Abstract

A linearly/circularly polarized heterodyne light beam coming from a heterodyne light source with an electro-optic modulator in turn enters a modified Twyman–Green interferometer to measure the surface plane of a GRIN lens. Two groups of periodic sinusoidal segments recorded by a fast complementary metal-oxide semiconductor camera are modified, and their associated phases are derived with the unique technique. The data are substituted into the special equations derived from the Fresnel equations, and the refractive index can be obtained. When the processes are applied to other pixels, the full-field refractive-index distribution can be obtained similarly. Its validity is demonstrated.

© 2010 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  6. Y. F. Chao and K. Y. Lee, “Index profile of radial gradient index lens measured by imaging ellipsometric technique,” Jpn. J. Appl. Phys. 44, 1111–1114 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  14. IEEE, “Standard for terminology and test methods for analog-to-digital converters,” IEEE Std. 1241-2000 (IEEE, 2000), pp. 25–29.
  15. Q. Dou, H. Ma, G. Jia, Z. Chen, K. Cao, and T. Zhang, “Study on measurement of linear electro-optic coefficient of a minute irregular octahedron cBN wafer,” Opt. Laser Technol. 39, 647–651 (2007).
    [CrossRef]

2009 (3)

2008 (3)

2007 (1)

Q. Dou, H. Ma, G. Jia, Z. Chen, K. Cao, and T. Zhang, “Study on measurement of linear electro-optic coefficient of a minute irregular octahedron cBN wafer,” Opt. Laser Technol. 39, 647–651 (2007).
[CrossRef]

2006 (2)

Y. Youk and D. Y. Kim, “A simple reflection-type two-dimensional refractive index profile measurement technique for optical waveguides,” Opt. Commun. 262, 206–210(2006).
[CrossRef]

D. Vazquez, E. Acosta, G. Smith, and L. Garner, “Tomographic method for measurement of the gradient refractive index of the crystalline lens. II the rotationally symmetrical lens,” J. Opt. Soc. Am. A 23, 2551–2565 (2006).
[CrossRef]

2005 (1)

Y. F. Chao and K. Y. Lee, “Index profile of radial gradient index lens measured by imaging ellipsometric technique,” Jpn. J. Appl. Phys. 44, 1111–1114 (2005).
[CrossRef]

2004 (1)

2001 (1)

M. Ray, S. K. Sarkar, A. Basuray, and N. SoodBiswas, “Measurement of refractive index profile of GRIN glasses,” Proc. SPIE 4417, 483–488 (2001).
[CrossRef]

Acosta, E.

Basuray, A.

M. Ray, S. K. Sarkar, A. Basuray, and N. SoodBiswas, “Measurement of refractive index profile of GRIN glasses,” Proc. SPIE 4417, 483–488 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 42–43.

Cai, L.

Cao, K.

Q. Dou, H. Ma, G. Jia, Z. Chen, K. Cao, and T. Zhang, “Study on measurement of linear electro-optic coefficient of a minute irregular octahedron cBN wafer,” Opt. Laser Technol. 39, 647–651 (2007).
[CrossRef]

Chao, Y. F.

Y. F. Chao and K. Y. Lee, “Index profile of radial gradient index lens measured by imaging ellipsometric technique,” Jpn. J. Appl. Phys. 44, 1111–1114 (2005).
[CrossRef]

Chen, Q.

Chen, Y. L.

H. C. Hsieh, Y. L. Chen, W. T. Wu, and D. C. Su, “Method for measuring the refractive index distribution of a GRIN lens with heterodyne interferometry,” Proc. SPIE 7390, 73900G(2009).
[CrossRef]

Y. L. Chen and D. C. Su, “A method for determining full-field absolute phases in the common-path heterodyne interferometer with an electro-optic modulator,” Appl. Opt. 47, 6518–6523 (2008).
[CrossRef] [PubMed]

Chen, Z.

Q. Dou, H. Ma, G. Jia, Z. Chen, K. Cao, and T. Zhang, “Study on measurement of linear electro-optic coefficient of a minute irregular octahedron cBN wafer,” Opt. Laser Technol. 39, 647–651 (2007).
[CrossRef]

Cobb, M. J.

Dong, X.

Dou, Q.

Q. Dou, H. Ma, G. Jia, Z. Chen, K. Cao, and T. Zhang, “Study on measurement of linear electro-optic coefficient of a minute irregular octahedron cBN wafer,” Opt. Laser Technol. 39, 647–651 (2007).
[CrossRef]

Dragomir, N. M.

N. M. Dragomir, X. M. Goh, and A. Roberts, “Three-dimensional refractive index reconstruction with quantitative phase tomography,” Microsc. Res. Tech. 71, 5–10(2008).
[CrossRef]

Garner, L.

Goh, X. M.

N. M. Dragomir, X. M. Goh, and A. Roberts, “Three-dimensional refractive index reconstruction with quantitative phase tomography,” Microsc. Res. Tech. 71, 5–10(2008).
[CrossRef]

Hsieh, H. C.

H. C. Hsieh, Y. L. Chen, W. T. Wu, and D. C. Su, “Method for measuring the refractive index distribution of a GRIN lens with heterodyne interferometry,” Proc. SPIE 7390, 73900G(2009).
[CrossRef]

Jia, G.

Q. Dou, H. Ma, G. Jia, Z. Chen, K. Cao, and T. Zhang, “Study on measurement of linear electro-optic coefficient of a minute irregular octahedron cBN wafer,” Opt. Laser Technol. 39, 647–651 (2007).
[CrossRef]

Kim, D. Y.

Y. Youk and D. Y. Kim, “A simple reflection-type two-dimensional refractive index profile measurement technique for optical waveguides,” Opt. Commun. 262, 206–210(2006).
[CrossRef]

Lee, K. Y.

Y. F. Chao and K. Y. Lee, “Index profile of radial gradient index lens measured by imaging ellipsometric technique,” Jpn. J. Appl. Phys. 44, 1111–1114 (2005).
[CrossRef]

Li, T.

Li, X.

Liu, Z.

Ma, H.

Q. Dou, H. Ma, G. Jia, Z. Chen, K. Cao, and T. Zhang, “Study on measurement of linear electro-optic coefficient of a minute irregular octahedron cBN wafer,” Opt. Laser Technol. 39, 647–651 (2007).
[CrossRef]

Oven, R.

Ray, M.

M. Ray, S. K. Sarkar, A. Basuray, and N. SoodBiswas, “Measurement of refractive index profile of GRIN glasses,” Proc. SPIE 4417, 483–488 (2001).
[CrossRef]

Roberts, A.

N. M. Dragomir, X. M. Goh, and A. Roberts, “Three-dimensional refractive index reconstruction with quantitative phase tomography,” Microsc. Res. Tech. 71, 5–10(2008).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007), pp. 203–208.

Sarkar, S. K.

M. Ray, S. K. Sarkar, A. Basuray, and N. SoodBiswas, “Measurement of refractive index profile of GRIN glasses,” Proc. SPIE 4417, 483–488 (2001).
[CrossRef]

Smith, G.

SoodBiswas, N.

M. Ray, S. K. Sarkar, A. Basuray, and N. SoodBiswas, “Measurement of refractive index profile of GRIN glasses,” Proc. SPIE 4417, 483–488 (2001).
[CrossRef]

Su, D. C.

H. C. Hsieh, Y. L. Chen, W. T. Wu, and D. C. Su, “Method for measuring the refractive index distribution of a GRIN lens with heterodyne interferometry,” Proc. SPIE 7390, 73900G(2009).
[CrossRef]

Y. L. Chen and D. C. Su, “A method for determining full-field absolute phases in the common-path heterodyne interferometer with an electro-optic modulator,” Appl. Opt. 47, 6518–6523 (2008).
[CrossRef] [PubMed]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007), pp. 203–208.

Vazquez, D.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 42–43.

Wu, W. T.

H. C. Hsieh, Y. L. Chen, W. T. Wu, and D. C. Su, “Method for measuring the refractive index distribution of a GRIN lens with heterodyne interferometry,” Proc. SPIE 7390, 73900G(2009).
[CrossRef]

Wu, Y.

Xi, J.

Xu, Y.

Ye, P.

Yin, C.

Youk, Y.

Y. Youk and D. Y. Kim, “A simple reflection-type two-dimensional refractive index profile measurement technique for optical waveguides,” Opt. Commun. 262, 206–210(2006).
[CrossRef]

Zhang, M.

Zhang, T.

Q. Dou, H. Ma, G. Jia, Z. Chen, K. Cao, and T. Zhang, “Study on measurement of linear electro-optic coefficient of a minute irregular octahedron cBN wafer,” Opt. Laser Technol. 39, 647–651 (2007).
[CrossRef]

Zheng, Y.

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

Y. F. Chao and K. Y. Lee, “Index profile of radial gradient index lens measured by imaging ellipsometric technique,” Jpn. J. Appl. Phys. 44, 1111–1114 (2005).
[CrossRef]

Microsc. Res. Tech. (1)

N. M. Dragomir, X. M. Goh, and A. Roberts, “Three-dimensional refractive index reconstruction with quantitative phase tomography,” Microsc. Res. Tech. 71, 5–10(2008).
[CrossRef]

Opt. Commun. (1)

Y. Youk and D. Y. Kim, “A simple reflection-type two-dimensional refractive index profile measurement technique for optical waveguides,” Opt. Commun. 262, 206–210(2006).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

Q. Dou, H. Ma, G. Jia, Z. Chen, K. Cao, and T. Zhang, “Study on measurement of linear electro-optic coefficient of a minute irregular octahedron cBN wafer,” Opt. Laser Technol. 39, 647–651 (2007).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

M. Ray, S. K. Sarkar, A. Basuray, and N. SoodBiswas, “Measurement of refractive index profile of GRIN glasses,” Proc. SPIE 4417, 483–488 (2001).
[CrossRef]

H. C. Hsieh, Y. L. Chen, W. T. Wu, and D. C. Su, “Method for measuring the refractive index distribution of a GRIN lens with heterodyne interferometry,” Proc. SPIE 7390, 73900G(2009).
[CrossRef]

Other (3)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007), pp. 203–208.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 42–43.

IEEE, “Standard for terminology and test methods for analog-to-digital converters,” IEEE Std. 1241-2000 (IEEE, 2000), pp. 25–29.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the modified Twyman–Green interferometer.

Fig. 2
Fig. 2

Configuration of the heterodyne light source.

Fig. 3
Fig. 3

(a) Sampled interference signal as V < V π and (b) corresponding modified interference signal with lengthened period.

Fig. 4
Fig. 4

Refractive-index contour of the GRIN lens with this method.

Fig. 5
Fig. 5

Refractive-index contour of the same GRIN lens with our previous method.

Equations (17)

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E 1 = 1 2 ( e i π f t e i π f t ) .
E r 1 = A N ( 0 ° ) · Q 2 ( 45 ° ) · M · Q 2 ( 45 ° ) · R B S · E 1 · e i ϕ d 1 = ( 1 0 0 0 ) 1 2 ( 1 i i 1 ) ( r m 0 0 r m ) 1 2 ( 1 i i 1 ) ( e i ϕ r / 2 0 0 e i ϕ r / 2 ) 1 2 ( e i π f t e i π f t ) e i ϕ d 1 = i r m e i ( π f t ϕ d 1 ϕ r / 2 ) 2 ( 1 0 ) ,
E t 1 = A N ( 0 ° ) · R B S · G · E 1 · e i ϕ d 2 = ( 1 0 0 0 ) ( e i ϕ r / 2 0 0 e i ϕ r / 2 ) ( r 0 0 r ) 1 2 ( e i π f t e i π f t ) e i ϕ d 2 = r e i ( π f t + ϕ d 2 ϕ r / 2 ) 2 ( 1 0 ) .
I A = | E r 1 + E t 1 | 2 = I 01 + γ 1 · cos ( 2 π f t + ϕ 1 ) = 1 2 { r 2 + r m 2 2 r r m cos [ 2 π f t + π 2 ( ϕ d 1 ϕ d 2 + ϕ r ) ] } ,
ϕ 1 = π 2 ( ϕ d 1 ϕ d 2 + ϕ r ) .
E r 2 = ( A N ( 0 ° ) · Q 2 ( 45 ° ) · M · Q 2 ( 45 ° ) · R B S · Q 1 ( 45 ° ) · E 1 ) · e i ϕ d 1 = e i 2 π f t + i 2 r m e i ( ϕ d 1 π f t + ϕ r / 2 ) ( 1 0 ) ,
E t 2 = ( A N ( 0 ° ) · R B S · G · Q 1 ( 45 ° ) · E 1 ) · e i ϕ d 2 = e i 2 π f t i 2 r e i ( ϕ d 2 π f t ϕ r / 2 ) ( 1 0 ) ,
I B = | E r 2 + E t 2 | 2 = I 02 + γ 2 · cos ( 2 π f t + ϕ 2 ) = A · cos ( 2 π f t ) + B · sin ( 2 π f t ) + C ,
A = r r m sin ( ϕ d 1 ϕ d 2 + ϕ r ) and B = 1 2 ( r m 2 r 2 ) .
ϕ 2 = tan 1 ( B A ) = cot 1 [ 2 r r m sin ( ϕ d 1 ϕ d 2 + ϕ r ) ( r m 2 r 2 ) ] .
r = cos ϕ 1 + cos 2 ϕ 1 + cot 2 ϕ 2 cot ϕ 2 r m ,
r = n 1 n + 1 .
n = cot ϕ 2 r m cos ϕ 1 + r m cos 2 ϕ 1 + cot 2 ϕ 2 cot ϕ 2 + r m cos ϕ 1 r m cos 2 ϕ 1 + cot 2 ϕ 2 .
I c ( t k ) = I 0 [ 1 + γ cos ( 2 π V V π T t k + ψ ) ] .
I ( t k ) = A 0 · cos ( 2 π f t k ) + B 0 · sin ( 2 π f t k ) + C 0 = A 0 2 + B 0 2 · cos ( 2 π f t k + ψ ) + C 0 ,
ψ = tan 1 ( B 0 A 0 ) .
Δ n = | ( n ϕ 1 ) · Δ ϕ 1 | + | ( n ϕ 2 ) · Δ ϕ 2 | = r m | sin ϕ 1 sin 2 ϕ 2 Δ ϕ 1 + 2 cos ϕ 1 Δ ϕ 2 | | cos 2 ϕ 1 + cot 2 ϕ 2 cos ϕ 1 | cos 2 ϕ 1 + cot 2 ϕ 2 [ cos ϕ 2 + r m sin ϕ 2 ( cos ϕ 1 cos 2 ϕ 1 + cot 2 ϕ 2 ) ] 2 .

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