Abstract

We develop a theoretical description for polarizers that goes beyond the paraxial approximation. By combining existing theories for fields with nonplanar wavefronts, we are able to derive a simple power series expansion expressing the electric field of a light beam after a polarizer as a linear function of the field and its spatial derivatives evaluated before the polarizer. The first few terms of such expansion are explicitly given, and their physical meaning is discussed.

© 2009 Optical Society of America

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  1. J. N. Damask, Polarization Optics in Telecommunications, 1st. ed. (Springer, 2004).
  2. L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
    [CrossRef] [PubMed]
  3. W. L. Erikson and S. Singh, Phys. Rev. E 49, 5778 (1994).
    [CrossRef]
  4. K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
    [CrossRef]
  5. H. P. Urbach and S. F. Pereira, Phys. Rev. Lett. 100, 123904 (2008).
    [CrossRef] [PubMed]
  6. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
    [CrossRef]
  7. M. O. Scully and M. S. Zubairy, Phys. Rev. A 44, 2656 (1991).
    [CrossRef] [PubMed]
  8. Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
    [CrossRef] [PubMed]
  9. B. Sick, B. Hecht, and L. Novotny, Phys. Rev. Lett. 85, 4482 (2000).
    [CrossRef] [PubMed]
  10. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
    [CrossRef] [PubMed]
  11. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
    [CrossRef]
  12. R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  13. M. Xiao, J. Opt. Soc. Am. A 14, 2977 (1997).
    [CrossRef]
  14. Y. Fainman and J. Shamir, Appl. Opt. 23, 3188 (1984).
    [CrossRef] [PubMed]
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics1st ed. (Cambridge U. Press, 1995).
  16. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  17. For nonideal polarizers Eq. generalizes to êμ(κ⃗)P-->d0êP(êP*⋅êμ(κ⃗))+d1êQ(êQ*⋅êμ(κ⃗)), where 0⩽di⩽1(i=0,1) are the so-called diattenuation factors , and êQ(κ⃗)=q⃗⊥/|q⃗⊥| with q⃗⊥=q̂−k̂(k̂⋅q̂), where p̂*⋅q̂=0. By using this rule, that reduces to Eq. for d1=0, all the results presented in this Letter may be straightforwardly extended to imperfect polarizers.

2008

H. P. Urbach and S. F. Pereira, Phys. Rev. Lett. 100, 123904 (2008).
[CrossRef] [PubMed]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

2007

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[CrossRef]

2003

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

2002

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

2001

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef] [PubMed]

2000

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

B. Sick, B. Hecht, and L. Novotny, Phys. Rev. Lett. 85, 4482 (2000).
[CrossRef] [PubMed]

1997

1994

W. L. Erikson and S. Singh, Phys. Rev. E 49, 5778 (1994).
[CrossRef]

1991

M. O. Scully and M. S. Zubairy, Phys. Rev. A 44, 2656 (1991).
[CrossRef] [PubMed]

1990

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

1984

1975

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef] [PubMed]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef] [PubMed]

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Cicchitelli, L.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

Damask, J. N.

J. N. Damask, Polarization Optics in Telecommunications, 1st. ed. (Springer, 2004).

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Erikson, W. L.

W. L. Erikson and S. Singh, Phys. Rev. E 49, 5778 (1994).
[CrossRef]

Fainman, Y.

Friberg, A. T.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[CrossRef]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, Phys. Rev. Lett. 85, 4482 (2000).
[CrossRef] [PubMed]

Hora, H.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

Kaivola, M.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[CrossRef]

Keitel, C. H.

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Lindfors, K.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics1st ed. (Cambridge U. Press, 1995).

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef] [PubMed]

B. Sick, B. Hecht, and L. Novotny, Phys. Rev. Lett. 85, 4482 (2000).
[CrossRef] [PubMed]

Pereira, S. F.

H. P. Urbach and S. F. Pereira, Phys. Rev. Lett. 100, 123904 (2008).
[CrossRef] [PubMed]

Postle, R.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

Priimagi, A.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[CrossRef]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Phys. Rev. A 44, 2656 (1991).
[CrossRef] [PubMed]

Setala, T.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[CrossRef]

Shamir, J.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Shevchenko, A.

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[CrossRef]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Sick, B.

B. Sick, B. Hecht, and L. Novotny, Phys. Rev. Lett. 85, 4482 (2000).
[CrossRef] [PubMed]

Singh, S.

W. L. Erikson and S. Singh, Phys. Rev. E 49, 5778 (1994).
[CrossRef]

Urbach, H. P.

H. P. Urbach and S. F. Pereira, Phys. Rev. Lett. 100, 123904 (2008).
[CrossRef] [PubMed]

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics1st ed. (Cambridge U. Press, 1995).

Xiao, M.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef] [PubMed]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Phys. Rev. A 44, 2656 (1991).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt. Soc. Am. A

Nat. Photonics

K. Lindfors, A. Priimagi, T. Setala, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[CrossRef]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Opt. Commun.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Phys. Rev. A

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

M. O. Scully and M. S. Zubairy, Phys. Rev. A 44, 2656 (1991).
[CrossRef] [PubMed]

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef] [PubMed]

Phys. Rev. E

W. L. Erikson and S. Singh, Phys. Rev. E 49, 5778 (1994).
[CrossRef]

Phys. Rev. Lett.

H. P. Urbach and S. F. Pereira, Phys. Rev. Lett. 100, 123904 (2008).
[CrossRef] [PubMed]

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

B. Sick, B. Hecht, and L. Novotny, Phys. Rev. Lett. 85, 4482 (2000).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Other

J. N. Damask, Polarization Optics in Telecommunications, 1st. ed. (Springer, 2004).

For nonideal polarizers Eq. generalizes to êμ(κ⃗)P-->d0êP(êP*⋅êμ(κ⃗))+d1êQ(êQ*⋅êμ(κ⃗)), where 0⩽di⩽1(i=0,1) are the so-called diattenuation factors , and êQ(κ⃗)=q⃗⊥/|q⃗⊥| with q⃗⊥=q̂−k̂(k̂⋅q̂), where p̂*⋅q̂=0. By using this rule, that reduces to Eq. for d1=0, all the results presented in this Letter may be straightforwardly extended to imperfect polarizers.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics1st ed. (Cambridge U. Press, 1995).

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Figures (1)

Fig. 1
Fig. 1

Pictorial representation of the transformation (4). The input polarization unit vector e ̂ μ ( κ ) is projected by the polarizer into the output vector e ̂ P ( κ ) ( e ̂ P * e ̂ μ ( κ ) ) , where, by definition, e ̂ P ( κ ) p | p | , with p = p ̂ p and p = k ̂ ( k ̂ p ̂ ) .

Equations (20)

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A ( X , Z ) = μ = 1 2 d 2 κ 2 π e ̂ μ ( κ ) A ̃ μ ( κ , Z ) e i κ X ,
A ̃ μ ( κ , Z ) = A ̃ μ ( κ , 0 ) exp ( i κ z Z )
A ̃ μ ( κ , 0 ) = d 2 X 2 π e ̂ μ ( κ ) A ( X , 0 ) e i κ X .
e ̂ μ ( κ ) e i κ X P e ̂ P ( e ̂ P * e ̂ μ ( κ ) ) e i κ X ,
A ( X , Z ) = d 2 κ 2 π e ̂ P ( κ ) μ = 1 2 α μ * ( κ ) A ̃ μ ( κ , Z ) e i κ X ,
A ( X , Z ) = μ = 1 2 d 2 κ 2 π e ̂ μ ( κ ) A ̃ μ ( κ , Z ) e i κ X ,
A ̃ μ ( κ , Z ) = ν = 1 2 P μ ν ( κ ) A ̃ ν ( κ Z ) ,
A ( X , Z + Z ) = μ = 1 2 d 2 κ 2 π e ̂ μ ( κ ) A ̃ μ ( κ , Z ) e i κ z Z e i κ X = μ = 1 2 d 2 κ 2 π e ̂ μ ( κ ) A ̃ μ ( κ , Z + Z ) e i κ X ,
A ( X , Z ) = d 2 X G ( X X ) A ( X , Z ) ,
G ( X ) = ( 2 π ) 1 d 2 κ 2 π G ̃ ( κ ) e i κ X .
G ̃ ( κ ) = n , m = 0 κ x n κ y m n ! m ! | n + m κ x n κ y m G ̃ ( κ ) | κ = 0 .
G ( X X ) = n , m = 0 G ̃ n m i n + m n + m X n Y m δ ( X X ) ,
A ( X , Z ) = n , m = 0 G ̃ n m i n + m n + m X n Y m A ( X , Z ) .
A ( ρ , ζ ) = n , m = 0 G ̃ n m i n + m ( θ 0 2 ) n + m n + m ξ n η m A ( ρ , ζ ) ,
G ̃ 00 = [ | a | 2 a b * 0 a * b | b | 2 0 0 0 0 ] ,
G ̃ 10 = [ 0 0 | a | 2 0 0 a * b | a | 2 a b * 0 ] ,
G ̃ 01 = [ 0 0 a b * 0 0 | b | 2 a * b | b | 2 0 ] ,
G ̃ 20 = [ 1 + | b | 4 a b * | b | 2 0 a * b | b | 2 | a b | 2 0 0 0 | a | 2 ] ,
G ̃ 11 = [ Δ | b | 2 1 + Δ a b * 0 1 + Δ a * b Δ | a | 2 0 0 0 Δ ] ,
G ̃ 02 = [ | a b | 2 a b * | a | 2 0 a * b | a | 2 1 + | a | 4 0 0 0 | b | 2 ] ,

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