Study on stress birefringence measurement of uni-axial crystal Xiaohong Wei, Liqun Chai, Qiang Li, Bo Gao

Chengdu Fine Optics and Engineering Research Center, Chengdu 610041, China

ABSTRACT

Residual stress birefringence in crystal will affect frequency conversion efficiency and beam quality. In this paper the distribution characteristics of inherent stress birefringence in crystal is analyzed, through delicate adjustment the optical axis is oriented and qualitative results obtained for KDP crystals are presented and discussed by imaging digital stress measurement instrument, and the stress gradient distribution is calculated, also the effect of deviation from optical axis on the measured stress distribution results is discussed. Keywords: Stress birefringence, KDP, stress birefringence gradient

1. INTRODUCTION

Optical glasses are optically isotropic in the relaxed condition, i.e. the refractive index is equally large in all spatial directions. However, mechanical stresses induced by material or production lead to deformations in the material structure and thus to different particle densities along axes. As the propagation velocity of light depends upon the density of the material, this sort of change in the microstructure leads to a direction-dependent change of the refractive index. The medium therefore becomes birefringent under stress, which is referred as stress birefringence (SBR). Apart from optically isotropic materials there are also many naturally occurring optically anisotropic materials, also known as birefringent materials, e.g. crystals such as calcite or KH2PO4 (KDP). Birefringence is a characteristic property of many transparent crystals depending sensitively upon the crystalline structure [1]. It is exhibited by twenty of the thirty-two crystal classes [2]. A measurement of this property is, therefore, a useful tool for studying these crystals[3]. Due to their interesting electrical and optical properties, structural phase transitions, and ease of crystallization, KDP has been the subject of a wide variety of investigations for over 40 years [4]. Today KDP is widely used to control the parameters of laser light such as pulse length, polarization, and frequency through the first- and the second-order electro-optic effects [5]. Efficient operation of electro-optic devices such as Pockels cells and frequency converters requires crystals with high degree of perfection. In particular, internal strains in the crystals can lead to beam depolarization and wavefront error [6,7], generate spatial variations in the refractive index tensor [8] through the stress-optic effect as well as inhomogeneity [9] in the unit normal to the c plane of the crystal. Exact determination of stress birefringence and its spatial distribution is therefore of great importance to the manufacture of optical materials and components. In this paper we present a polarization method which can measure the birefringence of the whole image area of a uni-axial crystal simultaneously. First we briefly review Jones vectors and matrices, then the experimental results are presented, and the accuracy of this method is discussed.

2. CHARACTERISTIC OF THE TRANSMITTED LIGHT

The most common method to measure the birefringence of a sample is simply to place both the sample under test and a quarter waveplate between crossed linear polarizers (one polarizer and one analyzer), and observe the change of the transmitted light while rotating the analyzer, as shown in Fig. 1.

polarizer sample 1/4 plate

Fig.1 Schematic diagram of stress birefringence measurement method.

analyzer detector

6th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Optical Test and Measurement Technology and Equipment, edited by Yudong Zhang, Libin Xiang, Sandy To,

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Since the light source used is polarized, first we briefly review the Jones vector and matrix. The input linearly polarized light written in the form of a Jones vector is

⎥⎦

⎤⎢⎣

⎡=

01

inputE (1)

A sample with a fast axis in the vertical direction is written in the form of Jones matrix as

⎥⎦

⎤⎢⎣

⎡=

)exp(001

0 δiGsample

(2) Where δ is the phase lead in y-axis. When δ < 0, the fast axis is in y-axis; when δ > 0, the fast axis is in x-axis. For a general case where the fast axis is rotated an angle θ with respect to horizontal, the Jones matrix can be written as

)2

exp(2cos

2tan12sin

2tan

2sin2

tan2cos2

tan1

2cos

cossinsincos

)exp(001

cossinsincos

0

δ

θδθδ

θδθδδ

θθθθ

δθθθθ

θθθ

iii

ii

iRGRG samplesample

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−

−−=

⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⋅⋅= −

(3) When θ=45°, the Jones matrix can be simplified as

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

12

tan2

tan1

2cos δ

δδ

i

iGsample

(4) Thus the transmitted light can be expressed as

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎦

⎤⎢⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−⋅⎥

⎦

⎤⎢⎣

⎡=⋅⋅=

2sin

2cos

01

12

tan2

tan1

2cos

001

δ

δ

δ

δδ

i

i

iEGGE inputsampleplateoutput

(5) From eq.(5) we know that the light transmitted through the quarter-wave plate is linear polarized and the angle between the light vector and the fast axis α equals δ/2, thus the phase difference or the phase retardation is given by

Btλ )π/(2/ == δα (6) where t = thickness, B = birefringence, and λ= wavelength. But different from normal optical material, the presence of an electric field and a strain field in KDP crystal leads to changes in the components of the refractive index tensor n. The combined linear electro-optic and stress-optic effect is described by

jijjiji pErn ε+=Δ )/1( 2

(7) where E is the electric field in the medium , ε is the strain tensor, and r and p are the electro-optic and elastic-optic tensors, respectively. In order to differentiate the elastic-optic effect induced stress from the electro-optic induced stress, the polarization direction of the light source should be oriented at the optical axis to avoid the double refraction of O (ordinary) and E (extraordinary)-ray.

3. EXPERIMENTAL RESULTS

We use the Ilis digital instrument to automatically measure the stress birefringence of KDP crystal. The temperature in the laboratory is 22℃, with 0.5℃/hr temperature vibration, the humidity is about 50%, the clear aperture is 108 mm × 108 mm, and the image resolution is 0.106mm/pixel. By carefully adjust the sample to make the polarization direction of the light as close to the optical axis as possible. Figure 2 and Fig. 3 show the stress distribution of a KDP type-I crystal and a switch crystal.

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Fig. 2 Stress birefringence distribution of a KDP type-I crystal. The sample size is 70×70×10mm3.

Fig. 3 Stress birefringence distribution of a KDP switch crystal. The sample size is 90×25×10mm3.

In order to see the stress variation rate, we define the stress gradient as below, and the stress gradient graphs are shown in Fig 4.

⎪⎪⎩

⎪⎪⎨

⎧

+=

∂∂

=∂

∂=

22 ),(),(),(

),(),(),(),(

yxyxyx

yyxSyx，

xyxSyx

yxxy

yx

σσσ

σσ

(8) where S(x,y) is stress with nm/cm as its unit, σx, σy are the stress gradient in the x and y direction, respectively, the unit is nm/cm2, and σxy is the overall stress gradient.

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(a)

(b)

(c)

Fig. 4 Stress gradient distributions: (a) σx, (b) σy, (c) σxy .

There are some other intrinsic problems in this method. One of them is the birefringent walk off, i.e., the difference in displacement between the E-ray and the O-ray can be large. Suppose the angle between the polarization direction and the optical axis is θ, the refractive index of E-ray can be calculated from Sellmeier equation:

2

2

2

2

2

cossin)(

1

oee nnnθθ

θ+=

(9) the refractive index difference between E-ray and the O-ray is

oe nnn −=Δ )(θ (10) The wave length of the light source is 592nm, let the sample thickness L=1cm, then the optical pass difference (OPD) Δ= Δn • L induced by double refraction of E-ray and the O-ray is shown in Fig. 4 and table 1.

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Fig. 5 OPD induced by double refraction of E-ray and O-ray.

4. CONCLUSIONS

Birefringence is a characteristic property of many transparent crystals depending sensitively upon the crystalline structure. A measurement of this property is, therefore, a useful tool for studying these crystals. In this paper by carefully adjustment the KDP crystal the inner stress birefringence and stress gradient distribution is obtained to better understand the stress characteristic of KDP crystal. Also the effect of deviation from optical axis on the measured stress distribution results is analyzed.

REFERENCES

1. M. Born and E. Wolf, Principles of Optics, Pergamon, New York, 1970. 2. J. F. Nye, Physical Properties of Crystals, Clarendon, Oxford, 1960. 3. D. J. Benard and W. C. Walker, Rev. Sci. Instrum. 47, 122 ,1976. 4. L. N. Rashkovich, KDP-Family Single Crystals, Adam Hilger, New York, 1991. 5. F. Zerike and J. E. Midwinter, Applied Non-Linear Optics, Wiley, New York,1973. 6. Keith B Doyle, Victor L Genberg, Gregory J Michels. Numerical methods to compute optical errors due to stress

birefringence [J]. Proc. SPIE(S0277-786X)，2002，34：4769-4778. 7. YIN Shao-hui，JIN Song，ZHU Ke-jun，et al. Stress Analysis of Compression Molding of Aspherical Glass

Lenses Using Finite Element Method [J]. Opto-Electronic Engineering，2010，37(10)：111-115. 8. SCHOTT Technical Information [Z]. TIE-27 Stress in Optical Glass，2004. 9. WANG Shen-lai，WANG Bo，ZHANG Guang-hui，et al. Study on important of the homogeneity of KDP crystals

by annealing [J]. High Power Laser and Particle Beams，2004，16(4)：437-440.

Table 1 Variation of Δn and Δnl induced by θ.

θ Δn Δn • L 0.05° 3×10

-8 0.3nm

0.1° 1×10-7 1nm

0.2° 5×10-7 5nm

0.5° 3×10-6 30nm

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