Birefringence of Muscovite Mica Plate: Temperature Effect in the Ultraviolet and Visible Spectrum

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1 Birefringence of Muscovite Mica Plate: Temperature Effect in the Ultraviolet and Visible Spectrum Xu Zhang 1,*, Fuquan Wu, Limei Qi, Xia Zhang, Dianzhong Hao 1. School of Phsics Science and Engineering, Tongji Universit, Shanghai, 9, China. Shandong Provincial Ke Laborator of Laser Polarization and Information Technolog, College of Phsics and Engineering, Qufu Normal Universit, Qufu, 73165, China * Corresponding author: zhanguq@163.com Abstract We developed a method to measure the phase retardation and birefringence of muscovite mica plate in the temperature range of 3K to 358K within the spectrum of 3 to 7 nm. The phase retardation data is gained through the standard transmission ellipsometr using spectroscopic ellipsometer. With the phase retardation and thickness of the mica plate we can calculate its birefringence dispersion. Our results give abundant phase retardation and birefringence data of muscovite mica in the ultraviolet and visible spectrum from 3K to 358K. From the eperimental data, the phase retardation and birefringence will drop down at the fied wavelength when the temperature rises. The accurac of the birefringence of mica plate is better than Kewords : Phase retardation, Birefringence, Muscovite mica wave plate, Temperature effect OCIS codes: (1.13) Ellipsometr and polarimetr; (1.55) Phase measurement, (1.1) Instrumentation, measurement, and metrolog. 1 Introduction Muscovite mica is widel used in polarization technolog [1,] and as substrate in technological and biological applications [3-5]. Because of its low thickness mica is the ideal crstal to manufacture the zero order wave plate which is not sensitive to the wavelength [6]. On the contrar the designed multiple-order plate can onl work at a single wavelength because a slight shift from the designed wavelength will cause a large variation to the phase retardation of the plate [7,8]. But the birefringence dispersion of mica plate is not alwas the same as to different samples [1,9-1] which makes the determination of different mica sample s birefringence dispersion essential. In this work, we develop a method to determine the phase retardation and birefringence dispersion of the mica plate between 3 and 7 nm. Before the measurement we coated the mica plate to increase the transmittance because there are multiple coherent reflections inner the mica plate which have great influence on the phase retardation (shown in part.). Using the spectroscopic ellipsometer we gain the phase retardation of the mica plate. With the thickness and phase retardation of the mica plate we can calculate its birefringence dispersion. Since the birefringence dispersion of mica plate under different temperatures ma be used in its applications, we measured the birefringence dispersion in the temperature range of 3K to 358K. Measuring principle.1 Ellipsometr The standard ellipsometr includes the reflection ellipsometr and the transmission

2 ellipsometr. Ellipsometric angles and are two important parameters in the ellipsometr method. As to the isotropic medium, the relationship between the ellipsometr parameters and jp the reflection (or transmission) matri j is [13,14]: s jp tan ep( i ). (1) j s As to the reflection ellipsometr, jp and j s denote the p- and s-polarized comple reflection coefficients( jp, s rp, s ). To transmission ellipsometr, the denote the p- and s-polarized comple transmission coefficients ( jp, s tp, s ). But to the anisotropic medium, the generalized ellipsometr should be used. The reflection (or jpp jsp transmission) matri is not diagonalized, it is jps j. The relationship between the ss ellipsometr parameters and this matri is [13,14]: jps ps tan ps ep( i ps) j pp jsp sp tansp ep( isp). () jss jpp pp tan pp ep( i pp) jss jp The reflection (or transmission) matri can be diagonalized as j when the optic ais is s parallel or vertical to the incident plane [14,15]. Under this condition, the standard ellipsometr can still be used to the anisotropic medium. As to the standard transmission ellipsometr, tp t tant ep( i t ), (3) t s where t t ep( i ) ( tp is the phase shift of the p vibration), ts ts ep( i ts) ( ts is the p p tp 1 phase shift of the s vibration), t tan t, t tp ts is the phase retardation between the p vibration and s vibration. When a linearl polarized light is incident normall upon a mica wave plate, it will be decomposed into o light ( s vibration) and e light ( p vibration). The refractive inde is n o and n e respectivel. Because the two lights have different speeds ( v e v o ) in the mica plate, after the passed through the mica plate with the thickness of d the phase retardation between them is [1]: ( n n ) d /, (4) e

3 where ( ne n ) is the birefringence at the wavelength. Muscovite mica is biaial crstal. In the manufacture of mica plate, from the propert of the cleavage plane we know ne no n [1,16,17]. Then the birefringence of the mica wave plate is e n, n n n n. From the above discussion, the standard transmission ellipsometr can be used to measure the phase retardation of the mica wave plate.. Multiple coherent reflections in mica plate [18] There are multiple coherent reflections within the mica plate because of its little thickness and ecellent flatness which is shown in Fig. 1. The fast and slow aes of the mica plate are set as the and ais separatel and the propagation direction is set as z ais. I is the incident light intensit, R 1, R, R 3. are the reflected light and T 1, T, T 3. are the transmitted light. I R1 R R 3 T 1 T T 3 z d Fig.1 Multiple coherent reflections in mica wave plate, d is the width of the mica plate. ep( i ) As we all know the Jones matri of the mica wave plate is ep( i ) without the consideration of the multiple reflections within it. Considering the multiple reflections, the Jones matri should be (shown in Appendi A): t ep( i ) 1r ep( i ) Jm t ep( i ), (5) 1r ep( i ) n n n where r, t n n n n, nd n n n, r, t n n n n, nd. Then J can m be set as 1 Jm ep( i) ep( i ). (6) The final form of equation (6) is the Jones matri of the mica plate considering the multiple reflections within it. From this Jones matri we can see is the comple amplitude ratio of the transmitted light at the and direction and is the phase retardation between them ( ). From calculation we can gain (shown in Appendi A):

4 (1 r1 )(1 r1 ) tan (1 r1 )(1 r1 ) tan arctan (1 r1 )(1 r1 ) (1 r1 )(1 r1 ) tan tan. (7) In order to displa the effect of the multiple reflections within mica plate, we calculated the phase retardation curve of a mica plate (a quarter wave plate at 633nm) in the spectrum of 55-7nm. The reflection inde of mica wave plate before coated is about 5.18%. Then from equation (7) we can gain Fig. (a). The red line is the result without the consideration of multiple reflections. If we set the reflection inde as 1.1%, we can gain Fig. (b). It is quite evident that the oscillations will be reduced a lot with the decrease of the reflection inde which enables us to gain more accurate phase retardation data. So we should coat the mica wave plate before we use the standard transmission ellipsometr to measure its phase retardation and birefringence. Fig. (a) The phase retardation of the mica quarter wave plate (reflection inde =5.18%) 3 Eperiment Fig. (b) The phase retardation of the mica quarter wave plate (reflection inde =1.1%) We evaporate broadband anti-reflective film on the mica wave plate. The film coefficient is Sub/Al O 3^1/WD1^/MgF ^1/Air. The anti-reflective film can increase the transmission in the ultraviolet and visible spectrum and suppress the multiple coherent reflections within the plate (shown in part.). 3.1 Phase retardation measurement The UVISEL spectroscopic phase modulated ellipsometer (USPME, shown in Fig. 3(a)) made b the French Jobin Yvon corporation is used in the eperiment to measure the phase retardation of the mica wave plate. L is laser, P the polarizer, S the measured mica wave plate, M the modulator, A the analzer and D the detector. L P S M A D 5H Fig. 3(a) Optical sstem for measuring z the phase retardation and birefringence of mica plate. z

5 Temperature regulating device S A / Fig. 3(b) Temperature regulating device on the sample table As to the USPME, the polarizer P, modulator M and analzer A have two configurations [19]. Configuration I is PM 45, M, A 45 and configuration Ⅱ is PM 45, M 45, A 45. In configuration I, the final detected intensit collected b the detector is I( t) I[ I Issin ( t) Iccos ( t)], (8) with I 1, Is sin sin, Ic sin cos. (9) While in configuration II, the final detected intensit collected b the detector is I( t) I[ I Issin ( t) Iccos ( t)], (1) with I, I sin sin, I cos. (11) 1 s In configuration I we can measure accuratel over the full 36 range (theoretical equations (9) give tan and sin ). But when is near 45 we will suffer from accurac problems. As to the configuration II, we can measure accuratel over the full 9 range, but suffer from indetermination in the range 9 7. Since we want to gain the precise phase retardation of the mica plate, we use configuration I in the eperiment. We developed the following eperiment procedures: Step 1, Polarizer P, modulator M and analzer A are aligned in orientations with P, M, A 45. Then place another analzer A on the sample table shown in Fig. 3(b). Adjust A to make P and A crossed. (We add another analzer A because the angles of M and A are fied in configuration I.) Step, Place the measured mica plate between P and A, adjust the plate until minimum intensit (within the measurement noise level) is detected which means the optic ais of the measured plate is parallel or vertical to the transmission ais of the polarizer P. In the mean time the first harmonic component of the modulated intensit R and the second harmonic component R should be zero (within the measurement noise level) which enables us to use the ellipsometric angles and in the standard ellipsometric configurations to measure a biaiall anisotropic slab (the mica wave plate) [15]. Step 3, Set P 9, then adjust A to make P and A crossed again. If the first and the second harmonic components of the modulated intensit are still zero(within the measurement noise level), then the measured plate is vertical to the incident light. Step 4, Take A awa and adjust P 45 (configuration I). Install the temperature regulating device on the sample table shown in Fig. 3(b). We adjust the temperature from 3K to 93K as to mica plate sample 1 with a temperature interval of 1K and 98K to 358K to mica plate c

6 sample also with a temperature interval of 1K. Set the spectral range (3 7 nm) and wavelength interval of the UVISEL ellipsometer, the retardation of the wave plate can be gained from the outputted data. Step 5, Then adjust P 45 to repeat the measurement in step 4. This step is used to check the p and s transmissions are same or not which can be used to judge the standard transmission ellipsometr can be applied or not. If p and s transmissions are different, there must be miscuts in the preparation of the wave plate. Then analsis schemes must invoke the full anisotropic generalized ellipsometr algorithm [14]. From our eperiment results the p and s transmissions agree well with each other. We set the wavelength interval in this eperiment as 1 nm. The interval can be set as 5 nm, 1 nm and so on. So we can gain more data of the phase retardation when the interval is smaller. 3. Thickness measurement and birefringence calculation In order to gain the birefringence of the mica plate, we should know the thickness of the plate from equation (4). Our mica plate samples were measured at different points b a high precise digital micrometer with a resolution ratio of.1μm at the temperature of 93K. The mean value of the sample 1 is 33.8 μm and 33.1 μm as to sample. From the distribution of thickness values for the two mica plates, the uncertaint in thickness values is about. μm. Since the thickness of the mica plate d T at temperature T is []: d d [ ( T T )], (1) T 6 where d is the thickness of the mica plate at the temperature of 93K, T is 93K. From equation (4) and (1), the birefringence of the mica plate at temperature T is: ( ne n ) T. (13) dt Based on this equation we can stud the temperature effect of mica plate s birefringence. 4 Results and discussions 4.1 Results analsis The phase retardation of muscovite mica plate sample 1 is shown in Fig. 4 (a). Compared with Fig., the oscillations have been eliminated b the coating of the anti-reflective film which enhances the accurac of our phase retardation data. From Fig. 4 (a) when the temperature rises, the phase retardation drops down at the fied wavelength. Since the thickness of mica plate will increase due to the heat epansion so the birefringence should decrease with the rise of the temperature at the fied wavelength which is verified b Fig. 4(b). So the effect of the birefringence is the main factor in the temperature effect of the phase retardation of muscovite mica plate. With the phase retardation data, we can calculate the birefringence of mica from equation (13) in the temperature range of 3K to 93K as to sample 1. The birefringence data of sample 1 are listed in Table 1. From Fig. 4(b) (drawn from the data in Table 1) we can find the birefringence decreases with the rise of the temperature at the fied wavelength as mentioned above. When the temperature is fied, the birefringence will go up with the increase of the wavelength. Oscillations of birefringence eist at some wavelengths which is in accordance with the data in reference [1].

7 Fig.4 (a) Phase retardation of the mica plate of thickness 33.8μm Fig.4(b) Birefringence of the mica plate of thickness 33.8μm The phase retardation of muscovite mica plate sample is shown in Fig. 5(a). Its birefringence data are listed in Table and shown in Fig. 5(b). From the figures we can see the variet law of phase retardation and birefringence of sample is same to sample 1. Fig.5 (a) Phase retardation of the mica plate of thickness 33.1μm Fig.5 (b) Birefringence of the mica plate of thickness 33.1μm 3 Table 1 Birefringence ( ne n ) 1 of muscovite mica plate sample 1 of thickness 33.8μm from 3K to 93K. (nm) 3K 33K 43K 53K 63K 73K 83K 93K

8 Table Birefringence ( ne n ) 1 of muscovite mica plate sample of thickness 33.1μm from 98K to 358K. (nm) 98K 38K 318K 38K 338K 348K 358K

9

10 4. Error analsis The error of the mica plate s birefringence mainl arises from the error in the determination of the mica plate s thickness and the error of the wavelength and phase retardation. From equation (13), the maimum error of ( ne n) is ( ne n ) d (14) dt dt dt The error of the mica plate s birefringence as to mica plate sample 1 in the visible spectrum is shown in the following. Since the uncertaint in mica plate s thickness value is. μm, the maimum value of dt d is in the 4 7 nm spectrum (the maimum value of is gained at 56nm from our data). As to the USPME, the accurac of the wavelength is.1 nm. So the maimum value of d is (the maimum value of is gained at 4nm). Through repeated measurement, we know that the maimum error of is.1 o, so that the maimum value of is (when is 7nm) in the 4 7 nm spectrum. d From the above discussion the maimum error of ( ne n) as to mica plate sample 1 in the visible spectrum is less than Using the same method the maimum error of ( ne n) is less than in the ultraviolet spectrum. As to mica plate sample, the maimum error of ( ne n) is less than in the visible spectrum and in the ultraviolet spectrum. 5 Conclusion Using the standard transmission ellipsometr, we have demonstrated a method to measure the phase retardation and birefringence of muscovite mica plate. For two mica plate samples, we gained the phase retardation and birefringence dispersion in the temperature range of 3K to 93K and 98K to 358K respectivel in the spectrum of 3 to 7 nm. From error analsis the data of mica birefringence have an accurac of better than in the 3 7 nm spectrum. The phase retardation and birefringence drop down when the temperature rises at the fied wavelength from the eperimental results. The method described above can also be used to measure the phase retardation and birefringence of the other uniaial or some biaial crstals. Acknowledgement This stud was supported b the National Youth Natural Science Foundation of China (No ). References [1] Bennett J.M., Bennett H.E. Polarization. In: Driscoll WG, editor. Handbook of optics, Chap. 1. New York: McGraw-Hill, p

11 [] Evelina A. Bibikova, and Natalia D. Kundikova, Properties of an adjustable quarter-wave sstem under conditions of multiple beam interference, Appl. Opt. 5(9), (13). [3] W. H. Briscoe, S. Timtuss, F. Tiberg, R. K. Thomas, D. J. McGillivra, and J. Klein, Boundar lubrication under water, Nature 444, (6). [4] T. Fukuma, Y. Ueda, S. Yoshioka, and H. Asakawa, Atomicscale distribution of water molecules at the mica-water interface visualized b three-dimensional scanning force microscop, Phs. Rev. Lett. 14, 1611 (1). [5] Annunziata Savoia, Marco Siano, Domenico Paparo, and Lorenzo Marrucci, Nonlocal optical second harmonic generation from centrosmmetric birefringent crstals: the case of muscovite mica, J. Opt. Soc. Am. B, 8(4), (11). [6] Ignacio Moreno, JoséV. Carrión, JoséLuis Martínez, Pascuala García-Martínez, María M. Sánchez-López, and Juan Campos, Optical retarder sstem with programmable spectral retardance, Opt. Lett., 39(19), (14). [7] N. N. Nagib, S. A. Khodier, and H. M. Sidki, Retardation characteristics and birefringence of a multiple-order crstalline quartz plate, Opt. Laser Technol. 35, (3). [8] M. Emam-Ismail, Spectral variation of the birefringence, group birefringence and retardance of a gpsum plate measured using the interference of polarized light, Opt. Laser Technol. 41(5), (9). [9] Matthew J. Romerein, Jeffre N. Philippson, Robert L. Brooks, and Ralph C. Shiell Calibration method using a single retarder to simultaneousl measure polarization and full characterize a polarimeter over a broad range of wavelengths, Appl. Opt., 5(8), (11). [1] Shurcliff W.A. Polarized light. Cambridge, MA: Harvard Universit Press, 196. p. 99. [11] I. N. Shklarevskii, T. I. Korneeva, and A. N. Razanov, An interferometric method for determining the refractive inde of mica, J. Appl. Spectrosc. 4(1), 65-67(1966). [1] M.S. El-Bahrawi, N.N. Nagib, S.A. Khodier, and H.M. Sidki, Birefringence of muscovite mica, Opt. Laser Technol. 3, (1998). [13] Volodmr Tkachenko, Antigone Marino, Francesco Vita, D'Amore F, De Stefano L, Malinconico M, Rippa M, and Abbate G, Spectroscopic ellipsometr stud of liquid crstal and polmeric thin films in visible and near infrared, Eur. Phs. J. E 14, (4). [14] M. Schubert, Another centur of ellipsometr, Annalen der Phsik 15, (6).

12 [15] H. Touir, M. Stchakovsk, R. Ossikovski, and M. Warenghem, Coherent and incoherent interference modelling and measurement of anisotropic multilaer stacks using conventional ellipsometr, Thin Solid Films, , (4). [16] S.Y. El-Zaiat, Interferometric determination of refraction and dispersion of a birefringent material: numerical procedure, Opt. Laser Technol. 34, 15-1 (). [17] Emre Coşkun, Serhat Özder, and Erhan Tiraki, The Paul wavelet algorithm: an approach to calculate the refractive inde dispersion of a dielectric film from transmittance spectrum, Appl. Phs. B 113, 43-5 (13). [18] Xu Zhang, Fuquan Wu, Xia Zhang, Dianzhong Hao, and Limei Qi, Measure the phase retardation and birefringence of the mica wave plate using the spectroscopic ellipsometer, Acta Photonica Sinica, 39(11), 5-3(1). [19] Xu Zhang, Fuquan Wu, Limei Qi, Xia Zhang, and Dianzhong Hao, Phase retardation and birefringence of the crstalline quartz plate in the ultraviolet and visible spectrum, [arxiv: ]. [] Zhang He. Snthetic mica. Shanghai: Science and Technolog Press, 196, p. 14. (in Chinese). Appendi A: We set T i and T i as the transmitted light at the and direction separatel. As to the direction, the composed light is 4 T T T t t r ep( i ) t r ep( i4 ), (A1) 1 3 where r n n, n n t n nd n n,, with n is the refractive inde of the incident medium, n is the refractive inde at the direction in the mica wave plate, is the wavelength of the incident light. So the whole composed light at direction is t 1 r ep( i ). (A) Using the same deduction method, the whole composed light at direction is t. (A3) 1 re( i ) Considering the multiple reflections, the Jones matri should be

13 t 1r ep( i ) ep( i ) Jm t ep( i ) 1r ep( i ) t ep( i ) 1r ep( i ). (A4) tep( i ) 1r ep( i ) Equation (A4) is equation (5). If we make t ep( i ) t ep( i ) ep( i), ep( i),, 1 r ep(, then J i ) 1 r ep( i ) m 1 can be set as Jm ep( i) ep( i ). (A5) Equation (A5) is equation (6). From calculation we gain 1 r1 tan tan 1 r1 1 r1 tan tan 1 r1. (A6) Then tan tan (1 r1 )(1 r1 ) tan (1 r1 )(1 r1 ) tan tg 1 tan tan (1 r )(1 r ) (1 r )(1 r ) tan tan, (A7) (1 r1 )(1 r1 ) tan (1 r1 )(1 r1 ) tan So we can gain equation (7): arctan (1 r1 )(1 r1 ) (1 r1 )(1 r1 ) tan tan.