Abstract

The common definition of the Brewster angles for dielectric and magnetic achiral materials are the angles at which the vertically and horizontally polarized reflection coefficients vanish. We examine broader definitions of the Brewster conditions for waves that are incident on a free-space–chiral interface. Besides the common definition, the Brewster angles have been defined as the angles at which the polarizations of the reflected waves are independent of the polarizations of the incident waves. We consider total transmission of incident plane waves that satisfy the Brewster conditions at a free-space–chiral medium planar interface. In this case we determine the polarization of the incident wave for which the reflected vertically and horizontally polarized waves vanish simultaneously. Thus with this definition of the Brewster conditions the polarization of the reflected wave is undefined. The conditions for the excitation of surface waves are considered. The characteristic polarizations that are the same for the reflected and incident waves are also examined subject to the Brewster conditions. Potential applications of this analysis are to experimentally determine the chiral or geotropic measure of the medium and to identify and characterize biological and chemical materials through their optical activity in real time. Several independent measurements can be taken with the same polarimetric instrument to avoid false identifications. Since measurements can be conducted in the reflection mode they can be nonintrusive.

© 2010 Optical Society of America

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References

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  1. A. Lakhtakia, “Would Brewster recognize today’s Brewster angle?” Optics News 15, 14–18 (1989).
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  2. A. Lakhtakia and J. R. Diamond, “Reciprocity and the concept of the Brewster wavenumber,” Int. J. Infrared Millim. Waves 12, 1167–1174 (1991).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  4. C. Johnk, Engineering Electromagnetic Fields and Waves, 2nd ed. (Wiley, 1988).
  5. J. Lekner, Theory of Reflection (Nijhoff/Kluwer, 1987).
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  8. S. Bassiri, C. H. Papas, and N. Engheta, “Electromagnetic wave propagation through a dielectric–chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
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  9. I. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).
  10. J. Lekner, “Brewster angles in reflection by uniaxial crystals,” J. Opt. Soc. Am. A 10, 2059–2064 (1993).
    [CrossRef]
  11. J. Lekner, “Optical properties of nonisotropic chiral media,” Pure Appl. Opt. 5, 417–443 (1996).
    [CrossRef]
  12. J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  13. J. Kong, Theory of Electromagnetic Waves (Wiley, 1975).
  14. A. Gevorgyan, “Diffraction enhancement and suppression of plane of polarization in chiral photonic crystals,” Tech. Phys. 52, 75–82 (2007).
    [CrossRef]
  15. R. Azzam and N. Bashara, Ellipsometry and Polarized Light (Elsevier, 1977).
  16. M. Potter, Mathematical Methods in Physical Sciences (Prentice Hall, 1978).
  17. M. Silverman, Waves and Grains (Princeton Univ. Press, 1998).
  18. E. Bahar “Reflection and transmission matrices at a free-space–chiral interface based on the invariant constitutive relations for gyrotropic media and the Drude–Born–Federov constitutive relations,” J. Opt. Soc. Am. A 26, 1834–1838 (2009).
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  19. M. Silverman, N. Ritchie, G. Cushman, and B. Fished, “Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium,” J. Opt. Soc. Am. A 5, 1852–1862 (1988).
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  20. E. Bahar, “Optimum electromagnetic wave excitations of complex media characterized by positive or negative refractive indices and by chiral properties,” J. Opt. Soc. Am. B 24, 2807–2813 (2007).
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  21. E. Bahar, “Characterization of natural and artificial optical activity by the Mueller matrix for oblique incidence, total internal reflection, and Brewster angle,” J. Opt. Soc. Am. B 25, 1294–1302 (2008).
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  22. E. Bahar, “Roadmaps for the use of Mueller matrix measurements to detect and identify biological and chemical materials through their optical activity: potential applications in biomedicine, biochemistry, security, and industry,” J. Opt. Soc. Am. B 26, 364–470 (2009).
    [CrossRef]

2009

2008

2007

1996

J. Lekner, “Optical properties of nonisotropic chiral media,” Pure Appl. Opt. 5, 417–443 (1996).
[CrossRef]

1993

1991

A. Lakhtakia and J. R. Diamond, “Reciprocity and the concept of the Brewster wavenumber,” Int. J. Infrared Millim. Waves 12, 1167–1174 (1991).
[CrossRef]

1989

1988

1986

Azzam, R.

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (Elsevier, 1977).

Bahar, E.

Bashara, N.

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (Elsevier, 1977).

Bassiri, S.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Cushman, G.

Diamond, J. R.

A. Lakhtakia and J. R. Diamond, “Reciprocity and the concept of the Brewster wavenumber,” Int. J. Infrared Millim. Waves 12, 1167–1174 (1991).
[CrossRef]

Engheta, N.

Fished, B.

Gevorgyan, A.

A. Gevorgyan, “Diffraction enhancement and suppression of plane of polarization in chiral photonic crystals,” Tech. Phys. 52, 75–82 (2007).
[CrossRef]

Johnk, C.

C. Johnk, Engineering Electromagnetic Fields and Waves, 2nd ed. (Wiley, 1988).

Kong, J.

J. Kong, Theory of Electromagnetic Waves (Wiley, 1975).

Lakhtakia, A.

A. Lakhtakia and J. R. Diamond, “Reciprocity and the concept of the Brewster wavenumber,” Int. J. Infrared Millim. Waves 12, 1167–1174 (1991).
[CrossRef]

A. Lakhtakia, “Would Brewster recognize today’s Brewster angle?” Optics News 15, 14–18 (1989).
[CrossRef]

Lekner, J.

J. Lekner, “Optical properties of nonisotropic chiral media,” Pure Appl. Opt. 5, 417–443 (1996).
[CrossRef]

J. Lekner, “Brewster angles in reflection by uniaxial crystals,” J. Opt. Soc. Am. A 10, 2059–2064 (1993).
[CrossRef]

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, 1987).

Lindell, I.

I. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Papas, C. H.

Potter, M.

M. Potter, Mathematical Methods in Physical Sciences (Prentice Hall, 1978).

Ritchie, N.

Sihvola, A.

I. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Silverman, M.

Stratton, J.

J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tretyakov, S.

I. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Viitanen, A.

I. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Int. J. Infrared Millim. Waves

A. Lakhtakia and J. R. Diamond, “Reciprocity and the concept of the Brewster wavenumber,” Int. J. Infrared Millim. Waves 12, 1167–1174 (1991).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Optics News

Pure Appl. Opt.

J. Lekner, “Optical properties of nonisotropic chiral media,” Pure Appl. Opt. 5, 417–443 (1996).
[CrossRef]

Tech. Phys.

A. Gevorgyan, “Diffraction enhancement and suppression of plane of polarization in chiral photonic crystals,” Tech. Phys. 52, 75–82 (2007).
[CrossRef]

Other

R. Azzam and N. Bashara, Ellipsometry and Polarized Light (Elsevier, 1977).

M. Potter, Mathematical Methods in Physical Sciences (Prentice Hall, 1978).

M. Silverman, Waves and Grains (Princeton Univ. Press, 1998).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

C. Johnk, Engineering Electromagnetic Fields and Waves, 2nd ed. (Wiley, 1988).

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, 1987).

I. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

J. Kong, Theory of Electromagnetic Waves (Wiley, 1975).

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Figures (1)

Fig. 1
Fig. 1

Physical interpretation of the cross-polarized reflection coefficients.

Equations (53)

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D = ϵ ( E + β x E ) , B = μ ( H + β x H ) ,
D = ϵ E j ω g H , B = μ H + j ω g E .
[ E r V E r H ] = [ R V V R V H R H V R H H ] [ E i V E i H ] = R [ E i V E i H ] .
j f 2 T 01 H H T 10 V V tan 2 θ 1 = R V H = R H V .
T 01 H H T 10 V V = 4 cos θ 0 cos θ 1 ( Z 0 cos θ 0 + Z 1 cos θ 1 ) ( Y 0 cos θ 0 + Y 1 cos θ 1 ) .
T 01 H H T 10 V V = T 01 V V T 10 H H .
R H V = f T 01 H H R V V T 10 V V tan 2 θ 1 2 j ,
R V H = f T 01 V V R H H T 10 H H tan 2 θ 1 2 j .
[ R V V R V H R H V R H H ] [ E i V E i H ] = [ E r V E r H ] = [ 0 0 ] .
R V V R H H = R V H R H V .
E i V E i H = R V H R V V = R H H R H V .
sin θ B V = ( 1 Z 1 2 Z 0 2 ) ( 1 ϵ 0 2 ϵ 1 2 ) ,
sin θ B H = ( 1 Y 1 2 Y 0 2 ) ( 1 μ 0 2 μ 1 2 ) ,
( Z 0 cos θ 0 Z 1 cos θ 1 Z 0 cos θ 0 + Z 1 cos θ 1 ) ( Y 0 cos θ 0 Y 1 cos θ 1 Y 0 cos θ 0 + Y 1 cos θ 1 ) = [ f 4 cos θ 0 cos θ 1 tan 2 θ 1 2 ( Z 0 cos θ 0 + Z 1 cos θ 1 ) ( Y 0 cos θ 0 + Y 1 cos θ 1 ) ] 2 .
[ ( Z 0 cos θ 0 ) 2 ( Z 1 cos θ 1 ) 2 ] [ ( Y 0 cos θ 0 ) 2 ( Y 1 cos θ 1 ) 2 ] = 4 f 2 cos 2 θ 0 cos 2 θ 1 tan 4 θ 1 .
g ( θ 0 ) ( Z 0 2 cos θ 0 ) 2 ( Z 1 cos θ 1 ) 2 = Z 0 2 ( 1 sin 2 θ 0 ) Z 1 ( 1 k 0 2 sin 2 θ 0 k 1 2 ) = Z 0 2 [ ( 1 Z 1 2 Z 0 2 ) ( 1 ϵ 0 2 ϵ 1 2 ) sin 2 θ 0 ] .
g ( θ 0 ) = g ( θ B V + Δ V ) = g ( θ B V ) + Δ V g ( θ B V ) .
g ( θ B V ) = d g d θ 0 ( θ 0 = θ B V ) = ( Z 0 2 ( 1 ϵ 0 2 ϵ 1 2 ) 2 sin θ B V cos θ B V ) .
Δ V = g ( θ 0 ) g ( θ B V ) .
h ( θ 0 ) Y 0 2 cos 2 θ 0 Y 1 2 cos 2 θ 1 = Y 0 2 [ ( 1 Y 1 2 Y 0 2 ) ( 1 μ 0 2 μ 1 2 ) sin 2 θ 0 ] .
Δ V = ( 4 f 2 k 0 4 k 1 4 ) sin 4 θ B V ( 1 sin 2 θ B V ) ( 1 ϵ 0 2 ϵ 1 2 ) sin 2 θ B V [ ( 1 Y 1 2 Y 0 2 ) ( 1 μ 0 2 μ 1 2 ) sin 2 θ V B ] [ 1 k 0 2 sin 2 θ B V k 1 ]
Δ H = ( 4 f k 0 4 k 1 4 ) sin 4 θ B H ( 1 sin 2 θ B H ) ( 1 μ 0 2 μ 1 2 ) sin 2 θ B H [ ( 1 Z 1 2 Z 0 2 ) ( 1 ϵ 0 2 ϵ 1 2 ) sin 2 θ B H ] [ 1 k 0 2 sin 2 θ B H k 1 2 ] .
E r V E r H = R V V R H V = R V H R H H
| R H H | | R H V | = | R V H | | R V V | .
det R 1 = 1 det R = 1 R V V R H H R H V R V H = 0
1 R V V = 0 or 1 R H H = 0 .
( Z 0 cos θ 0 + Z 1 cos θ 1 ) ( Y 0 cos θ 0 + Y 1 cos θ 1 ) ,
cos 2 θ 0 = 1 sin 2 θ 0 = [ ( 1 Z 1 2 Z 0 2 ) ( 1 ϵ 0 2 ϵ 1 2 ) ]
cos 2 θ 1 = 1 sin 2 θ 1 = ( Z 0 2 Z 1 2 ) cos 2 θ 0 .
Z 0 cos θ 0 + Z 1 cos θ 1 = k 0 cos θ 0 ϵ 0 + k 1 cos θ 1 ϵ 1 = 0 .
R V V T V V = cos θ 1 cos θ 0 = Z 0 Z 1 .
R V V = 0 , Z 0 cos θ 0 Z 1 cos θ 1 = 0 .
cos 2 θ 0 = ( 1 Y 1 2 Y 0 2 ) ( 1 μ 0 2 μ 1 2 ) ,
cos 2 θ 1 = Y 0 2 Y 1 2 cos 2 θ 0 = ( Y 0 2 Y 1 2 1 ) ( 1 μ 0 2 μ 1 2 ) .
Y 0 cos θ 0 + Y 1 cos θ 1 = 0 ,
R H H T H H = Y 1 cos θ 1 Y 0 cos θ 0 = 1
Y 0 cos θ 0 Y 1 cos θ 1 = 0
[ R V V R V H R H V R H H ] [ E i V E i H ] = R E i = [ E r V E r H ] = E r = λ E i .
( R + λ I ) E i = [ ( R V V λ ) R V H R H V ( R H H λ ) ] [ E i V E i H ] = 0
det ( R λ I ) = 0 .
( R V V λ ) ( R H H λ ) = R H V R V H .
λ 2 λ ( R V V + R H H ) + R V V R H H R H V R V H = 0 .
λ = R V V + R H H 2 ± [ ( R V V + R H H 2 ) 2 ( R V V R H H R H V R V H ) ] 1 2 = R V V + R H H 2 ± R V V R H H 2 [ 1 + 4 R H V R V H ( R V V R H H ) 2 ] 1 2 .
λ 1 R V V + R H V R V H ( R V V R H H ) ,
λ 2 R H H + R H V R V H ( R H H R V V ) .
[ ( R V V λ 1 ) R V H R H V ( R H H λ 1 ) ] [ E i 1 V E i 1 H ] = 0 .
E i 1 V E i 1 H = R V H λ 1 R V V = λ 1 R H H R H V .
[ ( R V V λ 2 ) R V H R H V ( R H H λ 2 ) ] [ E i 2 V E i 2 H ] = 0 .
E i 2 V E i 2 H = R V H λ 2 R V V = λ 2 R H H R H V .
[ R V V R V H R H V R H H ] [ R V H λ 1 R V V ] = [ R V V R V H + R V H ( λ 1 R V V ) R H V R V H + R H H ( λ 1 R V V ) ] = λ 1 [ R V H λ 1 R V V ] .
R E i 1 = λ 1 E i 1 = E r 1 .
R E i 2 = λ 2 E i 2 = E r 2 .
[ R V V R V H R H V R H H ] [ E i 1 V E i 2 V E i 1 H E i 2 H ] = [ E i 1 V E i 2 V E i 1 H E i 2 H ] [ λ 1 0 0 λ 2 ] = [ λ 1 E i 1 V λ 2 E i 2 V λ 1 E i 1 H λ 2 E i 2 H ] = [ E r 1 V E r 2 V E r 1 H E r 2 H ] .

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