Abstract

This study develops a method for determining the chiral parameter and the refractive index of an isotropic chiral medium using chiral reflection equations and critical angle phenomena. Linearly polarized light propagates back and forth in a parallelogram prism between two parallel compartments with chiral solutions. A beam splitter then divides the light that emerges from the prism into a reflected light beam and a transmitted light beam. The two beams pass through a compensator and an analyzer, respectively, to cause phase compensation and interference of s and p polarizations. The phase difference between the two interference signals are initially optimized by a suitable optical arrangement and subsequently measured by heterodyne interferometry. Additionally, the refractive index of the solution is determined from the critical angle that occurred at the discontinuity of the phase difference between the two interference signals. These results are substituted into derived equations to calculate the chiral parameter. The approach has the merits of both common-path interferometry and heterodyne interferometry.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  8. C. Chou, Y. C. Huang, C. M. Feng, and M. Chang, “Amplitude sensitive optical heterodyne and lock-in technique on small optical rotation angle of chiral liquid,” Jpn. J. Appl. Phys. 36, 356-359 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. T. C. Preston and N. D. Jones, “Simple liquid-core waveguide polarimetry,” Appl. Phys. Lett. 89, 253509 (2006).
    [CrossRef]
  12. F. Vollmer and P. Fischer, “Ring-resonator-based frequency-domain optical activity measurements of a chiral liquid,” Opt. Lett. 31, 453-455 (2006).
    [CrossRef]
  13. M. H. Chou and D. C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36, 7104-7106 (1997).
    [CrossRef]
  14. M. H. Chiu, J. Y. Lee, and D. C. Su, “Complex refractive-index measurement based on Fresnel's equations and the uses of heterodyne interferometry,” Appl. Opt. 38, 4047-4052 (1999).
    [CrossRef]
  15. M. P. Silverman and J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromag. Waves Appl. 6, 587-601 (1992).
    [CrossRef]
  16. M. P. Silverman and J. Badoz, “Large enhancement of chiral asymmetry in light-reflection near critical angle,” Opt. Commun. 74, 129-133 (1989).
    [CrossRef]
  17. R.C.Weast, ed., Handbook of Chemistry and Physics, 61st ed. (CRC Press, 1981), pp. D227-D270 and E-418.
  18. N. Berova, K. Nakanishi, and R. W. Woody, Circular Dichroism: Principles and Applications, 2nd ed. (Wiley, 2000), pp. 30-31.

2006

T. C. Preston and N. D. Jones, “Simple liquid-core waveguide polarimetry,” Appl. Phys. Lett. 89, 253509 (2006).
[CrossRef]

F. Vollmer and P. Fischer, “Ring-resonator-based frequency-domain optical activity measurements of a chiral liquid,” Opt. Lett. 31, 453-455 (2006).
[CrossRef]

1999

1997

M. H. Chou and D. C. Su, “Improved technique for measuring small angles,” Appl. Opt. 36, 7104-7106 (1997).
[CrossRef]

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, and C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314-321 (1997).
[CrossRef]

C. Chou, Y. C. Huang, C. M. Feng, and M. Chang, “Amplitude sensitive optical heterodyne and lock-in technique on small optical rotation angle of chiral liquid,” Jpn. J. Appl. Phys. 36, 356-359 (1997).
[CrossRef]

1994

H. J. King, C. Chou, H. Chang, and Y. C. Huang, “Concentration measurements in chiral media using optical heterodyne polarimeter,” Opt. Commun. 110, 259-262 (1994).
[CrossRef]

T. W. King, G. L. Cote, R. McNichols, and M. J. Goetz, “Multispectral polarimetric glucose detection using a single pockels cell,” Opt. Eng. 33, 2746-2753 (1994).
[CrossRef]

1993

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “On the Maxwell-Garnett model of chiral composites,” J. Mater. Res. 8, 917-922 (1993).
[CrossRef]

1992

M. P. Silverman, J. Badoz, and B. Briat, “Chiral reflection from a naturally optically active medium,” Opt. Lett. 17, 886-888(1992).

G. L. Cote, M. D. Fox, and R. B. Northrop, “Noninvasive optical polarimetric glucose sensing using a true phase measurement technique,” IEEE Trans. Biomed. Eng. 39, 752-756 (1992).
[CrossRef]

M. P. Silverman and J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromag. Waves Appl. 6, 587-601 (1992).
[CrossRef]

1990

1989

M. P. Silverman and J. Badoz, “Large enhancement of chiral asymmetry in light-reflection near critical angle,” Opt. Commun. 74, 129-133 (1989).
[CrossRef]

1988

1986

Badoz, J.

M. P. Silverman, J. Badoz, and B. Briat, “Chiral reflection from a naturally optically active medium,” Opt. Lett. 17, 886-888(1992).

M. P. Silverman and J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromag. Waves Appl. 6, 587-601 (1992).
[CrossRef]

M. P. Silverman and J. Badoz, “Light-reflection from a naturally optically-active birefringent medium,” J. Opt. Soc. Am. A 7, 1163-1173 (1990).

M. P. Silverman and J. Badoz, “Large enhancement of chiral asymmetry in light-reflection near critical angle,” Opt. Commun. 74, 129-133 (1989).
[CrossRef]

Berova, N.

N. Berova, K. Nakanishi, and R. W. Woody, Circular Dichroism: Principles and Applications, 2nd ed. (Wiley, 2000), pp. 30-31.

Briat, B.

Chang, H.

H. J. King, C. Chou, H. Chang, and Y. C. Huang, “Concentration measurements in chiral media using optical heterodyne polarimeter,” Opt. Commun. 110, 259-262 (1994).
[CrossRef]

Chang, J. G.

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, and C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314-321 (1997).
[CrossRef]

Chang, M.

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, and C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314-321 (1997).
[CrossRef]

C. Chou, Y. C. Huang, C. M. Feng, and M. Chang, “Amplitude sensitive optical heterodyne and lock-in technique on small optical rotation angle of chiral liquid,” Jpn. J. Appl. Phys. 36, 356-359 (1997).
[CrossRef]

Chiu, M. H.

Chou, C.

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, and C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314-321 (1997).
[CrossRef]

C. Chou, Y. C. Huang, C. M. Feng, and M. Chang, “Amplitude sensitive optical heterodyne and lock-in technique on small optical rotation angle of chiral liquid,” Jpn. J. Appl. Phys. 36, 356-359 (1997).
[CrossRef]

H. J. King, C. Chou, H. Chang, and Y. C. Huang, “Concentration measurements in chiral media using optical heterodyne polarimeter,” Opt. Commun. 110, 259-262 (1994).
[CrossRef]

Chou, M. H.

Cote, G. L.

T. W. King, G. L. Cote, R. McNichols, and M. J. Goetz, “Multispectral polarimetric glucose detection using a single pockels cell,” Opt. Eng. 33, 2746-2753 (1994).
[CrossRef]

G. L. Cote, M. D. Fox, and R. B. Northrop, “Noninvasive optical polarimetric glucose sensing using a true phase measurement technique,” IEEE Trans. Biomed. Eng. 39, 752-756 (1992).
[CrossRef]

Cushman, G. M.

Feng, C. M.

C. Chou, Y. C. Huang, C. M. Feng, and M. Chang, “Amplitude sensitive optical heterodyne and lock-in technique on small optical rotation angle of chiral liquid,” Jpn. J. Appl. Phys. 36, 356-359 (1997).
[CrossRef]

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, and C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314-321 (1997).
[CrossRef]

Fischer, P.

Fisher, B.

Fox, M. D.

G. L. Cote, M. D. Fox, and R. B. Northrop, “Noninvasive optical polarimetric glucose sensing using a true phase measurement technique,” IEEE Trans. Biomed. Eng. 39, 752-756 (1992).
[CrossRef]

Goetz, M. J.

T. W. King, G. L. Cote, R. McNichols, and M. J. Goetz, “Multispectral polarimetric glucose detection using a single pockels cell,” Opt. Eng. 33, 2746-2753 (1994).
[CrossRef]

Huang, Y. C.

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, and C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314-321 (1997).
[CrossRef]

C. Chou, Y. C. Huang, C. M. Feng, and M. Chang, “Amplitude sensitive optical heterodyne and lock-in technique on small optical rotation angle of chiral liquid,” Jpn. J. Appl. Phys. 36, 356-359 (1997).
[CrossRef]

H. J. King, C. Chou, H. Chang, and Y. C. Huang, “Concentration measurements in chiral media using optical heterodyne polarimeter,” Opt. Commun. 110, 259-262 (1994).
[CrossRef]

Jones, N. D.

T. C. Preston and N. D. Jones, “Simple liquid-core waveguide polarimetry,” Appl. Phys. Lett. 89, 253509 (2006).
[CrossRef]

King, H. J.

H. J. King, C. Chou, H. Chang, and Y. C. Huang, “Concentration measurements in chiral media using optical heterodyne polarimeter,” Opt. Commun. 110, 259-262 (1994).
[CrossRef]

King, T. W.

T. W. King, G. L. Cote, R. McNichols, and M. J. Goetz, “Multispectral polarimetric glucose detection using a single pockels cell,” Opt. Eng. 33, 2746-2753 (1994).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “On the Maxwell-Garnett model of chiral composites,” J. Mater. Res. 8, 917-922 (1993).
[CrossRef]

Lee, J. Y.

McNichols, R.

T. W. King, G. L. Cote, R. McNichols, and M. J. Goetz, “Multispectral polarimetric glucose detection using a single pockels cell,” Opt. Eng. 33, 2746-2753 (1994).
[CrossRef]

Nakanishi, K.

N. Berova, K. Nakanishi, and R. W. Woody, Circular Dichroism: Principles and Applications, 2nd ed. (Wiley, 2000), pp. 30-31.

Northrop, R. B.

G. L. Cote, M. D. Fox, and R. B. Northrop, “Noninvasive optical polarimetric glucose sensing using a true phase measurement technique,” IEEE Trans. Biomed. Eng. 39, 752-756 (1992).
[CrossRef]

Preston, T. C.

T. C. Preston and N. D. Jones, “Simple liquid-core waveguide polarimetry,” Appl. Phys. Lett. 89, 253509 (2006).
[CrossRef]

Ritchie, N.

Silverman, M. P.

Su, D. C.

Varadan, V. K.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “On the Maxwell-Garnett model of chiral composites,” J. Mater. Res. 8, 917-922 (1993).
[CrossRef]

Varadan, V. V.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “On the Maxwell-Garnett model of chiral composites,” J. Mater. Res. 8, 917-922 (1993).
[CrossRef]

Vollmer, F.

Woody, R. W.

N. Berova, K. Nakanishi, and R. W. Woody, Circular Dichroism: Principles and Applications, 2nd ed. (Wiley, 2000), pp. 30-31.

Appl. Opt.

Appl. Phys. Lett.

T. C. Preston and N. D. Jones, “Simple liquid-core waveguide polarimetry,” Appl. Phys. Lett. 89, 253509 (2006).
[CrossRef]

IEEE Trans. Biomed. Eng.

G. L. Cote, M. D. Fox, and R. B. Northrop, “Noninvasive optical polarimetric glucose sensing using a true phase measurement technique,” IEEE Trans. Biomed. Eng. 39, 752-756 (1992).
[CrossRef]

J. Electromag. Waves Appl.

M. P. Silverman and J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromag. Waves Appl. 6, 587-601 (1992).
[CrossRef]

J. Mater. Res.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “On the Maxwell-Garnett model of chiral composites,” J. Mater. Res. 8, 917-922 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

C. Chou, Y. C. Huang, C. M. Feng, and M. Chang, “Amplitude sensitive optical heterodyne and lock-in technique on small optical rotation angle of chiral liquid,” Jpn. J. Appl. Phys. 36, 356-359 (1997).
[CrossRef]

Opt. Commun.

H. J. King, C. Chou, H. Chang, and Y. C. Huang, “Concentration measurements in chiral media using optical heterodyne polarimeter,” Opt. Commun. 110, 259-262 (1994).
[CrossRef]

C. M. Feng, Y. C. Huang, J. G. Chang, M. Chang, and C. Chou, “A true phase sensitive optical heterodyne polarimeter on glucose concentration measurement,” Opt. Commun. 141, 314-321 (1997).
[CrossRef]

M. P. Silverman and J. Badoz, “Large enhancement of chiral asymmetry in light-reflection near critical angle,” Opt. Commun. 74, 129-133 (1989).
[CrossRef]

Opt. Eng.

T. W. King, G. L. Cote, R. McNichols, and M. J. Goetz, “Multispectral polarimetric glucose detection using a single pockels cell,” Opt. Eng. 33, 2746-2753 (1994).
[CrossRef]

Opt. Lett.

Other

R.C.Weast, ed., Handbook of Chemistry and Physics, 61st ed. (CRC Press, 1981), pp. D227-D270 and E-418.

N. Berova, K. Nakanishi, and R. W. Woody, Circular Dichroism: Principles and Applications, 2nd ed. (Wiley, 2000), pp. 30-31.

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Figures (4)

Fig. 1
Fig. 1

Measuring the phase difference associated with reflections of light between two chiral solutions separated by a parallelogram prism: H 1 , half-wave plate; EO, electro-optic modulator; BS, beam splitter; C 1 and C 2 , compensators; AN 1 and AN 2 , analyzers; D 1 and D 2 , detectors.

Fig. 2
Fig. 2

Geometry of chiral reflection. k t is the wave vector of the predicted transmitted ray in the absence of chirality.

Fig. 3
Fig. 3

ψ versus θ i of (a) 50% saccharose solution, and (b) 50% glucose solution, respectively.

Fig. 4
Fig. 4

Calculated curves of Δ g versus θ i of (a) 50% saccharose solution, and (b) 50% glucose solution, respectively.

Tables (1)

Tables Icon

Table 1 Experimental Results and the Corresponding Reference Data

Equations (40)

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D = ε [ E + χ × E ] ,
B = μ [ H + χ × H ] ,
D = ε [ E + if     ( k × E ) / n k 0 ] ,
B = μ [ H + if     ( k × H ) / n k 0 ] ,
E i = ( cos α sin α ) .
E i = ( exp ( i ω t 2 ) 0 0 exp ( i ω t 2 ) ) ( cos α sin α ) = ( cos α × exp ( i ω t 2 ) sin α × exp ( i ω t 2 ) ) .
E 1 = 1 2 ( cos 2 β cos β sin β cos β sin β sin 2 β ) ( t p 0 0 t s ) ( r 11 r 12 r 21 r 22 ) 2 ( t p 0 0 t s ) ( cos α × exp ( i ω t 2 ) sin α × exp ( i ω t 2 ) ) = 1 2 [ A 1 exp ( i ω t / 2 + i δ 1 ) + A 2 exp ( i ω t / 2 + i δ 2 ) ] ,
r 11 [ ( n / n 0 ) 2 cos θ 1 q 1 ] / [ ( n / n 0 ) 2 cos θ 1 + q 1 ] ,
r 22 ( cos θ 1 q 1 ) / ( cos θ 1 + q 1 ) ,
r 12 = r 21 = i [ ( n / n 0 ) 2 ( z + z ) cos θ 1 ] [ ( cos θ 1 + q 1 ) { ( n / n 0 ) 2 cos θ 1 + q 1 } ] ,
q 1 = [ ( n / n 0 ) 2 sin 2 θ 1 ] 1 / 2 ,
z ± = cos θ ±
z + z 2 n 0 g sin 2 θ 1 / n 2 q 1 ,
θ 1 = 60 ° + sin 1 ( sin θ i n 0 ) ,
A 1 = t p cos α [ t p ( r 11 2 + r 12 r 21 ) cos β ] 2 [ t s r 21 ( r 11 + r 22 ) sin β ] 2 ,
A 2 = t s sin α [ t p r 12 ( r 11 + r 22 ) cos β ] 2 + [ t s ( r 12 r 21 + r 22 2 ) sin β ] 2 ,
δ 1 = arg [ t p ( r 11 2 + r 12 r 21 ) + t s r 21 ( r 11 + r 22 ) tan β ] ,
δ 2 = arg [ t s ( r 22 2 + r 12 r 21 ) + t p r 12 ( r 11 + r 22 ) cot β ] ,
I 1 = | E 1 | 2 = 1 2 [ A 1 2 + A 2 2 + 2 A 1 A 2 cos ( ω t + ϕ 1 ) ] ,
ϕ 1 = δ 1 δ 2 = arg [ t p ( r 11 2 + r 12 r 21 ) + t s r 21 ( r 11 + r 22 ) tan β ] arg [ t s ( r 22 2 + r 12 r 21 ) + t p r 12 ( r 11 + r 22 ) cot β ] .
ϕ 1 arg [ t p r 11 2 + t s r 21 ( r 11 + r 22 ) tan β ] arg [ t s r 22 2 + t p r 12 ( r 11 + r 22 ) cot β ] .
E 2 = 1 2 ( cos 2 ( β ) cos ( β ) sin ( β ) cos ( β ) sin ( β ) sin 2 ( β ) ) ( t p 0 0 t s ) ( r 11 r 12 r 21 r 22 ) 2 ( t p 0 0 t s ) ( cos α × exp ( i ω t 2 ) sin α × exp ( i ω t 2 ) ) = 1 2 [ B 1 exp ( i ω t / 2 + i δ 3 ) + B 2 exp ( i ω t / 2 + i δ 4 ) ] ,
B 1 = t p cos α [ t p ( r 11 2 + r 12 r 21 ) cos β ] 2 [ t s r 21 ( r 11 + r 22 ) sin β ] 2 ,
B 2 = t s sin α [ t p r 12 ( r 11 + r 22 ) cos β ] 2 + [ t s ( r 12 r 21 + r 22 2 ) sin β ] 2 ,
δ 3 = arg [ t p ( r 11 2 + r 12 r 21 ) t s r 21 ( r 11 + r 21 ) tan β ] ,
δ 4 = arg [ t s ( r 22 2 + r 12 r 21 ) t p r 12 ( r 11 + r 22 ) cot β ] .
I 2 = | E 2 | 2 = 1 2 [ B 1 2 + B 2 2 + 2 B 1 B 2 cos ( ω t + ϕ 2 ) ] ,
ϕ 2 = δ 3 δ 4 = arg [ t p ( r 11 2 + r 12 r 21 ) t s r 21 ( r 11 + r 22 ) tan β ] arg [ t s ( r 22 2 + r 12 r 21 ) t p r 12 ( r 11 + r 22 ) cot β ] .
ϕ 2 arg [ t p r 11 2 t s r 21 ( r 11 + r 22 ) tan β ] arg [ t s r 22 2 t p r 12 ( r 11 + r 22 ) cot β ] .
ψ = ϕ 1 ϕ 2
ψ = tan 1 [ ( F 1 tan β + F 2 cot β ) g 1 F 1 F 2 g 2 ] tan 1 [ ( F 1 tan β + F 2 cot β ) g 1 F 1 F 2 g 2 ] ,
F 1 = sin 2 θ 1 cos θ 1 { ( cos θ 1 + q 1 ) [ ( n / n 0 ) 2 cos θ 1 q 1 ] + ( cos θ 1 q 1 ) [ ( n / n 0 ) 2 cos θ 1 + q 1 ] } n 0 q 1 ( cos θ 1 + q 1 ) 2 [ ( n / n 0 ) 2 cos θ 1 q 1 ] 2 ,
F 2 = sin 2 θ 1 cos θ 1 { ( cos θ 1 + q 1 ) [ ( n / n 0 ) 2 cos θ 1 q 1 ] + ( cos θ 1 q 1 ) [ ( n / n 0 ) 2 cos θ 1 + q 1 ] } n 0 q 1 ( cos θ 1 q 1 ) 2 [ ( n / n 0 ) 2 cos θ 1 + q 1 ] 2 .
ψ tan 1 [ ( F 1 tan β + F 2 cot β ) g ] tan 1 [ ( F 1 tan β + F 2 cot β ) g ] 2 tan 1 [ ( F 1 tan β + F 2 cot β ) g ] .
g tan ( ψ / 2 ) × ( F 1 tan β + F 2 cot β ) 1 .
n = n 0 sin θ 1 c = n 0 sin [ 60 ° + sin 1 ( sin θ i c n 0 ) ] ,
[ θ s ] = θ s ( deg ) C v L ( dm ) = ( 2 π / λ ) g ref × L ( m ) × ( 180 / π ) C v L ( dm ) = ( 36 / λ ) g ref C v ,
Δ n = | n 0 cos θ 1 c | Δ θ 1 = | 2 cos θ i c 1 + 3 tan [ 60 ° + sin 1 ( θ i c n 0 ) ] | Δ θ i ,
g ψ 2 F 1 tan β .
Δ g = | g ψ Δ ψ | + | g F 1 F 1 θ i Δ θ i | + | g F 1 F 1 n Δ n | ,

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