Spherical angular spectrum and the fractional order Fourier transform

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Spherical angular spectrum and the fractional order Fourier transform

Pierre Pellat-Finet, Pierre-Emmanuel Durand, Eric Fogret

To cite this version:

Pierre Pellat-Finet, Pierre-Emmanuel Durand, Eric Fogret. Spherical angular spectrum and the frac-

tional order Fourier transform. Optics Letters, Optical Society of America - OSA Publishing, 2006,

31 (23), pp.3429-3431. �hal-00907972�

Spherical angular spectrum and the fractional order Fourier transform

Pierre Pellat-Finet, Pierre-Emmanuel Durand, and Éric Fogret

Groupe d’Optique Théorique et Appliquée, Université de Bretagne Sud, B.P. 92116, 56321 Lorient Cedex, France Received June 12, 2006; revised September 4, 2006; accepted September 13, 2006;

posted September 15, 2006 (Doc. ID 71928); published November 9, 2006

The notion of a spherical angular spectrum leads to the decomposition of the field amplitude on a spherical emitter into a sum of spherical waves that converge onto the Fourier sphere of the emitter. Unlike the usual angular spectrum, the spherical angular spectrum is propagated as the field amplitude, in a way that can be expressed by a fractional order Fourier transform.

© 2006 Optical Society of America

OCIS codes: 070.2580, 070.2590

.

According to a scalar theory of diffraction, the field amplitude transfer from a plane emitter P to a plane receiver Q at a distance D can be written as the con- volution product

¹

U

_Q共

s

兲

= 冕

_R2

h共 s − r

兲U_P共

r

兲d

r ,

共1兲

where

␭

is the wavelength, U

_P共

r

兲

the field amplitude at the point r on P, U

_Q共

s

兲

the field amplitude on Q, and h the spatial impulse function associated with the field propagation. Apart from a constant phase factor, and in the limit of a second-order approxima- tion, the expression for h is

h共 r

兲

=

共i/␭D兲exp关−共i

␲ /␭D兲r

²兴, 共2兲

where r

²

=

储

r

储²

.

The same transfer can be expressed as a simple product in the Fourier space if we introduce the an- gular spectra

¹

A

_P

of the field on P and A

_Q

on Q. If F

_␲/2关UP兴

denotes the Fourier transform of U

_P

, we de- fine the angular spectrum on P by

A

_P共⌽兲

= 1

␭²

F

_␲/2关UP兴

冉 ^⌽

␭

冊

= 1

␭²

冕

_R2

exp 冉 ²ⁱ

^␭

^{␲ ⌽ · r} 冊 ^U

^P共

^r

^兲d

^{r ,}

^共3兲

where ⌽ =␭ F ( F is a spatial frequency and ⌽ an an- gular spatial frequency). We define

⌽=储⌽储, and we

use

H共⌽兲 = F

_␲/2关h兴共⌽/␭兲

= exp关共i ␲ D/␭兲⌽

²兴

,

共4兲

so that Eq. (1) leads us to

A

_Q共⌽兲

= H共⌽兲A

P共⌽兲. 共5兲

The former descriptions of the field transfer are summed up in the following diagram:

共6兲

Diagram (6) is not commutative: to pass from U

_P

to A

_Q

we perform a Fourier transform followed by a multiplication, or a convolution product followed by a Fourier transform; the operation order cannot be in- verted.

We shall introduce the notion of spherical angular spectrum and explain how diagram (6) becomes one that is commutative. We refer to spherical emitters and receivers: they are portions of spheres with axial symmetry. A spherical surface A has a vertex

⍀

and a center of curvature C: its radius of curvature is R

_A

=

⍀C

and can be positive or negative according to the surface convexity. A point M on A is represented by the vector r =

⍀m, where

m is the orthogonal projec- tion of M onto the plane tangent to A at

⍀.

U

_A共

r

兲

de- notes the field amplitude at M on A, and we define the spherical angular spectrum on A as

S

_A共⌽兲

=

共1/␭²兲F_␲/2关U_A兴共⌽/␭兲. 共7兲

We invert Eq. (7) and obtain

U

_A共

r

兲

= 冕

_R2

exp 冉 ^{− 2i}

␭

^␲ ⌽ · r 冊 ^S

^{A共⌽兲d⌽, 共8兲}

so that the field amplitude on A is the sum of elemen- tary field amplitudes exp共−2i ␲ ⌽ · r /

␭兲, weighted by

the spherical angular spectrum. To interpret such a field amplitude, we introduce the Fourier sphere F of A: its vertex is C, and its center of curvature is

⍀

(Fig. 1). Its radius of curvature is R

_F

=C⍀ = −R

_A

.

We know

²

that the field amplitudes on A and F are related by a Fourier transform [as Eq. (10) will

Fig. 1. Spherical wave converging at the point R

_A

⌽ on the Fourier sphere F generates a field amplitude on A equal to exp共−2i␲⌽ · r /␭兲.

December 1, 2006 / Vol. 31, No. 23 / OPTICS LETTERS 3429

0146-9592/06/233429-3/$15.00 © 2006 Optical Society of America

show]. If ␦ denotes the Dirac distribution, a luminous point located at R

_A

⌽ on F is represented by

␦

共

s − R

_A

⌽兲, up to a dimensional factor, and generates a field amplitude on A proportional to exp共−2i ␲ ⌽ · r /

␭兲. We use the reciprocity theorem³

and conclude that exp共−2i ␲ ⌽ · r /

␭兲

represents the amplitude of the field generated on A by the spheri- cal wave converging at the point R

_A

⌽ on F. The wave can also be seen as a tilted spherical wave as indi- cated in Fig. 1.

We examine now how the spherical angular spec- trum is propagated. The field amplitude transfer from the spherical emitter A to the spherical receiver B at distance D can be written as

⁴

U

_B共

s

兲

= i

␭D

exp 冋 ^{− i ␲}

^␭

^s

²

冉 ^{D 1 + R 1}

B

冊 册

⫻

冕

_R2

exp 冉 ²ⁱ

␭D

^␲ s · r 冊

⫻

exp 冋 ^{i ␲}

^␭

^r

²

冉 ^{R 1}

A

− 1

D 冊 册 ^U

^A共

^r

^兲d

^{r .}

^共9兲

If B =F (the Fourier sphere of A), Eq. (9) reduces to U

_F共

s

兲

=

共i/␭R_A兲F_␲/2关U_A兴共

s /␭R

_A兲, 共10兲

and we obtain

S

_F共⌽兲

= 1

␭²

F

_␲/2关U_F兴

冉 ^⌽

^␭

冊 ^{= iR}

^␭A

^F

^{␲/2关SA兴}

冉 ^R

^A␭

^⌽ 冊 ^.

^共11兲

The transfer of the spherical angular spectrum from an emitter to its Fourier sphere is expressed by a Fourier transform, as well as the field amplitude.

If A and B are concentric spheres, then D= R

_A

− R

_B

. We introduce ␬ = R

_B

/ R

_A

and the function h

_BA

such that

h

_BA共

r

兲

= i

␭D

exp关共− i ␲␬ /␭D兲r

²兴. 共12兲

Equation (9) then becomes

U

_B共

s

兲

= 冕

_R2

h

_BA

冉 ^{s ␬ − r} 冊 ^U

^A共

^r

^兲d

^{r =}

^关hBA

^{ⴱ U}

^A兴

冉 ^{s ␬} 冊

^共13兲

and involves a convolution product. We conclude that the spherical angular spectra on A and B are such that

S

_B共⌽兲

= ␬ exp关共i ␲␬ D/␭兲⌽

²兴S_A共

␬ ⌽兲.

共14兲

The propagation between two concentric spheres acts as a linear filter.

For arbitrary emitter A and receiver B we intro- duce the sphere A ⬘ concentric to A that passes by the center of B, and the sphere B ⬘ concentric to B that passes by the center of A (Fig. 2). Clearly B ⬘ ^{is the} Fourier sphere of A ⬘ , so that the field transfer from A to B can be decomposed into a linear filtering from A to A ⬘ followed by a Fourier transform from A ⬘ ^{to B} ⬘ and a linear filtering from B ⬘ ^to B. The radius of A ⬘ ^is

opposite of the radius of B ⬘ ^{: R}

A

⬘ = R

_A

− D− R

_B

= −R

_B

_⬘ . We use ␬ ⬘ ^=R

A

⬘ / R

_A

, ␬ ⬙ ^{= R}

B

/ R

_B

_⬘ , D ⬘ ^{=D +R}

B

, and D ⬙

= D− R

_A

, and we deduce from Eqs. (11) and (14) that S

_B共⌽兲

= iR

_A

_⬘ ␬ ⬙

␭

␬ ⬘ ^exp 冉 ^{i ␲␬ ⬙}

^␭

^{D ⬙}

^⌽2

冊 ^冕

_R²

^exp 冉 ^{i ␲ D}

^␭

^{␬ ⬘}

^⌽

⬘ ^⬘

²

冊

⫻

exp 冉 ^{2i ␲␬}

␭

␬ ^⬙ ⬘ ^R

^A

^{⬘ ⌽ · ⌽} ⬘ 冊 ^S

^A共⌽

^⬘

^兲d⌽

^{⬘ .}

^共15兲

Equation (15) has the same form as Eq. (9): quadratic phase factors in front of and inside the integral, Fou- rier kernel. Unlike the usual angular spectrum, the spherical angular spectrum propagates as the field amplitude.

Since we know

^2,4

that the field transfer from a spherical emitter A to a spherical receiver B can be represented by a fractional order Fourier transform F

_␣

[Eq. (16) provides a definition], the similarity be- tween Eqs. (15) and (9) leads us to express the trans- fer of the spherical angular spectrum by using a frac- tional Fourier transform. For that purpose, we define the fractional Fourier transform of order ␣ (a com- plex number) of the two-dimensional function f by

⁵

F

_␣关f兴共

␳ ⬘

^兲

⁼ ie

^−i␣

sin ␣ ^{exp共− i} ␲␳ ⬘

²

^{cot ␣}

^兲

⫻

冕

_R2

exp共− i ␲␳

²

cot ␣

兲

⫻

exp 冉 ^{2i sin ␲␳ ⬘ ␣ · ␳} 冊 ^{f共 ␳}

^兲d

^{␳ .}

^共16兲

The standard Fourier transform is obtained by set- ting ␣ = ␲ / 2 in Eq. (16).

Equation (9) can be expressed as a fractional Fou- rier transform of order ␣ as follows. We choose

␣

共−

␲

⬍

␣

⬍

␲

兲

and ⑀ (an auxiliary parameter) such that

cot

²

␣ =

共D

+ R

_B兲共R_A

− D兲

D共D − R

_A

+ R

_B兲

, ␣ D

艌

0,

共17兲

⑀

²

= D共D + R

_B兲

共R_A

− D兲共D − R

_A

+ R

_B兲

, ⑀ R

_A⬎

0.

共18兲

(We assume that ␣ and ⑀ are real numbers. We refer to a recent publication

⁶

for handling complex param- Fig. 2. Field transfer from A to B can be decomposed into a linear filtering from A to A ⬘ followed by a Fourier trans- form from A ⬘ ^{to B} ⬘ and another linear filtering from B ⬘ ^to B .

3430 OPTICS LETTERS / Vol. 31, No. 23 / December 1, 2006

eters.) We define scaled variables and scaled field am- plitudes on A and on B by

␳ = r

共␭

⑀ R

_A兲^1/2

, V

_A共

␳

兲

= U

_A关共␭

⑀ R

_A兲^1/2

␳

兴, 共19兲

␴ =

共cos

␣ + ⑀ sin ␣

兲

s

共␭

⑀ R

_A兲^1/2

,

V

_B共

␴

兲

= U

_B

冋 ^cos

^共␭

^{⑀ ␣ R +}

^A

^⑀

^兲1/2

^{sin ␴ ␣} 册 ^.

^共20兲

Equation (9) becomes

V

_B共

␴

兲

= e

^i␣共cos

␣ + ⑀ sin ␣

兲F_␣关V_A兴共

␴

兲 共21兲

and shows how the scaled field amplitude on B is re- lated to the fractional Fourier transform of order ␣ of the scaled field amplitude on A.

Since we are looking for a commutative diagram, and since the composition law of fractional Fourier transforms demands that F

_␣

ⴰ F

_␲/2

= F

_␲/2

ⴰ F

_␣

, we ex- pect the transfer of the spherical angular spectrum from A to B to be expressed by a fractional Fourier transform whose order should be equal to ␣ . We prove that this holds true indeed. Since ␣ has been defined with respect to the field amplitude transfer, we only have to define scaled variables and scaled an- gular spectra

⌺_A

and

⌺_B

. According to Eq. (15), the angular spatial frequencies are ⌽ ⬘ ^{on A and ⌽ on B,} and we choose

␾ ⬘ ⁼ 冉 ^{⑀ R}

^␭A

冊

^1/2

^{⌽ ⬘ , ␾ =} 冉 ^{⑀ R}

^␭A

冊

^1/2

^{cos ␣ + ⌽ ⑀ sin ␣ ,}

共22兲

⌺A共

␾ ⬘

^兲

^{= S}

A关共␭/

⑀ R

_A兲^1/2

␾ ⬘

^{兴, 共23兲}

⌺B共

␾

兲

= S

_B关共␭/

⑀ R

_A兲^1/2共cos

␣ + ⑀ sin ␣

兲

␾

兴. 共24兲

Thus Eq. (15) can be written as

⌺_B共

␾

兲

=

关e^i␣

/共cos ␣ + ⑀ sin ␣

兲兴F_␣关⌺_A兴共

␾

兲. 共25兲

The transfer of the spherical angular spectrum can be expressed by a fractional order Fourier transform, as well as the field amplitude transfer, with the same order.

From the definitions of scaled field amplitudes V

_A

and V

_B

and scaled angular spectra

⌺_A

and

⌺_B

, we de- duce

⌺_A共

␾ ⬘

^兲

⁼

^共

^{⑀ R}

A

/␭兲V ˆ

A共

␾ ⬘

^{兲, 共26兲}

⌺_B共

␾

兲

=

兵

⑀ R

_A

/关␭共cos ␣ + ⑀ sin ␣

兲²兴其V

ˆ

B共

␾

兲, 共27兲

where V ˆ

A

and V ˆ

B

are the Fourier transforms of V

_A

and V

_B

. Equation (25) leads to

V ˆ

B共

␾

兲

= e

^i␣共cos

␣ + ⑀ sin ␣

兲F_␣关V

ˆ

A兴共

␾

兲, 共28兲

which corresponds to Eq. (21).

We come to the following commutative diagram, which is applied to scaled field amplitudes and their Fourier transforms [a factor e

^i␣共cos

␣ + ⑀ sin ␣

兲

has been omitted]:

共29兲

Obviously diagram (6) cannot be reduced to dia- gram (29). We conclude that the notion of angular spectrum, as it is usually defined,

¹

does not fit in with fractional Fourier optics.

In conclusion, we remark that the notion of spheri- cal angular spectrum can be used to solve propaga- tion problems in the framework of a scalar theory of diffraction: for example, it can be applied to image formation.

⁷

It offers a substitute for the usual angu- lar spectrum and affords the application of the frac- tional order Fourier transform in solving these prob- lems.

P. Pellat-Finet (pierre-pellat-finet@univ-ubs.fr) and É. Fogret are also with the Département d’Optique, UMR CNRS 6082, École Nationale Supérieure des Télécommunications de Bretagne, Brest, France.

References

1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

2. P. Pellat-Finet and G. Bonnet, Opt. Commun. 111 , 141 (1994).

3. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

4. P. Pellat-Finet, Opt. Lett. 19 , 1388 (1994).

5. V. Namias, J. Inst. Math. Appl. 25 , 241 (1980).

6. P. Pellat-Finet and E. Fogret, Opt. Commun. 258 , 103 (2006).

7. P. Pellat-Finet and P. E. Durand, C. R. Phys. 7 , 457 (2006).

December 1, 2006 / Vol. 31, No. 23 / OPTICS LETTERS 3431