Abstract

A flexible optical system able to perform the fractional Fourier transform (FRFT) almost in real time is presented. In contrast to other FRFT setups the resulting transformation has no additional scaling and phase factors depending on the fractional orders. The feasibility of the proposed setup is demonstrated experimentally for a wide range of fractional orders. The fast modification of the fractional orders, offered by this optical system, allows to implement various proposed algorithms for beam characterization, phase retrieval, information processing, etc.

© 2009 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2007

2006

A. A. Malyutin, "Tunable Fourier transformer of the fractional order," Quantum Electron. 36, 79-83 (2006).
[CrossRef]

I. Moreno, C. Ferreira, and M. M. Sánchez-López, "Ray matrix analysis of anamorphic fractional Fourier systems," J. Opt. A: Pure and Applied Optics 8, 427-435 (2006), http://stacks.iop.org/1464-4258/8/427.
[CrossRef]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, "Optical system design for orthosymplectic transformations in phase space," J. Opt. Soc. Am. A 23, 2494-2500 (2006), http://josaa.osa.org/abstract.cfm?URI=josaa-23-10-2494.
[CrossRef]

2003

1998

1996

1994

1993

Alieva, T.

Bastiaans, M. J.

Bernet, S.

Calvo, M. L.

Crabtree, K.

Davis, J. A.

Dorsch, R. G.

Ferreira, C.

Fürhapter, S.

Jesacher, A.

Lohmann, A. W.

Malyutin, A. A.

A. A. Malyutin, "Tunable Fourier transformer of the fractional order," Quantum Electron. 36, 79-83 (2006).
[CrossRef]

Maurer, C.

Mendlovic, D.

Moreno, I.

I. Moreno, C. Ferreira, and M. M. Sánchez-López, "Ray matrix analysis of anamorphic fractional Fourier systems," J. Opt. A: Pure and Applied Optics 8, 427-435 (2006), http://stacks.iop.org/1464-4258/8/427.
[CrossRef]

I. Moreno, J. A. Davis, and K. Crabtree, "Fractional Fourier transform optical system with programmable diffractive lenses," Appl. Opt. 42, 6544-6548 (2003).
[CrossRef] [PubMed]

Nemes, G.

Ozaktas, H. M.

Ritsch-Marte, M.

Rodrigo, J. A.

Sahin, A.

Sánchez-López, M. M.

I. Moreno, C. Ferreira, and M. M. Sánchez-López, "Ray matrix analysis of anamorphic fractional Fourier systems," J. Opt. A: Pure and Applied Optics 8, 427-435 (2006), http://stacks.iop.org/1464-4258/8/427.
[CrossRef]

Schwaighofer, A.

Seigman, A. E.

Zalevsky, Z.

Appl. Opt.

J. Opt. A: Pure and Applied Optics

I. Moreno, C. Ferreira, and M. M. Sánchez-López, "Ray matrix analysis of anamorphic fractional Fourier systems," J. Opt. A: Pure and Applied Optics 8, 427-435 (2006), http://stacks.iop.org/1464-4258/8/427.
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Quantum Electron.

A. A. Malyutin, "Tunable Fourier transformer of the fractional order," Quantum Electron. 36, 79-83 (2006).
[CrossRef]

Other

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley&Sons, NY, USA, 2001).

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, "Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms," Opt. Lett. 20, 1181-1183 (1995). URL http://ol.osa.org/abstract.cfm?URI=ol-20-10-1181.
[CrossRef] [PubMed]

J. A. Rodrigo, "First-order optical systems in information processing and optronic devices," Ph.D. thesis, Universidad Complutense de Madrid (2008).

Supplementary Material (4)

» Media 1: MOV (2340 KB)     
» Media 2: MOV (2429 KB)     
» Media 3: MOV (2929 KB)     
» Media 4: MOV (3573 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

(a) FRFT optical setup. P input and P output are the input and output planes. The free-space intervals z are fixed. Normalized operation curves z · px,y for the generalized lenses L1 and L2 are displayed in (b) and (c), respectively.

Fig. 2.
Fig. 2.

Experimental setup. The amplitude and phase distributions of the input signal are generated by means of SLM1 (CRL-XGA2, 1024×768 pixels) and SLM2, correspondingly. SLM2 and SLM3 (Holoeye LCR-2500, 1024×768 pixels) implement L1 and L2 lens, respectively. Output signal is registered by a CCD camera (XGA, 4.6 µm pixel size). The optical path z is set at 50 cm. SLM performance at phase-only modulation is reached by using a λ/2 wave plate (WP).

Fig. 3.
Fig. 3.

Input signals HG3,2 (a) and LG+ 4,1 (d) that coincide with their FRFT(0°, 0°) for each case. Their transformations under symmetric FRFT(γ, γ) (b) (Media 1) and (e) (Media 2), as well as under antisymmetric FRFT(γ, -γ) (c) (Media 3) and (f) (Media 4) are displayed, correspondingly. The FRFT operations for angle interval γ ∊ [90°, 270°] were stored as a video file with 30 fps. Units in axis x and y are in mm.

Fig. 4.
Fig. 4.

Numerical simulation (first and second row) and experimental results (third row, Media 4) corresponding to LG4,1 + transformation under antisymmetric FRFT(γ, -γ).

Fig. 5.
Fig. 5.

Transformation of the image ∣LG4,1 +∣ under symmetric FRFT(γ, γ). Numerical simulation (first and second row) and experimental results (third row).

Fig. 6.
Fig. 6.

Transformation of ∣LG4,1 +∣ image under FRFT for γy = γx + k45° with k = 1,2,3. Numerical simulation (first and second row) and experimental results (third row).

Equations (9)

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F γ x , γ y ( r o ) = exp [ i ( γ x + γ y ) / 2 ] i sin γ x sin γ y ∫∫ f ( x i , y i ) exp { [ ( x o 2 + x i 2 ) cot γ x 2 x i x o csc γ x ] }
× exp { [ ( y o 2 + y i 2 ) cot γ y 2 y i y o csc γ y ] } d x i d y i ,
p x , y ( 1 ) = z 1 ( 1 cot ( γ x , y / 2 ) / 2 ) ,
p x , y ( 2 ) = 2 z 1 ( 1 sin γ x , y ) ,
Ψ j ( x , y ) = exp ( λz p x ( j ) x 2 ) exp ( λz p y ( j ) y 2 ) ,
F γ x , γ y ( x o , y o ) = 1 2 λ z sin γ x sin γ y ∫∫ f i ( x i , y i ) exp { 2 λz [ ( x o 2 + x i 2 ) cot γ x 2 x i x o csc γ x ] }
× exp { 2 λz [ ( y o 2 + y i 2 ) cot γ y 2 y i y o csc γ y ] } d x i d y i .
H G m , n ( r ; w ) = 2 1 / 2 H m ( 2 π x w ) H n ( 2 π y w ) 2 m m ! w 2 n n ! w exp ( π w 2 r 2 ) ,
L G p , l ± ( r ; w ) = w 1 ( p ! ( p + l ) ! ) 1 / 2 [ 2 π ( x w ± i y w ) ] l L p l ( 2 π w 2 r 2 ) exp ( π w 2 r 2 )

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