## Programmable two-dimensional optical fractional Fourier processor

Optics Express, Vol. 17, Issue 7, pp. 4976-4983 (2009)

http://dx.doi.org/10.1364/OE.17.004976

Acrobat PDF (1330 KB)

### 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

## 1. Introduction

2. 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]

*f*(

**r**

_{i}) for parameters

*γ*and

_{x}*γ*, which are known as transformation angles, is defined as

_{y}**r**

_{i,o}= (

*x*,

_{i,o}*y*) are the input and output spatial coordinates, respectively [1]. Note that the kernel of FRFT is separable with respect to

_{i,o}*x*andy coordinates. For angles

*γ*=

_{x}*γ*= 0 the FRFT corresponds to the identity transformation, whereas for

_{y}*γ*= 0 and

_{x}*γ*=

_{y}*π*it reduces to image reflection. Meanwhile for

*γ*=

_{x}*γ*=

_{y}*π*/2 the Fourier transform is obtained. The cases

*γ*=

_{x}*γ*

_{y}=

*γ*and

*γ*= -

_{x}*γ*=

_{y}*γ*correspond to the symmetric and antisymmetric FRFTs, respectively [3

3. 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]

*γ*,

_{x}*γ*) can be understood as a generalization of the Fourier transform. It is usual to define the transformation angle as

_{y}*γ*=

*qπ*/2, where

*q*is called fractional order. For instance, the order q = 4 leads to the self-imaging case meanwhile

*q*= - 1 corresponds to the inverse Fourier transform. The properties and applications of the FRFT are discussed in detail for example in [1].

4. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transform and their optical implementation,” J. Opt. Soc. Am. A **10**, 1875–1881 (1993). [CrossRef]

5. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional order Fourier transform,” J. Opt. Soc. Am. A **10**, 2181–2186 (1993). [CrossRef]

6. A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical Implementations of Two-Dimensional Fractional Fourier Transforms and Linear Canonical Transforms with Arbitrary Parameters,” Appl. Opt. **37**, 2130–2141 (1998), http://ao.osa.org/abstract.cfm?URI=ao-37-11-2130. [CrossRef]

7. 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]

8. A. A. Malyutin, “Tunable Fourier transformer of the fractional order,” Quantum Electron. **36**, 79–83 (2006). [CrossRef]

9. 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]

3. 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]

*i*) distances between lenses and input-output planes are fixed;

*ii*) the change of the transformation parameters is only achieved by means of power variation of the lenses;

*iii*) the change of transformation parameters does not produce additional scaling of the transformed field;

*iv*) the number of lenses used in the setup satisfying the above mentioned conditions is minimal. It has been shown that such flexible systems for the FRFT [3

3. 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]

10. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express **15**, 2190–2203 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-5-2190. [CrossRef] [PubMed]

11. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A **24**, 3135–3139 (2007), http://josaa.osa.org/abstract.cfm?URI=josaa-24-10-3135. [CrossRef]

12. G. Nemes and A. E. Seigman, “Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A **11**, 2257–2264 (1994). [CrossRef]

**23**, 2494–2500 (2006), http://josaa.osa.org/abstract.cfm?URI=josaa-23-10-2494. [CrossRef]

10. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express **15**, 2190–2203 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-5-2190. [CrossRef] [PubMed]

11. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A **24**, 3135–3139 (2007), http://josaa.osa.org/abstract.cfm?URI=josaa-24-10-3135. [CrossRef]

## 2. Flexible system design for FRFT

*z*between them, see Fig. 1(a), where the last lens and first one are identical (L

_{3}= L

_{1}). Each generalized lens, L

_{1}and L

_{2}, is an assembled set of two crossed (at angle

*π*/2) cylindrical lenses with variable lens power given by:

**23**, 2494–2500 (2006), http://josaa.osa.org/abstract.cfm?URI=josaa-23-10-2494. [CrossRef]

_{1}and L

_{2}, respectively. From Eq. (2) it follows that this optical setup performs FRFT for the angle interval

*γ*∊ [

_{x,y}*π*/2,3

*π*/2], see Fig. 1(b) and 1(c). This interval is sufficient for a large list of the FRFT applications including adaptive filtering, beam characterization, phase space tomography, etc. Nevertheless, the entire interval

*γ*∊ (0,2

_{x,y}*π*) can be also covered, due to the relation

*F*

^{γx+π,γy+π}(

**r**

_{o}) =

*F*

^{γx,γy}(-

**r**

_{o}).

*F*

^{γx,γy}(

**r**

_{o}) can be derived from input signal

*f*(

**r**

_{i}) applying the phase modulation function associated with each generalized lens L

_{j}(

*j*= 1,2):

**23**, 2494–2500 (2006), http://josaa.osa.org/abstract.cfm?URI=josaa-23-10-2494. [CrossRef]

*s*

^{2}= 2

*λz*, which is independent of the transformation angles

*γ*and

_{x}*γ*. Notice that the described algorithm is used for the numerical simulation of this FRFT setup.

_{y}*f*(

*x*,

_{i}*y*) generation we also apply a transmissive SLM (SLM1) which modulates the amplitude of a collimated Nd:YAG laser beam with wavelength

_{i}*λ*= 532 nm. Thus the amplitude distribution ∣

*f*(

*x*,

_{i}*y*)∣ is implemented on SLM1 that is projected on SLM2 by using a 4-f lens system, where its phase distribution arg[

_{i}*f*(

*x*,

_{i}*y*)] is addressed together with the phase arg[Ψ

_{i}_{1}(

*x*,

*y*)] associated to the first generalized lens. At the distance corresponding to the optical path

*z*(in our case

*z*= 50 cm) the SLM3 is located, which implements the generalized lens L

_{2}. The intensity distribution of the output signal is registered by a CCD camera that is placed at the distance corresponding to the optical path z from the SLM3, see Fig. 2. Each SLM is connected to the same PC and the alignment between them is reached digitally which is limited by the pixel size (19 µm in our case). Thus position stages for the SLM alignment are not required. We have developed a customized software for the system control able to change the fractional orders at almost real time and to store the measured FRFT power spectra as a video file.

## 3. Experimental results

*r*

^{2}=

*x*

^{2}+

*y*

^{2}, Hm is the Hermite polynomial,

*w*is the beam waist, and

*L*is the Laguerre polynomial with radial index

^{l}_{p}*p*and azimuthal index

*l*. In contrast to the HG modes, the LG ones are vortex beams which carry Orbital Angular Momentum (OAM):

*l*

*h*̄ per photon. The HG and LG modes are eigenfunctions of the symmetric FRFT [

*γ*=

_{x}*γ*=

_{y}*γ*, Ec. (4)] for

*w*= 0.73 mm). Moreover a HG mode also does not change under the antisymmetric FRFT (

*γ*= -

_{x}*γ*=

_{y}*γ*), meanwhile a LG mode is transformed into intermediate HG-LG modes with fractional OAM:

*l*

*h*̄ sin 2

*γ*, [14

14. T. Alieva and M. J. Bastiaans, “Orthonormal mode sets for the two-dimensional fractional Fourier transformation,” Opt. Lett. **32**, 1226–1228 (2007), http://ol.osa.org/abstract.cfm?URI=ol-32-10-1226. [CrossRef] [PubMed]

*γ*= (2

*k*+ 1)

*π*/4 to the LG

^{±}

_{p,l}(

**r**;

*w*) mode, where

*k*is an integer whereas

*p*= min(

*m*,

*n*) and

*I*= ∣

*m*-

*n*∣, the HG

_{m,n}(

**r**;

*w*) one (rotated at angle ±

*π*/4) is obtained. Therefore it is easy to test the FRFT experimental setup using the HG and LG modes because we know exactly their transformation for any angle. The system characterization has been done with the HG and LG modes for different indices. Here we demonstrate the experimental results, Fig. 3, for HG

_{3,2}3(a) and LG

^{+}

_{4,1}3(d) modes.

*γ*= 135° is displayed in Fig. 3(b) and 3(e) for HG

_{3,2}and LG

_{4,1}

^{+}, respectively. Meanwhile the antisymmetric FRFT at angle

*γ*= 135° is shown in Fig. 3(c) and 3(f) for HG

_{3,2}and LG

^{+}

_{4,1}, correspondingly. These transformations have been recorded as video at real time for the angle interval 7 s [90°, 270°] with step of 1° and frame rate 30 fps, see Fig. 3. As it has been expected, we observe that the intensity distributions of HG

_{3,2}and LG

_{4,1}

^{+}during the transformation are almost constant except for the antisymmetric FRFT of LG

_{4,1}

^{+}where intermediate HG-LG modes are obtained.

_{4,1}

^{+}mode under antisymmetric FRFT is also displayed in Fig. 4 for

*γ*= 225° +

*k*11.3°, where

*k*= 0, …,4. First and second rows (intensity and phase distribution) correspond to numerical simulation of the FRFT setup whereas experimental results are displayed at the third row, see Fig. 4. Notice that the phase distributions here and further correspond to the simulation of the FRFT system with three lenses [see Fig. 1(a)].

15. A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express **15**, 5801–5808 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-9-5801. [CrossRef] [PubMed]

*γ*∣ ≠ ∣

_{x}*γ*∣ we consider ∣LG

_{y}_{4,1}

^{+}∣ as an input signal. In contrast to the previous input signal LG

_{4,1}

^{+}, the ∣LG

_{4,1}

^{+}∣ is an image with a constant phase distribution. Let us first consider the symmetric FRFT of ∣LG

_{4,1}

^{+}∣ for

*γ*= 90° +

*k*22.5° with

*k*=0, …, 4, Fig. 5. Since ∣LG

_{4,1}

^{+}∣ is not an eigenfunction for the FRFT, as it is the case for LG

_{4,1}

^{+}, the intensity distribution is changing with variation of the fractional order. Notice that for FRFT(90°, 90°) it reduces to the Fourier transform of the input signal, whereas the FRFT(180°, 180°) leads to self-imaging. For the rest of angles, see Fig. 5, the results are similar (except for scaling) to the ones obtained under Fresnel diffraction of the input signal.

_{4,1}

^{+}∣ under the FRFT for

*γ*=

_{y}*γ*+

_{x}*k*45° with

*k*= 1, 2 and 3, see Fig. 6. The FRFT(

*α*,

*γ*) and FRFT(

_{y}*γ*,

_{x}*α*) when

*α*is set at 90° and 270° correspond to the direct/inverse Fourier transforms along the

*x*and

*y*axis, respectively. It is demonstrated in Fig. 6 for the case FRFT(90°, 135°) and FRFT(225°, 270°). For the rest of transformation angles intermediate images are obtained, Fig. 6. The comparison of the experimental and numerically simulated results again demonstrates the feasibility of the proposed setup.

16. D. Mendlovic, R. G. Dorsch, A. W. Lohmann, Z. Zalevsky, and C. Ferreira, “Optical illustration of a varied fractional Fourier-transform order and the Radon-Wigner display,” Appl. Opt. **35**, 3925–3929 (1996), http://ao.osa.org/abstract.cfm?URI=ao-35-20-3925. [CrossRef] [PubMed]

## 4. Conclusions

## Acknowledgments

## References

1. | H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, |

2. | 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. |

3. | J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for orthosymplectic transformations in phase space,” J. Opt. Soc. Am. A |

4. | D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transform and their optical implementation,” J. Opt. Soc. Am. A |

5. | A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional order Fourier transform,” J. Opt. Soc. Am. A |

6. | A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical Implementations of Two-Dimensional Fractional Fourier Transforms and Linear Canonical Transforms with Arbitrary Parameters,” Appl. Opt. |

7. | I. Moreno, J. A. Davis, and K. Crabtree, “Fractional Fourier transform optical system with programmable diffractive lenses,” Appl. Opt. |

8. | A. A. Malyutin, “Tunable Fourier transformer of the fractional order,” Quantum Electron. |

9. | I. Moreno, C. Ferreira, and M. M. Sánchez-López, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A: Pure and Applied Optics |

10. | J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express |

11. | J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A |

12. | G. Nemes and A. E. Seigman, “Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A |

13. | J. A. Rodrigo, “First-order optical systems in information processing and optronic devices,” Ph.D. thesis, Uni-versidad Complutense de Madrid (2008). |

14. | T. Alieva and M. J. Bastiaans, “Orthonormal mode sets for the two-dimensional fractional Fourier transformation,” Opt. Lett. |

15. | A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express |

16. | D. Mendlovic, R. G. Dorsch, A. W. Lohmann, Z. Zalevsky, and C. Ferreira, “Optical illustration of a varied fractional Fourier-transform order and the Radon-Wigner display,” Appl. Opt. |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(120.4820) Instrumentation, measurement, and metrology : Optical systems

(140.3300) Lasers and laser optics : Laser beam shaping

(200.4740) Optics in computing : Optical processing

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: December 18, 2008

Revised Manuscript: January 27, 2009

Manuscript Accepted: February 13, 2009

Published: March 16, 2009

**Citation**

Jose Augusto Rodrigo, Tatiana Alieva, and Maria L. Calvo, "Programmable two-dimensional optical fractional Fourier processor," Opt. Express **17**, 4976-4983 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-4976

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### References

- 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, 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]
- D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transform and their optical implementation," J. Opt. Soc. Am. A 10, 1875-1881 (1993). [CrossRef]
- A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional order Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993). [CrossRef]
- A. Sahin, H. M. Ozaktas, and D. Mendlovic, "Optical Implementations of Two-Dimensional Fractional Fourier Transforms and Linear Canonical Transforms with Arbitrary Parameters," Appl. Opt. 37, 2130-2141 (1998), http://ao.osa.org/abstract.cfm?URI=ao-37-11-2130. [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]
- 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, "Gyrator transform: properties and applications," Opt. Express 15, 2190-2203 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-5-2190. [CrossRef] [PubMed]
- J. A. Rodrigo, T. Alieva, and M. L. Calvo, "Experimental implementation of the gyrator transform," J. Opt. Soc. Am. A 24, 3135-3139 (2007), http://josaa.osa.org/abstract.cfm?URI=josaa-24-10-3135. [CrossRef]
- G. Nemes and A. E. Seigman, "Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics," J. Opt. Soc. Am. A 11, 2257-2264 (1994). [CrossRef]
- J. A. Rodrigo, "First-order optical systems in information processing and optronic devices," Ph.D. thesis, Universidad Complutense de Madrid (2008).
- T. Alieva and M. J. Bastiaans, "Orthonormal mode sets for the two-dimensional fractional Fourier transformation," Opt. Lett. 32, 1226-1228 (2007), http://ol.osa.org/abstract.cfm?URI=ol-32-10-1226. [CrossRef] [PubMed]
- A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, "Wavefront correction of spatial light modulators using an optical vortex image," Opt. Express 15, 5801-5808 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-9-5801. [CrossRef] [PubMed]
- D. Mendlovic, R. G. Dorsch, A. W. Lohmann, Z. Zalevsky, and C. Ferreira, "Optical illustration of a varied fractional Fourier-transform order and the Radon—Wigner display," Appl. Opt. 35, 3925-3929 (1996), http://ao.osa.org/abstract.cfm?URI=ao-35-20-3925. [CrossRef] [PubMed]

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