plane wave incident on a spiral phase plate leavesith local phase ��r, �� � m�, where m is a positive oregative integer. Spiral phase plates have been in-trumental in several fundamental investigations ofaves and photons. For example, a photon passing
hrough such a plate acquires angular orbital mo-entum m�1 which has been confirmed by experi-ent.2 Spiral phase plates have been used with
onlinear crystals to create twisted solitons.3 Sev-ral important applications have also been proposed.spiral phase plate with m � 1 in the Fourier plane
f an optical spatial filtering system could be used asn element for two-dimensional fringe demodulationo create the Hilbert transform of the input func-ion.4,5 Optical beams with angular momentum
� 1 have been recognized as potential systems for-bit quantum computation.6,7 Incorporated in a
aser system, a spiral phase element results in a hol-ow beam8 and has also been proposed as a method ofmproving the beam quality.9
Several methods have been used to make a spiralhase plate. One uses holography; the far-field dif-raction pattern of a hologram with a dislocation haspiral phase, the number m being equal to the Burg-r’s vector of the dislocation.10 A given holograman be used in principle at any wavelength but isnevitably limited in efficiency. A second method is
plate made by photolithography and having thick-ess h�r, �� � m���n 1�k0, where n is the refractive
ndex of the plate material and k0 � 2��. Such alate can have high transmission efficiency but is
The authors are with the Department of Physics, Technion-srael Institute of Technology, 32000 Haifa, Israel. S. Lipson’s-mail address is [email protected] 3 September 2003; revised manuscript received 3 Feb-
xact only for the wavelength for which it is designed.omewhat similar is the use of a liquid-crystal spatial
ight modulator programmed to give the requiredhase pattern.11 Another method utilizes a modeonverter designed from two cylindrical lenses thatransform the Hermite–Gaussian modes characteris-ic of a laser cavity into a pure Laguerre–Gaussianode with the required angular momentum. Thisethod requires fine control over the laser cavity
djustment.12
In this paper we describe the very simple construc-ion of a spiral phase plate that has high efficiencynd is adjustable, so that it can be used at any wave-ength in the wavelength region where the materialransmits. In addition, various values of m can bechieved with the same plate. The plate is con-tructed from a parallel-sided transparent plate witholished surfaces in which a crack is induced startingt one edge and terminating close to its center. Thelate is mounted in a rigid frame and, by use of a setcrew, one edge of the crack is twisted relative to thether. If the elastic limit of the plate material is notxceeded, the twisting process is controllable and re-ersible �Fig. 1�.Suppose that the plate has thickness d0. One side
f the crack is normal to the incident light, and thether side is twisted to a small angle �, so that fromnell’s law the angle of refraction on the twisted side
s � and the optical path difference between the twoides of the crack is:
� d0��n � 1� �cos�� � ��
cos ��
ncos ��
� 12
d0�2�1 � n1�.
or n � 1.5, this yields � 1�6d0�2, so that the phasetep has the value 2m� when this is equal to m.
or d0 � 3 mm and � 0.6 �m, this translates to �2 �.2 m 103; for m � 1, � � 0.035 rad � 2.0 deg.Construction of the plate utilized cracks found
long the raw edge where a sheet of 3-mm optical-uality plexiglass had been cut with a guillotine. Auitable crack, normal to the surface, was chosen for
Fig. 1. Construction of the spiral phase plate.
ig. 2. Sagnac interferometer used for the investigation. L1 im-ges the phase plate onto the CCD camera and L2 creates a refer-nce wave front with the same curvature, so that basically straightringes are obtained. Removing L2 creates the spiral fringe pat-ern.
phase plate, which was cut with lateral dimensions50 mm square from the region surrounding the
rack such that its termination was in the center.his plate was mounted on a post containing an ad-
usting screw to control �. The plate was put in aagnac interferometer, in which the two counter-ropagating waves were separated by �15 mm �Fig.� so that both beams passed through the plate, withne centered on the end of the crack and the othervoiding it. The near-field interferogram was ob-
ig. 3. Near-field interferograms with a plane reference wave.a� m � 1, �b� m � 2, and �c� m � 3.
scpware
aeititCswpervw
taS
R
1
1
1
1
F�
erved, showing a clear dislocation at the end of therack, by which � was adjusted to give exactly 2m�hase shift; then light scattered by the crack itselfas not observable. Three examples �m � 1, 2, 3�re shown in Fig. 3. When the imaging lens in theeference wave, L2 in Fig. 2 was removed, interfer-nce between the transmitted spiral phase wave and
ig. 4. Near-field interferograms with a spherical reference wave.a� m � 1, �b� m � 2 and �c� m � 3.
spherical reference was recorded, showing thequivalent spiral interferograms �Fig. 4�.13 In thenterferograms for m � 3 it can already be observedhe the twist is bending the plate significantly �i.e., �s a function of the radius�, so that a fourth disloca-ion appears a short distance from the main one.onstruction from glass plates was less successful,ince the crack had to be produced by sawing; other-ise, the bending strain would cause the crack toropagate. The width of the saw cut created consid-rable scattering, which reduced the quality of theesults. In experiments on other types of plastic,alues of m as large as 11 have been achieved, butith poorer optical quality.
This device was developed as part of an experimen-al project in the Senior Student Laboratory. Thessistance of Roman Vander, Modi Hirschhorn, andhmuel Hoida is gratefully acknowledged.
eferences1. M. Padgett and L. Allen, “Light with a twist in its tail,” Con-
temp. Phys. 41, 275–285 �2000�.2. J. Courtal, K. Dholakia, D. A. Robertson, L. Allen, and M. J.
Padgett, “Measurement of the rotational frequency shift im-parted to a light beam possessing orbital angular momentum,”Phys. Rev. Lett. 80, 3217–3219 �1999�.
3. T. Carmon, R. Uzdin, C. Pigier, Z. Musslimani, M. Segev, andA. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev.Lett. 87, 143901–143904 �2001�.
4. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demod-ulation of two-dimensional fringe patterns. I. General back-ground of the spiral phase quadrature transform,” J. Opt. Soc.Am. A 18, 1862–1870 �2001�.
5. K. G. Larkin, “Natural demodulation of two-dimensional fringepatterns. II. Stationary-phase analysis of the spiral phasequadrature transform,” J. Opt. Soc. Am. A 18, 1871–1881�2001�.
6. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglementof the orbital angular momentum states of photons,” Nature�London� 412, 313–316 �2001�.
7. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J.Haglin, and R. E. Williams, “Geometric phase associated withmode transformations of optical beams bearing orbital angularmomentum,” Phys. Rev. Lett. 90, 203901–203904 �2003�.
8. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, andJ. P. Woerdman, “Helical-wave-front laser-beams producedwith a spiral phaseplate,” Opt. Commun. 112, 321–327 �1994�.
9. R. Oren, N. Davidson, A. A. Friesem, and E. Hasman,“Continuous-phase elements can improve laser beam quality,”Opt. Lett. 25, 939–941 �2000�.
0. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White,“Generation of optical-phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 �1992�.
1. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt.Lett. 28, 872–874 �2003�.
2. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, andJ. P. Woerdman, “Astigmatic laser mode converters and trans-fer of orbital angular momentum,” Opt. Commun. 96, 123–132�1993�.
3. M. Harris, C. A. Hill, and J. M. Vaughan, “Optical helices andspiral interference fringes,” Opt. Commun. 106, 161–166�1994�.
