Adjustable spiral phase plate Carmel Rotschild, Shachaf Zommer, Shulamit Moed, Oren Hershcovitz, and Stephen G. Lipson
A spiral phase retarder 共r, 兲 ⫽ m has been constructed with use of a deformed cracked plexiglass plate. By changing the degree of deformation, the retarder can be adjusted for use at any wavelength, and the value of the phase step 2m at ⫽ 2 can be chosen. © 2004 Optical Society of America OCIS codes: 100.5090, 260.3160, 350.6980, 230.6120.
A plane wave incident on a spiral phase plate leaves with local phase 共r, 兲 ⫽ m, where m is a positive or negative integer. Spiral phase plates have been instrumental in several fundamental investigations of waves and photons. For example, a photon passing through such a plate acquires angular orbital momentum mប1 which has been confirmed by experiment.2 Spiral phase plates have been used with nonlinear crystals to create twisted solitons.3 Several important applications have also been proposed. A spiral phase plate with m ⫽ 1 in the Fourier plane of an optical spatial filtering system could be used as an element for two-dimensional fringe demodulation to create the Hilbert transform of the input function.4,5 Optical beams with angular momentum m ⬎ 1 have been recognized as potential systems for m-bit quantum computation.6,7 Incorporated in a laser system, a spiral phase element results in a hollow beam8 and has also been proposed as a method of improving the beam quality.9 Several methods have been used to make a spiral phase plate. One uses holography; the far-field diffraction pattern of a hologram with a dislocation has spiral phase, the number m being equal to the Burger’s vector of the dislocation.10 A given hologram can be used in principle at any wavelength but is inevitably limited in efficiency. A second method is a plate made by photolithography and having thickness h共r, 兲 ⫽ m兾共n ⫺ 1兲k0, where n is the refractive index of the plate material and k0 ⫽ 2兾. Such a plate can have high transmission efficiency but is
exact only for the wavelength for which it is designed. Somewhat similar is the use of a liquid-crystal spatial light modulator programmed to give the required phase pattern.11 Another method utilizes a mode converter designed from two cylindrical lenses that transform the Hermite–Gaussian modes characteristic of a laser cavity into a pure Laguerre–Gaussian mode with the required angular momentum. This method requires fine control over the laser cavity adjustment.12 In this paper we describe the very simple construction of a spiral phase plate that has high efficiency and is adjustable, so that it can be used at any wavelength in the wavelength region where the material transmits. In addition, various values of m can be achieved with the same plate. The plate is constructed from a parallel-sided transparent plate with polished surfaces in which a crack is induced starting at one edge and terminating close to its center. The plate is mounted in a rigid frame and, by use of a set screw, one edge of the crack is twisted relative to the other. If the elastic limit of the plate material is not exceeded, the twisting process is controllable and reversible 共Fig. 1兲. Suppose that the plate has thickness d0. One side of the crack is normal to the incident light, and the other side is twisted to a small angle ␣, so that from Snell’s law the angle of refraction on the twisted side is  and the optical path difference between the two sides of the crack is:
冋
⌬ ⫽ d 0 共n ⫺ 1兲 ⫹ The authors are with the Department of Physics, TechnionIsrael Institute of Technology, 32000 Haifa, Israel. S. Lipson’s e-mail address is
[email protected]. Received 3 September 2003; revised manuscript received 3 February 2004; accepted 18 February 2004. 0003-6935兾04兾122397-03$15.00兾0 © 2004 Optical Society of America
⯝⫺
cos共␣ ⫺ 兲 n ⫺ cos  cos 
册
1 d 0␣ 2共1 ⫺ n ⫺1兲. 2
For n ⫽ 1.5, this yields ⌬ ⫽ 1兾6d0␣2, so that the phase step has the value 2m when this is equal to m. 20 April 2004 兾 Vol. 43, No. 12 兾 APPLIED OPTICS
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Fig. 1. Construction of the spiral phase plate.
For d0 ⫽ 3 mm and ⫽ 0.6 m, this translates to ␣2 ⫽ 1.2 m 10⫺3; for m ⫽ 1, ␣ ⫽ 0.035 rad ⫽ 2.0 deg. Construction of the plate utilized cracks found along the raw edge where a sheet of 3-mm opticalquality plexiglass had been cut with a guillotine. A suitable crack, normal to the surface, was chosen for
Fig. 3. Near-field interferograms with a plane reference wave. 共a兲 m ⫽ 1, 共b兲 m ⫽ 2, and 共c兲 m ⫽ 3.
Fig. 2. Sagnac interferometer used for the investigation. L1 images the phase plate onto the CCD camera and L2 creates a reference wave front with the same curvature, so that basically straight fringes are obtained. Removing L2 creates the spiral fringe pattern. 2398
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a phase plate, which was cut with lateral dimensions ⬃50 mm square from the region surrounding the crack such that its termination was in the center. This plate was mounted on a post containing an adjusting screw to control ␣. The plate was put in a Sagnac interferometer, in which the two counterpropagating waves were separated by ⬃15 mm 共Fig. 2兲 so that both beams passed through the plate, with one centered on the end of the crack and the other avoiding it. The near-field interferogram was ob-
a spherical reference was recorded, showing the equivalent spiral interferograms 共Fig. 4兲.13 In the interferograms for m ⫽ 3 it can already be observed the the twist is bending the plate significantly 共i.e., ␣ is a function of the radius兲, so that a fourth dislocation appears a short distance from the main one. Construction from glass plates was less successful, since the crack had to be produced by sawing; otherwise, the bending strain would cause the crack to propagate. The width of the saw cut created considerable scattering, which reduced the quality of the results. In experiments on other types of plastic, values of m as large as 11 have been achieved, but with poorer optical quality. This device was developed as part of an experimental project in the Senior Student Laboratory. The assistance of Roman Vander, Modi Hirschhorn, and Shmuel Hoida is gratefully acknowledged. References
Fig. 4. Near-field interferograms with a spherical reference wave. 共a兲 m ⫽ 1, 共b兲 m ⫽ 2 and 共c兲 m ⫽ 3.
served, showing a clear dislocation at the end of the crack, by which ␣ was adjusted to give exactly 2m phase shift; then light scattered by the crack itself was not observable. Three examples 共m ⫽ 1, 2, 3兲 are shown in Fig. 3. When the imaging lens in the reference wave, L2 in Fig. 2 was removed, interference between the transmitted spiral phase wave and
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