Abstract

Based on the generalized Huygens–Fresnel diffraction integral (Collins’ formula), the propagation equation of Hermite–Gauss beams through a complex optical system with a limiting aperture is derived. The elements of the optical system may be all those characterized by an ABCD ray-transfer matrix, as well as any kind of apertures represented by complex transmittance functions. To obtain the analytical expression, we expand the aperture transmittance function into a finite sum of complex Gaussian functions. Thus the limiting aperture is expressed as a superposition of a series of Gaussian-shaped limiting apertures. The advantage of this treatment is that we can treat almost all kinds of apertures in theory. As application, we define the width of the beam and the focal plane using an encircled-energy criterion and calculate the intensity distribution of Hermite–Gauss beams at the actual focus of an aperture lens.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Vicari and F. Bloisi, “Matrix representation of axisymmetric optical systems including spatial filters,” Appl. Opt. 28, 4682–4686 (1989).
    [CrossRef]
  2. X. Y. Tao, N. R. Zhou, and B. D. Lü, “Recurrence propagation equation of Hermite-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Optik 114, 113–117 (2003).
    [CrossRef]
  3. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  4. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  5. E. Feldheim, “Équations intégrales pour les polynomes d'Hermite à une et plusieurs variables, pour les polynomes de Laguerre, et pour les fonctions hypergéométriques les plus générales,” Proc. Kon. Ned. Akad. v. Wetensch. 43, 224–239 (1940).
  6. J. J. Wen and A. Breazeale, “A diffraction beam field expressed as the superpositon of Gaussion beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
    [CrossRef]
  7. P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999).
    [CrossRef]
  8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Courier Dover Publications, 1964).

2003

X. Y. Tao, N. R. Zhou, and B. D. Lü, “Recurrence propagation equation of Hermite-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Optik 114, 113–117 (2003).
[CrossRef]

1999

1989

1988

J. J. Wen and A. Breazeale, “A diffraction beam field expressed as the superpositon of Gaussion beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

1970

1966

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1940

E. Feldheim, “Équations intégrales pour les polynomes d'Hermite à une et plusieurs variables, pour les polynomes de Laguerre, et pour les fonctions hypergéométriques les plus générales,” Proc. Kon. Ned. Akad. v. Wetensch. 43, 224–239 (1940).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Courier Dover Publications, 1964).

Bloisi, F.

Breazeale, A.

J. J. Wen and A. Breazeale, “A diffraction beam field expressed as the superpositon of Gaussion beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Collins, S. A.

Feldheim, E.

E. Feldheim, “Équations intégrales pour les polynomes d'Hermite à une et plusieurs variables, pour les polynomes de Laguerre, et pour les fonctions hypergéométriques les plus générales,” Proc. Kon. Ned. Akad. v. Wetensch. 43, 224–239 (1940).

Greene, P. L.

Hall, D. G.

Kogelnik, H.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Lü, B. D.

X. Y. Tao, N. R. Zhou, and B. D. Lü, “Recurrence propagation equation of Hermite-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Optik 114, 113–117 (2003).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Courier Dover Publications, 1964).

Tao, X. Y.

X. Y. Tao, N. R. Zhou, and B. D. Lü, “Recurrence propagation equation of Hermite-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Optik 114, 113–117 (2003).
[CrossRef]

Vicari, L.

Wen, J. J.

J. J. Wen and A. Breazeale, “A diffraction beam field expressed as the superpositon of Gaussion beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Zhou, N. R.

X. Y. Tao, N. R. Zhou, and B. D. Lü, “Recurrence propagation equation of Hermite-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Optik 114, 113–117 (2003).
[CrossRef]

Appl. Opt.

J. Acoust. Soc. Am.

J. J. Wen and A. Breazeale, “A diffraction beam field expressed as the superpositon of Gaussion beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. Opt. Soc. Am.

Opt. Express

Optik

X. Y. Tao, N. R. Zhou, and B. D. Lü, “Recurrence propagation equation of Hermite-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Optik 114, 113–117 (2003).
[CrossRef]

Proc. IEEE

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Proc. Kon. Ned. Akad. v. Wetensch.

E. Feldheim, “Équations intégrales pour les polynomes d'Hermite à une et plusieurs variables, pour les polynomes de Laguerre, et pour les fonctions hypergéométriques les plus générales,” Proc. Kon. Ned. Akad. v. Wetensch. 43, 224–239 (1940).

Other

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Courier Dover Publications, 1964).

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.

Schematic representation of an optical system without apertures.

Fig. 2.
Fig. 2.

Schematic representation of an optical system with limiting apertures.

Fig. 3.
Fig. 3.

Schematic representation of the aperture lens.

Fig. 4.
Fig. 4.

Intensity distribution of TEM11-mode Hermite–Gauss beam at the actual focus.

Fig. 5.
Fig. 5.

Intensity distribution of TEM00-mode Hermite–Gauss beam for both methods.

Tables (1)

Tables Icon

Table 1. Set of Coefficients u, ε Used in Representing the Aperture Transmittance

Equations (53)

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

Ui(xi,yi)=Kiexp[j(π/λqix)xi2]Hm(2xiωix)×exp[j(π/λqiy)yi2]Hn(2yiωiy),
1qix,y=1Rix,yjλπωix,y2,
U(x,y)=Kijk2πBxByexp(jkL)exp[j(π/λqix)xi2]×Hm(2xiωix)exp[j(π/λBx)][Dxx22xxi+Axxi2]dxi×exp[j(π/λqiy)yi2]Hn(2yiωiy)exp[j(π/λBy)][Dyy22yyi+Ayyi2]dyi,
U(x,y)=Kif(x)g(y),
f(x)=[jk2πBxexp(jkL)]12exp[j(π/λqix)xi2]Hm(2xiωix)×exp[j(π/λBx)][Dxx22xxi+Axxi2]dxi,
g(y)=[jk2πByexp(jkL)]12exp[j(π/λqiy)yi2]Hn(2yiωiy)×exp[j(π/λBy)][Dyy22yyi+Ayyi2]dyi.
1πexp[(yt)2]Hm(tλ)dt=(11λ2)m/2Hm(yλ21),
f(x)=[ωixωxexp(jkL)]1/2exp[j(12+m)δx]×exp[j(π/λqx)x2]Hm(2xωx).
g(y)=[ωiyωyexp(jkL)]1/2exp[j(12+n)δy]×exp[j(π/λqy)y2]Hn(2yωy),
qx,y=Ax,yqix,y+Bx,yCx,yqix,y+Dx,y,1qix,y=1Rix,yjλπωix,y2,
1qx,y=1Rx,yjλπωx,y2,
ωx,yωix,yexp(jδx,y)=Ax,y+Bx,yqix,y,
δx,y=arctan[λBx,yπωix,y2(Ax,y+Bx,y/Rix,y)],
1Rx,y=(ωix,yωx,y)2[Ax,yCx,y+1Rix,y(Ax,yDx,y+Bx,yCx,y)+Bx,yDx,yqix,yqix,y*].
U(x,y)=Ki(ωixωiyωxωy)1/2exp{j[kL+(12+m)δx+(12+n)δy]}×exp[j(π/λqx)x2]Hm(2xωx)×exp[j(π/λqy)y2]Hn(2yωy).
T=T(x)T(y)=k=1Mukexp(εkx2)l=1Nvlexp(φly2),
UA(x,y)=KAexp[j(π/λqAx)x2]Hm(2xωAx)×exp[j(π/λqAy)y2]Hn(2yωAy),
KA=Ki(ωixωiyωAxωAy)1/2exp{j[kL1+(12+m)δAx+(12+n)δAy]},
qAx,y=A1x,yqix,y+B1x,yC1x,yqix,y+D1x,y,1qix,y=1Rix,yjλπωix,y2,
1qAx,y=1RAx,yjλπωAx,y2,
ωAx,yωix,yexp(jδAx,y)=A1x,y+B1x,yqix,y,
δAx,y=arctan[λB1x,yπωix,y2(A1x,y+B1x,y/Rix,y)],
1RAx,y=(ωix,yωAx,y)2[A1x,yC1x,y+1Rix,y(A1x,yD1x,y+B1x,yC1x,y)+B1x,yD1x,yqix,yqix,y*],
M1x,y=[A1x,yB1x,yC1x,yD1x,y],
U(x,y)=KAexp(jkL2)k=1Muk(jk2πB2x)1/2exp[(εk+jπ/λqAx)xA2]Hm(2xAωAx)exp[j(π/λB2x)][D2xx22xxA+A2xxA2]dxA×l=1Nvl(jk2πB2y)1/2exp[(φl+jπ/λqAx)yA2]Hn(2yAωAy)×exp[j(π/λB2y)][D2yy22yyA+A2yyA2]dyA,
U(x,y)=Kexp(jπD2xλB2xx2)exp(jπD2yλB2yy2)×k=1MukQx(12ωAx2Qx2)m/2exp[(π/λB2xQx)2x2]Hm(Pxx)×l=1NvlQy(12ωAy2Qy2)n/2exp[(π/λB2yQy)2y2]Hn(Pyy),
Qx2=jπλqAx+jπA2xλB2x+εkQy2=jπλqAy+jπA2yλB2y+φl,
Px=jπλB2xQx(ωAx2Qx2/21)1/2,
Py=jπλB2yQy(ωAy2Qy2/21)1/2,
K=KA(jk2B2xB2y)exp(jkL2),
M2x,y=[A2x,yB2x,yC2x,yD2x,y].
M=[ABCD]=[s/ff+s1/f1].
T(x,y)=k=1Mukexp(εkx2a2)l=1Nulexp(εly2a2).
qix,y=ikω02/2.
Ei(xi,yi)=Kiexp[(xi2+yi2)/ω02]Hm(2xiω0)Hn(2yiω0).
E(x,y,z)=Kiexp(jkz)(jπλB)exp(jπDλBx2)exp(jπDλBy2)k=1MukQx,k(12ω02Qx,k2)m/2exp[(πλBQx,k)2x2]Hm(Px,kx)×l=1NulQy,l(12ω02Qy,l2)n/2exp[(πλBQy,l)2y2]Hn(Py,ly),
Qx,k2=1ω02+jπAλB+εka2Qy,l2=1ω02+jπAλB+εla2,
Px,k=jπλBQx,k(ω02Qx,k2/21)1/2,
Py,l=jπλBQy,l(ω02Qy,l2/21)1/2,
I(x,y,z)=|E(x,y,z)|2.
ωωωωI(x,y,z)dxdyI(x,y,z)dxdy=0.8.
S11xS11y=0.8,
S11x,y=k=110l=110ξx,y[πerf(ηx,yω)2exp(ηx,y2ω2)ηx,yω]ηx,y3/k=110l=110ξx,yπηx,y3,
ξx=(k2B)ukul*Qx,kQx,l*(12Qx,k)1/2(12Qx,l*)1/2Px,kPx,l*,
ξy=(k2B)ukul*Qy,kQy,l*(12Qy,k)1/2(12Qy,l*)1/2Py,kPy,l*,
Qx,k2=1ω02+jπAλB+εka2,Qy,l2=1ω02+jπAλB+εla2,
Px,k=jπλBQx,k(ω02Qx,k2/21)1/2,
Py,l=jπλBQy,l(ω02Qy,l2/21)1/2,
ηx2=(ikD2B+k24B2Qx,k2)+(ikD2B+k24B2Qx,l2)*,
ηy2=(ikD2B+k24B2Qy,k2)+(ikD2B+k24B2Qy,l2)*,
erf(y)=2π0yexp(t2)dt.
Γ(z)=0tz1etdt(z>0),
γ(a,x)=0xetta1dt(a>0),

Metrics