Abstract

The offset Fourier transform (offset FT), offset fractional Fourier transform (offset FRFT), and offset linear canonical transform (offset LCT) are the space-shifted and frequency-modulated versions of the original transforms. They are more general and flexible than the original ones. We derive the eigenfunctions and the eigenvalues of the offset FT, FRFT, and LCT. We can use their eigenfunctions to analyze the self-imaging phenomena of the optical system with free spaces and the media with the transfer function exp[j(h2x2+h1x+h0)] (such as lenses and shifted lenses). Their eigenfunctions are also useful for resonance phenomena analysis, fractal theory development, and phase retrieval.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. N. Bracewell, The Fourier Integral and Its Applications (McGraw-Hill, Boston, Mass., 2000).
  2. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  3. H. M. Ozaktas, M. A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).
  4. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  5. K. B. Wolf, “Canonical transforms,” in Integral Transforms in Science and Engineering, K. B. Wolf, ed. (Plenum, New York, 1979), Chap. 9, pp. 381–416.
  6. S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
    [CrossRef] [PubMed]
  7. D. F. V. James, G. S. Agarwal, “The generalized Fresnel transform and its applications to optics,” Opt. Commun. 126, 207–212 (1996).
    [CrossRef]
  8. S. C. Pei, J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
    [CrossRef]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  10. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).
  11. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  12. P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
    [CrossRef]
  13. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner distribution and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
    [CrossRef] [PubMed]
  14. S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
    [CrossRef]
  15. S. Abe, J. T. Sheridan, “Corrigenda to ‘Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach’,” J. Phys. A 27, 7937–7938 (1994).
    [CrossRef]
  16. T. Alieva, A. M. Barbe, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, 211–215 (1997).
    [CrossRef]
  17. S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. Press, Cambridge, UK, 1981).
  18. S. C. Pei, J. J. Ding, Department of Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, are preparing a manuscript to be called “Properties and applications of the eigenfunctions of the linear canonical transform.”
  19. M. L. Mehta, “Eigenvalues and eigenvectors of the finite Fourier transform,” J. Math. Phys. 28, 781–785 (1987).
    [CrossRef]
  20. T. Alieva, M. J. Bastiaans, “Finite-mode analysis by means of intensity information on fractional optical systems,” J. Opt. Soc. Am. A 19, 481–484 (2002).
    [CrossRef]
  21. T. Alieva, M. J. Bastiaans, “Mode analysis in optics through fractional transforms,” Opt. Lett. 24, 1206–1208 (1999).
    [CrossRef]
  22. T. Alieva, M. J. Bastiaans, “Powers of transfer matrices determined by means of eigenfunctions,” J. Opt. Soc. Am. A 16, 2413–2418 (1999).
    [CrossRef]

2002 (2)

S. C. Pei, J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

T. Alieva, M. J. Bastiaans, “Finite-mode analysis by means of intensity information on fractional optical systems,” J. Opt. Soc. Am. A 19, 481–484 (2002).
[CrossRef]

1999 (2)

1997 (1)

T. Alieva, A. M. Barbe, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, 211–215 (1997).
[CrossRef]

1996 (1)

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

1995 (1)

1994 (6)

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner distribution and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Corrigenda to ‘Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach’,” J. Phys. A 27, 7937–7938 (1994).
[CrossRef]

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

1989 (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

1987 (1)

M. L. Mehta, “Eigenvalues and eigenvectors of the finite Fourier transform,” J. Math. Phys. 28, 781–785 (1987).
[CrossRef]

1980 (1)

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

Abe, S.

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Corrigenda to ‘Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach’,” J. Phys. A 27, 7937–7938 (1994).
[CrossRef]

Agarwal, G. S.

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

Alieva, T.

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Barbe, A. M.

T. Alieva, A. M. Barbe, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, 211–215 (1997).
[CrossRef]

Bastiaans, M. J.

Bonnet, G.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Integral and Its Applications (McGraw-Hill, Boston, Mass., 2000).

Ding, J. J.

S. C. Pei, J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

S. C. Pei, J. J. Ding, Department of Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, are preparing a manuscript to be called “Properties and applications of the eigenfunctions of the linear canonical transform.”

Goodman, J. W.

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

James, D. F. V.

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

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Lipson, H.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. Press, Cambridge, UK, 1981).

Lipson, S. G.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. Press, Cambridge, UK, 1981).

Lohmann, A. W.

Mehta, M. L.

M. L. Mehta, “Eigenvalues and eigenvectors of the finite Fourier transform,” J. Math. Phys. 28, 781–785 (1987).
[CrossRef]

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]

Ozaktas, H. M.

Pei, S. C.

S. C. Pei, J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

S. C. Pei, J. J. Ding, Department of Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, are preparing a manuscript to be called “Properties and applications of the eigenfunctions of the linear canonical transform.”

Pellat-Finet, P.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Sheridan, J. T.

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
[CrossRef] [PubMed]

S. Abe, J. T. Sheridan, “Corrigenda to ‘Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach’,” J. Phys. A 27, 7937–7938 (1994).
[CrossRef]

Wolf, K. B.

K. B. Wolf, “Canonical transforms,” in Integral Transforms in Science and Engineering, K. B. Wolf, ed. (Plenum, New York, 1979), Chap. 9, pp. 381–416.

Zalevsky, Z.

H. M. Ozaktas, M. A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Appl. Opt. (1)

IEEE Trans. Signal Process. (2)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

S. C. Pei, J. J. Ding, “Eigenfunctions of linear canonical transform,” IEEE Trans. Signal Process. 50, 11–26 (2002).
[CrossRef]

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. Math. Phys. (1)

M. L. Mehta, “Eigenvalues and eigenvectors of the finite Fourier transform,” J. Math. Phys. 28, 781–785 (1987).
[CrossRef]

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

J. Phys. A (3)

S. Abe, J. T. Sheridan, “Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach,” J. Phys. A 27, 4179–4187 (1994).
[CrossRef]

S. Abe, J. T. Sheridan, “Corrigenda to ‘Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach’,” J. Phys. A 27, 7937–7938 (1994).
[CrossRef]

T. Alieva, A. M. Barbe, “Self-fractional Fourier functions and selection of modes,” J. Phys. A 30, 211–215 (1997).
[CrossRef]

Opt. Commun. (2)

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

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

Opt. Lett. (2)

Optik (Stuttgart) (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

Other (6)

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. Press, Cambridge, UK, 1981).

S. C. Pei, J. J. Ding, Department of Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, are preparing a manuscript to be called “Properties and applications of the eigenfunctions of the linear canonical transform.”

R. N. Bracewell, The Fourier Integral and Its Applications (McGraw-Hill, Boston, Mass., 2000).

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

K. B. Wolf, “Canonical transforms,” in Integral Transforms in Science and Engineering, K. B. Wolf, ed. (Plenum, New York, 1979), Chap. 9, pp. 381–416.

H. M. Ozaktas, M. A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Lens shifted upward with distance x0.

Fig. 2
Fig. 2

Medium whose width is h2x2+h1x+h0.

Fig. 3
Fig. 3

Optical system with a shifted lens and a prism.

Fig. 4
Fig. 4

Spherical mirror pair system (the centers of the two mirrors are aligned).

Fig. 5
Fig. 5

Spherical mirror pair system (the centers of the two mirrors are not aligned).

Equations (153)

Equations on this page are rendered with MathJax. Learn more.

GF(ω)=FT[g(x)]=12π-exp(-jωx)g(x)dx.
Ga(u)=OFα[g(x)]=1-j cot α2π1/2-expj2u2 cot α-jux csc α+j2x2 cot αg(x)dx.
G(a,b,c,d)(u)=OF(a,b,c,d)[g(x)]=1j2πb1/2-expj2dbu2-jubx+j2abx2g(x)dxwhenb0,
G(a,b,c,d)(u)=OF(a,b,,c,d)[g(x)]=d exp(jcdu2/2)g(du)  whenb=0,
FT[g(x)]=jOF(0,1,-1,0)[g(x)],
OFα[g(x)]=[exp(jα)]1/2OF(cos α,sin α,-sin α,cos α)[g(x)].
GFτ,η(ω)=FTτ,η[g(x)]=12π-exp[-j(ω-τ)(x-η)]g(x)dx.
g(x)=IFTτ,η[GFτ,η(ω)]=12π-exp[j(x-η)(ω-τ)]GFτ,η(ω)dω.
Gατ,η(u)=OFα,τ,η[g(x)]=1-j cot α2π1/2exp(jηu)-×expj2(u-τ)2 cot α-j(u-τ)x csc α+j2x2 cot αg(x)dx,
G(a,b,c,d)τ,η(u)=OF(a,b,c,d,τ,η)[g(x)]=1j2πb1/2exp(jηu)-×expj2db(u-τ)2-ju-τbx+j2abx2g(x)dxwhenb0,
G(a,b,c,d)τ,η(u)=OF(a,0,c,d,τ,η)[g(x)]=d expjηu+j2cd(u-τ)2g(du-dτ),whenb=0,
ad-bc=1.
GFτ,η(ω)=exp[j(ω-τ)η]GF(ω-τ),
Gατ,η(u)=exp(jηu)Gα(u-τ),
G(a,b,c,d)τ,η(u)=exp(jηu)G(a,b,c,d)(u-τ).
OFβ,τ2,η2{OFα,τ1,η1[g(x)]}=exp(jφ)OFα+β,τ3,η3[g(x)],
τ3=τ2+τ1 cos α+η1 sin α,
η3=η2-τ1 sin α+η1 cos α,
φ=sin(2β)4(τ12-η12)+τ1η1 sin2 β+(τ1 sin β-η1 cos β)τ2.
OF(a2,b2,c2,d2,τ2,η2){OF(a1,b1,c1,d1,τ1,η1)[f(x)]}=exp(jϕ)OF(e,f,g,h,r,s)[f(x)],
efgh=a2b2c2d2a1b1c1d1,
rs=a2b2c2d2τ1η1+τ2η2,
ϕ=-a2c22τ12-b2c2τ1η1-b2d22η12-(τ1c2+η1d2)τ2.
abcdtω+τη
a2b2c2d2a1b1c1d1tω+τ1η1+τ2η2=efghtω+rs,
{OF(a,b,c,d,τ,η)[f(x)]}-1=expj2cdτ2-jadτη+j2abη2×OF(d,-b,-c,a,-dτ+bη,cτ-aη)[f(x)].
h(x)=exp[j(h2x2+h1x+h0)].
ϕm(x)=exp(-x2/2)Hm(x),
FT(ϕm(x))=(-j)mϕm(ω).
n=-δ(x-p2π),x2π-1/2,sec hxπ2.
OFα[ϕm(x)]=exp(-jmα)ϕm(x),
Φ(x)=p=0apϕpM+q(x),
q=0, 1,, M-1,aps are constants,
Eigenfunctions:A exp-(1+iρ)x22σ2Hmxσ,
σ2=2|b|[4-(a+b)2]1/2,ρ=sgn(b)(a-d)[4-(a+d)2]1/2.
Eigenvalues:λ=[exp(-jα)]1/2 exp(-iαm),
α=cos-1a+d2=sin-1sgn(b)2[4-(a+d)2]1/2.
Eigenfunctions:  n=0Anδ(x-(4nπ|c|-1+h)1/2)+m=0Bmδ(x+(4mπ|c|-1+h)1/2),
Eigenvalues:λ=exp(jch/2).
Eigenfunctions:  expjd-a4bx2-expj(x-q)22ρg(q)dq,
g(x)=n=0Cn exp[jx(4nπ|b|-1+h)1/2]+m=0Dm exp[-jx(4mπ|b|-1+h)1/2],
0h<4π/|b|,Cns,Dms,andρcan be arbitrary real-valued constants.
Eigenvalues:λ=exp(-jbh/2).
Eigenfunctions:  expj2ρx2-expj2h(x-q)2g(q)dq,
σg(σx)=λg(x),
ρ=-2scs(d-a)+[(a+d)2-4]1/2,
h=sb[(a+d)2-4]1/2,
σ={a+d±[(a+d)2-4]1/2}/2,
s=sgn[σ-σ-1].
Eigenvalues:λ=σg(σx)/g(x).
OC[f(x)]=exp(jφ)OB(OA{OB-1[f(x)]}),
OA[e(x)]=λe(x),
OC{OB[e(x)]}=exp(jφ)OB[OA(OB-1{OB[e(x)]})]=exp(jφ)OB{OA[e(x)]}=λ exp(jφ)OB[e(x)].
FTτ,η[g(x)]=j exp(-jτη)OF(0,1,-1,0,τ,η)[g(x)].
OB[g(x)]=exp(jh2x)g(x-h1),
OB-1[g(x)]=exp(-jh1h2)exp(-jh2x)g(x+h1).
j exp(-jτη)OF(0, 1, -1, 0, τ, η)[f(x)]=exp(jφ)OF(1, 0, 0, 1, h1, h2)(jOF(0, 1, -1, 0, 0, 0)×{exp(-jh1h2)OF(1,0,0,1,-h1,-h2)[f(x)]}).
01-10tω+τη=01-10tω+-h1-h2+h1h2=01-10tω+h1-h2h1+h2.
h1=(τ+η)/2,h2=(η-τ)/2.
OB[g(x)]=expj2(η-τ)xgx-τ+η2.
j exp(-jτη)OF(0,1,-1,0,τ,η)[f(x)]=j exp(jφ-jh12+jh1h2-jh1h2)×OF(0,1,-1,0,τ,η)[f(x)],
φ=(τ-η)2/4.
FT[E(x)]=λE(x),
Eτ,η(t)=expj2(η-τ)xEx-τ+η2,
Fτ,η[Eτ,η(x)]=λ expj4(τ-η)2Eτ,η(x).
exp(-x2/2)h=0ahH4h+q(x),
q=0, 1, 2, 3,ahs can be arbitrary real-valued constants,
Φq,τ,ηa0,a1,a2,(x)=expjη-τ2x-(2x-η-τ)28×h=0ahH4h+qx-η+τ2,
q=0, 1, 2, 3,ahs are free to choose,
(-j)q exp[j(τ-η)2/4],q=0, 1, 2, 3.
|Eτ,η(x)|=|E(x)|whenτ=-η.
τ=ηor,more generally,(τ-η)2=2Nπ,
OF(a,b,c,d,τ,η)[f(x)]=exp(jφ)OF(1,0,0,1,h1,h2)(OF(a,b,c,d, 0,0)×{exp(-jh1h2)OF(1,0,0,1,-h1,-h2)[f(x)]}).
abcdtω+τη=abcdtω+-h1-h2+h1h2=abcdtω+(1-a)h1-bh2-ch1+(1-d)h2.
h1=(1-d)τ+bη2-a-d,
h2=cτ+(1-a)η2-a-d(ifa+d2),
OB[g(x)]=expjcτ+(1-a)η2-a-dx×gx-(1-d)τ+bη2-a-d.
OF(a,b,c,d,τ,η)[f(x)]=expjφ+j(ch12+dh1h2)-jac2h12+bch1h2+bd2h22-jh1h2OF(a,b,c,d,τ,η)[f(x)],
φ=ac2-ch12+(bc-d+1)h1h2+bd2h22=-cτ2+2(1-d)τη+bη22(2-a-d).
OF(a,b,c,d)[E(x)]=λE(x),
Eτ,η(x)=expjcτ+(1-a)η2-a-dx×Ex-(1-d)τ+bη2-a-d(ifa+d2),
λτ,η=expj-cτ2-2(1-d)τη+bη22(2-a-d)λ.
OF(1,0,c,1,τ,η)[g(x)]=exp[jηx+jc(x-τ)2/2]g(x-τ).
λEc,τ,η(x)=exp[jηx+jc(x-τ)2/2]Ec,τ,η(x-τ)(τ0),
Ec,τ,η(x+τ)=λ-1 exp[jη(x+τ)+jcx2/2]Ec,τ,η(x)(τ0),
n=0Anδx+ηc-4nπ|c|+h1/2+m=0Bmδx+ηc+4mπ|c|+h1/2,
0h<4π/|c|,AnsandBms can be arbitrary real-valued constants.
n=-Anδ(x-2nπη-1+h),
handAns can be arbitrary real-valued constants.
abcdtω+τη=a2b2c2d210c11d2-b2-c2a2tω+-d2τ2+b2η2c2τ2-aη2+τ1η1+τ2η2,
abcd=1+c1b2d2-c1b22c1d221-c1b2d2,
τη=a2b2c2d2τ1η1+-a+1-b-c-d+1τ2η2.
b2=±|bc1-1|,
d2=a-d2c1b2,
c1 can be an arbitrary real-valued constant, except that sign(c1)=-sign(b).
a2=0,η2=0.
c2=-b2-1,τ2can be an arbitrary real-valued constant,
τ1η1=d2-b2b2-10τ+(a-1)τ2η+cτ2.
Eτ,η(x)=expjd22b2(x-τ2)2×-exp-jb2(x-τ2)qg(q)dq,
g(x+τ1)=λ-1 exp[jη1(x+τ1)+jc1x2/2]g(x),
λ exp(jφ),
φ=τ1η1+b2d2η12+(τ1c2+η1d2)τ2+τ1(1+τ1b2d2)2τ12+(1+τ1b2d2)τ1η1+(τ1c2+η1d2)(b2η1+τ2)+c2d2τ222.
OFαE(x)=λE(x),
Eτ,η(x)=expj-τ cot(α/2)+η2x×Ex-τ+η cot(α/2)2,
λτ,η=expjcot(α/2)4(τ2+η2)-jτη2λ.
-ϕm(x)ϕn(x)¯dx=Cmδm,n.
f(x)=n=0anϕn(x),andan=-f(x)ϕn(x)¯dx-|ϕn(x)|2dx,
|ϕn(x)|=|ϕn(-x+x0)|for alln,
ϕn(x)=expjη-τ2x-x-τ+η22×Hnx-τ+η2,
n=0, 1, 2, 3,.
{a, b, c, d, τ, η}={1, zλ/2π, 0, 1, 0, 0}.
{a, b, c, d, τ, η}={1, 0, -2π/fλ, 1, 0, 0}.
{a, b, c, d, τ, η}={1, 0, 0, 1, 0, -2π(n-1)σ/λ},
σ=bottom-(width of the prism)height of the prism,
n=refractive index of the prism.
{a, b, c, d, τ, η}={1, 0, -2π/fλ, 1, 0, 2πx0/fλ}.
w(x)=h2x2+h1x+h0,xis not too large,
abcd=104π(n-1)h2/λ1,
τη=02π(n-1)h1/λ.
Do(x)=exp(jφ)OF(a,b,c,d,τ,η)×[Di(x)],φis some constant phase,
OF(a,b,c,d,τ,η)[g(x)]=OFAN{OFAN-1[OFA3(OFA2{OFA1[g(x)]})]}.
|σG(a1,b1,c1,d1,τ1,η1)(σ(u-τ+σ-1τ1))|=|G(a,b,c,d,τ,η)(u)|whenσ=a1/a=b1/b.
a1=a,b1=b,τ1=τ,
a1=a,b1=b.
a1:b1=a:b,τ1=a1τ/a.
a1:b1=a:b.
1z2k-1011z1k-10110-kf-11tω+0kx0f-1+0-k(n-1)σ=1-(z1+z2)f-1(z1+z2)k-1-kf-11tω+(z1+z2)x0f-1-(n-1)z2σk[x0f-1-(n-1)σ], (k=2π/λ).
abcd=1-(z1+z2)f-1(z1+z2)k-1-kf-11,
τη=(z1+z2)x0f-1-(n-1)z2σk[x0f-1-(n-1)σ].
a1=1-(z1+z2)f-1,b1=(z1+z2)k-1,
τ1=(z1+z2)x0f-1-(n-1)z2σ,
a1b1=2πλ[(z1+z2)-1-f1-1],
|D0(x)|=|λρDi(σ(x-s))|,
ρ=a1ff-z1-z2,
s=(z1+z2)x0f-(n-1)z2σ-τ1ρ,
Di(x)=A exp(jy0x)exp-x22Hm(x-x0),
τ=x0(1-cos α)-y0 sin α,
η=x0 sin α-y0(1-cos α),
a/b=cot α.
FAR(x,y)=rAFAI(x, y),FBR(x, y)=rBFBI(x, y),
F1(x, y)=exp[j(2kD+ϕ)]rBOFy(a,b,c,d,τy,ηy)×{OFx(a,b,c,d,τx,ηx)[Fi(x, y)]},
abcd=21-DRA1-DRB-12DkDRB-12k1RA+1RB-2DRARB-DRA2+D2RA2RB21-DRA1-DRB-1,
τxηx=2x0DRB-12kx0RB-1(1-RA-1D),τyηy=2y0DRB-12ky0RB-1(1-RA-1D),
ϕ=2kD(x02+y02)RB2DRA-1.
Fi(x, y)=f1(x)f2(y),
OFx(a,b,c,d,τx,ηx)[f1(x)]=ζ1f1(x),
OFy(a,b,c,d,τy,ηy)[f2(y)]=ζ2f2(y),
F1(x, y)=exp[j(2kD+ϕ)]ζ1ζ2rBFi(x, y).
Fn(x, y)=exp[j(2knD+nϕ)]ζ1ζ2rAn-1rBnFi(x, y).
arg{exp[j(2knD+nϕ)]ζ1ζ2rArB}=0,
FS(x, y)=n=1Fn(x, y)=exp[j(2kD+ϕ)]ζAζBrB1-exp[j(2kD+ϕ)]ζAζBrArBFi(x, y).
Xa,b[m]=N-1n=0N-1exp-j2πN(m-a)(n-b)x[n].
f(x)=n=0anϕn(x),

Metrics