Measurements of material refractive index with
a circular heterodyne interferometer
Zhi-Cheng Jian
a, Jiun-You Lin
b, Po-Jen Hsieh
a, and Der-Chin Su*
a aInstitute of Electro-Optical Engineering, National Chiao Tung University,
1001 Ta-Hsueh Road, Hsin-Chu 30050, Taiwan, R.O.C.
b
Department of Mechatronic Engineering, National Changhua University of Education,
No. 2, Shi-Da Road, Changhua City 20056, Taiwan, ROC.
ABSTRACT
When a light coming from a circularly polarized heterodyne light source incidents on an optical material, a phase difference between s- and p- polarization components of the reflected light occurs. This phase difference can be measured accurately with the heterodyne interferometry. The measured data are substituted into the special equations derived from Fresnel equations, the refractive index can be estimated. This method bears both merits of a common-path interferometer and a heterodyne interferometer. The refractive indices of three optical glasses and two birefringent crystals were measured to show the validity of this method.
Keyword: refractive index, circular heterodyne interferometer, isotropic material, birefringent crystal, Fresnel equations.
1. INTRODUCTION
Optical materials such as optical glasses or birefringent crystals are often used to fabricate optical components. Recently, some devices, for example, birefringent laser cavity filters1, poled-polymer electro-optic devices2, liquid-crystal spatial
light modulations3, and magneto-optic recording media4, have been used for many applications. To enhance their quality
and performance, it is necessary to determine their refractive indices accurately. There are several methods5-15 for
measuring the refractive index of an optical material. They are usually divided into two types: the transmission type measurement method5-10 and the reflection type measurement method11-15. In the former method, the phase variations of
the light beam transmitted through an optical material are measured. So, the accuracy of thickness, flatness and parallelism of the two opposite sides of materials are strongly required. Hence, the measurement processes become tedious. The latter method such as ellipsometric technique is related with the light intensity variations. Consequently, it is easily influenced by the stability of the light source, the scattering light, the internal reflection, etc., and its resolution will be decreased.
To overcome these drawbacks, a circular heterodyne interferometer for measuring the refractive indices of an isotropic material and a birefringent crystal is proposed in this paper. It utilizes a common-path heterodyne interferometric technique and Fresnel equations. When a light coming from a circularly polarized heterodyne light source16 incidents on
an optical material, a phase difference between s- and p- polarization components of the reflected light occurs. This phase difference can be measured accurately with the heterodyne interferometry. The measured data are substituted into the special equations derived from Fresnel equations, the refractive index can be estimated. This method bears both merits of a common-path interferometer and a heterodyne interferometer.
2. PRINCIPLE
The schematic diagram of this method is shown in Fig. 1. A light beam coming from a circularly polarized heterodyne light source is incident at θ onto an optical material OM located on a rotation stage. The light beam reflected from OM
passes through an analyzer ANt and enters a photo detector Dt. If the amplitude of the light detected by Dt is Et, then the
intensity measured by Dt is I =t Et 2. Here, It acts as a test signal. On the other hand, the electronic modulated signal
of the circularly heterodyne polarized light source is filtered and becomes the reference signal. Finally, these two signals are sent to a phase meter PM and the phase difference between them can be measured.
Fig. 1 Schematic structure for measuring the phase difference owing to reflection at an optical material. EO: electro-optic modulator; Q: quarter-wave plate; OM: optical material; ANt: analyzer; Dt: photodetector; LVA: linear voltage amplifier; FG: function generator;
PM: phase meter.
2.1 Circularly polarized heterodyne light source
The circularly polarized heterodyne light source consists of a linearly polarized laser light source, an electro-optic modulator EO and a quarter- wave plate Q as shown as shown in Fig. 1. EO is driven by a function generator FG and a linear voltage amplifier LVA. For convenience, the +z axis is chosen along the propagation direction and the y-axis is along the vertical direction. Let the laser light be horizontally linearly polarized, the fast axis of EO and Q be 45° and 0°
with respect to the x-axis, respectively. If an external saw tooth voltage signal with angular frequency ω and amplitude
2 λ
V , the half-voltage of EO, is applied to EO, then the phase retardation produced by EO can be expressed as ωt. The
Jones vector of the light coming from the circularly polarized heterodyne light source can be written as
Rotation Stage Dt ANt θ OM Q Laser EO FG Filter PM z y
Circularly polarized heterodyne light source
Ir It
). 2 exp( 1 2 1 ) 2 exp( 1 2 1 ) 2 sin( ) 2 cos( ) ( ) 0 ( 0 t i i t i i t t E t EO Q Ei ω ω ω ω ω − − + = − = ⋅ ⋅ ° = (1)
From Eq. (1), it is obvious that there is an angular frequency difference ω between the left- and the right- circular
polarizations of the light beam.
2.2 Phase difference between s- and p- polarizations of reflected light 2.21 Birefringent crystal
Fig. 2 Reflection at surface of an optical material. OA: optical axis.
Here OM is a birefringent crystal with the extraordinary index ne and the ordinary index no, and its optical axis is located
at α with the incident plane as shown in Fig. 2. Ifthe transmission axis of ANt is located at β with respect to the x-axis,
then we have
( )
(
)
(
)
, sin cos 2 sin sin cos 2 cos sin cos ) ( + − + = ⋅ ⋅ = ⋅ ⋅ = β β ω β β ω β β β β t r r t r r E r r r r AN E S AN E ss ps sp pp i ss ps sp pp i t (2)where S is the Jones matrix for OM, rpp and rss are the direct-reflection coefficients, and rps and rsp are the cross-reflection
coefficients, respectively. They can be expressed as
Incident plane Optical axis (OA)
α
2 1 5 2 6 1 A A A A A A rpp + + = , (3a) 2 1 3 4 2 1 ( ) A A A A A A rps = + − , (3b) 2 1 5 6 A A A A rsp + − = , (3c) and 2 1 4 2 3 1 A A A A A A rss + + = , (3d) where α θ θ cos )tan (sin2 1 C C A + = , (4a) 2 2 cos ) cos ( tan C Bn n B n A o o o + + = θ θ α , (4b) C C A + − = θ θ cos cos 3 , (4c) B n B n A o o + − = θ θ cos cos 4 , (4d) C n C n A o o + − = θ θ cos cos 2 2 5 , (4e) 2 2 6 cos cos C Bn C Bn A o o + − = θ θ , (4f) ) cos sin ( sin2 2 2 2 2 2 2 2 θ α α e o e o n n n n B = − + , (4g) θ 2 2 2 = −sin o n C . (4h) Hence, we have
(
)
[
+ ω +φ]
= = E I t It t 2 01 cos , (5) where 2 ) sin cos ( ) sin cos ( 2 2 0 β β β β sp ps ss pp r r r r I = + + + , (6) and(
)(
)
(
) (
)
+ − + + + = − 2 2 1 sin cos sin cos sin cos sin cos 2 tan β β β β β β β β φ ss ps sp pp ss ps sp pp r r r r r r r r . (7) 2.2.2 Isotropic materialForisotropic material, we have the relation ne = no = n. Then, Eq. (3) can be expressed as
θ θ θ θ 2 2 2 2 2 2 pp sin cos sin cos − + − − = n n n n r , (9) and θ θ θ θ 2 2 2 2 ss sin cos sin cos − + − − = n n r . (10)
The average intensity Io and the phase difference φ in Eqs. (5) ~ (7) can be rewritten as
2 ) sin ( ) cos ( 2 2 0 β β ss pp r r I = + , (11) and
(
(
)
)
(
)
− = − 2 2 1 sin cos 2 sin tan β β β φ ss pp ss pp r r r r , (12) respectively.2.3 Estimation of refractive index
The electronic signal generated by FG is filtered, and it uses as the reference signal. So the reference signal has the form of
Ir = I′
[
1+cos(
ω +t φ0)
]
. (13)Both the test signal Itand the reference signal Ir are sinusoidal signals. They are sent to a phase meter PM, and φ can be
measured accurately when φ0 is known.
2.3.1 Birefringent crystal
From Eqs. (3) ~ (5) and Eq. (7), we can see that φ depends on ne, no, α, θ, and β. In practical measurement processes, θ
and β are obtained from the direct angle readouts of the division mark of the rotation stage. Consequently, only three
factors ne, no and α should be solved. That is, we have
) , , ( α φ φ = ne no . (14)
Theoretically, the data of φ being corresponding to three different conditions should be measured. They are substituted
into Eq. (14) and ne, no and α can be obtained. However, these equations are very complicated, it is not easy solve them
directly. For easier operations and estimations, θ and β could be so chosen that Eq. (7) can be simplified. As the
condition β=0° is chosen, Eq. (7) can be rewritten as
− = − 2 2 1 2 tan ps pp ps pp r r r r φ . (15) It can be seen from Eqs. (3) and (4) that either rps or rsp equals zero as α equals either 0° or 90°, respectively. Hence,
under the condition β=0°, the optical axis OA of the birefrigent crystal can be rotated until the condition φ=0° is satisfied.
Then, the optical axis is located at either 0° or 90° with respect to the incidence plane.
Next, ANt is rotated so that β is nonzero, and Eq. (7) is rewritten as
− ⋅ = − β β β φ 1 2 2 2 2 sin cos 2 sin tan ss pp ss pp r r r r . (16) Now we consider two particular conditions:
(i) if α=0°, then θ θ θ θ 2 2 2 2 sin cos sin cos − + − − = o e o o e o pp n n n n n n r , (17a) and
θ θ θ θ 2 2 2 2 sin cos sin cos − + − − = o o ss n n r ; (17b) (ii) if α=90°, then θ θ θ θ 2 2 2 2 2 2 sin cos sin cos − + − − = o o o o pp n n n n r , (18a) and θ θ θ θ 2 2 2 2 sin cos sin cos − + − − = e e ss n n r . (18b) Since three unknowns (ne, no, and α) are to be solved, we need three equations. These can be obtained by measuring φ at
three different incident angles θ1, θ2, and θ3. Three corresponding phase differences φ1, φ2, and φ3 are obtained, and
represented as ) , , ( 1 1 φ α φ = ne no , (19a) ) , , ( 2 2 φ α φ = ne no , (19b) ) , , ( 3 3 φ α φ = ne no . (19c)
Two of Eqs. (19a)~(19c) are combined to form a set of simultaneous equations, and three sets are obtained. Any set of the simultaneous equations can be solved under either condition (i) or (ii), and so two corresponding pairs of solutions for (ne, no) are obtained. Therefore, there are six pairs for (ne, no). Among them, three pairs are derived under condition (i)
and form a group of solutions. The other three are derived under condition (ii) and form another group of solutions. Then, the justification of correct solutions can be achieved by the following approaches:
1. Rationality of the solution: In general, both ne and no are within the range 1 and 5. If any estimated data of ne and no is
not within this range, it is obvious that the estimated data may be incorrect.
2. Comparison between ne and no: Either a positive or negative crystal is tested, all three pairs of solutions of either group
should meet with only either ne >noor ne <no. If not, then that group is incorrect.
Hence, only one group of solutions is correct, and the corresponding data of α is the azimuth angle of its optical axis.
2.3.2 Isotropic material
From Eq.(10), it is easily seen that the case rss=0 should not exist. From Eq. (12) it is obvious that only when β is neither
0° nor 90°, then φ=0° as rpp=0. Under this condition, the incident angle is equivalent to the Brewster’s angle θB. Eqs. (9)
and (10) are substituted into Eq. (12), we obtain − − + − − = − θ θ θ β θ θ θ β θ θ φ 4 2 2 2 2 2 2 2 2 2 1 sin cos sin 2 2 cos ) cos sin sin 2 [( ) 2 sin ) cos ((sin tan n n n . (20) From Eq. (20), θBis capable to be determined with the experimental curve between φ and θ, so n can be evaluated by
3. EXPERIMENTS AND RESULTS
Fig. 3 Theoretical and experimental curves of φ versus θ for BK7, BaSF2, and SF2 as β=20°.
In order to demonstrate the feasibility of this method, the refractive indices of three glasses (BK7, BaSF2, and SF2), and two birefringent crystals (quartz and calcite) were measured. A He-Ne laser with 632.8 nm wavelength and an electro-optic modulator (PC/2; England Electro-Optics Development Ltd.) were used. The frequency of a sawtooth signal applied to the EO was 800 Hz. We used to a high-resolution rotation stage (PS-θ-90; Japan Chuo Precision Industrial
Company, Ltd.) with an angular resolution 0.005° to mount and rotate the test material, and a high- resolution
phasemeter with an angular resolution 0.01° to measure the phase difference. In addition, we used a personal computer to
record and analyze the data. Firstly, the refractive indices of three glasses were measured. For easier operation in glasses measurement, β=20° was chosen. The theoretical and experimental curves of φ versus θ for these three glasses are
shown in Fig 3. In this figure, the full curves represent the theoretical reference values which are obtained by introducing their reference refractive indices17 into Eq. (20), and the symbols “, ×, and ＋” represent the direct readouts of division
mark of rotation stage for BK7, BaSF2, and SF2, respectively. It is clear that these three curves show good correspondence.The Brewster angles of BK7, BaSF2, and SF2 were measured to be 56.574°, 58.942°, and 60.634°, their
refractive indices are 1.51508, 1.66046, and 1.77717, respectively.
Next, the refractive indices of two birefringent crystals were measured. The data of the three incident angles and their corresponding phase differences are listed in Tab. 1. These simultaneous equations are solved with 2-D Newton’s method and mathematics software “MATHEMATICA”. And two groups of solutions are calculated and summarized in Tab. 2. The right column represents the judged results according to the above approaches, and marks Ο and × mean the groups
of solutions are correct and incorrect. The measured data of (ne, no) and their averages for calcite and quartz are listed at
the first two rows in Tab. 3 and Tab. 4, respectively. α=90° exist in testing these two crystals.
90 60 30 0 -30 -60 -90 68 66 64 62 60 58 56 54 52 50 48 θ (degree) φ ( de gr ee ) BK7 BaSF2 SF2
Material θ1 θ2 θ3 φ1 φ2 φ3
Calcite 55° 60° 65° 24.52° -6.85° -25.85°
Quartz 55° 60° 65° 17.46° -24.40° -62.56°
Table 1 Experimental conditions and measurement results.
(ne, no) Material α (φ1, φ2 ) (φ2, φ3) (φ3 , φ1) Justification 0° (1.6695,1.5453) (0.5041,1.0007) (580.71,-22.545) × Calcite 90° (1.4333,1.6233) (1.4267,1.6144) (1.4333,1.6233) Ο 0° (1.5522,1.5627) (1.5128,1.4638) (1.5293,1.5132) × Quartz 90° (1.5552,1.5449) (1.5560,1.5243) (1.5647,1.5195) Ο
(φ1, φ2 ) (φ2, φ3) (φ3 , φ1) Average ne 1.4333 1.4267 1.4333 1.4311 no 1.6233 1.6144 1.6233 1.6203 ∆ne 9.977×10-4 1.196×10-3 6.178×10-4 9.371×10-4 ∆no 1.947×10-4 3.69×10-4 2.248×10-4 2.628×10-4 ∆α 0.0043° 0.0076° 0.0043° 0.0162°
Reference values from Ref. 19: (ne, no) are (1.4852,1.6559) at 627.8nm.
Table 3 Estimated results and their average for calcite.
(φ1, φ2 ) (φ2, φ3) (φ3 , φ1) Average ne 1.5552 1.5560 1.5647 1.5586 no 1.5449 1.5243 1.5195 1.5295 ∆ne 1.626×10-3 1.9763×10-3 1.046×10-3 1.549×10-3 ∆no 2.14×10-4 5.90×10-4 2.18×10-4 3.406×10-4 ∆α 0.1454° 0.0243° 0.0373° 0.069°
Reference values from Ref. 20: (ne, no) are (1.5518, 1.5428) at 627.8nm.
4. DISCUSSION
At first, measured resolutions of three glasses are discussed. From Eq. (20), we have
∆θ ≅ sin3θB⋅cosθB⋅tanβ ×∆φ , (21)
where ∆θ and ∆φ are the errors in the incident angle and the phase meter, respectively. Our experimental conditions are
substituted into Eq. (21), we can get ∆θ ≅ 0.001° . Substituting the data of ∆θ and θB into the equation
θ θ ∆⋅ =
∆n sec2 B , (22)
the measured resolutions of BK7, BaSF2, and SF2 are 5.7×10-5, 6.5×10-5, and 7.2×10-5, respectively.
Secondly, we discuss the measured resolutions of two birefringent crystals. From Eq. (7), we get φ α φ α = ∆ ∆ d d 1 , (23) o o e e n n n n ∂ ∆ ∂ + ∆ ∂ ∂ = ∆ 1 1 1 φ φ φ , (24) and o o e e n n n n ∂ ∆ ∂ + ∆ ∂ ∂ = ∆ 2 2 2 φ φ φ . (25) Eqs. (24) and (25) can be rewritten as
o e o e o o e n n n n n n n ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ ∆ ∂ ∂ + ∆ ∂ ∂ = ∆ 1 2 2 1 2 1 1 2 φ φ φ φ φ φ φ φ , (26) and o e o e e e o n n n n n n n ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ ∆ ∂ ∂ + ∆ ∂ ∂ = ∆ 1 2 2 1 2 2 1 1 φ φ φ φ φ φ φ φ , (27) where ∆α, ∆ne, and ∆no are the errors in α, ne, and no, and ∆φi and ∆φj are the errors in the phase differences at two
different incident angles θi and θj, respectively. Either i or j is an integer between 1 and 3, and i ≠ j. Considering the
angular resolution of the phase meter, the second-harmonic error, and the polarization- mixing error, ∆φ=∆φi=∆φj≅0.03°
can be estimated in our experiments18. Substituting this data and the experimental conditions into Eqs. (23), (26), and
(27), the corresponding data of ∆α, ∆ne, and ∆no of three sets of simultaneous equations and their averages are calculated
and listed at the last three rows in Tab. 3 and Tab. 4, respectively.
5. CONCLUSION
A novel method for determining the refractive indices of an optical material is presented with a common-path heterodyne interferometric technique and Fresnel equations. It has no drawbacks of the conventional methods. Besides, it has both merits of a common-path interferometer and a heterodyne interferometer. So, it has merits, such as, simple setup, high stability, easier operation and high-resolution.
5. ACKNOWLEDGEMENT
REFERENCES
1. X. Wang and J. Yao, “Transmitted and tuning characteristics of birefringent filters,” Appl. Opt. 31, 4505-4508 (1992).
2. J. F. Valley, J.W. Wu, and C. L. Valencia, “Heterodyne measurement of poling transient effects in electro-optic polymer thin films,” Appl. Phys. Lett. 57, 1084-1086 (1990).
3. I. Moreno, J. A. Davis, K. G. D’Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37, 3048-3052 (1998).
4. R. S. Weis and T. K. Gaylord, “Magnetooptic multilayered memory structure with a birefringent superstrate: a rigorous analysis,” Appl. Opt. 28, 1926-1930 (1989).
5. W. R. Hunter, “Optical constant of metals in the extreme ultraviolet I. A midified critical- angle technique for measuring the index of refraction of metals in the extreme ultraviolet,” J. Appl. Phys. 34, 15-19 (1964).
6. R. M. A. Azzam, “ Simple and direct determination of complex refractive index and thickness of unsupported or embedded thin thin films by combined reflection and transmission and ellipsometry at 45 ° angle of incidence,” J.
Opt. Soc. Am. 73, 1080-1082 (1983).
7. S. F. Noe, and H. E. Bennett, “Accurate null polarimetry for measuring the refractive index of transparent materials,” J. Opt. Soc. Am. A 10, 2076-2083 (1993).
8. R. P. Shukla, G. M. Perera, M. C. George, and P. Venkateswarlu, “Measurement of birefringence of optical materials using a wedged plate interferometer,” Opt. Commun. 78, 7-12 (1990).
9. M. H. Chiu, C. D. Chen, and D. C. Su, “Method for determining the fast axis and phase retardation of a wave plate,” J. Opt. Soc.Am. A 13, 1924-1929 (1996).
10. Y. C. Huang, C. Chou, and M. Chang, “Direct measurement of refractive indices of a linear birefringent retardation plate,” Opt. Commun. 133, 11-16 (1997).
11. M. Schubert, B. Rheinlander, J. A.Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsomery: determination of the dielectric function tensor from uniaxial TiO2.,” J. Opt.
Soc. Am. A 13, 875-883 (1996).
12. J. D. Hecht, A. Eifler, V. Riede, M. Schubert, G. Krauss, and V. Kramer, “Birefringence and reflectivity of single-crystal CdAl2Se4 by generalized ellipsometry,” Phys. Rev. B 57, 7037-7042 (1998).
13. G. E. Jellison, Jr., F. A. Modine, and L. A. Boatner, “The measurement of the optical functions of uniaxial materials using two-modulator generalized ellipsometry: rutile (TiO2),” Opt. Lett. 22, 1808-1810 (1997).
14. G. E. Jellison, Jr, and F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190-8198 (1997).
15. G. E. Jellison, Jr, and F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36, 8184-8189 (1997).
16. J. Y. Lee and D. C. Su, “A method for measuring Brewster’s angle by circularly polarized heterodyne interferometry,” J. Opt. 29, 349-353 (1998).
17. Schott Glass Technologies, Duryea, PA BK7 517642, BaSF2 664358 and SF2 648339 Data Sheets.
18. M. H. Chiu, J. Y. Lee, and D. C. Su, “Complex refractive-index measurement based on Fresnel’s equations and the uses of heterodyne interferometry,” Appl. Opt. 38, 4047-4052 (1999).
19. E. D. Palik, ed., Handbook of Optical Constants of Solids Ⅲ (Academic, New York, 1998), P. 708.
