Abstract

We have found the new eigenmodes, two-variable Hermite polynomials, exist in propagating plane waves in quadratic-index media and that a two-dimensional Talbot effect can be demonstrated with these modes.

© 2004 Optical Society of America

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  1. B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 88, 100404 (2002).
    [CrossRef]
  2. G. S. Agarwal, Opt. Commun. 119, 30 (1995).
    [CrossRef]
  3. A. W. Lohmann and J. O. Castaneda, Opt. Acta 30, 475 (1983).
    [CrossRef]
  4. F. Gori, Opt. Commun. 31, 4 (1979).
    [CrossRef]
  5. R. Sudol and B. J. Thompson, Opt. Commun. 31, 105 (1979).
    [CrossRef]
  6. R. J. Sudol and B. J. Thompson, Appl. Opt. 20, 1107 (1981).
    [CrossRef] [PubMed]
  7. L. Liu, J. Opt. Soc. Am. A 5, 1709 (1988).
    [CrossRef]
  8. P. Szwaykowski and J. Ojeda-Castaneda, Opt. Commun. 83, 1 (1991).
    [CrossRef]
  9. J. F. Clausre and S. Li, Phys. Rev. A 49, R2213 (1994).
    [CrossRef]
  10. C. J. Bordé, in Fundamental Systems in Quantum Optics, J. Dalibard, J. M. Raimond, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1992), p. 291.
  11. J. A. Amaud, Beam and Fiber Optics (Academic, New York, 1976).
  12. A. Erdèlyi, Higher Transcendental Functions, the Batemann Manuscript Project (McGraw-Hill, New York, 1953).
  13. H.-Y. Fan and X. Ye, Phys. Lett. A 175, 387 (1993).
    [CrossRef]
  14. H.-Y. Fan and H.-L. Lu, Opt. Lett. 28, 680 (2003).
    [CrossRef] [PubMed]
  15. H.-Y. Fan and J.-H. Chen, Phys. Lett. A 303, 311 (2002).
    [CrossRef]
  16. H.-Y. Fan and X. Ye, Phys. Rev. A 51, 3343 (1995).
    [CrossRef]
  17. H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
    [CrossRef]
  18. L. Yu, Y. Y. Lu, X. M. Zeng, M. Huang, M. Chen, W. Huang, and Z. Zhu, Opt. Lett. 23, 1158 (1998).
    [CrossRef]

2003 (1)

2002 (2)

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 88, 100404 (2002).
[CrossRef]

H.-Y. Fan and J.-H. Chen, Phys. Lett. A 303, 311 (2002).
[CrossRef]

1998 (1)

1995 (2)

H.-Y. Fan and X. Ye, Phys. Rev. A 51, 3343 (1995).
[CrossRef]

G. S. Agarwal, Opt. Commun. 119, 30 (1995).
[CrossRef]

1994 (2)

H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
[CrossRef]

J. F. Clausre and S. Li, Phys. Rev. A 49, R2213 (1994).
[CrossRef]

1993 (1)

H.-Y. Fan and X. Ye, Phys. Lett. A 175, 387 (1993).
[CrossRef]

1991 (1)

P. Szwaykowski and J. Ojeda-Castaneda, Opt. Commun. 83, 1 (1991).
[CrossRef]

1988 (1)

1983 (1)

A. W. Lohmann and J. O. Castaneda, Opt. Acta 30, 475 (1983).
[CrossRef]

1981 (1)

1979 (2)

F. Gori, Opt. Commun. 31, 4 (1979).
[CrossRef]

R. Sudol and B. J. Thompson, Opt. Commun. 31, 105 (1979).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, Opt. Commun. 119, 30 (1995).
[CrossRef]

Amaud, J. A.

J. A. Amaud, Beam and Fiber Optics (Academic, New York, 1976).

Arndt, M.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 88, 100404 (2002).
[CrossRef]

Bordé, C. J.

C. J. Bordé, in Fundamental Systems in Quantum Optics, J. Dalibard, J. M. Raimond, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1992), p. 291.

Brezger, B.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 88, 100404 (2002).
[CrossRef]

Castaneda, J. O.

A. W. Lohmann and J. O. Castaneda, Opt. Acta 30, 475 (1983).
[CrossRef]

Chen, J.-H.

H.-Y. Fan and J.-H. Chen, Phys. Lett. A 303, 311 (2002).
[CrossRef]

Chen, M.

Clausre, J. F.

J. F. Clausre and S. Li, Phys. Rev. A 49, R2213 (1994).
[CrossRef]

Erdèlyi, A.

A. Erdèlyi, Higher Transcendental Functions, the Batemann Manuscript Project (McGraw-Hill, New York, 1953).

Fan, H.-Y.

H.-Y. Fan and H.-L. Lu, Opt. Lett. 28, 680 (2003).
[CrossRef] [PubMed]

H.-Y. Fan and J.-H. Chen, Phys. Lett. A 303, 311 (2002).
[CrossRef]

H.-Y. Fan and X. Ye, Phys. Rev. A 51, 3343 (1995).
[CrossRef]

H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
[CrossRef]

H.-Y. Fan and X. Ye, Phys. Lett. A 175, 387 (1993).
[CrossRef]

Gori, F.

F. Gori, Opt. Commun. 31, 4 (1979).
[CrossRef]

Hackermüller, L.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 88, 100404 (2002).
[CrossRef]

Huang, M.

Huang, W.

Klauder, J. R.

H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
[CrossRef]

Li, S.

J. F. Clausre and S. Li, Phys. Rev. A 49, R2213 (1994).
[CrossRef]

Liu, L.

Lohmann, A. W.

A. W. Lohmann and J. O. Castaneda, Opt. Acta 30, 475 (1983).
[CrossRef]

Lu, H.-L.

Lu, Y. Y.

Ojeda-Castaneda, J.

P. Szwaykowski and J. Ojeda-Castaneda, Opt. Commun. 83, 1 (1991).
[CrossRef]

Petschinka, J.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 88, 100404 (2002).
[CrossRef]

Sudol, R.

R. Sudol and B. J. Thompson, Opt. Commun. 31, 105 (1979).
[CrossRef]

Sudol, R. J.

Szwaykowski, P.

P. Szwaykowski and J. Ojeda-Castaneda, Opt. Commun. 83, 1 (1991).
[CrossRef]

Thompson, B. J.

R. J. Sudol and B. J. Thompson, Appl. Opt. 20, 1107 (1981).
[CrossRef] [PubMed]

R. Sudol and B. J. Thompson, Opt. Commun. 31, 105 (1979).
[CrossRef]

Uttenthaler, S.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 88, 100404 (2002).
[CrossRef]

Ye, X.

H.-Y. Fan and X. Ye, Phys. Rev. A 51, 3343 (1995).
[CrossRef]

H.-Y. Fan and X. Ye, Phys. Lett. A 175, 387 (1993).
[CrossRef]

Yu, L.

Zeilinger, A.

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 88, 100404 (2002).
[CrossRef]

Zeng, X. M.

Zhu, Z.

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

A. W. Lohmann and J. O. Castaneda, Opt. Acta 30, 475 (1983).
[CrossRef]

Opt. Commun. (4)

F. Gori, Opt. Commun. 31, 4 (1979).
[CrossRef]

R. Sudol and B. J. Thompson, Opt. Commun. 31, 105 (1979).
[CrossRef]

P. Szwaykowski and J. Ojeda-Castaneda, Opt. Commun. 83, 1 (1991).
[CrossRef]

G. S. Agarwal, Opt. Commun. 119, 30 (1995).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (2)

H.-Y. Fan and X. Ye, Phys. Lett. A 175, 387 (1993).
[CrossRef]

H.-Y. Fan and J.-H. Chen, Phys. Lett. A 303, 311 (2002).
[CrossRef]

Phys. Rev. A (3)

H.-Y. Fan and X. Ye, Phys. Rev. A 51, 3343 (1995).
[CrossRef]

H.-Y. Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
[CrossRef]

J. F. Clausre and S. Li, Phys. Rev. A 49, R2213 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

B. Brezger, L. Hackermüller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger, Phys. Rev. Lett. 88, 100404 (2002).
[CrossRef]

Other (3)

C. J. Bordé, in Fundamental Systems in Quantum Optics, J. Dalibard, J. M. Raimond, and J. Zinn-Justin, eds. (Elsevier, Amsterdam, 1992), p. 291.

J. A. Amaud, Beam and Fiber Optics (Academic, New York, 1976).

A. Erdèlyi, Higher Transcendental Functions, the Batemann Manuscript Project (McGraw-Hill, New York, 1953).

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Equations (36)

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ikE0z=-12k22E0+12Q2x2+y2E0,
12Q2n2/n0, k0=ωc, k=ωcn0.
Hm,nζ,ζ*=l=0minm,nm!n!l!m-l!n-l!×-1lζm-lζ*n-l,
m,n=0tmtnm!n!Hm,nζ,ζ*=exp-tt+tζ+tζ*,
mHm-1,nζ,ζ*+Hm,n+1ζ,ζ*=ζ*Hm,nζ,ζ*,nHm,n-1ζ,ζ*+Hm+1,nζ,ζ*=ζHm,nζ,ζ*.
-1mexpξηd2zπznz*m×exp-z2+ξz-ηz*=Hm,nξ,η,
m,n=0Hm,nξ,ηHm,nρ,ksmtnm!n!=11-stexpsξρ+tηk-stρk-stξη1-st.
d2ζπHm,nζ,ζ*Hm,nζ,ζ**exp-ζ2=m!n!δm,mδn,n.
ζ|m,n=1m!n!Hm,nζ,ζ*exp-ζ2/2,
ζ|=00|m,n=0ambnm!n!Hm,nζ,ζ*exp-ζ2/2=00|exp-12ζ2-ab+aζ+bζ*,
X1=12a+a, P1=12ia-a, X2=12b+b, P2=12ib-b.
d2ζπ|ζζ|=1, d2ζdζ1dζ2.
|η=exp-½η2+ηa-η*b+ab|00,η=η1+iη2.
d2ηπ|ηη|=1, d2ηdη1dη2.
η|ζ=½expη*ζ-ηζ*/2.
Em,nx,y,z=ψm,nζ,ζ*exp-ikQz,
ζkQx+iy=ζ1+iζ2.
42ζ*ζψm,n+2k-ζ*ζψm,n=0.
2Hm,nζ*ζ-ζ*Hm,nζ*-ζHm,nζ+k-1Hm,n=0.
ζHm,nζ,ζ*=mHm-1,nζ,ζ*,ζ*Hm,nζ,ζ*=nHm,n-1ζ,ζ*.
2Hm,nζ*ζ-ζ*Hm,nζ*+nHm,n=0,2Hm,nζ*ζ-ζHm,nζ+mHm,n=0,
Em,nx,y,z=1m!n!exp-ζ2/2Hm,nζ,ζ*×exp-iQzm+n+1.
E0x,y,0=expiqx+isy.
q-η2kQ, sη1kQ;
E0x,y,0E0ζ1,ζ2,0=expiη2ζ2-η2ζ1=expη*ζ-ηζ*/2.
expη*ζ-ηζ*/2=m,nCm,nψm,n*ζ,ζ*,
d2ζπexpη*ζ-ηζ*/2ψm,nζ,ζ*=m,nCm,nd2ζπψm,nζ,ζ*ψm,n*ζ,ζ*=Cm,n
2d2ζπη|ζζ|m,n=2η|m,n=Cm,n.
η|=00|m,n=0iamibnm!n!Hm,n-iη*,iη,
Cm,n=2im+n1m!n!Hm,n-iη*,iηexp-η2/2.
Ex,y,z=m,nCm,nEm,nx,y,zEζ1,ζ2,z=2m,nim+n1m!n!Hm,n-iη*,iηHm,nζ,ζ*×exp-ζ2/2-η2/2exp-iQzm+n+1.
Eζ1,ζ2,z=1cos Qzexpη*ζ-ηζ*2 cos Qz-i2tan Qzζ2+η2.
Ex,y,z=1cos Qzexpisy+qxcos Qz-i2tan QzkQx2+y2+q2+s2kQ,
E0x,y,0=m,nEm,n cos2πmdxcos2πndy.
Ex,y,z=m,nEm,n cos4πmxd cos Qzcos4πnyd cos Qz×expi2tan Qz-kQx2+y2-4π2m2+n2kQd2.
exp-ikQtan Qz2π2m2+n2d2=1, π tan QzkQd2m2+n2 an integer,

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