angle are plotted in Fig. 5(a)-5(c) respectively. Figure 5(d) shows the two-dimensional common phase pattern ϕx(x,y). Figure 5(e) shows the polarization map. The left-hand elliptical polarizations are indicated in red and the right-hand elliptical polarizations are plotted in blue. In Fig. 5(f), the OAM spectrum has three orders of identical intensity and various linear polarization angles of 0, π/4, and π/2. No undesired orders exist in the OAM spectrum.
Fig. 5 The spiral phase and polarization modulation and the generated OAM spectrum. (a) The spiral common phase ϕx(φ). (b) The spiral amplitude ratio Ay(φ)/Ax(φ). (c) The spiral retardation ϕy(φ)- ϕx(φ). (d) The spiral common phase ϕx(x,y). (e) The polarization map. (f) The generated OAM spectrum (η-1:η0:η1 = 1:1:1). All OAM orders are linearly polarized with polarization angles of 0, π/4, and π/2 respectively indicated by double-end arrows.
3. Experimental demonstration
To demonstrate the tailoring of OAM spectrum with spiral complex field modulation, we set up an optical system that consists of a complex field generator and a hybrid conformal mapper as the OAM spectrum detector shown in Fig. 6.
Fig. 6 Experimental setup for demonstrating the OAM spectrum generation with spiral complex optical field. Section 1 and 2 are used to generate the required complex field modulation. Section 3 and 4 work as a hybrid conformal mapper to measure the OAM spectrum.
The laser source is an x-polarized He-Ne laser with a wavelength of 632.8 nm. The light is reflected by mirror M1 and a beam splitter, and hits section 1 of a spatial light modulator (SLM). A piece of opaque paper sandwiched between the beam splitters prevents the light beam from propagating in undesired directions. The phase of the light beam is modulated by section 1 of the SLM. After the phase modulation, the light beam is relayed by a folded 4f system (lens L1 and mirror M2) and transmits through another beam splitter and a quarter wave plate. The beam is then reflected by section 2 of the SLM and transmits through the quarter wave plate again. The fast axis of the quarter wave plate is aligned 45 degrees to the x-axis. The linear polarization azimuth angle of the light beam can thus be controlled by the phase of section 2 on a pixel by pixel level [24]. The linear polarizer P1 is aligned along the x-direction. Consequently, the amplitude of the light beam after the polarizer P1 can be modulated by the phase of section 2. Section 1 and 2 are utilized to generate the needed spiral phase and amplitude modulation. After the modulation, the beam is relayed by a 4f system (lens L2 and L3) to section 3 of the SLM. The spatial filter SF1 is used to filter out high-order diffracted light. Section 3 and 4 work as a high-resolution OAM sorter [25]. The phase pattern rendered by section 3 is a quadratic fan-out mapper that simultaneously accomplishes the log-polar transformation, fan-out beam copying and beam focusing. The phase pattern is mathematically expressed as:
The light beam from section 3 is redirected by a right-angle prism and propagates to section 4. The lens term functions as a physical lens and its focal length f is equal to the optical path length between section 3 and 4 so that the light beam from section 3 completes the Fraunhofer diffraction when it propagates to section 4.
The log-polar transformation term transforms the ring-shape intensity distribution of an OAM state to a straight line when the beam propagates to the Fourier plane [26,27]. Different OAM states are sorted into a set of parallel lines. The size and location of the transformed beam in the Fourier plane are regulated by the parameters d and p respectively. The fan-out term makes several copies of the unwrapped beam and placed them side by side to increase the sorting resolution [28,29].
The SLM panels used in the experimental demonstration are manufactured by Holoeye Photonics with a resolution of 1920 x 1080, a pixel pitch of 8 μm, and a fill factor of 87%. Since each SLM is utilized as two separate sections, the resolution of the area that can be utilized by the light beam is 960 x 1080. For the experimental demonstration, the parameters d, p and f are 0.1592mm, 1mm, and 172mm respectively. The fan-out term generates three copies of the unwrapped beam to triple the sorting resolution with bm = (1.329, 1, 1.329) and am = (-π/2, 0, -π/2) where m = −1, 0, and 1.
The phase pattern displayed on section 4 is a dual-phase corrector designed for removing the distorted phase brought by the conformal mapping as well as the abrupt phase jumps introduced from the beam copying process. The phase profile of the dual-phase corrector is given by:
where rect(x) ≡ 1 for |x| < 1/2 and 0 otherwise. The dual-phase corrector is a summation of (2N + 1) segments where each segment corrects the distorted phase of the corresponding copy of the unwrapped beam under the condition that the parameter L satisfies L = θf/(2π) [28,29]. The parameter ϕbc(m) removes the abrupt phase jumps between the neighboring copies of the unwrapped beam to ensure a smooth tilted plane wavefront. For the experimental demonstration, the compensating phases ϕbc(m) for the three copies are (0, 3π/2, 0), where m = −1, 0 and 1. After the phase correction, the light beam is relayed by another 4f system (lens L4 and L5) and focused by lens L6 to be detected by a camera.Figure 7 and Fig. 8 show the experimental results for the generation of OAM spectrum η-1:η0:η1 = 1:1:1 and η-1:η0:η1 = 2:1.5:1 with pure spiral phase modulation respectively. Experimental results are η-1:η0:η1 = 1:1.02:1.05 and η-1:η0:η1 = 1.94:1.46:1. The generation of the OAM spectrum η-1:η0:η1 = 2:1.5:1 with the spiral amplitude and phase modulation is shown in Fig. 9. As stated earlier, the advantage of the spiral amplitude and phase modulation is that theoretically the generated OAM spectrum has no undesired orders and the relative phases among these desired orders are also controllable. Figure 10 shows another example of the spiral amplitude and phase modulation. The generated OAM spectrum is η-2:η-1:η0:η1:η2 = 1:1.52:2.58:3.10:1.46.
Fig. 7 Generation of OAM spectrum (η-1:η0:η1 = 1:1:1) with spiral phase modulation. The experimental result is η-1:η0:η1 = 1:1.02:1.05.
Fig. 8 Generation of OAM spectrum (η-1:η0:η1 = 2:1.5:1) with spiral phase modulation. The experimental result is η-1:η0:η1 = 1.94:1.46:1.
Fig. 9 Generation of OAM spectrum (η-1:η0:η1 = 2:1.5:1) with spiral amplitude and phase modulation. The experimental result is η-1:η0:η1 = 1.95:1.51:1.
Fig. 10 Generation of OAM spectrum (η-2:η-1:η0:η1:η2 = 1:1.5:2.5:3:1.5) with spiral amplitude and phase modulation. The experimental result is η-2:η-1:η0:η1:η2 = 1:1.52:2.58:3.10:1.46.
The spiral complex field modulation is achieved with the use of SLMs in the paper. Therefore, the physical limit is determined by the specs of the SLMs such as resolution, panel size, fill factor, calibration, etc. As the number of desired OAM orders becomes large, the modulation patterns contain higher frequency components which could lead to aliasing if the resolution of the SLMs is insufficient.
The current experimental setup is incapable of demonstrating the OAM spectrum generation with spiral phase and polarization modulation shown in Fig. 5. Since the SLM is only responsive to the x-polarization component of the optical field, it cannot be utilized as a conformal mapper for both x- and y-polarization components simultaneously. If the hybrid conformal mapper is fabricated on refractive materials that are not polarization-sensitive, the experimental demonstration of OAM spectrum generation with spiral phase and polarization modulation can be accomplished.
4. Conclusions
In conclusions, we report the generation of OAM spectrum with spiral complex field modulation covering the pure spiral phase modulation, the spiral amplitude and phase modulation, and the spiral phase and polarization modulation. The pure spiral phase modulation requires an optimization algorithm to obtain the phase profile and it is increasingly difficult to maintain a high efficiency and accuracy when the number of OAM orders is large. The spiral amplitude and phase modulation requires no optimization. Not only the individual intensity but also the relative phases of the OAM orders among the spectrum can be accurately controlled. Experimental results demonstrate the validity of the two modulation schemes. The spiral phase and polarization modulation is the most efficient, most complex to implement, and capable of controlling the amplitude, phase and polarization of the individual OAM orders of the generated OAM spectrum. In the current paper, we have not been able to experimentally demonstrate the spiral phase and polarization modulation due to the polarization-sensitive nature of the SLM. Replacing the SLM-based OAM sorter with polarization insensitive OAM detection techniques can solve the problem.
Funding
National Natural Science Foundation of China (NSFC) (61505062, 91438108).
Acknowledgments
Chenhao Wan and Jian Chen have been supported by the China Scholarship Council (CSC).
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