Abstract

The analysis of many systems in optical communications and metrology utilizing Gaussian beams, such as free-space propagation from single-mode fibers, point diffraction interferometers, and interference lithography, would benefit from an accurate analytical model of Gaussian beam propagation. We present a full vector analysis of Gaussian beam propagation by using the well-known method of the angular spectrum of plane waves. A Gaussian beam is assumed to traverse a charge-free, homogeneous, isotropic, linear, and nonmagnetic dielectric medium. The angular spectrum representation, in its vector form, is applied to a problem with a Gaussian intensity boundary condition. After some mathematical manipulation, each nonzero propagating electric field component is expressed in terms of a power-series expansion. Previous analytical work derived a power series for the transverse field, where the first term (zero order) in the expansion corresponds to the usual scalar paraxial approximation. We confirm this result and derive a corresponding longitudinal power series. We show that the leading longitudinal term is comparable in magnitude with the first transverse term above the scalar paraxial term, thus indicating that a full vector theory is required when going beyond the scalar paraxial approximation. In spite of the advantages of a compact analytical formalism, enabling rapid and accurate modeling of Gaussian beam systems, this approach has a notable drawback. The higher-order terms diverge at locations that are sufficiently far from the initial boundary, yielding unphysical results. Hence any meaningful use of the expansion approach calls for a careful study of its range of applicability. By considering the transition of a Gaussian wave from the paraxial to the spherical regime, we are able to derive a simple expression for the range within which the series produce numerically satisfying answers.

© 2002 Optical Society of America

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References

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  1. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss and gain variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  2. D. C. O’Shea, Elements of Modern Optical Design (Wiley-Interscience, New York, 1985), pp. 247–252.
  3. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997), Chap. 2.
  4. G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  5. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  6. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
    [CrossRef]
  7. C. G. Chen, P. T. Konkola, R. K. Heilmann, G. S. Pati, M. L. Schattenburg, “Image metrology and system controls for scanning beam interference lithography,” J. Vac. Sci. Technol. B (to be published).
  8. M. H. Lim, J. Ferrera, K. P. Pipe, H. I. Smith, “A holo-graphic phase-shifting interferometer technique to measure in-plane distortion,” J. Vac. Sci. Technol. B 17, 2703–2706 (1999).
    [CrossRef]
  9. G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” in Extreme Ultraviolet Lithography, G. D. Kubiak, D. R. Kania, eds., Vol. 4 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 108–112.
  10. D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
    [CrossRef]
  11. D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
    [CrossRef]
  12. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, 1994), p. 978.
  13. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, 1994), pp. 737 [Eq. (6.631)(1)], 1062 [Eq. (8.972)(1)], 1087 [Eq. (9.220)(2)].
  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1 of Fortran Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 16.
  15. M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, UK, 1980), p. 752.

1999 (1)

M. H. Lim, J. Ferrera, K. P. Pipe, H. I. Smith, “A holo-graphic phase-shifting interferometer technique to measure in-plane distortion,” J. Vac. Sci. Technol. B 17, 2703–2706 (1999).
[CrossRef]

1979 (1)

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1972 (1)

1966 (1)

D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
[CrossRef]

1965 (1)

1964 (1)

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[CrossRef]

Agrawal, G. P.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, UK, 1980), p. 752.

Carter, W. H.

Chen, C. G.

C. G. Chen, P. T. Konkola, R. K. Heilmann, G. S. Pati, M. L. Schattenburg, “Image metrology and system controls for scanning beam interference lithography,” J. Vac. Sci. Technol. B (to be published).

Ferrera, J.

M. H. Lim, J. Ferrera, K. P. Pipe, H. I. Smith, “A holo-graphic phase-shifting interferometer technique to measure in-plane distortion,” J. Vac. Sci. Technol. B 17, 2703–2706 (1999).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1 of Fortran Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 16.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, 1994), p. 978.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, 1994), pp. 737 [Eq. (6.631)(1)], 1062 [Eq. (8.972)(1)], 1087 [Eq. (9.220)(2)].

Heilmann, R. K.

C. G. Chen, P. T. Konkola, R. K. Heilmann, G. S. Pati, M. L. Schattenburg, “Image metrology and system controls for scanning beam interference lithography,” J. Vac. Sci. Technol. B (to be published).

Kogelnik, H.

Konkola, P. T.

C. G. Chen, P. T. Konkola, R. K. Heilmann, G. S. Pati, M. L. Schattenburg, “Image metrology and system controls for scanning beam interference lithography,” J. Vac. Sci. Technol. B (to be published).

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lim, M. H.

M. H. Lim, J. Ferrera, K. P. Pipe, H. I. Smith, “A holo-graphic phase-shifting interferometer technique to measure in-plane distortion,” J. Vac. Sci. Technol. B 17, 2703–2706 (1999).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

O’Shea, D. C.

D. C. O’Shea, Elements of Modern Optical Design (Wiley-Interscience, New York, 1985), pp. 247–252.

Pati, G. S.

C. G. Chen, P. T. Konkola, R. K. Heilmann, G. S. Pati, M. L. Schattenburg, “Image metrology and system controls for scanning beam interference lithography,” J. Vac. Sci. Technol. B (to be published).

Pattanayak, D. N.

Pipe, K. P.

M. H. Lim, J. Ferrera, K. P. Pipe, H. I. Smith, “A holo-graphic phase-shifting interferometer technique to measure in-plane distortion,” J. Vac. Sci. Technol. B 17, 2703–2706 (1999).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1 of Fortran Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 16.

Rhodes, D. R.

D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
[CrossRef]

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, 1994), p. 978.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, 1994), pp. 737 [Eq. (6.631)(1)], 1062 [Eq. (8.972)(1)], 1087 [Eq. (9.220)(2)].

Schattenburg, M. L.

C. G. Chen, P. T. Konkola, R. K. Heilmann, G. S. Pati, M. L. Schattenburg, “Image metrology and system controls for scanning beam interference lithography,” J. Vac. Sci. Technol. B (to be published).

Smith, H. I.

M. H. Lim, J. Ferrera, K. P. Pipe, H. I. Smith, “A holo-graphic phase-shifting interferometer technique to measure in-plane distortion,” J. Vac. Sci. Technol. B 17, 2703–2706 (1999).
[CrossRef]

Sommargren, G. E.

G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” in Extreme Ultraviolet Lithography, G. D. Kubiak, D. R. Kania, eds., Vol. 4 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 108–112.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1 of Fortran Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 16.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1 of Fortran Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 16.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, UK, 1980), p. 752.

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997), Chap. 2.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. AP-14, 676–683 (1966).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Vac. Sci. Technol. B (1)

M. H. Lim, J. Ferrera, K. P. Pipe, H. I. Smith, “A holo-graphic phase-shifting interferometer technique to measure in-plane distortion,” J. Vac. Sci. Technol. B 17, 2703–2706 (1999).
[CrossRef]

Phys. Rev. A (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Proc. IEEE (1)

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[CrossRef]

Other (8)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, 1994), p. 978.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, San Diego, 1994), pp. 737 [Eq. (6.631)(1)], 1062 [Eq. (8.972)(1)], 1087 [Eq. (9.220)(2)].

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1 of Fortran Numerical Recipes, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 16.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, UK, 1980), p. 752.

D. C. O’Shea, Elements of Modern Optical Design (Wiley-Interscience, New York, 1985), pp. 247–252.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997), Chap. 2.

C. G. Chen, P. T. Konkola, R. K. Heilmann, G. S. Pati, M. L. Schattenburg, “Image metrology and system controls for scanning beam interference lithography,” J. Vac. Sci. Technol. B (to be published).

G. E. Sommargren, “Phase shifting diffraction interferometry for measuring extreme ultraviolet optics,” in Extreme Ultraviolet Lithography, G. D. Kubiak, D. R. Kania, eds., Vol. 4 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 108–112.

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Figures (8)

Fig. 1
Fig. 1

Plot of |Ez(1)|2 as a function of x and y for ω0=1.3 μm, λ = 351.1 nm, and z=1 mm. Ez(1) represents the first nonfundamental Hermite–Gaussian mode. Note the mirror symmetry of the two humps about the x axis.

Fig. 2
Fig. 2

Plot of the ratio of the vector field intensity |Ex|2+|Ez|2 to the scalar intensity |Ex|2 for z=1 mm, ω0=1.3 μm, and λ=351.1 nm (f=0.043). A 1% deviation is registered near the edge. The inset shows the scalar intensity profile |Ex|2 sampled over the same 0.2-mm × 0.2-mm region.

Fig. 3
Fig. 3

Plot of the ratio of the vector field intensity |Ex|2+|Ez|2 to the scalar intensity |Ex|2 for z=3.6 μm, ω0=0.2 μm, and λ=351.1 nm (f=0.28). A >40% deviation is registered near the edge. The inset shows the scalar intensity profile |Ex|2 sampled over the same 5-μm × 5-μm region.

Fig. 4
Fig. 4

Plots for z=20 mm, ω0=1.3 μm, and λ= 351.1 nm (f=0.043), the solid curve representing |Ex(0)|2, the dotted curve representing |Ex(0)+f2Ex(2)|2, and the dashed curve representing |Ex(0)+f2Ex(2)+f4Ex(4)|2.

Fig. 5
Fig. 5

Schematic showing the various coordinate measures used in deriving the range of applicability zt for the series expansion. Beyond the range, paraxial optics breaks down, and the series expansion approach becomes invalid.

Fig. 6
Fig. 6

Plots for ω0=1.3 μm, λ=351.1 nm, and z=zt/2=0.5 mm. (a) Numerically obtained scalar field intensity |Ex|2. (b) The thin solid, dotted, dashed, and thick solid curves represent differences from the numerical intensity by |Ex(0)|2, |Ex(0)+f2Ex(2)|2, and |Ex(0)+f2Ex(2)+f4Ex(4)|2 and the spherical result, respectively. Results are normalized to |Ex|max2. (c) The thin solid, dotted, dashed, and thick solid curves represent deviations from the numerical phase by Ex(0), Ex(0)+f2Ex(2), and Ex(0)+f2Ex(2)+f4Ex(4) and for the spherical result, respectively.

Fig. 7
Fig. 7

Same as Fig. 6 but for z=zt=1 mm.

Fig. 8
Fig. 8

Same as Fig. 6 but for z=2zt=2 mm.

Equations (60)

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Ex(r)=-+Ax(p, q)exp[ik(px+qy+mz)]dpdq,
Ey(r)=-+Ay(p, q)exp[ik(px+qy+mz)]dpdq,
Ez(r)=--+pm Ax(p, q)+qm Ay(p, q)×exp[ik(px+qy+mz)]dpdq.
(2+k2)E(r)=0
m=(1-p2-q2)1/2ifp2+q21i(p2+q2-1)1/2ifp2+q2>1.
Ex(x, y, 0)=exp-x2+y22ω02,
Ey(x, y, 0)=0.
Ax(p, q)=k2π2-+Ex(x, y, 0)exp[-ik(px+qy)]dxdy=12πf2exp-p2+q22f2,
Ay(p, q)=k2π2-+Ey(x, y, 0)exp[-ik(px+qy)]dxdy=0,
Ex(r)=01f2exp-b22f2exp(ikmz)J0(kρb)b db,
Ez(r)=-0ixρ1f2×exp-b22f2exp(ikmz)J1(kρb) b21-b2db,
Ex(r)=011f2exp-b22f2exp(ikmz)J0(kρb)b db,
Ez(r)=-01ixρ1f2×exp-b22f2exp(ikmz)J1(kρb) b21-b2db.
exp(ikmz)=exp(ikz1-b2)=πkz21/2n=01n!kzb22nHn-1/2(1)(kz),
H-1/2(1)(kz)=2πkz1/2exp(ikz),
Hn-1/2(1)(kz)=2πkz1/2exp(ikz)i-nm=0n-1-12ikzm×(n+m-1)!m!(n-m-1)!,n1,
Ez(r)=-ixρ1f2πkz21/2n=01n!kz2nHn-1/2(1)(kz)Tn(ρ),
Tn(ρ)=01exp-b22f2b2n+21-b2 J1(kρb)db.
11-b2=1+12 b2+38 b4+ .
Tn(ρ)=01exp-b22f2b2n+2×1+12 b2+38 b4+J1(kρb)dbTn(1)(ρ)+Tn(2)(ρ)+,
Tn(1)(ρ)=01exp-b22f2b2n+2J1(kρb)db,
Tn(2)(ρ)=1201exp-b22f2b2n+4J1(kρb)db,
Tn(1)(ρ)=0exp-b22f2b2n+2J1(kρb)db=(2f)2n+3n!2ρ22ω021/2exp-ρ22ω02Ln1ρ22ω02,
Tn(2)(ρ)=120exp-b22f2b2n+4J1(kρb)db=12 (2f)2n+5(n+1)!2ρ22ω021/2×exp-ρ22ω02Ln+11ρ22ω02,
Ez(r)Ezi(r)+Ezii(r)+,
Ezi(r)=-ixρ1f2πkz21/2n=01n!kz2nHn-1/2(1)(kz)Tn(1)(ρ),
Ezii(r)=-ixρ1f2πkz21/2n=01n!kz2nHn-1/2(1)(kz)Tn(2)(ρ),
Ezi(r)=-ixfω0exp-ρ22ω02exp(ikz)×1+n=1-izlnLn1ρ22ω02×m=0n-1(n+m-1)!m!(n-m-1)!-12ikzm,
Ezi(r)fEz(1)(r)+f3-iz/l(1+iz/l)2×L21ρ22ω02(1+iz/l)Ez(1)(r)+O(f 5),
Ez(1)(r)=-ixω0exp(ikz)(1+iz/l)2exp-ρ22ω02(1+iz/l).
Ezii(r)=-ixf3ω0exp-ρ22ω02exp(ikz)L11ρ22ω02+n=1(n+1)-izlnLn+11ρ22ω02×m=0n-1(n+m-1)!m!(n-m-1)!-12ikzm=f311+iz/l L11ρ22ω02(1+iz/l)Ez(1)(r)+O(f 5).
Ez(r)=fEz(1)(r)+f3Ez(3)(r)+O(f 5),
Ez(3)(r)=11+iz/l L11ρ22ω02(1+iz/l)-iz/l(1+iz/l)2 L21ρ22ω02(1+iz/l)Ez(1)(r).
Ex(r)=Ex(0)(r)+f2Ex(2)(r)+f4Ex(4)(r)+O(f 6),
Ex(0)(r)=exp(ikz)1+iz/lexp-ρ22ω02(1+iz/l),
Ex(2)(r)=-iz/l(1+iz/l)2 L2ρ22ω02(1+iz/l)Ex(0)(r),
Ex(4)(r)=-3iz/l(1+iz/l)4L4ρ22ω02(1+iz/l)+ρ28ω02 L31ρ22ω02(1+iz/l)Ex(0)(r),
ω(z)=ω0(1+z2/l2)1/2.
Ex(x, y, 0)=12exp-x2+y22ω02,
Ey(x, y, 0)=12exp-x2+y22ω02,
2ik z+T2Ex(0)(r)=0,
Ez(1)(r)=iω0xEx(0)(r),
2ik z+T2Ex(2)(r)=-kl 2Ex(0)(r)z2,
Ez(3)(r)=iω0Ex(2)(r)x+il Ez(1)(r)z,
ϕpar(ρ, z)=kz-arctanzl+ρ2z2ω02l(1+z2/l2).
Ex(r)=-i lzr2exp-klρ22r2exp(ikr).
ϕsph(ρ, z)=-π2+kr=-π2+kρ2+z2.
Δϕ(ρ, z)=ϕpar(ρ, z)-ϕsph(ρ, z).
ρt(z)=l2+l2z21/2.
l2+l2zt21/2=ω01+zt2l21/2.
zt=l(2-f2)+4+f42f21/22lf=2k2ω03,
|fEz(1)(x, y, z)|2=f2x2ω02(1+z2/l2)2×exp-x2+y2ω02(1+z2/l2)
|fEz(1)|max2=f2e(1+z2/l2)
xmax=ω0(1+z2/l2)1/2.
|fEz(1)|max2f2ezt2/l2=f42e0.184f4.
|f 2Ex(2)(ρ, zt)|2
=f4zt2/l2(1+zt2/l2)3L2ρ22ω02(1+izt/l)2
×exp-ρ2ω02(1+zt2/l2)
|f 2Ex(2)|max24exp(4) f40.073f4
ρmax2ω0zt/l=22ω0f.

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