## Abstract

We investigate output mode purities of Laguerre–Gaussian (LG) beams generated from four typical simultaneous amplitude and phase modulation methods with phase-only spatial light modulators (SLMs). Numerical simulations supposing the practical SLM, i.e., stepwise phase modulation with a pixelated device, predict an output mode purity of beyond 0.969 for the LG beams of less than radially and azimuthally fifth order. Experimental results of generating LG beams are also shown to demonstrate the effects of the simultaneous phase and amplitude modulation.

© 2008 Optical Society of America

Owing to the development of the phase modulation spatial light modulator (SLM), various higher-order light beams such as Laguerre–Gaussian (LG), Bessel, and Airy beams are now available via holographic methods [1, 2, 3, 4]. However, in spite of the merits of light utilization efficiency, flexibility, and controllability offered by the SLM, output beam quality is limited in the phase-only holographic beam generation, e.g., mode purities of holographically generated LG beams are at most $\sim 0.85$ [1, 2] for radially higher-order LG modes. To further improve the LG mode purity, it is necessary to control light’s amplitudes as well as phases, i.e., to control the complex amplitude [5, 6].

The phase-only complex-amplitude modulation scheme for low-resolution devices, such as SLM, originates from the report of Kirk and Jones [7]. However, Kirk’s method, which encodes slowly varying amplitude information by modulating a high-frequency phase carrier, was regarded as suitable for a high-resolution phase modulation device. Instead of Kirk’s method, there are various alternative approaches working with SLM [8, 9]. Above all, Arrizón *et al.* [9] also demonstrated LG beam generation to confirm that their methods work correctly with the SLM. However, a quantitative discussion of the output LG beam purity achieved by the phase-only complex-amplitude modulation is presently unavailable (to our knowledge).

In this Letter we investigate output LG mode purities achieved by the above four methods for complex-amplitude modulation [7, 8, 9] through numerical simulations reflecting the feature of a practical SLM device, i.e., stepwise phase modulation at each pixelated electrode. As a result, the mode purity can be beyond 0.969 for the LG modes of up to radially and azimuthally the fifth order, indicating more than 10% improvement of mode purity from that achieved by the phase-only modulation of a top-hat input beam. Experimentally generated LG beams using the SLM are also presented to demonstrate the effects of simultaneous phase and amplitude modulation.

We start from the definition of normalized complex amplitude ${u}_{p}^{l}$ for the LG mode of radially $p\text{th}$ and azimuthally $l\text{th}$ orders (referred to as ${\mathrm{LG}}_{p}^{l}$ mode in the following). Applying the paraxial and scalar wave approximations, ${u}_{p}^{l}$ is expressed as follows at the beam waist [1, 2] in the cylindrical coordinates $(r,\varphi ,z)$, where *z* is the coordinate for the beam propagation direction:

We briefly review four typical phase-only modulation methods for complex-amplitude modulation. These four methods commonly employ a high-frequency phase carrier. We choose a blazed phase grating (BPG) pattern as the phase carrier, a choice that can enhance light utilization efficiency and can separate output light from unmodulated zeroth-order light. The BPG is aligned repeatedly in the *x* direction on the SLM surface and specified by an integer value *d* denoting the pixel number involved in a unit BPG structure. Thus the BPG pattern ${\varphi}_{i}^{\mathrm{BG}}$ is expressed as follows with respect to the *x*-directional pixel position *i* on the SLM surface:

*i*and

*j*denote the pixel position in the

*x*and

*y*directions, respectively), is determined by

*ψ*and ${\varphi}^{\mathrm{BG}}$ in a different way for each phase modulation method. Here we adopt the notation $f(\Psi ;A)$ for the phase value

*Ψ*modulated with respect to a real amplitude

*A*through a specific relationship

*f*. Following this notation, ${\Phi}_{ij}$ is expressed as ${\Phi}_{ij}=f({\psi}_{ij}+{\varphi}_{i}^{\mathrm{BG}};{A}_{ij})$ in methods 1, 2, and 3, while ${\Phi}_{ij}={\psi}_{ij}+f({\varphi}_{i}^{\mathrm{BG}};{A}_{ij})$ in method 4. For methods 1 and 4,

*f*can be commonly given in terms of the function ${\mathrm{sinc}}^{-1}\left(x\right)$ as the inverse function of $\mathrm{sinc}\left(x\right)=\mathrm{sin}\left(\pi x\right)\u2215\left(\pi x\right)$, i.e., $f(\Psi ;A)=\Psi [1-{\mathrm{sinc}}^{-1}\left(A\right)]$, where

*A*is rescaled to satisfy $0\u2a7dA\u2a7d1$. Explicit expressions of

*f*for methods 2 and 3 can be found in [9]. In the practical experiments, the value of ${\Phi}_{ij}$ is wrapped into the interval of $[0,2\pi ]$ when displayed on the SLM. Figure 1 exhibits examples of holographic phase patterns for generating the ${\mathrm{LG}}_{2}^{1}$ beam via the four complex-amplitude modulation methods.

The mode purity of a holographically generated LG beam is calculated as an inner product between ${u}_{p}^{l}$ of a required LG mode and that of the holographic output. The holographic output is assumed to be given as a Fourier transformation of the complex-amplitude distribution on the SLM surface, where the complex-amplitude distribution is expressed as a two-dimensional array of $N\times N$ complex values. It is noted that the beam radius ${w}_{0}$ (in this Letter, ${w}_{0}=50\text{\hspace{0.17em} pixels}$) in Eq. (1) should be replaced with $N\u2215\left(2\pi {w}_{0}\right)$ to obtain ${u}_{p}^{l}$ of the required mode on the observing plane. Moreover, each pixel is expressed with $n\times n$ lattice points subdividing the pixel homogeneously in the *x* and *y* directions to take account of the pixelated electrode structure of the SLM. In this Letter, we choose $n=5$ and $N=\mathrm{20,000}$, meaning that we suppose an area of $\mathrm{4,000}\times \mathrm{4,000}\text{\hspace{0.17em} pixels}$. Practically, the SLM device has $792\times 600\text{\hspace{0.17em} pixels}$, and only the central circular regime of the radius of $290\text{\hspace{0.17em} pixels}$ is illuminated by a top-hat input beam. To reproduce this situation, we calculate the complex-amplitude distribution by multiplying $\mathrm{exp}\left(i{\Phi}_{ij}\right)$ to a top-hat amplitude profile, of which fringe is smoothed to present a ${\mathrm{cos}}^{2}$ profile. Finally, the output mode purity is calculated on $2500\times 2500$ points around the first-order diffraction direction.

Figure 2 exhibits the behavior of output mode purities $\left(\eta \right)$ varying the BPG’s pitch *d* for the ${\mathrm{LG}}_{2}^{1}$ beam generated via four methods. We notice from Fig. 2 the tendency that methods 1 and 2 have to achieve larger *η* for large *d*, while methods 3 and 4 do so for small *d*. We also note that method 4, which has been considered not to work with SLM, achieves a larger *η* for $d\u2a7d4$, although the decrease of *η* at larger *d* is more prominent for method 4 than for the other methods. These observations on *η* are also true for other LG beams of $p,l\u2a7d5$ except for minor differences in the behavior of *η*.

Table 1 lists the maximum *η* values for five typical LG beams achieved by four complex-amplitude modulation methods. We also show in Table 1 the maximum *η* values derived through the present calculation method assuming phase-only modulation of a top-hat input beam [2]. In Table 1, the maximum *η* tends to decrease as *p* and/or *l* increase; nevertheless, $\eta \u2a7e0.985$ can be achieved by methods 2, 3, and 4. Here we note that methods 3 and 4 achieve the maximum *η* for small *d* as shown in Fig. 2 and that BPG patterns of small *d* can separate the first-order light from the zeroth-order light with a larger angle. Thus, methods 3 and 4 are preferable for maximally avoiding the mixture of zeroth-order lights with first-order output lights, especially when generating widely distributed ${\mathrm{LG}}_{p}^{l}$ beams, i.e., those of larger *p* and *l*.

Figure 3 demonstrates the ${\mathrm{LG}}_{2}^{1}$ beam patterns via the phase-only [Fig. 3a] and the above four complex-amplitude modulation methods [Figs. 3b, 3c, 3d, 3e] with the SLM following the experimental procedures in [1, 2]. Each result in Figs. 3b, 3c, 3d, 3e was obtained under the BPG pitch *d* maximizing the output LG mode purity. Although the maximum output mode purities are similar for methods 2, 3, and 4 (Table 1), only Fig. 3c exhibits an irregular pattern that is not predicted from a numerical calculation. We consider that the deviation is introduced by nonideal properties of the practical SLM, such as “flyback region” [10], i.e., nonideal phase distribution owing to phase wrapping. We can also observe in Fig. 3b the typical property of method 1 that the output mode pattern suffers from polygonal deformation [9].

As stated in [2], it is generally difficult to evaluate the output LG mode purities from observed beam patterns with sufficient numerical precision to distinguish the small difference of *η* in Table 1. However, we can visually observe in Fig. 3 the effect of complex-amplitude modulation. Figure 3a exhibits an ${\mathrm{LG}}_{2}^{1}$ beam pattern generated through phase-only modulation of a top-hat input beam by optimizing the size of the holographic phase pattern with respect to the size of the input beam [1, 2]. The phase-only modulation scheme attaches the ideal phase profile of the LG mode to the output beam, although the amplitude profile of the output beam involves deviations from the ideal one. Resultantly, effects of the residual mode contents appear as multiple ring patterns in the surrounding area of the beam profile [Fig. 3a]. Contrarily, such rings disappear for complex-amplitude modulation methods [Figs. 3b, 3c, 3d, 3e], suggesting that the complex-amplitude modulation works properly to produce nearly ideal amplitude profiles.

In this Letter, we investigated mode purities of LG beams generated holographically applying phase-modulation procedures for encoding complex-amplitude distribution [7, 8, 9]. The mode purity was calculated as an inner product between the complex amplitude of the required LG mode and that of the holographic output beam derived as a Fourier transformation of a holographic phase pattern. As a result, the mode purity turned out to be beyond 0.969 for the LG modes of up to radially and azimuthally the fifth order, which is a more than 10% improvement compared to that achieved by the phase-only modulation of a top-hat input beam. Experimentally obtained beam patterns also demonstrated the effects of the complex amplitude modulation.

The authors are grateful to T. Hiruma, Y. Suzuki, T. Hara, and Y. Mizobuchi for their encouragement and also thank H. Toyoda and H. Itoh for helpful discussions and experimental support.

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