Abstract

A virtual source that generates a Hermite–Gauss wave of mode numbers m and n is introduced. An expression is obtained for this Hermite–Gauss wave. From this expression, the paraxial approximation and the first 3 orders of nonparaxial corrections for the corresponding paraxial Hermite–Gauss beam are determined. When both m and n are even, leading to maximum amplitude along the axis, the number of orders of nonvanishing nonparaxial corrections is found to be equal to m+n/2.

© 2003 Optical Society of America

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References

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  1. K. Iizuka, Elements of Photonics (Wiley-Interscience, New York, 2002), pp. 205–227, 237–249.
  2. C. J. R. Sheppard and T. Wilson, IEE J. Microwaves Opt. Acoust. 2, 105 (1978).
    [CrossRef]
  3. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  4. M. Couture and P. A. Belanger, Phys. Rev. A 24, 355 (1981).
    [CrossRef]
  5. T. Takenaka, M. Yokota, and O. Fukumitsu, J. Opt. Soc. Am. A 2, 826 (1985).
    [CrossRef]
  6. G. A. Deschamps, Electron. Lett. 7, 684 (1971).
    [CrossRef]
  7. L. B. Felsen, J. Opt. Soc. Am. 66, 751 (1976).
  8. S. Y. Shin and L. B. Felsen, J. Opt. Soc. Am. 67, 699 (1977).
  9. S. R. Seshadri, Opt. Lett. 27, 1872 (2002).
    [CrossRef]
  10. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 1034, formulas 8.956.6 and 8.956.7.

2002

1987

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

1985

T. Takenaka, M. Yokota, and O. Fukumitsu, J. Opt. Soc. Am. A 2, 826 (1985).
[CrossRef]

1981

M. Couture and P. A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

1978

C. J. R. Sheppard and T. Wilson, IEE J. Microwaves Opt. Acoust. 2, 105 (1978).
[CrossRef]

1977

1976

1971

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Belanger, P. A.

M. Couture and P. A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Couture, M.

M. Couture and P. A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Felsen, L. B.

Fukumitsu, O.

T. Takenaka, M. Yokota, and O. Fukumitsu, J. Opt. Soc. Am. A 2, 826 (1985).
[CrossRef]

Gradshteyn, S.

S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 1034, formulas 8.956.6 and 8.956.7.

Iizuka, K.

K. Iizuka, Elements of Photonics (Wiley-Interscience, New York, 2002), pp. 205–227, 237–249.

Miceli, Jr., J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Ryzhik, I. M.

S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 1034, formulas 8.956.6 and 8.956.7.

Seshadri, S. R.

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, IEE J. Microwaves Opt. Acoust. 2, 105 (1978).
[CrossRef]

Shin, S. Y.

Takenaka, T.

T. Takenaka, M. Yokota, and O. Fukumitsu, J. Opt. Soc. Am. A 2, 826 (1985).
[CrossRef]

Wilson, T.

C. J. R. Sheppard and T. Wilson, IEE J. Microwaves Opt. Acoust. 2, 105 (1978).
[CrossRef]

Yokota, M.

T. Takenaka, M. Yokota, and O. Fukumitsu, J. Opt. Soc. Am. A 2, 826 (1985).
[CrossRef]

Electron. Lett.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

IEE J. Microwaves Opt. Acoust.

C. J. R. Sheppard and T. Wilson, IEE J. Microwaves Opt. Acoust. 2, 105 (1978).
[CrossRef]

J. Opt. Soc. Am. A

T. Takenaka, M. Yokota, and O. Fukumitsu, J. Opt. Soc. Am. A 2, 826 (1985).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Phys. Rev. A

M. Couture and P. A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Other

K. Iizuka, Elements of Photonics (Wiley-Interscience, New York, 2002), pp. 205–227, 237–249.

S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 1034, formulas 8.956.6 and 8.956.7.

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Equations (34)

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2x2+2y2+2z2+k2Fm,nx,y,z=-Sexmxmδxnynδyδz-zex,
Fm,nx,y,z=--F¯m,npx,py,z×exp-i2πpxx+pyydpxdpy,
F¯m,npx,py,z=--Fm,nx,y,z×expi2πpxx+pyydxdy
Fm,nx,y,z=--dpxdpy exp-i2πpxx+pyy×iSex/2ζ-i2πpxm-i2πpyn×expiζz-zex
ζ=k2-4π2px2+py21/2.
Fm,nx,y,z=expikz×iSex2kexp-ikzex--dpxdpy×exp-i2πpxx+pyy-i2πpxm-i2πpyn×exp-i2π2kpx2+py2z-zex.
Fm,nx,y,z=expikzmxmnyn×Sex exp-ikzex4πz-zexexpikx2+y22z-zex.
Fm,n,px,y,0=-1mHmx˜exp-x˜2×-1nHny˜exp-y˜2,
Hmv=-1mexpv2dm/dvmexp-v2,
Fm,n,px,y,0=w0m+nmxmnynexp-x2+y2w02.
zex=ikw02/2=ib,
Sex=-i4πb exp-kbw0m+n.
Fm,n,px,y,z=expikzmx˜mny˜n×q2zexp-q2zx˜2+y˜2,
Fm,n,px,y,z=expikzq2+m+nz-1m+n×exp-q2zx˜2+y˜2×Hmqzx˜Hnqzy˜,
q2z=1+izb-1.
w0/xFm,n,px,y,z=Fm+1,n,px,y,z,
w0/yFm,n,px,y,z=Fm,n+1,px,y,z.
Fm,nx,y,z=2πb exp-kbw0m+n--dpxdpy×exp-i2πpxx+pyy-i2πpxm-i2πpyn×expiζz-ib/ζ.
2/x2+2/y2+2/z2+k2Gx,y,z=-δxδyδz-ib
Gx,y,z=expikR/4πR,
R=x2+y2+z-ib21/2.
Fm,nx,y,z=-ib exp-kbw0m+nmxmnyn×expikRR.
Fm,nx,y,z=expikzmx˜mny˜nπw02--dpxdpy×exp-i2πpxx+pyy×exp-i2π2kpx2+py2z-ib×G¯px,py,z,
G¯px,py,z=1+s=1s=3kw0-2sG¯2spx,py,z,
G¯2px,py,z=124π2w02px2+py2-116q2z4π2w02px2+py22.
Fm,nx,y,z=expikzs=0s=3kw0-2sfm,n2sx,y,z,
fm,n2sx,y,z=q2+m+nz-1m+nexp-q2zx˜2+y˜2×-1sq2szgm,n2sx,y,z,
gm,n2s=t=st=2sa2s,tl=0l=tt!l!t-l!Hm+2t-lqzx˜×Hn+2lqzy˜,
a0,0=1,
a2,1=1/2,    a2,2=1/16.
a4,2=3/8,    a4,3=1/16,    a4,4=1/512,
a6,3=5/16,    a6,4=15/256,    a6,5=3/1024,    a6,6=1/4096×6.
Hm0=-1m/22m-1!/m/2-1!
Fm,n0,0,z=expikzq2+m+nz-1m+nHm0Hn0×s=0s=3kw0-2s-1sq2szj+s!s!j-s!,

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