Profile of Radial Gradient Index Lens Measured by ... - IOPscience

Japanese Journal of Applied Physics Vol. 44, No. 2, 2005, pp. 1111–1114 #2005 The Japan Society of Applied Physics

Index Profile of Radial Gradient Index Lens Measured by Imaging Ellipsometric Technique Yu Faye CHAO and Kan Yan L EE
Department of Photonics, Institute of Electro-Optical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, R.O.C. (Received July 7, 2004; accepted October 29, 2004; published February 8, 2005)

A simple three-intensity imaging ellipsometric technique is proposed for measurement of the refractive index profile (RIP) of a radial gradient index (GRIN) lens. According to our numerical analysis, the improved ellipsometric parameters measured by this method are valid as long as the azimuth deviation of the polarizer is less than 3 . A BK7 glass and a radial GRIN lens were measured. A fairly flat refractive index surface profile was obtained for the glass within a 0.3% error. The on-axis refractive index of the radial GRIN lens and its RIP are well fitted to the specifications provided by the manufacturer. Instead of the refracted-ray method, we provide a reflective and nondestructive measurement technique for measurement of the gradient index of a flat surface. [DOI: 10.1143/JJAP.44.1111] KEYWORDS: ellipsometry, imaging, refractive index profile, gradient index lens

1.

Introduction

The graded index structure has been widely used in fiber communication, in technologies such as the graded index fiber, and the gradient-index (GRIN) lens. The knowledge of the refractive index profile (RIP) of this graded index structure is important not only to assess its performance in optical devices, but also to control the quality of products.1) Since it is very important to be able to measure the RIP of a gradient index material, we suggest a nondestructive measurement technique based on imaging ellipsometry, which enables measurement of the refractive index profile. A number of measurement techniques are used to measure the RIP of gradient index materials, such as scanning refracted ray,2–5) imaging,6,7) and the interference method.8,9) The basic concept of the first two methods is the use of raytracing technique, while the third method is based on interference theory. All of these methods are in transmission mode for determination of the RIP of GRIN media. In this paper, we present a simple imaging ellipsometric technique, which is carried out in a reflective and nondestructive setup for measuring the RIP of a gradient index material. A charged couple device (CCD) camera system is used in a Polarizer–Sample–Analyzer (PSA) ellipsometer10) to form an imaging ellipsometer. We employed this simple ellipsometer for the measurement of the RIP of the BK7 glass and obtained a fairly flat surface. This gives us sufficient accuracy for measuring the refractive index of a flat GRIN lens. The on-axis refractive index (n0 ) and the RIP of the GRIN lens measured by this imaging ellipsometer are well fitted to the specifications provided by the manufacturer. Instead of using the refracted-ray method, we provide a reflective and nondestructive measurement technique for the measurement of the gradient index of a flat surface. 2.

Theory

The basic PSA imaging ellipsometer is constructed as shown in Fig. 1. The ellipsometric parameters
and  are defined as tan
ei ¼ rp =rs ;

ð1Þ

where rp and rs are the reflection coefficients in the planes


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Fig. 1. Schematic setup of the PSA imaging ellipsometer: Laser HeNe laser; P, polarizer; A, analyzer; S, sample; Expander, beam expander and CCD camera. The elliptically distributed intensity on the right is under various azimuth angles of A in the polar coordinates.

parallel (p) and perpendicular (s) to the incident plane, respectively. The measured intensity can be written as IðP; AÞ ¼ I0 ½sin2 P sin2 A þ tan2
cos2 P cos2 A þ 0:5 tan
cos  sin 2P sin 2A;

ð2Þ

where the azimuths P and A are the transmission axes of the polarizer and analyzer, respectively. When P ¼ 45 , the intensity measured at the detector is expressed simply as IðAÞ ¼ 0:5I0 ½sin2 A þ tan2
cos2 A þ tan
cos  sin 2A:

ð3Þ

The intensity distribution of the reflected light is in an elliptical form,11) such as shown on the right-hand side of Fig. 1, which can be expressed as IðAÞ ¼

L T cos2 ðA 
Þ þ sin2 ðA 
Þ: 2 2

ð4Þ

By comparing eqs. (3) and (4), one can easily prove the following: 1  R cos 2
tan2
¼ ; 1 þ R cos 2
ð5Þ tan 2
¼  cos  tan 2
: where R ¼ ðL  TÞ=ðL þ TÞ. Since there are only three unknowns, namely, L, T, and
, we deduce them by means of the three-intensity technique10) instead of using Fourier

1111

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Y. F. C HAO and K. Y. LEE

analysis. The three intensities are measured at A ¼ 0 , 60 , and 120 ; then the unknown parameters can be obtained through the following relations: pffiffiffi 3½Ið60Þ  Ið120Þ ; tan 2
¼ 2Ið0Þ  Ið60Þ  Ið120Þ ð6Þ 2Ið0Þ  Ið60Þ  Ið120Þ R cos 2
¼ : Ið0Þ þ Ið60Þ þ Ið120Þ Since the elliptical properties can be converted to the ellipsometric parameters
and , one can measure the refractive index through the ellipsometric ratio 12)  ¼ tan
ei

ð7Þ

and  n1 ¼ n0 tan  1 

4 sin2  ð1 þ Þ2

1 2

;

ð8Þ

where n1 and n0 are the refractive indices of the sample and air, respectively, while  is the angle of incidence. The refractive index can be calculated from the measured
and  by substituting them into eq. (8). Since a precise alignment is almost impossible to achieve, we analyze the deviation and improve the measurement accuracy on the basis of the assumption that the azimuth position of the polarizer deviates from the reference zero by , and the ellipsometric parameters
and  in eq. (5) are modified to be ½1  sinð2Þ½1  R cosð2
Þ tan
¼ ; ½1 þ sinð2Þ½1 þ R cosð2
Þ 2

ð9aÞ

let f ðÞ ¼

½1  sinð2Þ ½1 þ sinð2Þ

then 

½2 tan
cos  cos 2 ¼ tan 2
1  tan2
f ðÞ

ð9bÞ

for P ¼ 45 þ ; and tan2
¼

½1 þ sinð2Þ½1 þ R0 cosð2
0 Þ ; ½1  sinð2Þ½1  R0 cosð2
0 Þ

ð10aÞ

since f 0 ðÞ ¼ f 1 ðÞ; then ½2 tan
cos  cos 2 ¼ tan 2
0 1  tan2
f 0 ðÞ

ð10bÞ

for P ¼ 45 þ , where R0 and
0 are the parameters corresponding to eq. (9). By obtaining the product of eqs. (9a) and (10a), we can optimize the value of
to be independent of ;  can also be obtained by obtaining the ratio of eqs. (9a) and (10a). The accuracy of  can be improved by obtaining
ave ¼ ð180 
0 þ
Þ=2. To clarify the improved value of  with respect to the deviation of , we employ the two following methods of numerical analysis: (a) use
ave and the optimized value of
in eq. (5), and (b) take the average of ; while the value of
is measured under P ¼ 45 and 45 , respectively. Upon comparing the numerical results obtained from these two methods, as shown in Fig. 2, we can conclude that as long as  is less

Fig. 2. Numerical simulation for analyzing the system error of  as a function of the deviation of the polarizer, , under two improvement techniques, where  ¼ 124 . [
P ¼ 45 þ ,  P ¼ 45 þ , solid line: method (a) and dotted line: method (b)].

than 3 , the system errors of
and  will be well below the random error of the imaging ellipsometer. Since the optimizing process does improve the ellipsometric measurement accuracy,10) we can utilize this technique in imaging ellipsometry in order to have a sufficient accuracy to obtain the refractive index of a gradient medium. 3.

Experimental Procedures

The azimuth angles of the polarizer and analyzer were well aligned via the intensity ratio technique through two incident angles by following the procedures reported in ref. 13. After the alignment, the light beam (Melles Griot 05LPL-903-065,  ¼ 543:5 nm HeNe laser) was set at the incident angle of 70 then passed through a polarizer whose azimuth was set at 45 and 45 . The analyzer (A) was mounted on a motor controlled rotator and three radiances were obtained at the azimuths of the analyzer of 0 , 60 and 120 . The experimental setup is depicted in Fig. 1. The beam expander is composed of a microscope object lens and a convex lens, and it expands the beam by a factor of 10. Instead of using the conventional photodiode detector, we used a CCD camera (Starlight MX516) to record the intensities. It contains 512  290 pixels with a 16 bit grayscale format (i.e., 65500 grayscale) for each pixel whose size is 9:8  12:6 mm2 . All intensities were measured using the CCD camera and stored in a PC for calculation of the ellipsometric parameters. The Gaussian smoother filter was used in every image to suppress stray light. The parasitic error of beam deviation in rotating element ellipsometry is a well known problem. The measurements must be improved by combining the measurements taken under P ¼ 45 and 45 . The beam deviation is clearly apparent in this magnified PSA ellipsometry, and we corrected the parasitic error caused by a nonuniform dirt spot on a glass by matching the center of the contours of  under P ¼ 45 and 45 (Appendix). For a comparison of the surface profile, we measured the RIP of a BK7 wedge glass and a radialGRIN lens. The radial-GRIN lens, fabricated from an oxide glass by an ion-exchange method, was fabricated by the Nippon Sheet Glass (NSG) Corporation with the trade name SLW-180, and had the diameter of 1.8 mm.

Jpn. J. Appl. Phys., Vol. 44, No. 2 (2005)

Fig. 3. Imaging ellipsometric study of a BK7 glass: Upper: method (a) n ¼ 1:519  0:003; lower: method (b) n ¼ 1:520  0:003.

4.

Y. F. CHAO and K. Y. LEE

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Fig. 4. Imaging ellipsometric study of a flat GRIN lens: (a) Azimuth deviation () of the surface. (b) Measured refractive index profile (: x-axis, : y-axis, solid line: RIP provided by NSG).

Results

The values of
,  and  were deduced by combining eqs. (5), (6), (9) and (10) and using the three intensity technique. The refractive index of the medium can be obtained by substituting the measured
and  into eq. (8). In addition to using the optimized value of
, we also used the value of  obtained from the improved methods (a) and (b) to deduce the RIP of a BK7 glass. Both the RIPs of this glass are fairly flat, and their refractive indexes are 1.519 (0:003) and 1.520 (0:003). These results agree well with the refractive index provided by the manufacturer (Schott: optical glass, BK7, n ¼ 1:519), as shown in Figs. 3(a) and 3(b). This 103 accuracy of refractive index is sufficient for the measurement of the RIP of the gradient index media. The following specifications of the SLW-180 GRIN lens are provided by NSG: the on-axis refractive index (n0 ) is 1.614 and the gradient constant is 0.344 for  ¼ 550 nm. Both values are comparable to our measured results, which are 1.614 and 0.345, respectively. The most interesting aspect of this ellipsometric technique is that it can deduce the azimuth deviation () of the polarizer with respect to the surface, which can indicate whether  is less than 3 , because this is the largest angle for this technique to be valid, as shown in Fig. 4(a). Since the deviated azimuth is within our optimization condition, the RIP of the GRIN lens is well fitted to the expected profile, as shown in Fig. 4(b). It is known16) that any defocusing or tilting of the lens can cause polarization aberration, which may explain the higher degree of error at the outskirts. By assuming the RIP has a radial distribution, the maximum error of the measured RIP

Fig. 5. Refractive index contour of the GRIN lens: Solid line: RIP provided by NSG, dotted line: RIP measured by the imaging ellipsometric technique.

(0:004) is larger than the standard deviation measured in the BK7 glass, and Fig. 5 depicts the contour of the measured RIP. The asymmetry of this contour profile may be caused by a defect in the GRIN lens or a polarization aberration in the measurement system (i.e., beam expander). To study this problem further, we intend to install a higher magnification beam expander and a more stable light source.

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These measures are expected to improve the spatial resolution, and hence the refractive indices. 5.

Conclusions

This work suggests a photometric imaging ellipsometric technique with a minimum amount of measurements to obtain sufficient accurate ellipsometric parameters, and thus reduce the time needed for analyzing the RIP of an inhomogeneous medium such as a graded index fiber. Instead of using the refracted-ray method, which normally needs matching oil, we provide a reflective method, which is a nondestructive measurement technique, to measure the gradient index of a flat surface. Since deviation of the polarizer can indicate the surface structure, we can distinguish the morphology and homogeneity of the medium by means of this imaging ellipsometric technique. Acknowledgments This work is supported by the National Science Council (NSC) of the Republic of China under grant NSC92-2215E009-055. 1) W. J. Tomlinson: Appl. Opt. 19 (1980) 1127. 2) M. Young: Appl. Opt. 20 (1981) 3415. 3) R. Conde, C. Depeursinge, B. Gisin, N. Gisin and B. Groebli: Pure Appl. Opt. 5 (1996) 269. 4) N. Gisin, R. Passy and B. Perny: J. Lightwave Tech. 11 (1993) 1875. 5) N. H. Fontaine and M. Young: Appl. Opt. 38 (1999) 6836. 6) X. H. Sun, H. Ma, H. Ming, Z. Q. Zheng, J. W. Yang, Y. S. Zhang and J. P. Xie: Chin. Phys. Lett. 20 (2003) 374. 7) X. H. Sun, H. Ma, H. Ming, Z. Q. Zheng, J. W. Yang and J. P. Xie: Opt. Laser Tech. 36 (2004) 163. 8) P. W. Oliveira, H. Krug and H. Schmidt: Proc. SPIE 3136 (1997) 442. 9) Z. G. Liu, X. M. Dong, Q. G. Chen, C. Y. Yin, Y. Xu and Y. J. Zheng: Appl. Opt. 43 (2004) 1485. 10) Y. F. Chao, W. C. Lee, C. S. Hung and J. J. Lin: J. Phys. D 31 (1998) 1968. 11) E. Meyer, H. Frede and H. Knof: J. Appl. Phys. 38 (1967) 3682. 12) R. M. A. Azzam and N. M. Bashara: Ellipsometry and Polarized Light. (North-Holland, Amsterdam, 1992) Chap. 4, p. 274. 13) Y. F. Chao: U.S.A. Patent 5706088 (1998). 14) J. R. Zeidler, R. B. Kohles and N. M. Bashara: Appl. Opt. 13 (1974) 1938. 15) Y. F. Chao, M. W. Wang and Z. C. Ko: J. Phys. D 32 (1999) 2246. 16) R. A. Chipman and L. J. Chipman: Opt. Eng. 28 (1989) 100.

Appendix: A Dirt Spot on Glass Is Used to Correct the Beam Deviation in the PSA Imaging Ellipsometric Technique The parasitic error of beam deviation in the rotating element ellipsometry is a well known problem.14) Since we have to improve the ellipsometric measurements by using P

Fig. 6. Contour structure of  of a dirt spot on the glass: (a) P ¼ 45 ; and (b) P ¼ 45 .

at 45 and 45 , the beam deviation magnification by the beam expander in this imaging ellipsometric technique becomes a crucial problem that must be solved. We choose a nonuniform dirt spot on the wedge glass to correct this defect. Because the variation of  to the refractive index is more sensitive than
, we graphed the contours of  under P ¼ 45 and 45 , respectively, as shown in Fig. 6. The centers of these two distributions were clearly shifted from each other, which means that their incident angles were deviated by 0.04 ; this result is close to what has been measured previously.15) By matching the centers of these two distributions, we can correct the beam deviation problem in the imaging ellipsometric technique.