## Gyrator transform: properties and applications

Optics Express, Vol. 15, Issue 5, pp. 2190-2203 (2007)

http://dx.doi.org/10.1364/OE.15.002190

Acrobat PDF (300 KB)

### Abstract

In this work we formulate the main properties of the gyrator operation which produces a rotation in the twisting (position - spatial frequency) phase planes. This transform can be easily performed in paraxial optics that underlines its possible application for image processing, holography, beam characterization, mode conversion and quantum information.As an example, it is demonstrated the application of gyrator transform for the generation of a variety of stable modes.

© 2007 Optical Society of America

## 1. Introduction

2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and
transfer of orbital angular momentum,”
Opt. Commun. **96**, 123–132
(1993). [CrossRef]

3. E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and
nontransformed beams,” Opt. Commun. **83**, 123–135
(1991). [CrossRef]

4. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian
beams,” J. Opt. A.: Pure Appl. Opt. **6**, S157–S161
(2004). [CrossRef]

4. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian
beams,” J. Opt. A.: Pure Appl. Opt. **6**, S157–S161
(2004). [CrossRef]

5. G. F. Calvo, “Wigner representation and geometric
transformations of optical orbital angular momentum spatial
modes,” Opt. Lett. **30**, 1207–1209
(2005). [CrossRef] [PubMed]

6. R. Simon and K. B. Wolf, “Structure of the set of paraxial
optical systems,” J. Opt. Soc. Am. A **17**, 342–355
(2000). [CrossRef]

5. G. F. Calvo, “Wigner representation and geometric
transformations of optical orbital angular momentum spatial
modes,” Opt. Lett. **30**, 1207–1209
(2005). [CrossRef] [PubMed]

*x*,

*q*) and (

_{y}*y*,

_{qx}) [6

6. R. Simon and K. B. Wolf, “Structure of the set of paraxial
optical systems,” J. Opt. Soc. Am. A **17**, 342–355
(2000). [CrossRef]

*f*(

_{i}**r**

_{i}), associated in first order optics with complex field amplitude, can be written in the following form

**r**

^{t}*= (*

_{i,o}*x*,

_{i,o}*y*) indicates the input and output coordinates, respectively. Notice that

_{i,o}*t*stands for transposition operation. For α = 0 it corresponds to the identity transform, for α = π/2 it reduces to the Fourier transform with rotation of the coordinates at π/2, for α = π the reverse transform described by the kernel δ(

**r**

_{o}+

**r**

_{i}) is obtained, meanwhile for α = 3π/2 it corresponds to the inverse Fourier transform with rotation of the coordinates at π /2. For other angles α the kernel of the GT

*K*α(

*,*

*x*_{i}*,*

*y*_{i}*,*

*x*_{o}*y*) has a constant amplitude and a hyperbolic phase structure, which is shown in Fig. 1 for the angle α = π/4 and output coordinates

_{o}*x*=

_{o}*y*= 0 (see Fig. 1(a)) and 2

_{o}*x*=

_{o}*y*= 1 (see Fig. 1(b)).

_{o}**T**(α) [6

6. R. Simon and K. B. Wolf, “Structure of the set of paraxial
optical systems,” J. Opt. Soc. Am. A **17**, 342–355
(2000). [CrossRef]

**r**

^{t}= (

*x*,

*y*) is the ray position and

**q**

*= (*

^{t}*q*,

_{x}*q*) is the ray slope, and bold capital here and further indicates matrix notation.

_{y}8. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for
ortho-symplectic transformations in phase space,”
J. Opt. Soc. Am. A **23**, 2494–2500
(2006). [CrossRef]

8. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for
ortho-symplectic transformations in phase space,”
J. Opt. Soc. Am. A **23**, 2494–2500
(2006). [CrossRef]

## 2. Basic properties of the gyrator operation

9. T. Alieva, V. Lopez, F. Agullo Lopez, and L. B. Almeida, “The fractional Fourier transform in
optical propagation problems,” J. Mod.
Opt. **41**, 1037–1044
(1994). [CrossRef]

**T**(α)

**T**(β) =

**T**(α + β ). The inverse GT corresponds to the GT at angle -α. As it follows from Eq. (1) the inverse transform can be also written as

*R*

^{α}[

*R*

^{α}[

*f*(-

_{i}*x*,

_{i}*y*)](-

_{i}*x*,

_{o}*y*)] (

_{o}**r**) =

*f*(

_{i}**r**).

*f*at vector

_{i}**v**= (

^{t}*v*,

_{x}*v*) leads to the shift of its GT (for the angle α) at

_{y}**v**cosα and additional linear phase modulation:

**v͂**= (

^{t}*v*,

_{y}*v*), see appendix for more details. We observe that the shift of the amplitude of the GT ∣R

_{x}^{α}[

*f*(

_{i}**r**

_{i}-

**v**)](

**r**

_{o})∣ = ∣R

^{α}[

*f*(

_{i}**r**

_{i})](

**r**

_{o}-

**v**cosα)∣ is the same as for the case of two dimensional symmetric fractional FT at angle α [9

9. T. Alieva, V. Lopez, F. Agullo Lopez, and L. B. Almeida, “The fractional Fourier transform in
optical propagation problems,” J. Mod.
Opt. **41**, 1037–1044
(1994). [CrossRef]

*i*2

*π*

**k**

^{t}

**r**

_{i}) of the function

*f*(

_{i}**r**

_{i}) is also similar to the fractional FT case. It leads to the shift of its GT (for the angle α) at -

**k**͂ sin α and additional linear phase modulation:

**k**

^{t}= (

*k*,

_{x}*k*) and

_{y}**k**͂

^{t}= (

*k*,

_{y}*k*).

_{x}_{α}= sgn(sinα), σ

_{β}= sgn(sinβ),

*f*(

_{i}**Sr**

_{i}) corresponds to the GT at angle β of the initial function

*f*(

_{i}*r*) with additional scaling of the output coordinates and hyperbolic phase modulation. The scaling property for the GT is similar to one for the Fresnel transform or for the symmetrical fractional FT. Indeed during the Fresnel diffraction the change of the aperture scale leads to the observation of the same diffraction pattern (except of the corresponding scaling and chirp phase modulation) at another propagation distance. The principal difference is in the phase modulation which has hyperbolic form for the GT and chirp form for Fresnel or fractional FT transforms. Moreover in the case of GT there are two particular cases of scaling parameters

_{i}*s*=

_{x}*s*=

*s*

^{-1}

_{y}and

*s*=

_{x}*s*= -

*s*

^{-1}

_{y}when the expression Eq. (8) is significantly reduced.

*s*=

_{x}*s*=

*s*

^{-1}

_{y}the scaling does not change the transformation angle β = α, the output scaling is the same as the input one and there is no additional phase modulation

*s*=

_{x}*s*= -

*s*

^{-1}

_{y}then the angles relation reduces to cot β = -cotα and therefore β = π -α.

*s*= 1 we obtain the expression similar to Eq. (4).

## 3. Gyrator transform of selected functions

**v**= (

^{t}*v*,

_{x}*v*),

_{y}**k**

^{t}= 2π(

*k*,

_{x}*k*),

_{y}*a*> 0,

*b*and

*c*are real numbers, and

**r**

_{i}-

**v**) corresponds to the gyrator kernel as the output function,

*K*(

_{α}**r**

_{i}=

**v**,

**r**

_{o}), and therefore the product of hyperbolic and plane waves.

*c*. It is an important result because it means that GT can be used for localization of waves with hyperbolic phase front. For

*c*= tanα the plane wavefront,

*f*(

_{o}**r**

_{o}) = ∣sinα∣

^{-1}, is obtained at the output of the GT system. For other angles the hyperbolic wave transforms to the hyperbolic one. We underline only two particular cases, when the expressions for the GT of hyperbolic wave are simplified. Thus for the values of parameter

*c*= cotα and

*c*= (1+cotα)/(cotα-1) we obtain

*f*(

_{o}**r**

_{o}) = exp(

*iπ*(cotα-tanα)

*x*

_{o}*y*)/∣sinα∣ and

_{o}*f*(

_{o}**r**

_{o}) = exp(i2π

*x*

_{o}*y*)/∣sinα∣ respectively. Note that for

_{o}*c*= 0 (

*f*(

_{i}**r**

_{i}) = 1) the GT also corresponds to a hyperbolic wavefront as it is indicated at the third row of the Table 1.

*n*+1)/2 (

*f*(

_{o}**r**

_{o}) = exp

*iπ*

**r**

^{2}

_{o}/

*b*/

*ib*) and α =π

*n*(

*f*(

_{o}**r**

_{o}) = exp -

*iπb*

**r**

^{2}

_{o})), where

*n*is an integer.

*a*= 1 the additional phase shift vanishes and output function corresponds to the input function exp (-

*πr*

^{2}

_{o}). This result indicates that exp (-

*πr*

^{2}

_{o})is an eigenfunction of the GT for any transformation angle α.

10. M. Bastiaans and T. Alieva, “First-order optical systems with
unimodular eigenvalues,” J. Opt. Soc. Am.
A **23**, 1875–1883
(2006). [CrossRef]

*x*,

*y*) at π/4 and -π/4 correspondingly. From that follows (see reference [11

11. T. Alieva and M. Bastiaans, “Mode mapping in paraxial lossless
optics,” Opt. Lett. **30**, 1461–1463
(2005). [CrossRef] [PubMed]

*H*is the Hermite polynomial and

_{m}*w*is the beam waist, form the complete orthogonal set of eigenfunctions for the separable fractional FT for

*w*= 1 then the HG modes rotated at -π/4 form the set of the orthogonal eigenfunctions for the GT (row 7, Table 1).

3. E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and
nontransformed beams,” Opt. Commun. **83**, 123–135
(1991). [CrossRef]

4. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian
beams,” J. Opt. A.: Pure Appl. Opt. **6**, S157–S161
(2004). [CrossRef]

*HG*(

_{m,n}**r**;

*w*) for

*w*= 1 mode transforms into the helicoidal LG mode:

*L*is the Laguerre polynomial,

^{l}_{p}*p*= min(

*m*,

*n*) and

*l*= ∣

*m*-

*n*∣. The topological charge of the vortex mode is given by ±

*l*.

*HG*(

_{m,n}**r**;1) mode transforms to -

*LG*

^{-}

_{p,l}(

**r**;1) and -

*LG*

^{+}

_{p,l}(

**r**;1), respectively.

*f*(

_{i}**r**

_{i}) with periods

*k*

^{-1}

_{x},

*k*

^{-1}

_{y}can be written as a Fourier expansion

*l*=

*k*tanα

_{x}k_{y}_{l}, where

*l*is an integer. Then Eq. (19) is reduced to

*circ*(

*r*/ρ) (ρ = 1.6) for different transformation angles α = 0, 7π/36, π/4, 11π/36, π/2, (a-e) respectively. This image sequence Fig. 2(a-e) demonstrates the evolution from the input function Fig. 2(a) to its rotated Fourier transform obtained for α = π/2, Fig. 2(e). We observe how the rotational symmetry in the position (α = 0) and FT domain (α = π/2) changes to the rectangular one for other angles.

_{i}## 4. Gyrator transform applications

### 4.1. Hermite-Gaussian mode evolution under the gyrator transform

8. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for
ortho-symplectic transformations in phase space,”
J. Opt. Soc. Am. A **23**, 2494–2500
(2006). [CrossRef]

*nm*,

*w*= 0.73

*mm*, and spatial resolution 20

*μm*.

*HG*) modes of different orders to the helicoidal Laguerre-Gaussian (

_{m,n}*LG*) ones were considered [2

_{p,l}2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and
transfer of orbital angular momentum,”
Opt. Commun. **96**, 123–132
(1993). [CrossRef]

**6**, S157–S161
(2004). [CrossRef]

5. G. F. Calvo, “Wigner representation and geometric
transformations of optical orbital angular momentum spatial
modes,” Opt. Lett. **30**, 1207–1209
(2005). [CrossRef] [PubMed]

*HG*to

_{m,n}*LG*, and viceversa, is achieved when α =π/4+

_{p,l}*nπ*/2 (

*n*integer), meanwhile other modes are obtained for the rest of angle values if α ≠

*πn*/2.

*HG*mode of order

_{m,n}*m*= 1,

*n*= 0 to helicoidal

*LG*

_{p=0,l=1}is displayed for different values of angle α. The first and the second rows correspond to the intensity and phase distribution, respectively. The intensity distribution is normalized to the maximum intensity value of the input signal (α = 0), and phase values are represented for [-π,π ] region. Mode conversion from

*HG*

_{1,0}(α = 0) to

*LG*

^{±}

_{0,1}is obtained for α = π/4, 3π/4, 5π/4,7π/4, as it has been explained in Section 3. For α = π/2, π, 3π/2 the output mode corresponds to

*HG*

_{1,0}rotated at π/2, π, 3π/2 with additional phase shift exp(

*i*2α), respectively. Therefore the mode conversion from

*HG*

_{1,0}to

*HG*

_{0,1}is obtained for α = π/2, 3π/2. In Fig. 3(b) the intermediate modes obtained by the GT of

*HG*

_{1,0}for α = [0, π/4] are displayed. The modes obtained for every particular angle α are stable and possess fractional OAM [5

**30**, 1207–1209
(2005). [CrossRef] [PubMed]

### 4.2. Influence of scaling and shift properties to mode transformation

*s*=

_{x}*s*=

*s*

^{-1}

_{y}in order to avoid the change of the transformation angle and additional phase modulation, we observe (see Fig. 4) that for α = π/4 the transformation of the rotational symmetric intensity distribution typical for the LG mode into elliptical one (Fig. 4c, Fig. 4f). Figures 4 (a) and (d) correspond to the input signal

*HG*

_{1,0}scaled by

*s*= 1/2 and

*s*= 2, respectively. Figures 4 (b, c) and (e, f) are the corresponding output modes for α =π/5 and α =π/4. Therefore the GT of scaled HG mode is an alternative for generation of the elliptic LG beams [14

14. V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, “Elliptic Laguerre-Gaussian
beams,” J. Opt. Soc. Am. A **23**, 43–56,
(2006). [CrossRef]

*x*s we can apply the shifting theorem, Eq. (6), to obtain the output function. Notice that if the input signal is shifted at

_{i}**v**

^{t}= (

*v*,

_{x}*v*) the output signal is shifted at

_{y}**v**cosα and affected by an additional linear phase modulation. The GT at angle α = π/5 and α = π/4 of

^{t}*HG*

_{1,0}for different shifting parameters is displayed in Fig. 5. Figures 5 (b, c) and (e, f) are the output modes obtained from the

*HG*

_{1,0}mode shifted by

**v**= (1

^{t}*mm*,0) (Fig. 5a) and

**v**= (1

^{t}*mm*,-1

*mm*) (Fig. 5c), respectively.

### 4.3. Gyrator transform of HG mode composition

*n*+

*m*=

*const*) also produces a stable configuration after gyrator transformation. Thus for example the combination of

*HG*

_{3,0}and

*HG*

_{0,3}modes:

*HG*3 +

*HG*

_{0,3}(Fig. 6a) leads for α=π/4 to the odd Laguerre-Gaussian beams, which is the sum of two helicoidal LG modes with opposite OAMvalues:

*LG*

^{+}

_{0,3}+

*LG*

^{-}

_{0,3}(Fig. 6e). For other angles α =π/8, π/5, 2π/9 the intermediate modes are obtained (Fig. 6 b, c, d).

## Appendix

### Shift theorem for gyrator transform

**T**(α) as we have mentioned previously, Eq. (2) and (3). Therefore the Eq. (1) can be rewritten as follows:

*t*stands for transposition operation. In order to demonstrate the shift theorem it is suitable to apply this equation (21). Considering that the input function is affected by a shift which is indicated by means of the vector

**v**= (

^{t}*v*,

_{x}*v*), where

_{y}**u**=

**r**

_{i}-

**v**, we derive

*iπϕ*) is simplified as

**I**is a unity 2×2 matrix) have been used

**r**

_{o}is replaced by

**r**

_{o}-

**Xv**. Doing this we obtain the shift theorem as it was formulated in (6)

**v**͂ = (

*v*,

_{y}*v*). Finally, it is demonstrated that the shift of the function

_{x}*f*at vector

_{i}**v**= (

^{t}*v*,

_{x}*v*) leads to the shift of its GT (for the angle α) at vcosα and additional linear phase modulation.

_{y}### Scaling theorem for gyrator transform

*f*(

_{i}**Sr**

_{i}) =

*f*(

_{i}*s*, sy

_{x}x_{i}*y*). Therefore applying a change of variable

_{i}*x*´

_{i}=

*s*,

_{x}x_{i}*y*´

_{i}=

*s*for the equation Eq. (1), we obtain:

_{y}y_{i}*s*, then the last equation is rewritten as follows:

_{x}s_{y}_{α}= sgn(sinα), σ

_{β}= sgn(sinβ). In order to obtain this equation the definition of the GT and simple trigonometric relations have been used.We conclude that the GT at angle α of a scaled input function

*f*(

_{i}**Sr**

_{i}), corresponds to the GT at angle β of the initial function

*fi*(

**r**

_{i}) with additional scaling of the output coordinates and affected by a hyperbolic phase modulation.

### Gyrator transform of selected functions, Table 1

*c*= 0) (row 3, Table 1) is calculated as follows:

*x*´

_{o}=

*x*+

_{o}*k*sinα, and

_{y}*y*´

_{o}=

*y*+

_{o}*k*sinα and the Eq. (29) allow to calculate the GT of a plane wave (row 4, Table 1).

_{x}*f*(

_{i}**r**

_{i})= exp (γ

**r**

^{2}

_{i}, where γ = -π (

*a*+

*ib*) and

*a*≥0. According to Eq. (1) the GT of

*f*(

_{i}**r**

_{i}) = exp (γ

**r**

^{2}

_{i}) is given by

*y*. Therefore Re(γ) ≤ 0 must be satisfied. Using a change of variable

_{i}*t*=

*x*√γ, we again apply the Eq. (33) for integration with respect to

_{i}*x*and derive that

_{i}*d*= 1+(πcot(α)/γ)

^{2}. Then the GT of

*f*(

_{i}**r**

_{i}) = exp (γ

**r**

^{2}

_{i}is given by

## Acknowledgements

## References and links

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

2. | M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and
transfer of orbital angular momentum,”
Opt. Commun. |

3. | E. G. Abramochkin and V. G. Volostnikov, “Beam transformations and
nontransformed beams,” Opt. Commun. |

4. | E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian
beams,” J. Opt. A.: Pure Appl. Opt. |

5. | G. F. Calvo, “Wigner representation and geometric
transformations of optical orbital angular momentum spatial
modes,” Opt. Lett. |

6. | R. Simon and K. B. Wolf, “Structure of the set of paraxial
optical systems,” J. Opt. Soc. Am. A |

7. | K. B. Wolf, Geometric Optics on Phase Space, Springer-Verlag, Berlin (2004). |

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

9. | T. Alieva, V. Lopez, F. Agullo Lopez, and L. B. Almeida, “The fractional Fourier transform in
optical propagation problems,” J. Mod.
Opt. |

10. | M. Bastiaans and T. Alieva, “First-order optical systems with
unimodular eigenvalues,” J. Opt. Soc. Am.
A |

11. | T. Alieva and M. Bastiaans, “Mode mapping in paraxial lossless
optics,” Opt. Lett. |

12. | M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions, Frankfurt am Main, Germany (1984). |

13. | I. S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, Academic Press, NY, USA (1996). |

14. | V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs, and J. Turunen, “Elliptic Laguerre-Gaussian
beams,” J. Opt. Soc. Am. A |

**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 Optical Signal Processing

**History**

Original Manuscript: December 7, 2006

Revised Manuscript: January 16, 2007

Manuscript Accepted: January 22, 2007

Published: March 5, 2007

**Citation**

José A. Rodrigo, Tatiana Alieva, and María L. Calvo, "Gyrator transform: properties and applications," Opt. Express **15**, 2190-2203 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2190

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### References

- H. M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons, NY, USA (2001).
- M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123-132 (1993). [CrossRef]
- E. G. Abramochkin and V. G. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991). [CrossRef]
- E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A.: Pure Appl. Opt. 6, S157- S161 (2004). [CrossRef]
- G. F. Calvo, "Wigner representation and geometric transformations of optical orbital angular momentum spatial modes," Opt. Lett. 30, 1207-1209 (2005). [CrossRef] [PubMed]
- R. Simon and K. B. Wolf, "Structure of the set of paraxial optical systems," J. Opt. Soc. Am. A 17, 342-355 (2000). [CrossRef]
- K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, Berlin, 2004).
- J. A. Rodrigo, T. Alieva, M. L. Calvo, "Optical system design for ortho-symplectic transformations in phase space," J. Opt. Soc. Am. A 23, 2494-2500 (2006). [CrossRef]
- T. Alieva, V. Lopez, F. Agullo Lopez, L. B. Almeida, "The fractional Fourier transform in optical propagation problems, " J. Mod. Opt. 41, 1037-1044 (1994). [CrossRef]
- M. Bastiaans and T. Alieva, "First-order optical systems with unimodular eigenvalues," J. Opt. Soc. Am. A 23, 1875-1883 (2006). [CrossRef]
- T. Alieva and M. Bastiaans, "Mode mapping in paraxial lossless optics," Opt. Lett. 30, 1461-1463 (2005). [CrossRef] [PubMed]
- M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions, Frankfurt am Main, Germany (1984).
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products (Academic Press, NY, USA, 1996).
- V. V. Kotlyar, S. N. Khonina, A. A. Almazov, V. A. Soifer, K. Jefimovs and J. Turunen, "Elliptic Laguerre- Gaussian beams," J. Opt. Soc. Am. A 23, 43-56, (2006). [CrossRef]

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