Interlerometric analysis of stress-induced birefringence in a rotating glass disk

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1 Copyright 1998 Society of Photo-Optical Instrumentation Engineers. This paper was published in Proceedings of SPIE and is made available as an electronic reprint with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. Interlerometric analysis of stress-induced birefringence in a rotating glass disk Peter de Groot,* Ara Dergevorkian and Tod Erickson zygo, Laurel Brook Road, Middlefield, CT USA Centnpetal ABSTRACT forces in a rotating glass disk generate mechanical strains that are visible in polarized light. Dynamic stress birefringence in rotating disks has a practical effect on several classes of optical instruments, including flying-height testers for the data storage industry. We provide a model for stress-induced polarization effects, and describe an interferometric technique for mapping birefringence in a 100-mm diameter disk spinning at 12,000 rpm. The polarization interferometer employs a laser diode and a homodyne phase receiver to detect polarization-dependent phase shifts as small as I mrad at a data rate of 250 khz. 1. INTRODUCTION Birefringence in an amorphous glass material such as BK7 is normally on the order of 4 nm/cm or less. However, when subjected to mechanical stress, ordinary glass can exhibit much larger amounts of birefringence. An interesting example is a glass disk in rapid rotation. The centnpetal forces holding together a rotating disk engender a circularly symmetric stress pattern that changes the local material index. The resultant centripetal birefringence has tangential, radial and axial components that vary with radius and spin speed. A rapidly rotating glass disk is a central component in optical flying height test equipment for data storage read-write sliders. Flying height testers employ a glass disk in place of the actual magnetic media disk found in rigid disk drives, and thin-film interference provides information about the orientation and flying height of the slider. One way to measure flying height is by highspeed analysis of the intensity and polarization state of the reflected light.12 Polarization analysis is particularly sensitive to small changes in height between contact and 50 nm, where most modem sliders fly today. However, at the high rotation speeds typical of modern disk drives, it is important to include the effects of centripetal birefnngence in the measurement theory. Because of the importance of centnpetal birefnngence in flying height testing, we have studied the phenomenon in considerable detail. The complete theory may be found in the May 1998 edition of JOSA.3 The present paper compares this theory with recent experimental work, including a direct comparison of calculated index variations with measured values. Address correspondence to peterdzvgo.com or visit our website at Part of the SPIE Conference on Laser Interferometry IX: Techniques and Analysis. San Dieqo. California July 1998 SPIE Vol X/98/$1O.OO 227

2 2. POLARIZATION INTERFEROMETER Figure 1 shows the measuring instrument and geometry for our experiments with centnpetal birefringence. This is the same instrument that we use in a commercial flying height test product.4 A polarized 680-nm, 1OOmW multimode laser diode provides the source illumination via a single-mode fiber. The measurement beam passes through the glass at an internal incidence angle q$ with respect to the disk axis and at a skew angle with respect to a tangent line. After reflection from the underside of the disk, the beam passes back through the glass to an interferometnc receiver. Laser Interferometric Receiver Photodetectors Axis Wollaston Disk waveplate 2 4 Wollaston Fig. 1: Birefringence measurements on a rapidly rotating glass disk. Fig. 2: Detail of interferometric receiver shown in Fig.1. Figure 2 shows the interferometnc receiver in greater detail.5 The design is similar to several known types of homodyne receiver that employ a quarter-wave plate and Wollaston prisms to generate four signals in quadrature. Our receiver is unique in that the plate beam splitter is deliberately contrived to be partially polarizing. The polarization preference of the beam splitter makes it possible to simultaneously measure the intensity and relative phase of two orthogonal polarization states. A calibration procedure compensates for imperfections in the polarization components, resulting in an absolute phase-measurement accuracy of mrad. The receiver operates at 250 khz and has a phase noise of less than I mrad at this bandwidth. 3. MEASUREMENT THEORY When the disk is in rotation, the refractive index changes by small amounts (1O) according to the polarization of the light traversing the disk. This birefringence may be described by two relative indices with respect to the nominal index of the glass. The relative index En is for 228

3 polarization components within the plane of the disk, and Lfl is for light polarized parallel to the disk axis. The radial and tangential in-plane relative indices are defined as An. A detailed analysis of the effect of centnpetal birefringence shows the phase measured by the apparatus in Fig.2 is where 9,r+-aU (1.) a = kln0 sin(2)cos(b). (2.) =2kLn1 sin(ø) 2kLn [1 + cos(ø)j cos(2c) (3.). r2 2 U= (4.) rr The reflectivities r5, for bare glass are calculated from the Fresnel formula for the S and p polarization states, where p means parallel to the plane of incidence. The distance L is the thickness of the glass divided by the cosine of the angle 0, and the angular wavenumber k is 2z divided by the wavelength of the source light. For the purpose of analyzing disk birefringence, we acquire phase data at two different angles and solve for the two relative indices An0 and An±. Defining for positive and negative values of the same absolute value of we find that a = (5.) = f(e(-) + e())+ 71. (6.) Inverting Eqs.(2) and (3), 229

4 = klsin(2c)cos(q) (7) t + 2kLn [1 + cos(qj cos(2ç) 2kL sin(ø)2 (8.) For these measurements, it is best to choose values of C and 0 such that the denominators in Eqs.(7) and (8) are far from zero. 4. EXPERIMENT 1.6 ci) 1.4 C U) 0 0:: Ellipsometric phase 9(deg) 0.6 U) 1).5 C U) V CU a:: Angle Fig.3: Comparison of theory (upper plot) with experiment (lower plot) for the ellipsometric phase angle 9 in degrees as a function of skew angle and radial position. The internal incident angle 0 is 149. (deg) In a first experiment, we scanned a wide range of radii and angles to verify Eq.(1). For the theoretical calculations, the values of a and ci where calculated directly from stress equations for rotating disks6 and the stress-optic coefficients for BK7 glass. The data plotted in Fig.3 for 9 krpm show that the dependence of the measured phase 9 on radius and angle is reasonably well predicted by theory, although there are differences of as much as several 230

5 degrees of phase at some locations. These differences appear to be related to disk clampling stress together with gradual thermal changes in the disk. In a second experiment, we calculated the index variations An and An1 using Eqs.(5)(8) and data acquired at 12 krpm and = We then compared these results with theoretical values for birefringence using stress equations (Fig.4). These data show a satisfying agreement between experiment and theory, particularly with regard to the predominance of axial birefringence An1. E C) E C >< w0 z w I I I An - X RADIUS (mm) Fig. 4: Comparison of theory (solid lines) with experimental determination of the relative index changes (points). These data acquired at 12 krpm and 5. APPLICATIONS For us, the principal motivation for measuring centripetal birefnngence interlerometrically is to improve the accuracy of a polarizationbased flying height tester. Briefly, the flying height test principle involves a phase and intensity measurement of light reflected from a slider flying on the underside of the disk shown in Fig.1. The interference between the slider and the disk modifies the complex reflectivities of the s and p polarization components according to well-known thin film equations.7 The interferometric receiver (Fig.2) measures the relative phase 0 of the polarization components as well as the total intensity. A least-squares calculation infers the flying height from these data. Detailed analysis3 shows that our flying height tester will measure a phase 0 that differs from the zero-birefringence value by an amount 231

6 (I -I L9 = c a I Icos(9), (9.) IsJp) where are the intensities of the s and p polarization components of the reflected light. The and a values in Eq.(9) are the same ones that can be calculated by interferometric analysis of the bare glass disk and Eqs.(5)-(8). Consequently, our strategy for improving flying-height measurement accuracy in the presence of birefringence is to measure the c and a values just prior to loading the slider. The measurement involves a rapid measurement of the 0 value at two skew angles where is the angle at which the flying height is measured. This direct birefringence compensation algorithm works for all sources of circularly symmetric birefringence, including uniform clamping stress and slow heating and cooling of the disk. Other possible applications of the technique described in this paper include analysis of magneto-optic (MO) data storage systems. The injection molding process for plastic MO disks results in residual stress birefringence with a circular symmetry.8 Because the physical principle of Kerr-effect MO storage involves a small differential change in polarization, disk birefringence is an important issue.91 Quality control and R&D tools for MO disks employ an ellipsometnc geometry that is similar, but not identical, to that shown in Fig.1 h FURTHER WORK The equations presented in this paper are first-order approximations derived from the more complete theory put forth in ref.[3]. As rotation speeds for hard drives increase, it will prove useful to improve the algorithms by introducing higher-order terms. An additional improvement in the theory would be to take into account the variation in the angle as the measurement beam traverses the disk. This also introduces a higher-order term that may be important at very high speeds. REFERENCES AND NOTES 1 p de Groot, L. Deck, J. Soobitsky, J. Biegen, "Polarization interferometer for measuring the flying height of magnetic read-write heads," Opt. Left. j.(6), (1996). 2 US Patent No. 5,557,399. Additional US and Foreign Patents pending. P. de Groot "Birefringence in rapidly-rotating glass disks," J. Opt. Soc. Am. 15(5), (1998). See also US Patent No. 5,644,562. Pegasus 2OOO Flying Height Tester, manufactured by Zygo Corporation. US Patent No. 5,663,793. 6W. C. Young, "Roark's formulas for stress and strain" 6th Edition (McGraw-Hill, New York, 1989) p

7 7 M. Born and E. Wolf, "Principles of Optics" (Pergamon Press, 1980) p. 62 and p w. Siebourg, H. Schmid, F. M. Rateike, S. Anders, U. Grigg and H. LOwer, "Birefringence An important property of plastic substrates for magnetoopticai storage disks," Polymer Engineering and Science Q(18) (1990). 9 w. A. Challener and Thomas A. Rinehart, "Jones matrix analysis of magnetooptical media and readback systems," AppI. Opt. (18), (1987). 10 A. Takahashi, M. Mieda, Y. Murakami, K. Ohta, and H. Yamaoka ulnfluence of birefringence on the signal quality of magnetooptic disks using polycarbonate substrates," Appi. Opt. (14), (1988). M. Hone, "Simple birefringence measurement method for coated optical disks with a fixed incident angle ellipsometer," Appl. Opt. 4(25), (1995). 12 H. Fu, S. Sugaya, J. K. Erwin and M. Mansuripur "Measurement of birefringence for optical recording of disk substrates," Appi. Opt. (10) (1994). 233