Abstract

We present a simple formula to evaluate the relative focal shift in a circular-aperture lens system illuminated by a nonuniformly polarized (NUP) light wave. Specifically, it is shown that the relative focal shift is determined by the effective Fresnel number. The effective Fresnel number is equal to the product of the Fresnel number of the lens aperture and the parameter σ, which describes the uniformity of the polarization distribution of the NUP beam across the lens aperture. Some examples are given to illustrate the use of this approach. The influence of the polarization distribution of the incident NUP light wave on the polarization distribution in the axial points of the focused field is also presented.

© 2001 Optical Society of America

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  1. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  2. J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  3. Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
    [CrossRef]
  4. M. Martinez-Corral, J. Zapata-Rodriguez, P. Andres, E. Silvestre, “Effective Fresnel-number concept for evaluating the relative focal shift in focused beams,” J. Opt. Soc. Am. A 15, 449–455 (1998).
    [CrossRef]
  5. M. Martinez-Corral, J. Zapata-Rodriguez, P. Andres, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
    [CrossRef]
  6. C. J. R. Sheppard, K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
    [CrossRef]
  7. W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
    [CrossRef] [PubMed]
  8. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  9. P. L. Greene, D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999).
    [CrossRef] [PubMed]
  10. Q. Lu, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
    [CrossRef]
  11. J. M. Movilla, G. Piquero, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
    [CrossRef]
  12. F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
    [CrossRef]
  13. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
    [CrossRef]
  14. G. Piquero, J. M. Movilla, P. M. Mejias, R. Martinez-Herrero, “Beam quality of partially polarized beams propagating through lenslike birefringent elements,” J. Opt. Soc. Am. A 16, 2666–2668 (1999).
    [CrossRef]
  15. R. Martinez-Herrero, P. M. Mejias, J. M. Movilla, “Spatial characterization of general partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
    [CrossRef] [PubMed]
  16. G. P. Agrawal, E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019–2023 (2000).
    [CrossRef]
  17. R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, X. Nguyen Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443–451 (2000).
    [CrossRef]
  18. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23, 1–3 (1998).
    [CrossRef]
  19. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chap. 3.

2000 (3)

C. J. R. Sheppard, K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
[CrossRef]

G. P. Agrawal, E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019–2023 (2000).
[CrossRef]

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, X. Nguyen Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443–451 (2000).
[CrossRef]

1999 (3)

1998 (5)

1997 (1)

1995 (1)

Q. Lu, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
[CrossRef]

1992 (1)

Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
[CrossRef]

1984 (1)

1982 (1)

1981 (2)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Agrawal, G. P.

Andres, P.

M. Martinez-Corral, J. Zapata-Rodriguez, P. Andres, E. Silvestre, “Effective Fresnel-number concept for evaluating the relative focal shift in focused beams,” J. Opt. Soc. Am. A 15, 449–455 (1998).
[CrossRef]

M. Martinez-Corral, J. Zapata-Rodriguez, P. Andres, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

Boucher, V.

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, X. Nguyen Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443–451 (2000).
[CrossRef]

Carter, W. H.

Chevalier, R.

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, X. Nguyen Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443–451 (2000).
[CrossRef]

Dong, S.

Q. Lu, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
[CrossRef]

Dorkenoo, K. D.

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, X. Nguyen Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443–451 (2000).
[CrossRef]

Friese, E. J.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[CrossRef]

Greene, P. L.

Guattari, G.

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

Hall, D. G.

Heckenberg, N. R.

Kowalczyk, M.

M. Martinez-Corral, J. Zapata-Rodriguez, P. Andres, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

Larkin, K. G.

Li, Y.

Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
[CrossRef]

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Lu, Q.

Q. Lu, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chap. 3.

Martinez-Corral, M.

M. Martinez-Corral, J. Zapata-Rodriguez, P. Andres, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

M. Martinez-Corral, J. Zapata-Rodriguez, P. Andres, E. Silvestre, “Effective Fresnel-number concept for evaluating the relative focal shift in focused beams,” J. Opt. Soc. Am. A 15, 449–455 (1998).
[CrossRef]

Martinez-Herrero, R.

Mejias, P. M.

Movilla, J. M.

Nieminen, T. A.

Phu, X. Nguyen

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, X. Nguyen Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443–451 (2000).
[CrossRef]

Piquero, G.

G. Piquero, J. M. Movilla, P. M. Mejias, R. Martinez-Herrero, “Beam quality of partially polarized beams propagating through lenslike birefringent elements,” J. Opt. Soc. Am. A 16, 2666–2668 (1999).
[CrossRef]

J. M. Movilla, G. Piquero, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Rubinsztein-Dunlop, H.

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

Sheppard, C. J. R.

Silvestre, E.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Volle, R.

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, X. Nguyen Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443–451 (2000).
[CrossRef]

Weber, H.

Q. Lu, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
[CrossRef]

Wolf, E.

Zapata-Rodriguez, J.

M. Martinez-Corral, J. Zapata-Rodriguez, P. Andres, E. Silvestre, “Effective Fresnel-number concept for evaluating the relative focal shift in focused beams,” J. Opt. Soc. Am. A 15, 449–455 (1998).
[CrossRef]

M. Martinez-Corral, J. Zapata-Rodriguez, P. Andres, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (2)

Y. Li, “Focal shift formula for focused, apertured Gaussian beams,” J. Mod. Opt. 39, 1761–1764 (1992).
[CrossRef]

M. Martinez-Corral, J. Zapata-Rodriguez, P. Andres, M. Kowalczyk, “Analytical formula for calculating the focal shift in apodized systems,” J. Mod. Opt. 45, 1671–1679 (1998).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (5)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

R. Volle, V. Boucher, K. D. Dorkenoo, R. Chevalier, X. Nguyen Phu, “Local polarization state observation and third-order nonlinear susceptibility measurements by self-induced polarization state changes method,” Opt. Commun. 182, 443–451 (2000).
[CrossRef]

J. M. Movilla, G. Piquero, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

Q. Lu, S. Dong, H. Weber, “Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod,” Opt. Quantum Electron. 27, 777–783 (1995).
[CrossRef]

Other (1)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chap. 3.

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Figures (7)

Fig. 1
Fig. 1

Notation relating to the focusing of a nonuniformly polarized (NUP) beam.

Fig. 2
Fig. 2

Variation of the parameter σ with r0.

Fig. 3
Fig. 3

Distribution of the two electric components qx(ζ) (solid curve) and qy(ζ) (dotted curve) as a function of ζ: (a) n=2, (b) n=8, (c) n=50. κ=2 throughout.

Fig. 4
Fig. 4

Variation of the parameter σ values with n for three different κ values: κ=2 (solid curve), κ=4 (dashed curve), and κ=1.2 (dotted curve).

Fig. 5
Fig. 5

Axial intensity distributions for three different polarization distributions of the incident NUP beams. Solid curve: N=2, n=0, σ=1 and zmax/f=0.1915, curve: N=2.846, n=8, σ=0.703, and zmax/f=0.1955; dashed curve: N=3.873, n=50, σ=0.516, and zmax/f=0.1920. κ=2 throughout. (Note that for the three different polarization distributions, Neff=Nσ=2, and the relative focal shifts are nearly the same.)

Fig. 6
Fig. 6

Relative focal shifts zmax/f as a function of effective Fresnel number Neff at four different polarization distributions of the incident NUP beams (the solid curve is composed of four curves): (1) n=0, σ=1; (2) n=2, σ=0.926; (3) n=8, σ=0.703; and (4) n=50, σ=0.516. κ=2 throughout. The dotted curve is plotted by using Eq. (26).

Fig. 7
Fig. 7

Influence of the polarization distribution (lower figure) at the lens aperture on the polarization distributions (upper figure) along the axial points of the focused field. |γ0|=0.5, N=10, and κ=2, (a) n=2, (b) n=5, (c) n=50.

Equations (40)

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Jˆ(r1, r2)=Jxx(r1, r2)Jxy(r1, r2)Jyx(r1, r2)Jyy(r1, r2),
Jαβ(r1, r2)=Eα*(r1, t)Eβ(r2, t), α, β=x, y,
Jxx(r1, r2)=Ex*(r1)Ex(r2),
Jyy(r1, r2)=Ey*(r1)Ey(r2),
Jxy(r1, r2)=γ0Ex*(r1)Ey(r2)=Jyx*(r1, r2).
Jeq(r1, r2)=Jxx(r1, r2)+Jyy(r1, r2)=Ex*(r1)Ex(r2)+Ey*(r1)Ey(r2).
I(r)=Jxx(r, r)+Jyy(r, r)=Ex*(r)Ex(r)+Ey*(r)Ey(r).
Jαβ(r1, r2, z)=Jαβ(r1, r2)K*(r1, r1, z)×K(r2, r2, z)d2r1d2r2, α, β=x, y.
Jxx(r1, r2, z)+Jyy(r1, r2, z)=Jxx(r1, r2)K*(r1, r1, z)×K(r2, r2, z)d2r1d2r2+Jyy(r1, r2)K*(r1, r1, z)K(r2, r2, z)d2r1d2r2.
I(z)=[Jxx(r1, r2)+Jyy(r1, r2)]×K*(0, r1, z)K(0, r2, z)d2r1d2r2,
K(0, r, z)=exp(ikz)iλf(f+z) exp-iπ zr2λf(f+z),
I(z)=Ex*(z)Ex(z)+Ey*(z)Ey(z),
Eα(z)=exp(ikz)iλf(f+z) 0a02πEα(r, θ)×exp-i2π z2λf(f+z)r2r drdθ, α=x, y.
Eα(z)=2π exp(ikz)iλf(f+z) 0aE¯α(r)×exp-i2π z2λf(f+z)r2r dr,
E¯α(r)=12π 02πEα(r, θ)dθ
ζ=r2/a2,E¯α(r)=qα(ζ)
u=2πN z/f1+z/f,
Eα(u)=-iπN1-u2πN01qα(ζ)exp-i u2ζdζ,
I(u)=π2N21-u2πN201qx(ζ)exp-i u2ζdζ2+01qy(ζ)exp-i u2ζdζ2.
IN(u)=1-u2πN2 01qx(ζ)exp-i u2ζdζ2+01qy(ζ)exp-i u2ζdζ201qx(ζ)dζ2+01qy(ζ)dζ2.
exp-i u2ζ1-i u2ζ-u28ζ2.
IN(u)1-uπN+14 1π2N2-σ212u2,
σ=12 mx02σx2+my02σy2mx02+my021/2,
σα2=mα2mα0-mα1mα02,α=x, y.
mαl=01qα(ζ)ζl dζ,α=x, y,l=0, 1, 2,
dIN(u)/du=0.
umax=2πN1-π2σ2N2/12.
zmaxf=-1(π2/12)Neff2,
Neff=Nσ.
Ex(r, θ)=E0,br00,ar>b,
Ey(r, θ)=0,br0E0,ar>b,
I(r)=Ex*(r, θ)Ex(r, θ)+Ey*(r, θ)Ey(r, θ)=E02.
σ=4r08+4(1-r02)(1-r06)-3r08-3(1-r04)2r04+(1-r02)21/2,
Ex(r, θ)=E011+(κr2)n1/2Ey(r, θ)=E0(κr2)n1+(κr2)n1/2,ar0.
qx(ζ)=E011+(κζ)n1/2qy(ζ)=E0(κζ)n1+(κζ)n1/2,1ζ0.
Jxx(r)=E02 11+(κr2)n,
Jyy(r)=E02 (κr2)n1+(κr2)n,
Jxy(r)=γ0E02 (κr2)n1+(κr2)n=Jyx*(r).
P(r)=1-4 det Jˆ(tr Jˆ)21/2,
P(ζ)=1-(1-|γ0|2) 4(κζ)n[1+(κζ)n]21/2, 1ζ0.

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