Improved technique for measuring small angles

Improved technique for measuring small angles

Ming-Horng Chiu and Der-Chin Su

Based on the total-internal-reflection effect and heterodyne interferometry, an improved technique for measuring small angles is proposed. This technique not only expands the measurement range but it also improves measurement performances. Its validity is demonstrated. © 1997 Optical Society of America

Key words: Total-internal-reflection effect, heterodyne interferometry.

1. Introduction

There are several optical methods1–7_{for measuring}

small angles. They are sensitive and accurate, but their sizes are too large to be used in some space-limited areas.8 _{To solve this problem, Huang et}

al.8,9_{proposed an optical method for measuring the}

reflectance difference between two right-angle prisms based on the total-internal-reflection effect. Although it has some merits, as described by Huang

et al., there are some disadvantages, such as a small

measurable range, operation limited to a darkroom with a highly stable light source, and a careful es-timation to avoid influence from unnecessary reflec-tances at the entrance and the exit surfaces of the prisms. In this paper, an improved technique for measuring small angles by modifying Huang’s method is proposed. Although both of them are based on the total-internal-reflection effect, we measure the phase difference instead of the reflec-tance difference. And the phase difference can be extracted and measured accurately with hetero-dyne interferometry despite surrounding light and an unstable light source. Hence its performance is not affected by surrounding light noise. Moreover its validity is demonstrated.

2. Principle

The schematic diagram of this technique is shown in Fig. 1. Huang et al.’s optical system is modified by introducing an optical heterodyne source and

two analyzers before the detectors. The optical path arrangement is the same as that of Huang’s system. The light beam coming from a heterodyne optical source10_{with the frequency difference f}

be-tween s-~y-axis! and p-~x-axis! polarizations is in-cident on the beam splitter, BS, and is divided into two parts: the transmitted beam and the reflected beam. These two beams propagate nearly perpen-dicularly and are almost normally incident on the side surfaces of two right-angle prisms, P1 and P2, with the refractive index np placed on a rotary

stage, respectively. The beams are totally inter-nally reflected at the hypotenuse surfaces of the prisms. Then, they pass through analyzers, AN1 and AN2, which have the transmission axes at 45° with respect to the x axis and finally are detected by two detectors, D1 and D2. As shown in Fig. 2, if the rotary stage is rotated with a small angle u, which is defined positive as it is rotated clockwise, the two interference signals I1and I2detected by D1

and D2, respectively, are

I15 I10@1 1 V1cos~2pft 1 f11 fBS!#, (1)

I25 I20@1 1 V2cos~2pft 1 f2!#, (2)

where V1and V2are the visibilities of signals I1and

I₂, f₁ and f₂ are the phase differences between p and s polarizations due to the total internal reflec-tions within P1 and P2, respectively; fBS is the

phase difference between the p and s polarizations of the light reflected from BS with the coating layer of a refractive index n and an extinction coefficient

k. If a1 and a2 are the incident angles at the

hy-potenuse surfaces of prisms P1 and P2, according to

The authors are with the Institute of Electro-Optical Engineer-ing, National Chiao Tung University, 1001, Ta-Hsueh Road, Hsin-Chu, Taiwan.

Received 4 February 1997; revised manuscript received 19 May 1997.

0003-6935_{y97y287104-03$10.00y0} © 1997 Optical Society of America

Fresnel’s equation,11 f

1 and f2 can be written

as f15 2 tan21

H

@sin2_a 12 ~1ynp2!#1y2 tana1sina1

J

, (3) f25 2 tan21

H

@sin2_a 22 ~1ynp2!#1y2 tana2sina2

J

, (4) fBS5 tan21

S

bc2 ad bd1 ac

D

, (5) where a5 2v cos b, b5 cos2_{b 2 u}2_{2 v}2_, c5 2v cos b~n22 k22 2u2!, d5 ~n21 k2!2 3 cos2_{b 2 ~u}2_{1 v}2_!, 2u2_{5 n}2_{2 k}2_{2 sin}2_{b 1 @~n}2_{2 k}2_{2 sin}2_b!2 1 4n2 k2#1y2, 2v2_{5 2~m}2_{2 k}2_{2 sin}2_{b! 1 @~n}2_{2 k}2_{2 sin}2_b!2 1 4n2_k2_#1y2_, b 5 45° 1 u.

From the geometric relations of Fig. 2, we have

a15 45° 1 sin21

S

sinu np

D

, (6) a25 45° 2 sin21

S

sinu np

D

. (7)

From Eqs.~1! and ~2!, it is obvious that these two signals are sinusoidal with frequency f. When they are introduced into the phase meter, a phase differ-ence f between them can be obtained, and f can be expressed as

f 5 f12 f21 fBS. (8)

Substituting the dataf into Eqs. ~3!–~8!, we can cal-culate the rotation angleu. In our system the out-puts of the phase meter are sent to a personal computer, so that the result can be obtained in real time.

As the rotary stage is rotated, one of the incident angles at the hypotenuse surfaces of the prisms is increased and the other is decreased. Because this technique is based on the effect of the total internal reflection, this technique can be performed only if both of them are larger than the critical angle. Hence the allowable measurement range is from 2umaxtoumax, whereumaxshould satisfy

umax5 sin21

H

npsin

F

45°2 sin21

S

1

np

DGJ

. (9)

Substituting the latter experimental conditions, np5

1.51509, n5 3.57, and k 5 4.36, into Eqs. ~3!–~5! and ~8!, we can obtain the relation curves of f, ~f12 f2!,

andfBStou as shown in Fig. 3. It shows that in this

case the measurement range is between25.60° and 5.60°.

3. Experiments and Results

To show the feasibility of this technique, two right-angle prisms of BK7 glass with np 5 1.51509 and a

beam splitter with a chromium layer with n5 3.57 and k 5 4.36 are used, and they are mounted on a high-precision rotary stage~PS-u-90! with an angular resolution of 0.005° manufactured by Japan Chuo

Fig. 1. Schematic diagram for measuring small angles: PL, po-larizer; EOM, electro-optical modulator; AN, analyzer.

Fig. 2. Geometric relations between_{u and a}1ora2.

Fig. 3. Calculated curves of_{f versus u, ~f}12 f2!, and fBS.

Precision Industrial Company Ltd. The heterodyne optical source10_{with a wavelength of 632.8 nm and}

f5 2 kHz and a self-made phase meter with a

reso-lution of 0.01° are used. The experimental curves of f versus u for the angle measurement are shown in Fig. 4. In this figure, the circles represent the eval-uated values of the rotation angles, which are ob-tained by introducing the data off into Eqs. ~3!–~8!, and the dashed curve represents the direct angle readouts from the division mark of the rotary stage. It is clear that they have a good correspondence.

4. Discussion

From Eq. ~8!, we can obtain an expression for the measurement error:

Du 5 1

@df12 df2!ydu# 1 ~dfBS!ydu!

Df, (10) where Df is the angular resolution of the phase meter. In our experiments, the angular resolution of the phase meter is 0.01°. Consequently, the

rela-tion curve ofDu versus u can be obtained as shown in Fig. 5. Obviously, the error of the rotation angle becomes smaller asuuu increases; the best resolution is 8 3 1025° at uuu 5 5.6°. And the resolution of this technique is better than 1.6 3 1023° over the mea-surement range of25.60° # u # 5.60°.

Unlike many angle-measurement techniques, this approach is absolute: The phase difference is pre-cisely zero at zero angle. By using a higher-index glass, the angular range can be significantly ex-tended~e.g., for np5 1.8, umax' 20°!.

5. Conclusion

An improved technique for measuring small angles is proposed. Using heterodyne interferometry, it makes uses of the effect of total internal reflection in right-angle prisms as well as the phase-difference measurement between the two light beams passing through the prisms. This technique not only ex-pands the measurement range but also improves measurement performances. Its validity is demon-strated.

This study was supported in part by the National Science Council, Republic of China, under contract NSC85-2215-E-009-005.

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Fig. 5. Relation curves of_{Du versus u.} Fig. 4. Experimental curves of_{f versus u.}