Abstract

The Strehl ratio for focusing into biaxially birefringent media with small birefringence is investigated. An analytical expression for the Strehl ratio as a function of the polarization and birefringence parameters is derived in the paraxial limit from a generalization of the scalar aberration theory. The expression agrees well with results of a numerical calculation.

© 2004 Optical Society of America

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References

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  1. S. Stallinga, “Light distribution close to focus in biaxially birefringent media,” J. Opt. Soc. Am. A 21, 1785–1798 (2004).
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, UK, 1980).
  3. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  4. Selected Papers on Electromagnetic Fields in the Focal Region, J. J. Stamnes, ed., SPIE Milestone Series Vol. 168 (SPIE Optical Engineering Press, Bellingham, Wash., 2001).
  5. S. Stallinga, “Axial birefringence in high NA optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2858 (2001).
    [CrossRef]
  6. A. B. Marchant, “Cover sheet aberrations in optical recording,” in Optical Disk Systems and Applications, E. V. LaBudde, ed., Proc. SPIE421, 43–49 (1983).
    [CrossRef]
  7. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).
  8. M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
    [CrossRef]
  9. I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
    [CrossRef]
  10. M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
    [CrossRef]

2004 (1)

2001 (1)

Borg, H. J.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Born, M.

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

Coene, W. M.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Dekker, M. K.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[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, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Ichimura, I.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Kasami, Y.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Kawakubo, O.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Kuijper, M.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Marchant, A. B.

A. B. Marchant, “Cover sheet aberrations in optical recording,” in Optical Disk Systems and Applications, E. V. LaBudde, ed., Proc. SPIE421, 43–49 (1983).
[CrossRef]

Masuhara, S.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Meinders, E. R.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

Nakano, J.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Osato, K.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

Pfeffer, N.

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[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, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Schep, K.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

Spruijt, L.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

Stallinga, S.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

ter Meulen, J. M.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

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, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Ubbens, I. P.

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

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, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Wolf, E.

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

Yasuda, K.

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

J. Opt. Soc. Am. A (2)

Other (8)

A. B. Marchant, “Cover sheet aberrations in optical recording,” in Optical Disk Systems and Applications, E. V. LaBudde, ed., Proc. SPIE421, 43–49 (1983).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

M. K. Dekker, N. Pfeffer, M. Kuijper, W. M. Coene, E. R. Meinders, H. J. Borg, “Blue phase-change recording at high data densities and data rates,” in Optical Data Storage 2000, D. G. Stinson, R. Katayama, eds., Proc. SPIE4090, 28–35 (2000).
[CrossRef]

I. Ichimura, S. Masuhara, J. Nakano, Y. Kasami, K. Yasuda, O. Kawakubo, K. Osato, “On-groove phase-change optical recording for a capacity of 25 GB,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 168–177 (2001).
[CrossRef]

M. Kuijper, I. P. Ubbens, L. Spruijt, J. M. ter Meulen, K. Schep, “Groove-only recording under DVR conditions,” in Optical Data Storage 2001, T. Hurst, S. Kobayashi, eds., Proc. SPIE4342, 178–185 (2001).
[CrossRef]

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

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Selected Papers on Electromagnetic Fields in the Focal Region, J. J. Stamnes, ed., SPIE Milestone Series Vol. 168 (SPIE Optical Engineering Press, Bellingham, Wash., 2001).

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Figures (4)

Fig. 1
Fig. 1

Numerically calculated Strehl ratio (solid curve) and the analytical approximation of Eq. (63) (dashed line) as a function of NA for the birefringence parameters Δn=1.5×10-3 and Δn=4.0×10-3; for the linear polarization with minimum Strehl ratio; and for n¯=1.58, λ=405 nm, and a thickness d that scales such that dNA2=72.25 μm.

Fig. 2
Fig. 2

Numerically calculated Strehl ratio (left) and the analytical approximation of Eq. (63) (right) as a function of the retardation parameters dΔn and dΔn for a circular polarization and for n¯=1.58, NA=0.85, and λ=405 nm.

Fig. 3
Fig. 3

Numerically calculated Strehl-ratio (left) and the analytical approximation of Eq. (63) (right) as a function of the retardation parameters dΔn and dΔn for the linear polarization giving rise to a maximum Strehl ratio and for n¯=1.58, NA=0.85, and λ=405 nm.

Fig. 4
Fig. 4

Numerically calculated Strehl ratio (left) and the analytical approximation of Eq. (63) (right) as a function of the retardation parameters dΔn and dΔn for the linear polarization giving rise to a minimum Strehl ratio and for n¯=1.58, NA=0.85, and λ=405 nm. Values of the analytical Strehl ratio below 0.2 have been set equal to 0.2 (upper right corner of right-hand graph) in order to have roughly the same gray scales in both plots. This is already in the regime where the analytical approximation cannot be expected to give reasonable results.

Equations (72)

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I=1-(W2-W2),
n1=n¯(1-Δn/n¯)1/2n¯+Δn/2,
n2=n¯(1+Δn/n¯)1/2n¯-Δn/2,
n3=n¯(1-2Δn/n¯)1/2n¯+Δn,
pˆ=(cos θ cos ϕ, cos θ sin ϕ, -sin θ),
sˆ=(-sin ϕ, cos ϕ, 0),
kˆ=(sin θ cos ϕ, sin θ sin ϕ, cos θ),
Eα(rp)=E0πNij=1,2Fαj(rp)Aj,
Fαj(rp)=1πRd2ρl=1,2v^lαJljexp(ikrp),
v^1=cos ϕpˆ-sin ϕsˆ,
v^2=sin ϕpˆ+cos ϕsˆ.
J=JbirJen.
Jbir=R(χ)exp(iW+)00exp(iW-)R(-χ)=exp(iW+)cos2 χ+exp(iW-)sin2 χ[exp(iW+)-exp(iW-)]sin χ cos χ[exp(iW+)-exp(iW-)]sin χ cos χexp(iW+)sin2 χ+exp(iW-)cos2 χ.
R(χ)=cos χ-sin χsin χcos χ.
W±=kd2 cos θ [a±(b2+c2)1/2],
cos(2χ-2ϕ)=b/(b2+c2)1/2,
sin(2χ-2ϕ)=c/(b2+c2)1/2,
a=Δnsin2 θ-12Δnsin2 θ cos(2γ-2ϕ),
b=Δnsin2 θ+12Δn(1+cos2 θ)cos(2γ-2ϕ),
c=Δncos θ sin(2γ-2ϕ).
tp=2n0cos θ0n0cos θ+n¯cos θ0,
ts=2n0cos θ0n0cos θ0+n¯cos θ,
Jen=R(ϕ)tp00tsR(-ϕ)=tpcos2 ϕ+tssin2 ϕ(tp-ts)sin ϕ cos ϕ(tp-ts)sin ϕ cos ϕtscos2 ϕ+tpsin2 ϕ.
B=1cos θ0=(1-ρ2NA2/n02)-1/4,
Uα=U0j,l=1,2FαjFαl*AjAl*=U0μ=0,3IαμMμ,
U0=120n¯2π2E02N2,
Iαμ=12j,l=1,2(σμ)jlFαjFαl*,
Mμ=j,l=1,2(σμ)ljAjAl*.
Fjk=1πPd2ρ(VJ)jk(ρ),
F=VJ,
V=Bv1xv2xv1yv2y=R(ϕ)cos θ001/cos θR(-ϕ).
I=μ=0,3IμMμ=12μ=0,3Tr(σμFF)Mμ,
G=FJ0=JJ0=JJ0,
I=12μ=0,3Tr(σμGG)Mμ,
σμ=J0σμJ0=μ=0,3Sμνσν.
M¯μ=ν=0,3SνμMν
j=1,3M¯j2=M¯02=1.
I=12μ=0,3Tr(σμGG)M¯μ.
JJ0=exp(iK),
K=μ=0,3Wμσμ,
JJ0=1+iK-12K2,
G=σ0+iμ=0,3Wμσμ-12μ,ν=0,3WμWνσμσν.
GG=σ0-12μ,ν=0,3(WμWν-WμWν)(σμσν+σνσμ)=σ0-μ=0,3(Wμ2-Wμ2)σ0-2j=1,3(WjW0-WjW0)σj.
I=1-μ=0,3(Wμ2-Wμ2)-2j=1,3(WjW0-WjW0)M¯j=1-j=1,3[(Wj+M¯jW0)2-(Wj+M¯jW0)2],
M¯j=±WjW0-WjW0[j=1,3(WjW0-WjW0)2]1/2
I±=1-μ=0,3(Wμ2-Wμ2)±2j=1,3(WjW0-WjW0)21/2.
J=exp[i(τσ0+μvσ)]=exp(iτ)(cos μσ0+i sin μvσ),
J=exp(iW¯)R(χ)exp(iΔW/2)00exp(-iΔW/2)R(-χ)=exp(iW¯)[cos(ΔW/2)σ0+i sin(ΔW/2)vσ],
J0=R(γ)exp(ia/2)00exp(-ia/2)R(-γ)=cos(a/2)σ0+i sin(a/2)v0σ,
a=2πdΔnλ,
a=2πdΔnλ.
S=100001-2 sin2(a/2)sin2 (2γ)sin2(a/2)sin (4γ)sin asin (2γ)0sin2(a/2)sin (4γ)1-2 sin2(a/2)cos2 (2γ)-sin acos (2γ)0-sin asin (2γ)sin acos (2γ)cos a.
JJ0=exp(iW¯){[cos(ΔW/2)cos(a/2)+sin(ΔW/2)sin(a/2)vv0]σ0+i[sin(ΔW/2)cos(a/2)v-cos(ΔW/2)sin(a/2)v0+sin(ΔW/2)sin(a/2)v×v0]σ}.
W0=12[a-12acos(2γ-2ϕ)]ρNAn¯2,
W1=12 a[cos (2γ) cos(2γ-2ϕ)+cos(a/2)sinc(a/2)sin (2γ) sin(2γ-2ϕ)]×ρNAn¯2,
W2=12 a[sin (2γ) cos(2γ-2ϕ)+cos(a/2)sinc(a/2)cos (2γ) sin(2γ-2ϕ)]×ρNAn¯2,
W3=12 asin(a/2)sinc(a/2)sin(2γ-2ϕ)ρNAn¯2,
(W1+M¯1W0)2=196 (2acos (2γ)-M¯1a)2NAn¯4+124 a2cos2(a/2)sinc2(a/2)×sin2 (2γ)NAn¯4,
(W2+M¯1W0)2=196 (2asin (2γ)-M¯2a)2NAn¯4+124 a2cos2(a/2)sinc2(a/2)×cos2 (2γ)NAn¯4,
(W3+M¯3W0)2=196 M¯32a2NAn¯4+124×a2sin2(a/2)sinc2(a/2)NAn¯4.
M¯12+M¯22+M¯32=1,
M¯1cos (2γ)+M¯2sin (2γ)=M1cos (2γ)+M2sin (2γ)=cos (2) cos(2ξ-2γ),
I=1-196 {4[1+sinc2(a/2)]a2+a2-4aacos (2 )cos(2ξ-2γ)}NAn¯4.
Iax=1-112 a2NAn¯4,
Ilat=1-196 [22 cos (2) cos(2ξ-2γ)+sinc2(a/2)]a2NAn¯4,
Ilat,max=1-124sin2(a/2)NAn¯4,
σ0=1001,σ1=100-1,
σ2=0110,σ3=0-ii0.
σμ2=σ0,
σ0σj=σjσ0=σj,
σjσk=-σkσj=ijklσl,
μ=0,3(σμ)jl(σμ)lj=2δjjδll,

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