Abstract

We describe a matrix multiplication procedure for evaluating the pixelated version of the near-field pattern of a discrete, one- or two-dimensional input. We show that for an input with N×N pixels, in an area d×d, it is necessary to evaluate the Fresnel diffraction pattern at distances zd2/λN. Our numerical algorithm is also useful for evaluating the fractional Fourier transform by multiplying by a special phase matrix with fractional parameter ϵ. If the phase matrix is evaluated at ϵ=1, we find the discrete Fourier transform matrix.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 63–105.
  3. V. Arrizón, J. Ojeda-Castañeda, “Fresnel diffraction of substructured gratings: matrix description,” Opt. Lett. 20, 118–120 (1995).
    [CrossRef] [PubMed]
  4. V. Arrizón, J. G. Ibarra, J. Ojeda-Castañeda, “Matrix formulation of the Fresnel transform of complex transmittance gratings,” J. Opt. Soc. Am. A 13, 2414–2422 (1996).
    [CrossRef]
  5. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  6. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  7. M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780, 1786 (1971).
    [CrossRef]
  8. R. A. Roberts, C. T. Mullis, Digital Signal Processing (Addison-Wesley, Reading, Pa., 1987), pp. 105–112.
  9. G. B. Parrent, B. J. Thompson, Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1969), pp. 29–31.

1996

1995

1993

1980

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

1971

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780, 1786 (1971).
[CrossRef]

Arrizón, V.

Cathey, W. T.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 63–105.

Ibarra, J. G.

Lohmann, A. W.

Moshinsky, M.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780, 1786 (1971).
[CrossRef]

Mullis, C. T.

R. A. Roberts, C. T. Mullis, Digital Signal Processing (Addison-Wesley, Reading, Pa., 1987), pp. 105–112.

Namias, V.

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

Ojeda-Castañeda, J.

Parrent, G. B.

G. B. Parrent, B. J. Thompson, Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1969), pp. 29–31.

Quesne, C.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780, 1786 (1971).
[CrossRef]

Roberts, R. A.

R. A. Roberts, C. T. Mullis, Digital Signal Processing (Addison-Wesley, Reading, Pa., 1987), pp. 105–112.

Thompson, B. J.

G. B. Parrent, B. J. Thompson, Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1969), pp. 29–31.

J. Inst. Math. Appl.

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.

M. Moshinsky, C. Quesne, “Linear canonical transformations and their unitary representation,” J. Math. Phys. 12, 1772–1780, 1786 (1971).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Other

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 63–105.

R. A. Roberts, C. T. Mullis, Digital Signal Processing (Addison-Wesley, Reading, Pa., 1987), pp. 105–112.

G. B. Parrent, B. J. Thompson, Physical Optics Notebook (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1969), pp. 29–31.

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 (9)

Fig. 1
Fig. 1

Example of discrete spatial domain sampling of light distribution.

Fig. 2
Fig. 2

Object and Fresnel diffraction patterns at distance d2/λN from the object.

Fig. 3
Fig. 3

Simulated Fresnel diffraction of a square pupil. Traces through images of object and diffracted image from Fig. 2.

Fig. 4
Fig. 4

Object and Fresnel diffraction patterns of a circular aperture at distance azmin (a1) from the object.

Fig. 5
Fig. 5

Object and Fresnel diffraction patterns of a triangular aperture at distance azmin (a1) from the object.

Fig. 6
Fig. 6

Object and fractional Fourier transform patterns of a square aperture at distance zmin=d2/λN from the object, with ϵ=0.5.

Fig. 7
Fig. 7

Object and fractional Fourier transform patterns of a circular aperture at distance zmin=d2/λN from the object, with ϵ=0.5.

Fig. 8
Fig. 8

Fresnel patterns of a square aperture with replication factor Q=1 at multiples (a) 1.05 and (b) 1.3 of the minimum distance, z=d2/λN. The patterns were found using Eq. (13), which includes a dependence on Q.

Fig. 9
Fig. 9

Object and Fresnel transform patterns of a square aperture at distance 0.65zmin from object. Owing to use of a fraction of the valid z, the diffraction pattern is aliased. The patterns were found using Eq. (27), where a=0.65.

Equations (55)

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

Uˆ(x, z)=exp(jkz/2)jλz-U(ξ)expj πλz(ξ-x)2dξ,
Uˆ(x, z)=exp(jkz/2)jλzexpj πλz(x2)×-U(ξ)expj πλz(ξ2)×exp-j 2πλz(xξ)dξ.
Ur(ξ)=U(ξ)q=-QQδ(ξ-qd),
Uˆr(x, z)=Φ-Ur(ξ)expj πλz(x-ξ)2dξ=Φq=-QQ-U(ξ-qd)expj πλz(x-ξ)2dξ.
x=ξ-qd,
ξ=x+qd,
ξ2=x2+q2d2+2qdx,
Uˆr(x, z)=Φ-q=-QQU(x)×expj πλz[(x-qd)-x]2dx.
Uˆr(x, z)=Φq=-QQUˆ(x-qd, z).
Us(ξ)=n=0N-1Un dNδξ-n dN.
Uˆs(x, z)=Φ-n=0N-1Un dN×δξ-n dNexpj πλz(x-ξ)2dξ,
Uˆs(x, z)=Φn=0N-1Un dNexpj πλzx-n dN2.
Urs(ξ)=Us(ξ)q=-QQδ(ξ-qd)=n=0N-1Un dNδξ-n dN,q=-QQδ(ξ-qd).
Uˆrs(x, z)=Φq=-QQn=0N-1Un dN×expj πλzx-qd-n dN2.
Uˆrsm dN, z=Φq=-QQn=0N-1Un dN×expj πd2λzm-nN-q2=Φq=-QQn=0N-1Un dN×expj πd2N2λz(m-n-Nq)2.
z=zmin=d2/λN;
Uˆrsm dN, zmin
=Φq=-QQn=0N-1Un dNexpj πN(m-n)2exp(jπNq2).
q=-QQ exp(jπNq2)
=1:Nodd,Qeven-1:Nodd,Qodd(2Q+1):Neven.
Uˆrs(m)=Φn=0N-1U(n)expj πN(m-n)2.
Uˆrs(m)=Φ expj πNm2n=0N-1U(n)×exp-j 2πN(mn)expj πNn2,
Uˆrs(m1, m2)=Φn2=0N-1n1=0N-1U(n1, n2)×expj πN[(m1-n1)2+(m2-n2)2].
Uˆrs(m1, m2)=Φ expj πN(m12+m22)×n1=0N-1n2=0N-1U(n1, n2)×exp-j 2πN(m1n1+m2n2)×expj πN(n12+n22),
N=d2/λz.
fs=N/d,
fs,max=N/2d.
fz,max=d/2λz.
fs,maxfz,maxN2dd2λz.
zzmin=d2/λN.
Uˆrs(m)=Φn=0N-1U(n)expj πaN(m-n)2.
Fϵ[T](s)=T(t)exp-j (s2+t2)2 tan ϕexpj stsin ϕdt,
s=jx2πλztan ϕ1/2
t=jx2πλzcos ϕ sin ϕ1/2,
G(x)=T(t),
Fϵ[G](x)=G(x)expjπ (x2+x2 cos2 ϕ)λz×exp-j 2πxxλzdx.
U(x)=G(x)exp-jπ x2 sin2 ϕλz.
U(x)=G(x)exp-jπ x2λR,
0ϵ=sin ϕ=(z/R)1/21.
Us(x)=n=0N-1G(x)exp-jπ x2λRδx-n dN=n=0N-1Gn dNexp-jπ n2d2λRN2δx-n dN.
R=(d2/λN)ϵ2,
Us(x)=n=0N-1Gn dNexp-j πNn2ϵ2δx-n dN.
Uˆrs(m)=Φn=0N-1U(n)×expj πN(m-n)2exp-j πNn2ϵ2.
FDM=1N1W(1/2)m2W(1/2)(N-1)2W(1/2)n2W(1/2)(n-m)2W(1/2)[n-(N-1)]2W(1/2)(N-1)2W(1/2)[(N-1)-m]21,
W=expj 2πN.
FDMFDM*=FDM*FDM=I,
U=FDMU,
U=FDMUFDM
Pϵ=1000W-(1/2)n2ϵ2000W-(1/2)(N-1)2ϵ2.
Fϵ=1N1W(1/2)m2W(1/2)(N-1)2W(1/2)n2(1-ϵ2)W(1/2)[(n-m)2-n2ϵ2]W(1/2){[n-(N-1)]2-n2ϵ2}W(1/2)(N-1)2(1-ϵ2)W(1/2){[(N-1)-m]2-(N-1)2ϵ2}W-(1/2)(N-1)2ϵ2,
W=expj 2πN.
F(ϵ=1)
=1N1W(1/2)m2W(1/2)(N-1)21W(1/2)m2-nmW[(1/2)(N-1)2-n(N-1)]1W(1/2)m2-(N-1)mW-(N-1)2.
F(ϵ=1)=1N1111W-nmW-n(N-1)1W-(N-1)mW-(N-1)2
×1000W(1/2)m2000W(1/2)(N-1)2.

Metrics