Page 1

Gyrator Transform: properties and

applications

José A. Rodrigo, Tatiana Alieva, María L. Calvo

Abstract:

operation which produces a rotation in the twisting (position - spatial fre-

quency) phase planes. This transform can be easily performed in paraxial

optics that underlines its possible application for image processing, hologra-

phy, 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

In this work we formulate the main properties of the gyrator

Universidad Complutense de Madrid, Facultad de Ciencias Físicas,Ciudad Universitaria s/n,

Madrid 28040, Spain.

OCIS codes: (070.2590) Fourier transforms; (120.4820) Optical systems; (200.4740) Optical

processing; (140.3300) Laser beam shaping

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. 96, 123-132 (1993).

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

83, 123-135 (1991).

4. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A.: Pure Appl. Opt. 6, S157-

S161 (2004).

5. G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial

modes,” Opt. Lett. 30, 1207-1209 (2005).

6. R. Simon and K. B. Wolf , “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17, 342-355

(2000).

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

8. 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).

9. 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).

10. M. Bastiaans and T. Alieva, “First-order optical systems with unimodular eigenvalues,” J. Opt. Soc. Am. A 23,

1875-1883 (2006).

11. T. Alieva and M. Bastiaans, “Mode mapping in paraxial lossless optics,” Opt. Lett. 30, 1461-1463 (2005).

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. Je?movs and J. Turunen, “Elliptic Laguerre–

Gaussian beams,” J. Opt. Soc. Am. A 23, 43-56, (2006).

1. Introduction

Many interesting applications of the ?rst-order optical systems for information processing have

been proposed in the past decade. Some particular ?rst-order optical systems, performing frac-

tional Fourier transform, are used for shift-variant ?ltering, noise reduction, encryption [1].

Another ones serve as mode converters which permit to obtain the helicoidal vortex Laguerre-

Gaussian (LG) mode after the propagation of the Hermite-Gaussian (HG) beam through these

Page 2

Fig. 1. Graphical representation for the phase structure associated to the gyrator kernel for

α = π=4, xo= yo= 0 (a) and 2xo= yo= 1 (b). These ?gures (a) and (b) correspond to the

exponential argument of the kernel.

systems [2, 3, 4]. Besides LG modes, other stable modes carrying the fractional topological

charge [4, 5] can be obtained by generalized mode converter which can be described by the

gyrator transform (called in [6, 7] as a cross-gyrator). The gyrator transform as well as the frac-

tional Fourier transform belong to the class of the linear canonical integral transforms. But in

the contrast to the fractional Fourier transform (see for example [1] and references there in), the

gyrator operation is still little known for the optical community. The purpose of this paper is

to establish the main properties of the gyrator transform that opens the perspective of its appli-

cation for optical information processing together with fractional Fourier transform. Including

the gyrator transform in the list of image processing tools we enlarge a number of phase space

domains for more appropriate image representation, ?ltering operation, holographic recording

etc. As an example of gyrator action we demonstrate its application as a generator of stable

modes living on the main meridian of the Poincaré spheres [5].

Gyrator operation is mathematically de?ned as a linear canonical integral transform which

produces the rotation in position?spatial frequency planes (x;qy) and (y;qx) [6, 7] of phase

space. Thus the gyrator transform (GT) at parameter α, which will be called below as a rotation

angle, of a two-dimensional function fi(ri), associated in ?rst order optics with complex ?eld

amplitude, can be written in the following form

ZZ

=

sinα

fo(ro) = Rα[fi(ri)](ro) =

fi(xi;yi)Kα(xi;yi;xo;yo)dxidyi

?

1

ZZ

fi(xi;yi) expi2π(xoyo+xiyi)cosα ?(xiyo+xoyi)

sinα

?

dxidyi;

(1)

where rt

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(ro+ri) 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 Kα(xi;yi;xo;yo) has a constant amplitude and a hyperbolic phase

structure, which is shown in Fig. 1 for the angle α = π=4 and output coordinates xo= yo= 0

(see Fig. 1(a)) and 2xo= yo= 1 (see Fig. 1(b)).

Since the GT belongs to the class of the linear integral canonical transforms its kernel is

parametrized by the symplectic 4?4 matrix T(α) [6, 7]

?

where

X =

0cosα

i;o= (xi;o;yi;o) indicates the input and output coordinates, respectively. Notice that t

ro

qo

?

=

?

XY

X

?Y

??

ri

qi

?

= T(α)

?

ri

qi

?

;

(2)

?

cosα

0

?

;

Y =

?

0sinα

0sinα

?

;

(3)

which describes in the paraxial approximation the ray transformation in this system. Notice that

rt= (x;y) is the ray position and qt= (qx;qy) is the ray slope, and bold capital here and further

indicates matrix notation.

Based on the matrix formalism for ?rst-order lossless optical systems, it has been recently

shown [8] that the GT for the large range of angles α can be realized by an optimized ?exible

Page 3

optical system which contains only three generalized lenses with ?xed distance between them.

The angle α is changed by rotation of the cylindrical lenses which form the generalized lenses.

Other possibility is to use the spatial light modulator for variable lens performance. Explicit

equations for these generalized lenses as a function of the transformation angle α can be found

in [8].

2.Basic properties of the gyrator operation

In order to work properly with the GT and to design the corresponding optical system for its

experimental realization we need to know its basic properties. As in the case of the Fourier

transform (FT) or the fractional FT [9] the main theorems such as scaling, shift, modulation,

etc. have to be formulated.

From the equations Eq. (1) - (3) it is easy to see that the GT is periodic and additive with

respect to parameter α. The last can be proved directly by the multiplication of the ray trans-

formation matrices which parametrized the kernel: 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[fi(xi;yi)](xo;yo) = Rα[fi(?xi;yi)](?xo;yo);

and then Rα[Rα[fi(?xi;yi)](?xo;yo)](r) = fi(r).

ItisknownthattheParsevalrelationholdsforentireclassofthecanonicalintegraltransforms

and therefore for the gyrator operation which belongs to it. It is easy to demonstrated this

relation for the case of the gyrator operation, as it is shown in Eq. (5):

(4)

Z

Rα[fi(ri)](ro)(Rα[gi(re)](ro))?dro=

ZZ

fi(ri) exp

?

i2π(xoyo+xiyi)cosα ?(xiyo+xoyi)

sinα

?

i2π(xiyi?xeye)cosα

sinα

?i2π(ri?re)teIro

?

=

fi(ri)gi(ri)?dri;

?

dri

?

1

sin2α

Z

g?

i(re) exp

?

i2π?(xoyo+xeye)cosα +(xeyo+xoye)

sinα

?

i2π(xiyi?xeye)cosα

sinα

Z

dredro

=

1

sin2α

Z Z

Z

fi(ri)g?

i(re)exp

?

?

exp

sinα

!

drodridre

=

ZZ

fi(ri)g?

i(re)exp

?

δ(ri?re)dridre

(5)

where the matrixeI is given by

eI =

?

0

1

1

0

?

:

(6)

The shift of the function fiat vector vt= (vx;vy) leads to the shift of its GT (for the angle α) at

vcosα and additional linear phase modulation:

Rα[fi(ri?v)](ro) = exp?iπ(vxvysin2α ?2rt

where e vt= (vy;vx), see appendix for more details. We observe that the shift of the amplitude

oe vsinα)?Rα[fi(ri)](ro?vcosα);

(7)

of the GT jRα[fi(ri?v)](ro)j = jRα[fi(ri)](ro?vcosα)j is the same as for the case of two

dimensional symmetric fractional FT at angle α [9].

Page 4

The effect of plane wave modulation exp(?i2πktri) of the function fi(ri) is also similar to

the fractional FT case. It leads to the shift of its GT (for the angle α) at ?eksinα and additional

Rα[fi(ri)exp??i2πktri

where kt= (kx;ky) andekt= (ky;kx).

Rα[fi(Sri)](ro) =cosβ

cosαexp

linear phase modulation:

?](ro)=exp??iπ?kxkysin2α +2ktrocosα??Rα[fi(ri)]

?

ro+eksinα

?

;

(8)

Scaling theorem can be formulated in the following form (see appendix section):

i2πxoyo

1?

?cosβ

cosα

?2!

cotα

!

Rβ[fi(ri)]

?cosβ

cosαSro

?

; (9)

where

S =

?

sx

0

0

sy

?

and cotβ =cotα

sxsy

:

(10)

It means that the GT at angle α of the scaled function fi(Sri) corresponds to the GT at angle

β of the initial function fi(ri) 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 aper-

ture 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 scal-

ing parameters sx= s = s?1

y

and sx= s = ?s?1

reduced.

Thus if sx= 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

y

when the expression Eq. (9) is signi?cantly

Rα[fi(xis;yis?1)](ro) = Rα[fi(ri)](xos;yos?1):

(11)

This scaling property will be demonstrated in section 4.1 in application to generation of ellipti-

cal vortex beams.

If sx=s=?s?1

The Eq. (9) can be written as

Rα[fi(xis;?yis?1)](ro) = ?Rπ?α[fi(ri)]??xos;s?1yo

or using the additive property of the GT as

Rα[fi(xis;?yis?1)](ro) = ?R?α[fi(ri)]?xos;?s?1yo

In particular for s = 1 we obtain the expression similar to Eq. (4).

y

then the angles relation reduces to cotβ =?cotα and therefore β =π?α.

?;

?:

(12)

(13)

Rα[fi(xi;?yi)](ro) = ?R?α[fi(ri)](xo;?yo):

Gyrator transform of selected functions

(14)

3.

As it occurs for the Fourier transform the GT of only selected functions can be expressed an-

alytically. The fundamental functions: Dirac delta, 1, hyperbolic wave, plane wave, spherical

wave, Gaussian and Hermite-Gaussian mode and their GTs are displayed in Table 1. The fol-

lowing notations are used along the Table: vt= (vx;vy), kt= 2π(kx;ky), a > 0, b and c are real

numbers, and ℜ?π

4 is the operator of coordinate rotation at angle ?π=4.

Page 5

fi(ri)

fo(ro) = Rα[fi(ri)](ro)

?

sinαexp?i2πccotα?1

1

sinαexp(?i2πxoyotanα)

sinαexp(?i2π(xoyo+kxky)tanα)exp??

1

p

δ(ri?v)

1

sinαexpi2π(xoyo+vxvy)cosα?(vxyo+xovy)

sinα

?

exp(i2πcxiyi)

1

c+cotαxoyo

?; (c 6= ?cotα)

1

exp(?iktri)

exp??iπbr2

exp??πar2

HGm;n

1i

cosαktro

?

?

?

i

?

cos2α?b2sin2αexp

?

?

?iπ(1+b2)sin2α

cos2α?b2sin2αxoyo

exp

?

?iπbr2

o

cos2α?b2sin2α

?

i

?

4ri;1

1

p

cos2α+a2sin2αexpiπ(a2?1)sin2α

cos2α+a2sin2αxoyo

exp

?

?πar2

o

cos2α+a2sin2α

?

?

ℜ?π

?

eiα(n?m)HGm;n

Table 1. Selected functions and their gyrator transforms

?

ℜ?π

4ro;1

?

Let us consider in detail some particular cases from Table 1 (see appendix for intermediate

calculation). The ?rst row of Table 1 shows that the GT for δ(ri?v) corresponds to the gyrator

kernel as the output function, Kα(ri= v;ro), and therefore the product of hyperbolic and plane

waves.

Correspondingly the GT of a hyperbolic wave (see row 2, Table 1) transforms to Dirac func-

tion for angle such that cotα = ?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,

fo(ro) = (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 ex-

pressions for the GT of hyperbolic wave are simpli?ed. Thus for the values of parameter c =

cotα and c = (1+cotα)=(cotα ?1) we obtain fo(ro) = exp(iπ(cotα ?tanα)xoyo)=sinα

and fo(ro) = exp(i2πxoyo)=sinα respectively. Note that for c = 0 (fi(ri) = 1) the GT also

corresponds to a hyperbolic wavefront as it is indicated at the third row of the Table 1.

The gyrator transform of a plane wave (row 4, Table1) corresponds to a product of the plane

wave, with spatial frequency scaled by 1=cosα and the hyperbolic wave.

For the spherical wavefront (row 5, Table 1) its GT corresponds to a product of the spher-

ical wave, affected by the scaling factor and the hyperbolic wave. The hyperbolic contribu-

tion cancels for angles corresponding to position and rotated FT domains α = π(2n+1)=2

(fo(ro) = exp?iπr2

hyperbolic phase modulation. In the case a = 1 the additional phase shift vanishes and output

function corresponds to the input function exp??πr2

It has been shown in [10] that the GT at angle α can be represented as a fractional separable

Fourier transform at angles (α;?α) with rotation of the input and output coordinates (x;y) at

π=4 and ?π=4 correspondingly. From that follows (see reference [11]) that the eigenfunctions

for the GT are the eigenfunctions of the fractional FT rotated at angle ?π=4. Since the Hermite

Gaussian modes:

?p2πx

o=b?=ib) and α = πn (fo(ro) = exp??iπbr2

o

?), where n is an integer.

?. This result indicates that exp??πr2

The GT of a Gaussian function (row 6, Table 1) corresponds to the Gaussian function with

oo

?is

an eigenfunction of the GT for any transformation angle α.

HGm;n(r;w) = 21=2Hm

w

?Hn

?p2πy

w

?

p2mm!wp2nn!w

exp

?

?π

w2r2?

;

(15)

Page 6

where Hmis the Hermite polynomial and 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).

For α = ?π=4 the kernel of the GT is reduced to

K?π=4(xi;yi;xo;yo) = ?p2exp

?

?i2π

h

xoyo+xiyi?p2(xiyo+xoyi)

i?

:

(16)

In this case as it was shown (for example in [3, 4]) that the HGm;n(r;w) for w = 1 mode

transforms into the helicoidal LG mode:

s

max(m;n)!

LG?

p;l(r;w) = w?1

min(m;n)!

?p2π

?x

w?iy

w

??lLl

p

?2π

w2r2

?

exp

?

?π

w2r2?

;

(17)

where Llpis the Laguerre polynomial, p = min(m;n) and l = jm?nj. The topological charge of

the vortex mode is given by ?l.

For the transformation angle α = 3π=4; 5π=4 as it follows from Eq. (4) and Eq. (12)

HGm;n(r;1) mode transforms to ?LG?

Finally we consider the GT of periodic functions. It is well-known that a periodic function

fi(ri) with periods k?1

y

can be written as a Fourier expansion

p;l(r;1) and ?LG+

p;l(r;1), respectively.

x, k?1

fi(ri) =∑

n;m

an;mexp(?i2π(xikxn+yikym)):

(18)

Then the GT of a periodic function is given by

fo(ro) = Rα[fi(ri)](ro) =∑

n;m

an;mRα[exp(?i2π(xikxn+yikym))](ro):

(19)

Using the expression for the GT of a plane wave (row 4, Table 1) we derive that

fo(ro) =exp(?i2πxoyotanα)

sinα

∑

n;m

an;mexp(?i2πnmkxkytanα)exp

?

?i2πnkxxo+mkyyo

cosα

?

(20)

:

An interesting result is obtained for angles which satisfy the relation l = kxkytanαl, where l is

an integer. Then Eq. (20) is reduced to

fo(ro) =exp(?i2πxoyotanαl)

sinαl

fi

?

1

cosαlro

?

;

(21)

which can be considered as a Talbot effect for the gyrator transform.

Finally as an example, ?gure 2 shows the squared moduli (intensity distribution in the case

of optical realization) of the GT for the circle function circ(ri=ρ) (ρ = 1:6) for different trans-

formation 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.

Fig. 2. Intensity distributions corresponding to the GT of the circle function are displayed

for different transformation angles α = 0 (a),7π=36 (b), π=4 (c), 11π=36 (d), and π=2 (e).

Note that for α = π=2 the rotated Fourier transform is obtained.

Page 7

4.Gyrator transform applications

The above mentioned properties of the GT make it a useful tool for optical information process-

ing. The GT provides an image representation in a new phase-space domain which was not

explored yet for signal analysis and synthesis. In particular it can be used for hyperbolic wave

detection, shift-variant ?ltering, encryption, beam characterization, generation of stable modes

with speci?c properties. The application of the GT for all these tasks certainly demands exten-

sive studies. Here we will consider only the mode transformation under the GT. In particular

we will consider the gyrator transformation of the Hermite-Gaussian modes. There is a double

interest to this modes. First of all they appear as a natural modes in laser resonators of rectan-

gular symmetry and propagate in a free space without changing their intensity form. On the

other hand the HG modes forms a complete orthonormal set and therefore often are used as a

basis for image representation. The GT of the HG modes generates other stable modes, which

also propagate in free space without changing their intensity form and the knowledge of these

modes permits to represent any image in the corresponding GT domain.

4.1. Hermite-Gaussian mode evolution under the gyrator transform

Let us consider the evolution oh the HG mode (15) under the GT. Since the GT for different

angles can be performed by optical system constructed from three generalized lenses (assem-

bled set of cylindrical lenses) and two ?xed free space intervals [8] the numerical simulations of

the GT can follow this recipe. Using free space propagation algorithm under Fresnel diffraction

regime and phase modulation functions for the generalized lenses we calculated the output pat-

terns for the GT system. The parameters used in these numerical simulations are the following:

wavelength λ = 532nm, w = 0:73mm, and spatial resolution 20µm.

During last decade various optical schemes were proposed for the generation of the vortex

beams which carry the orbital angular momentum (OAM). Mostly the conversion of Hermite-

Gaussian (HGm;n) modes of different orders to the helicoidal Laguerre-Gaussian (LGp;l) ones

were considered [2]. It was also shown that it is possible to generate the stable modes with

fractional orbital angular momentum [4, 5]. The GT can be seen as a ?exible mode converter

where modes of the same order but different OAM are obtained by varying the angle α. As it

was indicated in the previous section the mode conversion from HGm;nto LGp;l, and viceversa,

is achieved when α = π=4+nπ=2 (n integer), meanwhile other modes are obtained for the rest

of angle values if α 6= πn=2.

In Figure 3 the mode conversion from HGm;nmode of order m = 1; n = 0 to helicoidal

LGp=0;l=1is displayed for different values of angle α. The ?rst and the second rows corre-

spond to the intensity and phase distribution, respectively. The intensity distribution is nor-

malized to the maximum intensity value of the input signal (α = 0), and phase values are

represented for [?π;π] region. Mode conversion from HG1;0(α = 0) to LG?

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 HG1;0rotated at π=2; π;3π=2 with additional phase

shift exp(i2α), respectively. Therefore the mode conversion from HG1;0to HG0;1is obtained

for α = π=2; 3π=2. In Fig. 3(b) the intermediate modes obtained by the GT of HG1;0for

α = [0; π=4] are displayed. The modes obtained for every particular angle α are stable and

possess fractional OAM [5]. In general the action of the GT is associated with the movement

along the main meridian of the Poincaré spheres.

0;1is obtained

4.2.In?uence of scaling and shift properties to mode transformation

Let us now consider how the scaling of the input HG mode affects on the mode generation.

Based on the scaling theorem (9) and choosing scaling parameters sx= s = s?1

avoid the change of the transformation angle and additional phase modulation, we observe (see

y

in order to

Page 8

Fig. 3. Intensity (up row) and phase (low row) of the GT of

different angles α. Figure (a) corresponds to transformation angle α = 0; π=4;

π=2; 3π=4; π; 5π=4;3π=2;7π=4. (b) Intermediate sequence between angle α = 0 and

α = π=4 is displayed. (2.5 MB) Movie: mode transformation for different angles α, where

the input mode is HG1;0.

HG1;0 mode for

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 HG1;0scaled 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].

Fig. 4. Intensity (up row) and phase (low row) for different angles of the GT of HG1;0

affected by scaling factors sx= s = s?1

y: s = 1=2 (a, b, c) and s = 2 (d, e ,f), respectively.

When the input function is not centred at the optical axis we can apply the shifting theorem,

Eq. (7), to obtain the output function. Notice that if the input signal is shifted at vt=(vx;vy) the

output signal is shifted at vtcosα and affected by an additional linear phase modulation. The

GT at angle α = π=5 and α = π=4 of HG1;0for different shifting parameters is displayed in

Fig. 5. Figures 5 (b, c) and (e, f) are the output modes obtained from the HG1;0mode shifted

by vt= (1mm;0) (Fig. 5a) and vt= (1mm;?1mm) (Fig. 5c), respectively.

Fig. 5. Intensity (up row) and phase (low row) of the GT (for the angle α) of HG1;0mode

shifted by vt= (1mm;0) (a, b, c) and vt= (1mm;?1mm) (d, e, f).

4.3.Gyrator transform of HG mode composition

UptillnowwehaveconsideredthetransformationofonlyoneHGmode.Neverthelessthecom-

position of HG modes of the same order (n+m = const) also produces a stable con?guration

after gyrator transformation. Thus for example the combination of HG3;0and HG0;3modes:

HG3;0+HG0;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 OAM values: LG+

angles α = π=8; π=5; 2π=9 the intermediate modes are obtained (Fig. 6 b, c, d).

0;3+LG?

0;3(Fig. 6e). For other

5.Conclusion

A little known operation for two-dimensional signal manipulation, called gyrator transform,

has been studied. The main properties of the GT such as shift, scaling, plane wave modulation,

Parseval theorem, and other relevant properties have been formulated. The GTs of the selected

functions have been also found. The gyrator operation promises to be a useful tool in image

processing, holography, beam characterization, quantum information, new mode generation,

etc. For example, here it has been demonstrated its application for stable mode generator. The

Fig. 6. Intensity (up row) and phase (low row) distributions for GT of the HG3;0+HG0;3

input mode are displayed for different angles α = 0 (input mode);π=8; π=5; 2π=9; π=4

(LG+

the input mode is HG3;0+HG0;3.

0;3+LG?

0;3mode). (2.8 MB) Movie: mode transformation for different angles α, where

Page 9

experimental scheme for optical implementation of the GT is now under construction. The pre-

liminary results, not displayed in this paper, demonstrate very good agreement with theoretical

predictions.

Acknowledgements

Spanish Ministry of Education and Science is acknowledged for ?nancial support, project TEC

2005-02180/MIC.

6.Appendix

In this appendix we demonstrate the shift and scaling theorems associated to the gyrator trans-

formation and present the main intermediate calculations for the GT of the selected functions

from the Table 1, which were discussed previously.

6.1.

The kernel of the GT (Eq. (1)) is parametrized by the symplectic matrix T(α) as we have

mentioned previously, Eq. (2) and (3). Therefore the Eq. (1) can be rewritten as follows:

ZZ

Here t stands for transposition operation. In order to demonstrate the shift theorem it is suitable

to apply this equation (22). Considering that the input function is affected by a shift which is

indicated by means of the vector vt= (vx;vy), where u = ri?v, we derive

1

sinα

1

sinα

1

sinα

The kernel exp(iπφ) is simpli?ed as

exp(iπφ) = exp?iπ(vtYXv+rt

Shift theorem for gyrator transform

fo(ro) = Rα[fi(ri)](ro) =

1

sinα

fi(ri) exp?iπ?rt

iY?1Xri?2rt

iY?1ro+rt

oXY?1ro

??dri:

(22)

fo(ro) =

ZZ

ZZ

ZZ

fi(ri?v) exp?iπ?rt

fi(u) exp?iπ?(u+v)tY?1X(u+v)?2(u+v)tY?1ro+rt

fi(u) exp(iπφ) du:

iY?1Xri?2rt

iY?1ro+rt

oXY?1ro

??dri

=

oXY?1ro

??du

(23)

=

o[(XY?1X?(Y?1)t?Y]v)??

exp?iπ?rt

iY?1Xri?2rt

iY?1(ro?Xv)+(ro?Xv)tXY?1(ro?Xv)??;

vtY?1Xu = ut?Y?1X?tv = utXt?Y?1?tv;

XtX+YtY = I:

(24)

where the following relations (I is a unity 2?2 matrix) have been used

XYt= YXt;

(25)

The equation (24) has two exponential functions. The ?rst one corresponds to an additional

phase factor that can be extracted from the integral in Eq. (23), and the second one corresponds

to the gyrator kernel where the coordinate rois replaced by ro?Xv. Doing this we obtain the

shift theorem as it was formulated in (7)

fo(ro) = Rα[fi(ri?v)](ro)

= exp?iπ(vtYXv+rt

o[(XY?1X?(Y?1)t?Y]v)?Rα[fi(ri)](ro?Xv)

= exp(iπ(vxvysin2α ?2roe vsinα))Rα[fi(ri)](ro?vcosα);

(26)

Page 10

where e v = (vy;vx). Finally, it is demonstrated that the shift of the function fiat vector vt=

modulation.

(vx;vy) leads to the shift of its GT (for the angle α) at vcosα and additional linear phase

6.2.Scaling theorem for gyrator transform

For the case of the scaling theorem the input function is affected by a scaling factor fi(Sri) =

fi(sxxi;syyi). Therefore applying a change of variable x0

(1), we obtain:

i= sxxi, y0

i= syyifor the equation Eq.

fo(ro) = Rα[fi(Sri)](ro)

=exp(i2πxoyocotα)

sxsysinα

ZZ

fi(x0

i;y0

i) exp

?

i2π

?

x0

iy0

i

cotα

sxsy

?

1

sinα

?x0

iyo

sx

+xoy0

i

sy

???

dx0

idy0

i:

(27)

The next step is to de?ne cotβ = cot(α)=sxsy; then the last equation is rewritten as follows:

?

=exp(i2πxoyocotα)sinβ

sxsysinα

fo(ro) =exp(i2πxoyocotα)

sxsysinα

ZZ

fi(x0

i;y0

i) expi2π

?

x0

iy0

icotβ ?

1

sinα

?x0

iyo

sx

?xosinβ

?

+xoy0

i

sy

???

dx0

idy0

i

exp

?i2πcotβyoxosin2β

?cosβ

sxsysin2α

?2!!

!

Rβ[fi(ri)]

sysinα;yosinβ

sxsinα

?

=cosβ

cosαexpi2πxoyocotα

1?

cosα

Rβ[fi(ri)]

?cosβ

cosαSri

;

(28)

where the de?nition of the GT and simple trigonometric relations have been used. We conclude

that the GT at angle α of a scaled input function fi(Sri), corresponds to the GT at angle β of

the initial function fi(ri) with additional scaling of the output coordinates and affected by a

hyperbolic phase modulation.

6.3.Gyrator transform of selected functions, Table 1

Here we present the main intermediates calculations for the GT of the functions from Table 1.

The GT of the Dirac delta function (row 1, Table 1) is obtained directly applying the proper-

ties of the δ-function.

The GT of hyperbolic wavefront (row 2, Table 1) and the constant function 1 in particular

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

fo(ro) = Rα[exp(i2πcxiyi)](ro)

=exp(i2πxoyocotα)

sinα

ZZ

Z

Z

exp

exp

?

?i2πxoyi

?

?i2π

(c+cotα)sin2α

i2π

?

xiyi(c+cotα)?

?

?i2πxoyi

sinα

xoyo

1

sinα(xiyo+xoyi)

?

yi(c+cotα)?

?

??

dxidyi

=exp(i2πxoyocotα)

sinα

=exp(i2πxoyocotα)

sinα

=exp(i2πxoyocotα)

sinα

exp

?

sinα

dyi

Z

exp

?

i2πxi

yi(c+cotα)?

yo

sinα

yo

sinα

??

dxi

exp

?

?

δ

??

dyi

;

(29)

Page 11

where we used that

δ (v) =

Z

exp(i2πvx) dx:

(30)

The last expression, Eq. (29), is simpli?ed using the trigonometric relations

Rα[exp(i2πcxiyi)](ro) =

1

sinαexp

?

i2πccotα ?1

c+cotαxoyo

?

:

(31)

A simple change of variable: x0o= xo+kysinα, and y0o= yo+kxsinα and the Eq. (30) allow

to calculate the GT of a plane wave (row 4, Table 1).

The next two functions correspond to a spherical wavefront and a Gaussian function, (row 5

and 6 Table 1, respectively) and can be derived as particular cases of the GT of function fi(ri)=

exp?γr2

fo(xo;yo) =

sinαexp(i2πxoyocotα)go(xo;yo);

where

ZZ

=

sinα

rπ

The last expression in Eq. (33) has been obtained using the following equation

i

?, where γ = ?π(a+ib) and a ? 0. According to Eq. (1) the GT of fi(ri) = exp?γr2

1

i

?

is given by

(32)

go(xo;yo) =

exp?γr2

exp?x2

?γ

i

?exp

?

?i2πxiyo

iγ?exp

i2π

?

xiyicotα ?

?

?π2

1

sinα(xiyo+xoyi)

exp?y2

xicotα ?

sinα

??

?

?i2πxiyo

dxidyi

Z

iγ?exp

exp?x2

?

dxi

Z

iγ?exp

xo

?

i2π

xicotα ?

xo

sinα

?

?

yi

?

dyi

=

Z

γ

??2?

exp

?

sinα

dxi:

(33)

Z

exp?µx2+βx?dx =

rπ

?µexp

?

?β2

4µ

?

;

(34)

where Re(µ) ? 0, for calculation the integral with respect to yi: Therefore Re(γ) ? 0 must be

satis?ed. Using a change of variable t = xipγ, we again apply the Eq. (34) for integration with

respect to xiand derive that

s

γdsin2α

where d = 1+(πcot(α)=γ)2. Then the GT of fi(ri) = exp?γr2

expi2π

1?

q

Finally the GT of the spherical wave and Gaussian function (row 5 and 6 of the Table 1) are

obtained from Eq. (36) for γ = ?πa and γ = ?iπb, respectively.

go(xo;yo) =

π2

γ2dexp

?

π2

?x2

o+y2

o

??

exp

?

?i2π

π2

dγ2sin2αxoyocotα

?

;

(35)

i

?is given by

fo(ro) =

??

1

cos2α+(γ=π)2sin2α

cos2α +(γ=π)2sin2α

?

xoyocotα

?

exp

γr2

o

cos2α +(γ=π)2sin2α

!

: (36)