Abstract

We present a new mathematical technique which can be used to determine the main refractive indices and the birefringence of an anisotropic layer by using a polarizing microscope in conoscopic illumination. The values of the birefringence for the yellow radiation of a Na lamp are determined here for a Carpathian quartz sample, but the technique can also be applied to the study of other uniaxial substances such as liquid crystals, model membranes or biological tissues. The validity of the proposed method was tested by comparing the results with those obtained with a Rayleigh interferometer and by using the technology of channeled spectra.

© 2008 Optical Society of America

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  1. L. A. Pajdzik and A. M. Glazer, "Three-dimensional birefringence imaging with a microscope tilting-stage. I. Uniaxial crystals," J. Appl. Cryst. 39, 326-337 (2006).
    [CrossRef]
  2. L. A. Pajdzik and A. M. Glazer, "Three-dimensional birefringence imaging with a microscope tilting stage. II. Biaxial crystals," J. Appl. Cryst. 39, 856-870 (2006).
    [CrossRef]
  3. M. A. Geday, W. Kaminsky, J. G. Lewis, and A. M. Glazer, "Images of absolute retardance L·Δn, using the rotating polariser method," J. Microsc. 198, 1-9 (2000).
    [CrossRef] [PubMed]
  4. W. Kaminsky, K. Claborn, and B. Kahr, "Polarimetric imaging of crystals," Chem. Soc. Rev. 33, 514-525 (2004).
    [CrossRef] [PubMed]
  5. Birefringence imaging system www.oxfordcryosystems.co.uk/downloads/Metripol brochure.pdf.
  6. A. Echalier, R. L. Glazer, V. Fülöp, and M. A. Geday, "Assessing crystallization droplets using birefringence," Acta Cryst. D 60, 696-702 (2004).
  7. M. Delibaş and D. O. Dorohoi, Practicum in Optics (in Romanian) (Ed. Univ. "Al. I. Cuza", Iaşi, 1999).
  8. G. Cone, Optics of anisotropic media (in Romanina) (Ed. Tehnica, Bucureşti, 1990).
  9. D. O. Dorohoi, Optics (in Romanian) (Ed. "Şt. Procopiu", Iaşi, 1995).
  10. V. Pop, Bases of Optics (in Romanian, (Ed. Univ. "Al. I. Cuza", Iaşi, 1988).
  11. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).
  12. L. Dumitraşcu, I. Dumitraşcu, D. O. Dorohoi, D. Dimitriu, M. Aflori, and G. Apreutesei, Complementary Physics for PHD students (in Romanian), vol. I, (Ed. TehnoPress, Iaşi, 2007).
  13. http://www.creative.com.
  14. http:// www.mackichan.com.
  15. V. Pop, D. O. Dorohoi, and E. Cringeanu, "A New method for determining birefringence dispersion," J. Macromol. Sci. Phys. B33, 373-385 (1994).
  16. D. O. Dorohoi and M. Postolache, "The birefringence dispersion of poly-(phenyl) methacrylic ester of cetyl oxy benzoic acid, determined from channeled spectra," J. Macromol. Sci. Phys. 40, 239-249 (2001).
    [CrossRef]
  17. I. Dumitrascu, L. Dumitrascu, and D. O. Dorohoi, "The influence of the external field on the birefringence of the nematic liquid crystalline layer," J. Optoelectron. Adv. M. 8, 1028-1032 (2006).
  18. I. Diaconu, N. M. Puica, D. O. Dorohoi, and M. Aflori, "Birefringence dispersion of N-(4-methoxybenzilidene-4-butylaniline (MBBA) determined from channeled spectra," Spetrochim. Acta A 68, 536-541 (2007).
    [CrossRef]
  19. D. A. M. Androne, L. Dumitrascu, I. Horga, and D. O. Dorohoi, "Main refractive indices and birefringence of feldspars from Romanian pergmatites," Bull. Politechnical Inst. Iasi 55,137-141 (2006).
  20. D. M. A. Androne, D. O. Dorohoi, and D. Timpu, "Physical methods of identification of the feldspars from granite pergmatites," Romanian J. Phys. 53, 263-269 (2008).

2008 (1)

D. M. A. Androne, D. O. Dorohoi, and D. Timpu, "Physical methods of identification of the feldspars from granite pergmatites," Romanian J. Phys. 53, 263-269 (2008).

2007 (1)

I. Diaconu, N. M. Puica, D. O. Dorohoi, and M. Aflori, "Birefringence dispersion of N-(4-methoxybenzilidene-4-butylaniline (MBBA) determined from channeled spectra," Spetrochim. Acta A 68, 536-541 (2007).
[CrossRef]

2006 (4)

D. A. M. Androne, L. Dumitrascu, I. Horga, and D. O. Dorohoi, "Main refractive indices and birefringence of feldspars from Romanian pergmatites," Bull. Politechnical Inst. Iasi 55,137-141 (2006).

I. Dumitrascu, L. Dumitrascu, and D. O. Dorohoi, "The influence of the external field on the birefringence of the nematic liquid crystalline layer," J. Optoelectron. Adv. M. 8, 1028-1032 (2006).

L. A. Pajdzik and A. M. Glazer, "Three-dimensional birefringence imaging with a microscope tilting-stage. I. Uniaxial crystals," J. Appl. Cryst. 39, 326-337 (2006).
[CrossRef]

L. A. Pajdzik and A. M. Glazer, "Three-dimensional birefringence imaging with a microscope tilting stage. II. Biaxial crystals," J. Appl. Cryst. 39, 856-870 (2006).
[CrossRef]

2004 (2)

W. Kaminsky, K. Claborn, and B. Kahr, "Polarimetric imaging of crystals," Chem. Soc. Rev. 33, 514-525 (2004).
[CrossRef] [PubMed]

A. Echalier, R. L. Glazer, V. Fülöp, and M. A. Geday, "Assessing crystallization droplets using birefringence," Acta Cryst. D 60, 696-702 (2004).

2001 (1)

D. O. Dorohoi and M. Postolache, "The birefringence dispersion of poly-(phenyl) methacrylic ester of cetyl oxy benzoic acid, determined from channeled spectra," J. Macromol. Sci. Phys. 40, 239-249 (2001).
[CrossRef]

2000 (1)

M. A. Geday, W. Kaminsky, J. G. Lewis, and A. M. Glazer, "Images of absolute retardance L·Δn, using the rotating polariser method," J. Microsc. 198, 1-9 (2000).
[CrossRef] [PubMed]

1994 (1)

V. Pop, D. O. Dorohoi, and E. Cringeanu, "A New method for determining birefringence dispersion," J. Macromol. Sci. Phys. B33, 373-385 (1994).

Aflori, M.

I. Diaconu, N. M. Puica, D. O. Dorohoi, and M. Aflori, "Birefringence dispersion of N-(4-methoxybenzilidene-4-butylaniline (MBBA) determined from channeled spectra," Spetrochim. Acta A 68, 536-541 (2007).
[CrossRef]

Androne, D. A. M.

D. A. M. Androne, L. Dumitrascu, I. Horga, and D. O. Dorohoi, "Main refractive indices and birefringence of feldspars from Romanian pergmatites," Bull. Politechnical Inst. Iasi 55,137-141 (2006).

Androne, D. M. A.

D. M. A. Androne, D. O. Dorohoi, and D. Timpu, "Physical methods of identification of the feldspars from granite pergmatites," Romanian J. Phys. 53, 263-269 (2008).

Claborn, K.

W. Kaminsky, K. Claborn, and B. Kahr, "Polarimetric imaging of crystals," Chem. Soc. Rev. 33, 514-525 (2004).
[CrossRef] [PubMed]

Cringeanu, E.

V. Pop, D. O. Dorohoi, and E. Cringeanu, "A New method for determining birefringence dispersion," J. Macromol. Sci. Phys. B33, 373-385 (1994).

Diaconu, I.

I. Diaconu, N. M. Puica, D. O. Dorohoi, and M. Aflori, "Birefringence dispersion of N-(4-methoxybenzilidene-4-butylaniline (MBBA) determined from channeled spectra," Spetrochim. Acta A 68, 536-541 (2007).
[CrossRef]

Dorohoi, D. O.

D. M. A. Androne, D. O. Dorohoi, and D. Timpu, "Physical methods of identification of the feldspars from granite pergmatites," Romanian J. Phys. 53, 263-269 (2008).

I. Diaconu, N. M. Puica, D. O. Dorohoi, and M. Aflori, "Birefringence dispersion of N-(4-methoxybenzilidene-4-butylaniline (MBBA) determined from channeled spectra," Spetrochim. Acta A 68, 536-541 (2007).
[CrossRef]

D. A. M. Androne, L. Dumitrascu, I. Horga, and D. O. Dorohoi, "Main refractive indices and birefringence of feldspars from Romanian pergmatites," Bull. Politechnical Inst. Iasi 55,137-141 (2006).

I. Dumitrascu, L. Dumitrascu, and D. O. Dorohoi, "The influence of the external field on the birefringence of the nematic liquid crystalline layer," J. Optoelectron. Adv. M. 8, 1028-1032 (2006).

D. O. Dorohoi and M. Postolache, "The birefringence dispersion of poly-(phenyl) methacrylic ester of cetyl oxy benzoic acid, determined from channeled spectra," J. Macromol. Sci. Phys. 40, 239-249 (2001).
[CrossRef]

V. Pop, D. O. Dorohoi, and E. Cringeanu, "A New method for determining birefringence dispersion," J. Macromol. Sci. Phys. B33, 373-385 (1994).

Dumitrascu, I.

I. Dumitrascu, L. Dumitrascu, and D. O. Dorohoi, "The influence of the external field on the birefringence of the nematic liquid crystalline layer," J. Optoelectron. Adv. M. 8, 1028-1032 (2006).

Dumitrascu, L.

I. Dumitrascu, L. Dumitrascu, and D. O. Dorohoi, "The influence of the external field on the birefringence of the nematic liquid crystalline layer," J. Optoelectron. Adv. M. 8, 1028-1032 (2006).

D. A. M. Androne, L. Dumitrascu, I. Horga, and D. O. Dorohoi, "Main refractive indices and birefringence of feldspars from Romanian pergmatites," Bull. Politechnical Inst. Iasi 55,137-141 (2006).

Echalier, A.

A. Echalier, R. L. Glazer, V. Fülöp, and M. A. Geday, "Assessing crystallization droplets using birefringence," Acta Cryst. D 60, 696-702 (2004).

Fülöp, V.

A. Echalier, R. L. Glazer, V. Fülöp, and M. A. Geday, "Assessing crystallization droplets using birefringence," Acta Cryst. D 60, 696-702 (2004).

Geday, M. A.

A. Echalier, R. L. Glazer, V. Fülöp, and M. A. Geday, "Assessing crystallization droplets using birefringence," Acta Cryst. D 60, 696-702 (2004).

M. A. Geday, W. Kaminsky, J. G. Lewis, and A. M. Glazer, "Images of absolute retardance L·Δn, using the rotating polariser method," J. Microsc. 198, 1-9 (2000).
[CrossRef] [PubMed]

Glazer, A. M.

L. A. Pajdzik and A. M. Glazer, "Three-dimensional birefringence imaging with a microscope tilting-stage. I. Uniaxial crystals," J. Appl. Cryst. 39, 326-337 (2006).
[CrossRef]

L. A. Pajdzik and A. M. Glazer, "Three-dimensional birefringence imaging with a microscope tilting stage. II. Biaxial crystals," J. Appl. Cryst. 39, 856-870 (2006).
[CrossRef]

M. A. Geday, W. Kaminsky, J. G. Lewis, and A. M. Glazer, "Images of absolute retardance L·Δn, using the rotating polariser method," J. Microsc. 198, 1-9 (2000).
[CrossRef] [PubMed]

Glazer, R. L.

A. Echalier, R. L. Glazer, V. Fülöp, and M. A. Geday, "Assessing crystallization droplets using birefringence," Acta Cryst. D 60, 696-702 (2004).

Horga, I.

D. A. M. Androne, L. Dumitrascu, I. Horga, and D. O. Dorohoi, "Main refractive indices and birefringence of feldspars from Romanian pergmatites," Bull. Politechnical Inst. Iasi 55,137-141 (2006).

Kahr, B.

W. Kaminsky, K. Claborn, and B. Kahr, "Polarimetric imaging of crystals," Chem. Soc. Rev. 33, 514-525 (2004).
[CrossRef] [PubMed]

Kaminsky, W.

W. Kaminsky, K. Claborn, and B. Kahr, "Polarimetric imaging of crystals," Chem. Soc. Rev. 33, 514-525 (2004).
[CrossRef] [PubMed]

M. A. Geday, W. Kaminsky, J. G. Lewis, and A. M. Glazer, "Images of absolute retardance L·Δn, using the rotating polariser method," J. Microsc. 198, 1-9 (2000).
[CrossRef] [PubMed]

Lewis, J. G.

M. A. Geday, W. Kaminsky, J. G. Lewis, and A. M. Glazer, "Images of absolute retardance L·Δn, using the rotating polariser method," J. Microsc. 198, 1-9 (2000).
[CrossRef] [PubMed]

Pajdzik, L. A.

L. A. Pajdzik and A. M. Glazer, "Three-dimensional birefringence imaging with a microscope tilting-stage. I. Uniaxial crystals," J. Appl. Cryst. 39, 326-337 (2006).
[CrossRef]

L. A. Pajdzik and A. M. Glazer, "Three-dimensional birefringence imaging with a microscope tilting stage. II. Biaxial crystals," J. Appl. Cryst. 39, 856-870 (2006).
[CrossRef]

Pop, V.

V. Pop, D. O. Dorohoi, and E. Cringeanu, "A New method for determining birefringence dispersion," J. Macromol. Sci. Phys. B33, 373-385 (1994).

Postolache, M.

D. O. Dorohoi and M. Postolache, "The birefringence dispersion of poly-(phenyl) methacrylic ester of cetyl oxy benzoic acid, determined from channeled spectra," J. Macromol. Sci. Phys. 40, 239-249 (2001).
[CrossRef]

Puica, N. M.

I. Diaconu, N. M. Puica, D. O. Dorohoi, and M. Aflori, "Birefringence dispersion of N-(4-methoxybenzilidene-4-butylaniline (MBBA) determined from channeled spectra," Spetrochim. Acta A 68, 536-541 (2007).
[CrossRef]

Timpu, D.

D. M. A. Androne, D. O. Dorohoi, and D. Timpu, "Physical methods of identification of the feldspars from granite pergmatites," Romanian J. Phys. 53, 263-269 (2008).

Acta Cryst. D (1)

A. Echalier, R. L. Glazer, V. Fülöp, and M. A. Geday, "Assessing crystallization droplets using birefringence," Acta Cryst. D 60, 696-702 (2004).

Bull. Politechnical Inst. Iasi (1)

D. A. M. Androne, L. Dumitrascu, I. Horga, and D. O. Dorohoi, "Main refractive indices and birefringence of feldspars from Romanian pergmatites," Bull. Politechnical Inst. Iasi 55,137-141 (2006).

Chem. Soc. Rev. (1)

W. Kaminsky, K. Claborn, and B. Kahr, "Polarimetric imaging of crystals," Chem. Soc. Rev. 33, 514-525 (2004).
[CrossRef] [PubMed]

J. Appl. Cryst. (2)

L. A. Pajdzik and A. M. Glazer, "Three-dimensional birefringence imaging with a microscope tilting-stage. I. Uniaxial crystals," J. Appl. Cryst. 39, 326-337 (2006).
[CrossRef]

L. A. Pajdzik and A. M. Glazer, "Three-dimensional birefringence imaging with a microscope tilting stage. II. Biaxial crystals," J. Appl. Cryst. 39, 856-870 (2006).
[CrossRef]

J. Macromol. Sci. Phys. (2)

V. Pop, D. O. Dorohoi, and E. Cringeanu, "A New method for determining birefringence dispersion," J. Macromol. Sci. Phys. B33, 373-385 (1994).

D. O. Dorohoi and M. Postolache, "The birefringence dispersion of poly-(phenyl) methacrylic ester of cetyl oxy benzoic acid, determined from channeled spectra," J. Macromol. Sci. Phys. 40, 239-249 (2001).
[CrossRef]

J. Microsc. (1)

M. A. Geday, W. Kaminsky, J. G. Lewis, and A. M. Glazer, "Images of absolute retardance L·Δn, using the rotating polariser method," J. Microsc. 198, 1-9 (2000).
[CrossRef] [PubMed]

J. Optoelectron. Adv. M. (1)

I. Dumitrascu, L. Dumitrascu, and D. O. Dorohoi, "The influence of the external field on the birefringence of the nematic liquid crystalline layer," J. Optoelectron. Adv. M. 8, 1028-1032 (2006).

Romanian J. Phys. (1)

D. M. A. Androne, D. O. Dorohoi, and D. Timpu, "Physical methods of identification of the feldspars from granite pergmatites," Romanian J. Phys. 53, 263-269 (2008).

Spetrochim. Acta A (1)

I. Diaconu, N. M. Puica, D. O. Dorohoi, and M. Aflori, "Birefringence dispersion of N-(4-methoxybenzilidene-4-butylaniline (MBBA) determined from channeled spectra," Spetrochim. Acta A 68, 536-541 (2007).
[CrossRef]

Other (9)

Birefringence imaging system www.oxfordcryosystems.co.uk/downloads/Metripol brochure.pdf.

M. Delibaş and D. O. Dorohoi, Practicum in Optics (in Romanian) (Ed. Univ. "Al. I. Cuza", Iaşi, 1999).

G. Cone, Optics of anisotropic media (in Romanina) (Ed. Tehnica, Bucureşti, 1990).

D. O. Dorohoi, Optics (in Romanian) (Ed. "Şt. Procopiu", Iaşi, 1995).

V. Pop, Bases of Optics (in Romanian, (Ed. Univ. "Al. I. Cuza", Iaşi, 1988).

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

L. Dumitraşcu, I. Dumitraşcu, D. O. Dorohoi, D. Dimitriu, M. Aflori, and G. Apreutesei, Complementary Physics for PHD students (in Romanian), vol. I, (Ed. TehnoPress, Iaşi, 2007).

http://www.creative.com.

http:// www.mackichan.com.

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Figures (4)

Fig. 1.
Fig. 1.

Electric field intensity components on the transmission directions of the polarizers.

Fig. 2.
Fig. 2.

Ordinary and extraordinary rays in the anisotropic plate crossed perpendicular to the optical axis.

Fig. 3.
Fig. 3.

Simplified optical schema of a polarizing microscope with a webcam

Fig. 4.
Fig. 4.

Interference rings for some Carpathians quartz plates.

Tables (3)

Tables Icon

Table 1. Rings radii and corresponding sin of the incidence angle.

Tables Icon

Table 2. Refractive indices and the birefringence values for quartz crystal estimated with Eq. (14).

Tables Icon

Table 3. Main refractive indices and birefringence of Carpathian Quartz determined by various methods.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

E x = E P cos β ; E y = E P sin β
E x , A = E P · cos α · cos β ; E y , A = E P · sin α · sin β
ϕ A = ϕ P cos 2 α cos 2 β + sin 2 α sin 2 β + sin 2 α sin 2 β cos 2 cos 2 β cos Δ Ψ
ϕ A = ϕ P cos 2 ΔΨ 2
ΔΨ = ( 2 k + 1 ) π ; k = 0 , 1 , 2 , 3 , ; ϕ A = 0
ΔΨ = 2 k π ; k = 0 , 1 , 2 , 3 , ; ϕ A = ϕ P
( Δ ) = ( IA ) + ( AC ) ( IB )
( IA ) = n ef h cos r efk , ( IB ) = n o h cos⁡ r ok and ( AC ) = AB sin i k
1 · sin i k = n o sin r ok
1 · sin i k = n ef sin r efk
( Δ ) = h ( n ef cos r efk n o cos r ok ) = h ( n ef 2 sin 2 i k n o 2 sin 2 i k )
n ef = n e n o n e 2 + ( n o 2 n e 2 ) sin⁡ 2 r efk n ef 2 = n o 2 + n e 2 n o 2 n e 2 sin 2 i k
( Δ ) h = ( n o 2 n o 2 n e 2 sin 2 i k n o 2 sin 2 i k ) = k λ o 2 h ; k = 0 , 1 , 2 , 3 ,
{ n o 2 n o 2 n e 2 sin 2 i k n o 2 sin 2 i k = k λ o 2 h ( k = 1 , 2 , 3 , ) n o 2 n o 2 n e 2 sin 2 i m n o 2 sin 2 i m = m λ o 2 h ( m = 1 , 2 , 3 , ; m k )

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