Abstract

We consider partial spatial coherence and partial polarization of purely evanescent optical fields generated in total internal reflection at an interface of two dielectric (lossless) media. Making use of the electromagnetic degree of coherence, we show that, in such fields, the coherence length can be notably shorter than the light’s vacuum wavelength, especially at a high-index-contrast interface. Physical explanation for this behavior, analogous to the generation of incoherent light in a multimode laser, is provided. We also analyze the degree of polarization by using a recent three-dimensional formulation and show that the field may be partially polarized at a subwavelength distance from the surface even though it is fully polarized farther away. The degree of polarization can assume values unattainable by beamlike fields, indicating that electromagnetic evanescent waves generally are genuine three-dimensional fields. The results can find applications in near-field optics and nanophotonics.

© 2011 Optical Society of America

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References

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    [CrossRef]
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  21. The form of the Fresnel coefficient for p polarization given in Eq.  is valid only for an evanescent wave. For propagating waves, the factor 2n˜2γ2+1 is absent in the numerator. However, the factor follows directly from the boundary conditions, if the (complex) vector p^2 related to the evanescent wave is normalized to unity. For example, in , the square-root factor is missing since in those treatments p^2 is not normalized; more precisely, p^2*·p^2≠1 but instead p^2·p^2=1. The form of tp given in Eq.  preserves the physical meaning of the Fresnel coefficients as the ratio of the complex amplitudes on the opposite sides of the surface. See also .
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    [CrossRef]
  23. The p-polarized amplitude of the evanescent wave looks different from that given in , but it is the same. The variances in the appearance originate from the differences in the Fresnel coefficients for the p-polarized light. However, for the Fresnel coefficient in Eq. , the energy density of the p-polarized evanescent wave is of the intuitive form given in Eq. . For the evanescent wave of , the energy density contains an additional factor, wp=|tp|2|Ep|2(2n˜2γ2+1). See also above.
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    [CrossRef]

2009

2007

L. Józefowski, J. Fiutowski, T. Kawalec, and H.-G. Rubahn, “Direct measurement of the evanescent-wave polarization state,” J. Opt. Soc. Am. B 24, 624–628 (2007).
[CrossRef]

K. Blomstedt, T. Setälä, and A. T. Friberg, “Arbitrarily short coherence length within finite lossless source regions,” Phys. Rev. E 75, 026610 (2007).
[CrossRef]

J. J. Gil, “Polarimetric characterization of light and media: physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

2006

2004

2003

2002

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

2000

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

1999

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

1988

1987

1986

Blomstedt, K.

K. Blomstedt, T. Setälä, and A. T. Friberg, “Arbitrarily short coherence length within finite lossless source regions,” Phys. Rev. E 75, 026610 (2007).
[CrossRef]

Brosseau, C.

C. Brosseau and A. Dogariu, “Symmetry properties and polarization description for an arbitrary electromagnetic wavefield,” in Vol.  49 of Progress in Optics, E.Wolf, ed. (Elsevier, 2006), p. 315.
[CrossRef]

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971), Sec. 7.1.3.

Carminati, R.

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

Carter, W. H.

Collier, R. J.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971), Sec. 7.1.3.

Courjon, D.

D. Courjon, Near-Field Microscopy and Near-Field Optics(Imperial College, 2003).

de Fornel, F.

F. de Fornel, Evanescent Waves: From Newtonian Optics to Atomic Optics (Springer–Verlag, 2001).

Dogariu, A.

C. Brosseau and A. Dogariu, “Symmetry properties and polarization description for an arbitrary electromagnetic wavefield,” in Vol.  49 of Progress in Optics, E.Wolf, ed. (Elsevier, 2006), p. 315.
[CrossRef]

Fiutowski, J.

Foley, J. T.

Friberg, A. T.

Gil, J. J.

J. J. Gil, “Polarimetric characterization of light and media: physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

Greffet, J.-J.

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Joulain, K.

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

Józefowski, L.

Kaivola, M.

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Kawalec, T.

Kim, K.

Lin, L. H.

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971), Sec. 7.1.3.

Lindfors, K.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejias, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer–Verlag, 2009).
[CrossRef]

Mejias, P. M.

R. Martínez-Herrero, P. M. Mejias, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer–Verlag, 2009).
[CrossRef]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Nussenzveig, H. M.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1998).

Piquero, G.

R. Martínez-Herrero, P. M. Mejias, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer–Verlag, 2009).
[CrossRef]

Rubahn, H.-G.

Setälä, T.

Shchegrov, A. V.

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Sipe, J. E.

Tervo, J.

Wolf, E.

Eur. Phys. J. Appl. Phys.

J. J. Gil, “Polarimetric characterization of light and media: physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. E

K. Blomstedt, T. Setälä, and A. T. Friberg, “Arbitrarily short coherence length within finite lossless source regions,” Phys. Rev. E 75, 026610 (2007).
[CrossRef]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
[CrossRef]

Phys. Rev. Lett.

R. Carminati and J.-J. Greffet, “Near-field effects in spatial coherence of thermal sources,” Phys. Rev. Lett. 82, 1660–1663 (1999).
[CrossRef]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85, 1548–1551 (2000).
[CrossRef] [PubMed]

T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in near fields of thermal sources: effects of surface waves,” Phys. Rev. Lett. 88, 123902 (2002).
[CrossRef] [PubMed]

Other

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971), Sec. 7.1.3.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

R. Martínez-Herrero, P. M. Mejias, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer–Verlag, 2009).
[CrossRef]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

F. de Fornel, Evanescent Waves: From Newtonian Optics to Atomic Optics (Springer–Verlag, 2001).

The form of the Fresnel coefficient for p polarization given in Eq.  is valid only for an evanescent wave. For propagating waves, the factor 2n˜2γ2+1 is absent in the numerator. However, the factor follows directly from the boundary conditions, if the (complex) vector p^2 related to the evanescent wave is normalized to unity. For example, in , the square-root factor is missing since in those treatments p^2 is not normalized; more precisely, p^2*·p^2≠1 but instead p^2·p^2=1. The form of tp given in Eq.  preserves the physical meaning of the Fresnel coefficients as the ratio of the complex amplitudes on the opposite sides of the surface. See also .

C. Brosseau and A. Dogariu, “Symmetry properties and polarization description for an arbitrary electromagnetic wavefield,” in Vol.  49 of Progress in Optics, E.Wolf, ed. (Elsevier, 2006), p. 315.
[CrossRef]

The p-polarized amplitude of the evanescent wave looks different from that given in , but it is the same. The variances in the appearance originate from the differences in the Fresnel coefficients for the p-polarized light. However, for the Fresnel coefficient in Eq. , the energy density of the p-polarized evanescent wave is of the intuitive form given in Eq. . For the evanescent wave of , the energy density contains an additional factor, wp=|tp|2|Ep|2(2n˜2γ2+1). See also above.

D. Courjon, Near-Field Microscopy and Near-Field Optics(Imperial College, 2003).

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1998).

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Figures (5)

Fig. 1
Fig. 1

Illustration of refraction of a plane wave at an interface between two homogeneous dielectric media. The plane of incidence is in the x z plane, and the boundary lies in the x y plane. The unit vectors k ^ 1 , 2 are related to the wave vectors, whereas s ^ 1 , 2 and p ^ 1 , 2 are the unit vectors associated with the s and p polarizations, respectively. Furthermore, θ 1 is the angle of incidence, θ 2 is the angle of refraction, and n 1 , 2 are the refractive indices. The unit vector u ^ z in the positive z direction is the normal to the surface.

Fig. 2
Fig. 2

Squared magnitudes of the Fresnel transmission coefficients t s (blue, solid) and t p (red, dashed) as a function of the angle of incidence θ 1 at glass–air surface ( n 1 = 1.5 , n 2 = 1 ). The peaks are centered at the critical angle θ c 41.8 ° emphasized with the vertical dashed line.

Fig. 3
Fig. 3

Illustration of (a) the 3D degree of polarization [ P 3 ( ω ) from Eq. (35)] and (b) the electromagnetic degree of coherence [ μ EM ( ω ) from Eq. (38)] of a single evanescent wave as a function of the angle of incidence θ 1 θ c 41.8 ° , for the glass–air interface ( n 1 = 1.5 , n 2 = 1 ). For each curve, the intensities of the incident s- and p- polarized components are equal, ϕ p p = ϕ s s , but the correlation coefficient is varied: | μ p s | = 0 (dotted line), | μ p s | = 0.5 (dashed line), and | μ p s | = 1 (solid line).

Fig. 4
Fig. 4

Behavior of the electromagnetic degree of coherence as a function of the lateral distance Δ x for the field consisting of (a) 10 s-polarized evanescent waves and (b) 10 p-polarized evanescent waves. The dashed, dotted–dashed, and dotted lines correspond, respectively, to heights z = 0 , z = λ 2 / 4 , and z = λ 2 / 2 above the silicon–air surface ( n 1 = 4 , n 2 = 1 ). The evanescent waves are mutually uncorrelated and of equal intensity at z = 0 . The solid lines correspond to continuums of evanescent waves.

Fig. 5
Fig. 5

Three-dimensional degree of polarization, P 3 ( z , ω ) , as a function of the distance z from the silicon–air ( n 1 = 4 , n 2 = 1 ) interface for fields consisting of a superposition of uncorrelated, equal-intensity evanescent waves. The blue dotted and red dotted–dashed curves correspond, respectively, to purely s-polarized and purely p-polarized superpositions of 10 waves. Furthermore, the dashed and solid green curves are, respectively, for fields composed of 10 waves and a continuum of waves having uncorrelated s- and p-polarized components of the same intensity.

Equations (57)

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W i j ( r 1 , r 2 , ω ) = E i * ( r 1 , ω ) E j ( r 2 , ω ) , ( i , j ) = ( x , y , z ) .
W j i * ( r 2 , r 1 , ω ) = W i j ( r 1 , r 2 , ω ) .
ϕ i j ( r , ω ) = W i j ( r , r , ω ) .
P 3 2 ( r , ω ) = 3 2 [ tr Φ 2 ( r , ω ) tr 2 Φ ( r , ω ) 1 3 ] ,
0 P 3 ( r , ω ) 1 .
μ EM ( r 1 , r 2 , ω ) = W ( r 1 , r 2 , ω ) F S ( r 1 , ω ) S ( r 2 , ω ) ,
A F 2 = tr ( A · A ) .
0 μ EM ( r 1 , r 2 , ω ) 1.
n 1 sin θ 1 = n 2 sin θ 2 ,
E 1 ( r , ω ) = ( E s s ^ 1 + E p p ^ 1 ) e i k 1 · r ,
E 2 ( r , ω ) = ( t s E s s ^ 2 + t p E p p ^ 2 ) e i k 2 · r ,
t s = 2 cos θ 1 cos θ 1 + i γ ,
t p = 2 n ˜ cos θ 1 2 n ˜ 2 γ 2 + 1 cos θ 1 + i n ˜ 2 γ ,
γ = n ˜ 1 ( n ˜ sin θ 1 ) 2 1 .
k 1 = k x 1 u ^ x + k 1 2 k x 1 2 u ^ z ,
k 2 = k x 2 u ^ x + i k x 2 2 k 2 2 u ^ z ,
k ^ i = k i k i · k i * ,
s ^ i = u ^ z × k ^ i | u ^ z × k ^ i | ,
p ^ i = k ^ i × s ^ i ,
E 2 ( r , ω ) = ( i γ sin 2 θ 1 + γ 2 t p E p t s E s sin θ 1 sin 2 θ 1 + γ 2 t p E p ) e i k 1 sin θ 1 x e k 1 γ z .
w ( z , ω ) = E 2 * ( r , ω ) · E 2 ( r , ω ) = ( w s + w p ) e 2 k 1 γ z ,
w i = | t i | 2 | E i | 2 , i = ( s , p ) ,
T s = u ^ z · S 2 s u ^ z · S 1 s ,
T p = u ^ z · S 2 p u ^ z · S 1 p .
S 2 s x = k 1 sin θ 1 2 μ 0 ω | t s | 2 | E s | 2 e 2 k 1 γ z ,
S 2 s y = S 2 s z = 0 ,
S 2 p x = k 1 sin θ 1 2 μ 0 ω n ˜ 2 ( sin 2 θ 1 + γ 2 ) | t p | 2 | E p | 2 e 2 k 1 γ z ,
S 2 p y = S 2 p z = 0 .
Λ = λ 2 n ˜ sin θ 1 ,
λ 2 n ˜ Λ λ 2 ,
W ( r 1 , r 2 , ω ) = W 0 ( ω ) e i k 1 sin θ 1 Δ x e k 1 γ ( z 1 + z 2 ) ,
W 0 ( ω ) = 1 χ 2 ( γ 2 | t p | 2 ϕ p p i γ χ t p * t s ϕ p s i γ sin θ 1 | t p | 2 ϕ p p i γ χ t s * t p ϕ s p χ 2 | t s | 2 ϕ s s χ sin θ 1 t s * t p ϕ s p i γ sin θ 1 | t p | 2 ϕ p p χ sin θ 1 t p * t s ϕ p s sin 2 θ 1 | t p | 2 ϕ p p ) ,
χ = sin 2 θ 1 + γ 2 .
Φ ( r , ω ) = W 0 ( ω ) e 2 k 1 γ z .
P 3 ( ω ) = 1 3 ( 1 | μ p s | 2 ) w p w s ( w p + w s ) 2 ,
μ p s = ϕ p s ϕ p p ϕ s s
P 3 ( ω ) = 1 4 + 3 4 | μ p s | 2 .
μ EM ( ω ) = 1 2 ( 1 | μ p s | 2 ) w p w s ( w p + w s ) 2 .
P 3 ( ω ) μ EM ( ω ) .
E n ( r , ω ) = ( i γ n sin 2 θ n + γ n 2 t n p E n p t n s E n s sin θ n sin 2 θ n + γ n 2 t n p E n p ) e i k 1 sin θ n x e k 1 γ n z ,
γ n = n ˜ 1 ( n ˜ sin θ n ) 2 1 ,
t n s = 2 cos θ n cos θ n + i γ n ,
t n p = 2 n ˜ cos θ n 2 n ˜ 2 γ n 2 + 1 cos θ n + i n ˜ γ n ,
E tot ( r , ω ) = n = 1 N E n ( r , ω ) .
W ( r 1 , r 2 , ω ) = m , n N W m n ( ω ) e i k 1 ( sin θ n x 2 sin θ m x 1 ) e k 1 ( γ m z 1 + γ n z 2 ) ,
W m n ( ω ) = 1 χ m χ n ( γ m γ n t m p * t n p ϕ p p m n i γ m χ n t m p * t n s ϕ p s m n i γ m sin θ n t m p * t n p ϕ p p m n i γ n χ m t m s * t n p ϕ s p m n χ m χ n t m s * t n s ϕ s s m n χ m sin θ n t m s * t n p ϕ s p m n i γ n sin θ m t m p * t n p ϕ p p m n χ n sin θ m t m p * t n s ϕ p s m n sin θ m sin θ n t m p * t n p ϕ p p m n ) ,
χ i = sin 2 θ i + γ i 2 , i = ( m , n ) .
Φ ( r , ω ) = m , n N W m n ( ω ) e i k 1 ( sin θ n sin θ m ) x e k 1 ( γ m + γ n ) z .
μ EM ( s ) ( r 1 , r 2 , ω ) = m , n N e i k 1 ( sin θ n sin θ m ) Δ x e 2 k 1 ( γ m + γ n ) z m , n N e 2 k 1 ( γ m + γ n ) z ,
μ EM ( p ) ( r 1 , r 2 , ω ) = m , n N χ m n 2 χ m m χ n n e i k 1 ( sin θ n sin θ m ) Δ x e 2 k 1 ( γ m + γ n ) z m , n N e 2 k 1 ( γ m + γ n ) z ,
χ i j = γ i γ j + sin θ i sin θ j , ( i , j ) = ( m , n ) ,
P 3 ( p ) ( z , ω ) = 3 m , n N χ m n 2 χ m m χ n n e 2 k 1 ( γ m + γ n ) z 2 m , n N e 2 k 1 ( γ m + γ n ) z 1 2 ,
P 3 ( s p ) ( z , ω ) = 3 m , n N ( χ m n 2 χ m m χ n n + 1 ) e 2 k 1 ( γ m + γ n ) z 8 m , n N e 2 k 1 ( γ m + γ n ) z 1 2 .
μ EM ( Δ x , ω ) = w s 1 2 + w p 1 2 + w s 2 2 + w p 2 2 + 2 [ w s 1 w s 2 + w p 1 w p 2 χ 12 2 / ( χ 11 χ 22 ) ] cos [ k 1 ( sin θ 2 sin θ 1 ) Δ x ] w s 1 + w p 1 + w s 2 + w p 2 ,
Δ x = M λ 2 2 n ˜ ( sin θ 2 sin θ 1 ) ,
l coh = λ 2 2 n ˜ ( sin θ 2 sin θ 1 ) .
2 n ˜ ( sin θ 2 sin θ 2 ) > 1 .

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