Abstract

The expression for the Wigner distribution (WD) in polar coordinates was derived, based on the decomposition of coherent and partially coherent fields on the orthogonal sets of Hermite–Gauss modes. This representation allows one to analyze easily the structure of the WD and to describe the field propagation through first-order optical systems, including the self-imaging phenomenon.

© 2000 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Calif., 1966).
  2. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
  3. M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  4. C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).
  5. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  6. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  7. D. F. V. James, G. S. Agarwal, “The generalised Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
    [CrossRef]
  8. G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
    [CrossRef]
  9. T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
    [CrossRef]
  10. T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998).
    [CrossRef]
  11. T. Alieva, M. J. Bastiaans, “Self-imaging in first-order optical systems,” in Optics and Optoelectronics: Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), Vol. 1, pp. 126–131.
  12. T. Alieva, M. J. Bastiaans, “Powers of transfer matrices determined by means of eigenfunctions,” J. Opt. Soc. Am. A 16, 2413–2418 (1999).
    [CrossRef]
  13. T. Alieva, A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).
    [CrossRef]
  14. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  15. M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution, W. Mecklenbräuker, F. Hlawatsch, eds. (Elsevier, Amsterdam, 1997).
  16. T. Alieva, F. Agullo-Lopez, “Diffraction analysis of random fractal fields,” J. Opt. Soc. Am. A 15, 669–674 (1998).
    [CrossRef]
  17. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 838.
  18. S. Y. Lee, H. H. Szu, “Fractional Fourier transforms, wavelet transforms, and adaptive neural networks,” Opt. Eng. 33, 2326–2329 (1994).
    [CrossRef]

1999 (2)

1998 (2)

T. Alieva, F. Agullo-Lopez, “Diffraction analysis of random fractal fields,” J. Opt. Soc. Am. A 15, 669–674 (1998).
[CrossRef]

T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998).
[CrossRef]

1997 (1)

T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

1996 (1)

D. F. V. James, G. S. Agarwal, “The generalised Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

1994 (1)

S. Y. Lee, H. H. Szu, “Fractional Fourier transforms, wavelet transforms, and adaptive neural networks,” Opt. Eng. 33, 2326–2329 (1994).
[CrossRef]

1993 (1)

1992 (2)

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

1982 (1)

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

1979 (1)

Agarwal, G. S.

D. F. V. James, G. S. Agarwal, “The generalised Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Agullo-Lopez, F.

Alieva, T.

T. Alieva, M. J. Bastiaans, “Powers of transfer matrices determined by means of eigenfunctions,” J. Opt. Soc. Am. A 16, 2413–2418 (1999).
[CrossRef]

T. Alieva, A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).
[CrossRef]

T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998).
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Diffraction analysis of random fractal fields,” J. Opt. Soc. Am. A 15, 669–674 (1998).
[CrossRef]

T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

T. Alieva, M. J. Bastiaans, “Self-imaging in first-order optical systems,” in Optics and Optoelectronics: Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), Vol. 1, pp. 126–131.

Barbé, A.

T. Alieva, A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).
[CrossRef]

T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998).
[CrossRef]

T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

Bastiaans, M. J.

T. Alieva, M. J. Bastiaans, “Powers of transfer matrices determined by means of eigenfunctions,” J. Opt. Soc. Am. A 16, 2413–2418 (1999).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

T. Alieva, M. J. Bastiaans, “Self-imaging in first-order optical systems,” in Optics and Optoelectronics: Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), Vol. 1, pp. 126–131.

M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution, W. Mecklenbräuker, F. Hlawatsch, eds. (Elsevier, Amsterdam, 1997).

Cincotti, G.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

Gomez-Reino, C.

C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

Gori, F.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 838.

James, D. F. V.

D. F. V. James, G. S. Agarwal, “The generalised Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Lee, S. Y.

S. Y. Lee, H. H. Szu, “Fractional Fourier transforms, wavelet transforms, and adaptive neural networks,” Opt. Eng. 33, 2326–2329 (1994).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Calif., 1966).

Mendlovic, D.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Nazarathy, M.

Ozaktas, H. M.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 838.

Santarsiero, M.

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

Shamir, J.

Szu, H. H.

S. Y. Lee, H. H. Szu, “Fractional Fourier transforms, wavelet transforms, and adaptive neural networks,” Opt. Eng. 33, 2326–2329 (1994).
[CrossRef]

Wolf, K. B.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

Int. J. Optoelectron. (1)

C. Gomez-Reino, “GRIN optics and its application in optical connections,” Int. J. Optoelectron. 7, 607–680 (1992).

J. Inst. Math. Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Mod. Opt. (1)

T. Alieva, A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Phys. A (2)

G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992).
[CrossRef]

T. Alieva, A. Barbé, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, L211–L215 (1997).
[CrossRef]

Opt. Commun. (2)

T. Alieva, A. Barbé, “Self-imaging in fractional Fourier transform systems,” Opt. Commun. 152, 11–15 (1998).
[CrossRef]

D. F. V. James, G. S. Agarwal, “The generalised Fresnel transform and its application to optics,” Opt. Commun. 126, 207–212 (1996).
[CrossRef]

Opt. Eng. (1)

S. Y. Lee, H. H. Szu, “Fractional Fourier transforms, wavelet transforms, and adaptive neural networks,” Opt. Eng. 33, 2326–2329 (1994).
[CrossRef]

Other (5)

M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution, W. Mecklenbräuker, F. Hlawatsch, eds. (Elsevier, Amsterdam, 1997).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 838.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, Calif., 1966).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

T. Alieva, M. J. Bastiaans, “Self-imaging in first-order optical systems,” in Optics and Optoelectronics: Theory, Devices and Applications, O. P. Nijhawan, A. K. Gupta, A. K. Musla, K. Singh, eds. (Narosa, New Delhi, 1998), Vol. 1, pp. 126–131.

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Equations (74)

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go(u)=RM[gi(x)](u)=-gi(x)KM(x, u)dx,
KM(x,u)
=(1/iB)exp[iπ(Ax2+Du2-2xu)/B]B0A exp(iπCu2/A)δ(x-Au)B=0
M=ABCD,
Go(u1,u2)=--Gi(x1, x2)KM(x1, u1)KM*(x2, u2)dx1dx2.
Φn(x)=(π2nλn!)-1/2 exp[-12(1+iβ)(x/λ)2]Hn(x/λ)
θ=arccos[12(A+D)],
λ2=2B[4-(A+D)2]-1/2,
β=(A-D)[4-(A+D)2]-1/2.
g(x)=n=0gnΦn(x),
gn=|gn|exp(iϕn)=g(x)Φn*(x)dx.
G(x1, x2)=n=0m=0gn,mΦn(x1)Φm*(x2),
gn,m=|gn,m|exp(iφn,m)=G(x1, x2)Φn*(x1)Φm(x2)dx1dx2.
Go(x1, x2)=n=0m=0×exp[-i(n-m)θ]gn,mΦn(x1)Φm*(x2),
Wg(x, k)=g(x+y/2)g*(x-y/2)exp(-2πiky)dy,
g(x)=1g*(0)Wg(x/2, k)exp(2πikx)dk,
G(x, y)=Wg(x, k)exp(2πiky)dk.
uku=ABCDxkx,
WgM(x, k)=WgM(Dx-Bk, Ak-Cx).
An=cos nθ+12(A-D)sin nθ/sin θ,
Bn=B sin nθ/sin θ,
Cn=C sin nθ/sin θ,
Dn=cos nθ-12(A-D)sin nθ/sin θ.
|go(u)|2=|RM[gi(x)](u)|2=Wgi(Dx-Bk, Ak-Cx)dk.
Wg(x, kx)=n=0m=0gn,mWΦnΦm(x, kx),
WΦnΦm(x, kx)=Φn(x+x/2)Φm*(x-x/2)×exp(-ikxx)dx,
gn,m=|gn,m|exp(iϕn,m)=gngm*=|gngm|exp[i(ϕn-ϕm)].
WΦnΦm(x, kx)=Φn(x+x/2)Φm*(x-x/2)exp(-ikxx)dx=(π2n+mn!m!)-1/2 exp[-(x2+(x)2/4)]×Hn(x+x/2)Hm*(x-x/2)exp(-ikxx)dx.
WΦnΦm(ρ, 0)=Φn(ρ+ρ/2)Φm*(ρ-ρ/2)dρ.
ΦnΦmWΦnΦm(ρ, α), fractionalFT rotationofWD,Rθ[Φn], Rθ[Φm]WRθ[Φn]Rθ[Φm](ρ, α)=WΦnΦm(ρ, α+θ),
WΦnΦm(ρ, θ)=Rθ[Φn(x)](ρ+ρ/2)×R-θ[Φm*(x)](ρ-ρ/2)dρ.
WΦnΦm(ρ, θ)=exp[-i(n-m)θ]Φn(ρ+ρ/2)×Φm*(ρ-ρ/2)dρ.
 exp(-x2)Hn(x+y)Hm(x+z)dx
=2mπn!zm-nLnm-n(-2yz)(nm),
WΦnΦm(ρ, θ)=exp[-i(n-m)θ](-1)n21+(m-n)/2n!m!×exp(-ρ2)ρm-nLnm-n(2ρ2).
Wg(ρ, θ)=2 exp(-ρ2)m=0nm|gn,m|2(m-n)/2n!m!×ρm-nLnm-n(2ρ2)cos[(m-n)θ+ϕm,n],
Wg(ρ)=2 exp(-ρ2)m=0|gm,m|cos(ϕm,m)Lm(2ρ2).
Wg(ρ)=WΦn(ρ)=2 exp(-ρ2)Ln(2ρ2).
G(x1, x2)=m=0gm,mΦm(x1)Φm*(x2).
gm,m=2σ1+σ1-σ1+σmfor0<σ1,
G(x1, x2)
=2σ exp-π2[σ(x1+x2)2+σ-1(x1-x2)2].
Wg(ρ, θ0)=Wg(ρ, 0).
(m-n)θ0=2πk.
m=n+ql,
Wg(ρ, θ)=2 exp(-ρ2)×l=0n=0|gn,n+ql|2ql/2n!(n+ql)!1/2×ρqlLnql(2ρ2)cos(qlθ+ϕn+ql,n).
Wg(ρ, θ)=2 exp(-ρ2)×l=0|gngn+ql|2ql/2n!(n+ql)!1/2×ρqlLnql(2ρ2)cos(qlθ+ϕn+ql,n),
g(x)=l=0gn+lqΦn+lq(x),
G(x1, x2)=n=0l=0gn,n+qlΦn(x1)Φn+ql*(x2).
WΦnΦm(x, kx)
=Φn(x+x/2)Φm*(x-x/2)exp(-ikxx)dx=(π2n+mλ2n!m!)-1/2×exp-x2+x2/4λ2Hnx+x/2λ
×Hm*x-x/2λexp[-i(kx+xβ/λ2)x]dx.
Wg(ρ,θ)=2 exp(-ρ2)×m=0nm|gn,m|2(m-n)/2×n!m! ρm-nLnm-n(2ρ2)×cos[(m-n)θ+ϕm,n],
cot θ=β+λ2kx/x=(A-D+2Bkx/x)[4-(A+D)2]-1/2,
ρ2=(1+β2)λ-2x2+λ2kx2+2βxkx=2[4-(A+D)2]-1/2×[Bkx2-Cx2+(A-D)xkx],
Wg(ρ)=2 exp(-ρ2)m=0|gm,m|cos(ϕm,m)Lm(2ρ2)
A=cos θ+β sin θ,
B=λ2 sin θ,
C=-[(β2+1)/λ2]sin θ,
D=cos θ-β sin θ,
G(x1, x2)=2σλexp-π2λ2[σ(x1+x2)2+σ-1(x1-x2)2+iβ(x1-x2)(x1+x2)],
Wg(ρ)=2 exp(-ρ2)exp(-σ2/2)J0(2ρσ),
gm,m=1,mn,
gm,m=0,m>n.
Wg(ρ)=2 exp(-ρ2)Ln1(2ρ2).
g(n)(u)=l=0gn+lqΦn+lq(u)=1kl=0k-1 exp[i2π(n+12)l/k]RMl[f(x)](u),
f(x)=q=0n-1l=0gn+lqΦn+lq(u).
Wf(x, k)=exp[-(px2+qk2+tkx)].
(D2-1)p+C2q-DCt=0,
B2p+(A2-1)q-ABt=0,
DBp+ACq-BCt=0.
q=-Bp/C,
t=(D-A)p/C.
Wf(x, k)=exp{-p[x2-Bk2/C+(D-A)kx/C]}.

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