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52 OPTOINFORMATICS’05 IMPACT OF TILT OF A PHASE DOE ON THE PROPERTIES OF THE LASER BEAMS MATCHED WITH THE ANGULAR HARMONICS BASIS Almazov A.A., Khonina S.N., Kotlyar V.V. Samara State Aerospace University, Image Processing Systems Institute, Russian Academy <strong>of</strong> Sciences, 151 Molodogvardejskaya, Samara 443001, Russia E-mail: AAASkald@yandex.ru, khonina@smr.ru We discuss the generation <strong>of</strong> singular laser beams using the phase diffractive optical elements (DOEs) [1] . Among laser modes with helical singularity there are well-known higher-order Gauss-Laguerre [2] and Bessel [3] modes. Those modes contain optical vortices [4] providing the presence <strong>of</strong> an orbital angular momentum. In this paper, we consider new types <strong>of</strong> laser beams with orbital angular momentum, namely, optical vortices "imbedded" in a plane or Gaussian beam. Impact <strong>of</strong> various types <strong>of</strong> disturbances (DOE tilt, system misalignment, and inclusion <strong>of</strong> transparent obstacles) on the properties <strong>of</strong> the resulting laser beams with optical vortices or angular harmonics [5] with varying amplitude components is numerically studied. In Ref. [6] generation and detection <strong>of</strong> laser beams with angular harmonics using phase DOEs was considered. Impact <strong>of</strong> some types <strong>of</strong> distortions on the quality <strong>of</strong> beam generation and detection was simulated. It was demonstrated that using phase DOEs it is possible to provide high accuracy in detection <strong>of</strong> the various-order angular harmonics in the laser beams: the ratio <strong>of</strong> signal in the diffraction order <strong>of</strong> the harmonics under detection to that in the empty diffraction order was about 10 5 -10 3 . In real optical systems, a frequently found distortion is that the DOE is not perpendicular to the optical axis. Here, two types <strong>of</strong> distortion take place simultaneously. First, the DOE is virtually "compressed" along a transverse coordinate, generating an astigmatic output beam. Second, an effect <strong>of</strong> increased DOE relief depth occurs, with the subsequent phase delay <strong>of</strong> the light transmitted (see Fig. 1). ϕ( x ) = 2π h( x) (1) λ ϕ ( ) Perpendicular incidence ( x) 2 ( α ) λ h π ′ x = (2) cos Oblique incidence Fig. 1. An illustration <strong>of</strong> the perpendicular and oblique incidence <strong>of</strong> light on the DOE
SAINT-PETERSBURG, October 17 – 20, 2005 53 The designations in Fig. 1 are as follows: h(x) is the function <strong>of</strong> the DOE relief depth, x is the transverse coordinate (for simplicity, a 1D case is shown). F r, ϕ = A r, ϕ exp iΦ r, ϕ is transformed as In the general case, the field ( ) ( ) ( ( )) ⎛ iΦ′ ( ) ( ) ( r, ϕ ) ( ) ⎟ ⎞ F накл r, ϕ = A′ r, ϕ exp⎜ , where A′ ( r,ϕ ), ( r,ϕ ) ⎝ cos α ⎠ distorted functions A ( r,ϕ ), Ф ( r,ϕ ) Ф′ are the astigmatically . Thus, with increasing angle <strong>of</strong> the DOE tilt, α, the DOE singularity order is changed. Note that generally speaking, the singularity order ceases to be integral in this case. Hence, while possessing the properties <strong>of</strong> an astigmatic laser beam, the light field generated with the tilted DOE, will simultaneously show peculiar properties, which are more pronounced for greater DOE tilt. In Ref. [7] the generation <strong>of</strong> optical vortices in illumination <strong>of</strong> a spiral phase plate with a plane or Gaussian beam was discussed. In the present paper, experimental studies <strong>of</strong> the generation <strong>of</strong> singular beams in oblique illumination <strong>of</strong> similar spiral phase plates are conducted. Figure 2b-e illustrates some results <strong>of</strong> the experiments in oblique illumination <strong>of</strong> the third-order phase plate (Fig. 2a) with a converging laser beam. a b c d e Fig. 2. Experimental results with oblique illumination <strong>of</strong> the third-order phase plate (a) under different angles from 5 grad to 35 grad (b-e) The work is financially supported by the CRDF (grant REC-SA-014-02), the presidential grant <strong>of</strong> Russian Federation NS-1007.2003.1 and the grant <strong>of</strong> the Russian Foundation for Basic Research 05-01-96505. 1. Methods for Computer Design <strong>of</strong> Diffractive Optical Elements, ed. Victor A. Soifer – John Wiley & Sons, Inc., New York, 2002, 765 p. 2. A.E. Siegman, Lasers, University Science, Mill Valley, CA, 1986. 3. K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, K. Dholakia, “Orbital angular momentum <strong>of</strong> a high-order Bessel light beam”, J. Opt. B: Quantum Semiclass. Opt., 4, S82–S89, 2002. 4. M.S. Soskin, M.S. Vasnetsov, Singular optics, Progress in Optics 42, E. Wolf ed., 2001. 5. Kotlyar V.V., Khonina S.N., Soifer V.A., “Light field decomposition in angular harmonics by means <strong>of</strong> diffractive optics”, J. Mod. Opt. 45(7), 1495-1506 (1998). 6. A. Almazov, S. N. Khonina, V. V. Kotlyar, Generation and selection <strong>of</strong> laser beams represented as a superposition <strong>of</strong> an arbitrary number <strong>of</strong> angular harmonics with diffractive optical elements, Optical Journal, v. 72, # 5, 2005. 7. V.V. Kotlyar, A.A. Almazov, S.N. Khonina, V.A. Soifer, H. Elfstrom, J. Turunen, Generation <strong>of</strong> phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate, J. Opt. Soc. Am. A, Vol. 22, No. 5, pp. (2005).
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