June 1, 2007 / Vol. 32, No. 11 / OPTICS LETTERS
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Universal generation of higher-order multiringed Laguerre–Gaussian beams by using a spatial light modulator Yoshiyuki Ohtake, Taro Ando,* Norihiro Fukuchi, Naoya Matsumoto, Haruyasu Ito , and Tsutomu Hara Hamamatsu Photonics K.K., Central Research Laboratory, Hirakuchi, Hamamatsu-City, 434-8601, Japan *Corresponding author:
[email protected] Received January 18, 2007; revised March 8, 2007; accepted March 9, 2007; posted March 14, 2007 (Doc. ID 79167); published April 27, 2007 Laguerre–Gaussian (LG) beams of various higher-order radial modes are generated by using a reflective phase-only liquid crystal on silicon (LCOS) spatial light modulator (SLM). Because of the LCOS SLM’s phase-modulation characteristic of a wide spatial bandwidth, a phase modulation scheme effectively generates higher-order LG beams of up to the fifth-order radial mode. We also perform correlation analyses between the observed and the theoretical two-dimensional mode profiles to universally obtain correlation coefficients of more than 0.946, which suggest mode generations of high quality. © 2007 Optical Society of America OCIS codes: 090.1760, 100.5090, 120.5060, 140.3300, 230.3720.
Since Allen et al. [1] pointed out that Laguerre– Gaussian (LG) beams carry orbital angular momenta, LG beams have attracted a great deal of attention in both optical and physical communities. Although there are radial and azimuthal indices assigning the modes of LG beams, so far only LG beams of higher-order azimuthal modes have been emphasized [2–4]. In contrast, reports on LG beams of higher-order radial modes, i.e., multiringed LG beams, have been few in number [5–10]. Nevertheless, the generation of multiringed LG beams is not only an issue of interest as a technique for controlling transverse light modes, but also is expected to provide a valuable tool for manipulating cold atoms [10,11]. In this Letter we report universal generation of higher-order multiringed LG beams with a reflective phase-only liquid crystal on silicon (LCOS) spatial light modulator (SLM) [12]. To obtain multiringed LG beams, rapid radial phase discontinuities must be placed at correct positions, making it difficult to generate multiringed LG beams, especially when the radial mode index becomes large. However, a LCOS SLM, which can manipulate light phases accurately with a wide spatial bandwidth, makes it possible to achieve accurate phase settings with sufficient quality for generating beams of complicated structure. Consequently, we obtain LG beams whose radial and azimuthal indices are up to 共p , l兲 = 共5 , 1兲. In the following, we denote an LG mode of radial and azimuthal mode indices 共p , l兲 as LGpl. To determine a phase pattern for generating the LGpl beam, we analyze an electric field amplitude upl of the LGpl beam propagating in the z direction and focusing at z = 0. In cylindrical coordinates 共r , , z兲, upl is expressed as the following under the scalar wave approximation [1]: 0146-9592/07/111411-3/$15.00
upl共r, ,z兲 = 共− 1兲p
冋
2
p!
共p + 兩l兩兲!
册
1/2
共冑2兲兩l兩 wz
exp共− 2兲
⫻ Lp兩l兩共22兲exp共− il兲exp共− i2z/zR兲 ⫻ exp关i共2p + 兩l兩 + 1兲tan−1共z/zR兲兴,
共1兲
where we introduce Rayleigh length zR and variable = r / wz with wz = 关2共z2 + zR2兲 / 共kzR兲兴1/2 as the beam radius at z (k is the wavenumber of the light). A radial profile of upl is influenced by a generalized Laguerre polynomial, Lp兩l兩共x兲: since equation Lp兩l兩共x兲 = 0 has p different positive roots, 兵xi其 共i = 1 , 2 , . . . , p兲, the LGpl beam has radial nodes at r0 = 0 and ri = wz共xi / 2兲1/2. Noting that Lp兩l兩共x兲 changes its sign as x passes over each xi, we arrive at a helical phase pattern 共r , 兲 with phase discontinuities at every ri to generate the LGpl beam by modulating a plane-wave input light, i.e.,
共r, 兲 = − l + 关− Lp兩l兩共2r2/w02兲兴
共2兲
with 共x兲 as a unit step function. In practical use, the phase pattern is restricted to the interval between 0 and 2. Figures 1(a) and 1(b) demonstrate phase patterns for LG33 and LG51, respectively. Figure 1(c) shows a schematic diagram of our experimental setup. Light emitted from a He–Ne laser is collimated, expanded, and irradiated onto a reflective phase-only LCOS SLM (800⫻ 600 pixels with pixel pitch of 20 m in both directions) through a soft aperture (opening diameter, 12 mm). Phase patterns are displayed on the SLM with an additional phase pattern that effectively cancels the total distortion of the optical system [12]. When displaying the phase patterns, we also correct the phase control voltages to compensate for the nonlinear response of liquid crystal to an electric field and the inhomogeneous thick© 2007 Optical Society of America
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OPTICS LETTERS / Vol. 32, No. 11 / June 1, 2007
Fig. 1. Phase patterns for generating (a) LG33 and (b) LG51 beams. Beams assigned with positive radial indices possess counterclockwise helical wavefronts when observed against propagating beams. (c) Schematic diagram of experimental setup. BS, beam splitter; SLM, phase-only reflective LCOS SLM; L, convex lens whose focal length is f = 400 mm.
ness distribution of the liquid crystal layer. Because of the corrections, such phase patterns as in Figs. 1(a) and 1(b) can be applied as they are without adding the blazed phase grating pattern commonly used for generating LG beams [3,9]. The output light, propagating vertically relative to the SLM surface, is extracted by a beam splitter (BS) and focused with convex lens L. The focused light pattern is observed by using a CCD after magnification through an objective lens (4X). We can theoretically estimate the output mode purity of our generation scheme by analyzing an electric field amplitude of light on the SLM surface because the mode content of output light is conserved during propagation. In the following, we derive an explicit expression of mode purity to show that the highest mode purity is achieved by adjusting the sizes of phase patterns relative to the incident light beam size. We start from the assumption that the incident light has a flat wavefront at the SLM surface; in other words, the SLM surface corresponds to z = 0. We also assume that the incident light has an almost flattop profile of radius R0 for simplicity. When the SLM displays the phase pattern for generating LGpl beams, the amplitude cqk of LGqk content in the output light is given as [9] cqk =
冕 冕 2
1
冑 R 0
d
0
R0
tion of a. Plotting with varying a, we notice that exhibits a maximum at a particular a depending on p and l. Here, since R0 ⬇ 290 pixels in the experiment, we can obtain high-quality LG modes by choosing w0 as 290/ a pixel for displaying phase patterns on the SLM. Observed light patterns are analyzed as follows. Calculating the LG mode pattern as Ipl = 兩upl兩2, we notice that Ipl is independent of and that its cross section, Ipl共x , y兲 in Cartesian coordinates 共x , y兲, presents a similar figure at every z position. Thus output beams can be analyzed by fitting observed mode patterns to Ipl共x , y兲. Here, Ipl contains adjustable parameters, i.e., beam radius wz, beam center position 共Ox , Oy兲, background B0, and intensity scale factor A0, all of which should be treated as fitting parameters. For practicality we perform nonlinear fitting calculations with a two-dimensional model function: Ipl共x,y兲 = A0关r02共x,y兲兴兩l兩 exp关− r02共x,y兲兴 ⫻ 关Lp兩l兩共r02共x,y兲兲兴2 + B0 ,
共5兲
where r02共x , y兲 = 2关共x − Ox兲2 + 共y − Oy兲2兴 / wz2. Correspondence between the observed patterns and models obtained through fitting calculations is semiquantitatively estimated with correlation coefficients. Figure 2 shows the observed mode patterns of LGpl 共p , l = 1 , 2 , 3兲 beams, while Fig. 3 illustrates the comparison between the observed mode patterns (dots) and fitted curves (solid curves) along with correlation coefficients between observed and fitted profiles R and theoretical mode purities . Here we stress that fitting calculations and correlation analyses are performed for two-dimensional profiles and that Fig. 3 exhibits only part of the total information for demonstrative purposes. As seen in Fig. 3, observed mode patterns correspond well to the theoretical patterns, and the experimentally determined correlation coefficients behave similarly to the theoretical mode purities. Strictly, the correlation coefficient is not a
rdr uqk共r, ,0兲* exp共− il兲
0
⫻ 兵2关Lp兩l兩共2r2/w02兲兴 − 1其,
共3兲
with the help of Eqs. (2) and exp关i共−x兲兴 = 2共x兲 − 1 ( * indicates the complex conjugate). Here we note that cqk becomes nonzero only when k = l, owing to the integration on . Thus, substituting Eq. (1), Eq. (3) finally gives us cql =
冋
p! 共p + 兩l兩兲!
⫻
册
1/2
兵2关Lp兩l兩共兲兴
共− 1兲p
冑2a − 1其,
冕
2a2
d兩l兩/2e−/2Lq兩l兩共兲
0
共4兲
where a = R0 / w0 with w0 as a beam radius of output mode on the SLM surface. The output mode content of the desired mode is defined by = 兩cpl兩2 as a func-
Fig. 2. Observed mode patterns of multiringed LG beams whose mode indices are up to three. Patterns are aligned so that each column consists of mode patterns of the same azimuthal index l while each row consists of those of the same radial index p.
June 1, 2007 / Vol. 32, No. 11 / OPTICS LETTERS
Fig. 3. Cross sections of observed mode patterns (dots) with corresponding theoretical curves (solid curves). A correlation coefficient between observed and fitted profiles 共R兲 and theoretical mode purity 共兲 are also shown for each mode. Results are arranged to correspond to Fig. 2.
Fig. 4. Interference patterns of generated beams and plane reference waves for (a) LG22 and (b) LG33.
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Figure 5(a) displays an observed mode pattern of the LG51 beam. Although this beam has a spatial fine structure with six rings, the observed mode pattern corresponds well 共R = 0.950兲 to the theoretical LG51 profile shown in Fig. 5(b), suggesting the usefulness of the present scheme for generating higher-order multiringed LG modes. In summary, we reported the generation of higherorder multiringed LG beams using a reflective phaseonly LCOS SLM. Because of the desirable phase modulation characteristics of the SLM, mode patterns of various multiringed LG beams are universally reproduced. The observed mode patterns and phase structures indicate that the generated beams are of LG modes. Although more detailed measurements and analyses are required for quantitative estimations [10,13], the flexibility to generate various modes without changing the optical setup is desirable for applications compared with previous methods such as mode conversions from Hermite– Gaussian modes [5–7] and fixed-phase elements [8–10]. We note that the generation of Bessel beams, which are also valuable for the transportation of cold atoms [10,11], is also possible by using a LCOS SLM, the results of which are not reported here. The authors are grateful to T. Hiruma, Y. Suzuki, and Y. Mizobuchi for their encouragement and also thank H. Toyoda and T. Inoue for valuable suggestions on holographic techniques. Part of this study was performed thanks to the Leading Projects of MEXT, Japan. References
Fig. 5. (a) Observed mode pattern and (b) cross-sectional profiles of observed (dots) and fitted (solid curve) mode pattern for LG51 beam.
quantitative benchmark of correspondence; nevertheless, beams generated through the present scheme are shown to have profiles close to ideal mode profiles, at least semiquantitatively. We also investigated the phase structures of obtained beams by observing the interference patterns of the obtained LG beams and the plane reference waves, using a Mach–Zehnder interferometer [9]. Figures 4(a) and 4(b) show the typical interference patterns of the LG22 and LG33 beams, respectively, where forklike patterns indicate phase structures of l = 2 and l = 3 modes. Similar phase structures are also confirmed for other beams, indicating that the obtained beams actually have helical phase structures. Optical manipulation experiments [3,4] using the obtained beams also demonstrate the transfer of angular momenta to microspheres, the results of which are not reported here.
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