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Highly sensitive angular measurement with a Michelson interferometer
69
Highly sensitive angular measurement with a Michelson interferometer Pan Shi* and Erik Stijns Vrije Universiteit Brussel, ALNA-TW, Pleinlaan 2, B-1050 Brussels, Belgium
Abstract. In order to measure small-angle rotations, the two mirrors of a Michelson interferometer are replaced by right-angle prisms. By increasing the distance between the two prisms and by means of an electronic fringe divider technique, the resolution can be increased to 10 5 degree, with a range of measurement from + 5 ° to - 5 °.
Keywords. Rotation measurement, angle measurement, Michelson interferometer, metrology.
1. Introduction Optical methods based on a Michelson interferometer are often used in angular measurements; they have a high sensibility but a limited range of measurement. In 1974 C h a p m a n built a very sensitive set-up with a resolution of 57 ms of arc per fringe and with a range of measurement from + 3 ° to - 3 * [1]. In our earlier balanced set-up we obtained a typical sensibility of 10 . 4 degree with a range of measurement from + 5 ° to - 5 ° [2-1. In this paper the sensibility is increased by a factor of 10. In order to do so we increased the size of the rotating table to separate the two prisms by a larger distance. We also built an electronic "fringe divider"; for one fringe we get four pulses in the counter. In this way we increased the sensibility by an order of magnitude up to 10-5 degree, as compared to [2], while covering the same range of measurement.
* Permanent address: Department of Physics, Dalian University of Technology, Dalian, P.R. China.
Elsevier Industrial Metrology 1 (1990) 69 74 0921-5956/90/$3.50 © 1990. Elsevier Science Publishers B.V.
70
Pan Shi, E. Stijns / Highly sensitive angular measurement
2. The principle Figure 1 shows the principle of our modified Michelson interferometer for measuring rotations. The two end mirrors are replaced by two right-angle prisms which are placed on the rotating table, separated by a distance L. These two prisms together with two fixed mirrors trace back the two laser beams, which form the interference pattern. When the table rotates the optical paths in the two arms of the Michelson interferometer change, and the interference pattern shifts. Counting the fringes at the detector gives, after calibration, the angle of rotation. In [2], we calculated the optical path difference Ap as a function of the angle of rotation 0 for the balanced set-up of Fig. 1: Ap = 2(x/2a + L) sin 0,
(1)
where a is the length of the side of the right angle prism, and L is the distance between the two prisms as shown in Fig. I. If we use another parameter r, the radius of the rotation (see Fig. 1), eqn. (1) can be simplified to Ap = 4r sin 0.
(2)
The number N of fringes passing the detector is N = 2Ap/2,
(3)
so we get 0 = sin- 1N2/8r.
(4)
Pan Shi was born in Liaoning, P.R. China, in 1949. He received the BS in
Physics from the Dalian Institute of Technology in 1982, and the MS in Applied Sciences from the Vrije Universiteit Brussel in 1986. He is currently working towards a PhD in Applied Optics at the Vrije Universiteit Brussel.
Erik Stijns studied Physics at the Vrije Universiteit Brussel, where he obtained his PhD in 1977. He is lecturer on Optics and Laser Physics; his research interests are in Applied Optics.
Pan Shi, E. Stijns / Highly sensitive angular measurement
[ M3
71
DETECTOR
•
/
-7 M1
M2 /////
/////
i
\ o
/7 /
a
Fig. 1. Principle of a Michelson interferometer for the measurement of small-angle rotations.
F r o m (4) we can see that for the same angle of rotation 0, the n u m b e r of fringes passing the detector, i.e. the sensibility, can be increased by increasing r (increasing L, a or both). Figure 2 shows the new set-up. We use a He Ne laser as light source. A spatial filter and lens are used to get an enlarged collimated light beam. This beam is split into two beams, which pass through prism 1 and prism 2, respectively, and which are then reflected back by fixed mirrors 1 and 2. After passing the beam splitter for a second time, they are recombined to form an interference pattern which is detected by detectors 1 and 2. Between mirror 3 and prism 2 we now put a phase plate. It consists of two glass plates put together with a small angle q$, as shown in Fig. 3. In order to obtain a phase difference of 90 °, it can be shown that q$ could be expressed in the form
~b = ~ / 2 d ,
(5)
where d is the thickness of the plate, n is its refractive index and 2 is the wavelength of the laser beam in the plate. When the enlarged beam passes through this phase plate, it is split into two sub-beams with a 90 ° difference in phase. These two beams form the interference pattern with the beam coming from the other arm of the Michelson interferometer. The electronic detection consists of two photodiodes which read the interference pattern as two sinusoidal signals with 90 ° phase difference; following amplification they pass t h r o u g h two shaping circuits into square waves. Using these two waves
Pan Shi, E. Stijns / Highly sensitive angular measurement
72
.•
AMP+Phase Reverser ] +Differentiat or+Logic
Output Pulses~
~ControlVoltage
Up-Down l Counter
M3
7
LASER
-7" M1 /////
M2 ///// Phase = Plate
\
e, 0
r
Fig. 2. Experimental set-up.
0T Fig. 3. Phase plate used to transform a laser beam into two sub-beams with a 90 degree phase difference.
Pan Shi, E. Stijns / Highly sensitive angular measurement
73
we get two other square waves by using a phase reverser. These four square waves with 90 ° phase difference between each other then pass through four differentiators to generate pulses. The square waves and pulses are then sent to two AND gates in such a way that, when the interference fringes shift in some direction, only one AND gate gives an output (high voltage). When the direction of rotation changes the two AND gates change their outputs. Finally the two outputs of the two AND gates pass another AND gate to respond with pulses for the u p - d o w n counter. The outputs of the two AND gates are also used to drive a flip-flop to get the control voltage of the counter, counting either up or down according to the direction of the rotation.
4. Experimental verification The set-up of Fig. 2 was used to verify the principle. Two identical right-angle prisms, with sides a - - 3 0 mm, were placed on the rotating table. In order to get 10 ÷ 5 pulses per degree, the separation of the two prisms should be r = 113.31 mm. Two photodiodes were positioned as shown in Fig. 4 to transform the interference pattern in a response of two sinusoidal signals with a 90 ° phase difference. An oscilloscope was used to check the phase difference of the two signals received by D1 and D2 and adjust it properly by slightly tuning the position of the phase plate with respect to the incident beam. We checked the repeatability of the set-up of Fig. 2, by measuring the maximum angle of the rotating table 36 times. A relative standard deviation a / N between 0.04526% and 0.06109% was obtained. In this experiment, we did not calibrate the set-up, only the repeatability was checked. From [2] we know indeed that the linearity of the set-up is adequate, so that when the set-up response has good linearity and repeatability, it can be used for accurate measurements. The overall accuracy of the actual set-up depends only on the accuracy with which the calibration factor can be determined, using either a highly sensitive autocollimator or gauge blocks.
Fig. 4. Two-photodiodearrangement to transform the response to the interferencepattern in two sinusoidal signals with a 90 degree phase difference.
74
Pan Shi, E. Stijns / Highly sensitive angular measurement
5. Conclusion A Michelson interferometer can be used to measure small-angle rotations in a simple and accurate method. The sensibility can be increased up to 10-s degree, with a range of measurement from + 5 ° to - 5 ° by increasing the distance between the two prisms complemented with an electronic fringe dividing technique.
References [1] Chapman, G.D., Interferometric angular measurement, Appl. Opt. 13 (1974) 1646. [-2] Pan Shi and E. Stijns, New optical method for measuring small-angle rotations, Appl. Opt. 27 (1988) 4342.
Highly sensitive angular measurement with a Michelson interferometer Pan Shi* and Erik Stijns Vrije Universiteit Brussel, ALNA-TW, Pleinlaan 2, B-1050 Brussels, Belgium
Abstract. In order to measure small-angle rotations, the two mirrors of a Michelson interferometer are replaced by right-angle prisms. By increasing the distance between the two prisms and by means of an electronic fringe divider technique, the resolution can be increased to 10 5 degree, with a range of measurement from + 5 ° to - 5 °.
Keywords. Rotation measurement, angle measurement, Michelson interferometer, metrology.
1. Introduction Optical methods based on a Michelson interferometer are often used in angular measurements; they have a high sensibility but a limited range of measurement. In 1974 C h a p m a n built a very sensitive set-up with a resolution of 57 ms of arc per fringe and with a range of measurement from + 3 ° to - 3 * [1]. In our earlier balanced set-up we obtained a typical sensibility of 10 . 4 degree with a range of measurement from + 5 ° to - 5 ° [2-1. In this paper the sensibility is increased by a factor of 10. In order to do so we increased the size of the rotating table to separate the two prisms by a larger distance. We also built an electronic "fringe divider"; for one fringe we get four pulses in the counter. In this way we increased the sensibility by an order of magnitude up to 10-5 degree, as compared to [2], while covering the same range of measurement.
* Permanent address: Department of Physics, Dalian University of Technology, Dalian, P.R. China.
Elsevier Industrial Metrology 1 (1990) 69 74 0921-5956/90/$3.50 © 1990. Elsevier Science Publishers B.V.
70
Pan Shi, E. Stijns / Highly sensitive angular measurement
2. The principle Figure 1 shows the principle of our modified Michelson interferometer for measuring rotations. The two end mirrors are replaced by two right-angle prisms which are placed on the rotating table, separated by a distance L. These two prisms together with two fixed mirrors trace back the two laser beams, which form the interference pattern. When the table rotates the optical paths in the two arms of the Michelson interferometer change, and the interference pattern shifts. Counting the fringes at the detector gives, after calibration, the angle of rotation. In [2], we calculated the optical path difference Ap as a function of the angle of rotation 0 for the balanced set-up of Fig. 1: Ap = 2(x/2a + L) sin 0,
(1)
where a is the length of the side of the right angle prism, and L is the distance between the two prisms as shown in Fig. I. If we use another parameter r, the radius of the rotation (see Fig. 1), eqn. (1) can be simplified to Ap = 4r sin 0.
(2)
The number N of fringes passing the detector is N = 2Ap/2,
(3)
so we get 0 = sin- 1N2/8r.
(4)
Pan Shi was born in Liaoning, P.R. China, in 1949. He received the BS in
Physics from the Dalian Institute of Technology in 1982, and the MS in Applied Sciences from the Vrije Universiteit Brussel in 1986. He is currently working towards a PhD in Applied Optics at the Vrije Universiteit Brussel.
Erik Stijns studied Physics at the Vrije Universiteit Brussel, where he obtained his PhD in 1977. He is lecturer on Optics and Laser Physics; his research interests are in Applied Optics.
Pan Shi, E. Stijns / Highly sensitive angular measurement
[ M3
71
DETECTOR
•
/
-7 M1
M2 /////
/////
i
\ o
/7 /
a
Fig. 1. Principle of a Michelson interferometer for the measurement of small-angle rotations.
F r o m (4) we can see that for the same angle of rotation 0, the n u m b e r of fringes passing the detector, i.e. the sensibility, can be increased by increasing r (increasing L, a or both). Figure 2 shows the new set-up. We use a He Ne laser as light source. A spatial filter and lens are used to get an enlarged collimated light beam. This beam is split into two beams, which pass through prism 1 and prism 2, respectively, and which are then reflected back by fixed mirrors 1 and 2. After passing the beam splitter for a second time, they are recombined to form an interference pattern which is detected by detectors 1 and 2. Between mirror 3 and prism 2 we now put a phase plate. It consists of two glass plates put together with a small angle q$, as shown in Fig. 3. In order to obtain a phase difference of 90 °, it can be shown that q$ could be expressed in the form
~b = ~ / 2 d ,
(5)
where d is the thickness of the plate, n is its refractive index and 2 is the wavelength of the laser beam in the plate. When the enlarged beam passes through this phase plate, it is split into two sub-beams with a 90 ° difference in phase. These two beams form the interference pattern with the beam coming from the other arm of the Michelson interferometer. The electronic detection consists of two photodiodes which read the interference pattern as two sinusoidal signals with 90 ° phase difference; following amplification they pass t h r o u g h two shaping circuits into square waves. Using these two waves
Pan Shi, E. Stijns / Highly sensitive angular measurement
72
.•
AMP+Phase Reverser ] +Differentiat or+Logic
Output Pulses~
~ControlVoltage
Up-Down l Counter
M3
7
LASER
-7" M1 /////
M2 ///// Phase = Plate
\
e, 0
r
Fig. 2. Experimental set-up.
0T Fig. 3. Phase plate used to transform a laser beam into two sub-beams with a 90 degree phase difference.
Pan Shi, E. Stijns / Highly sensitive angular measurement
73
we get two other square waves by using a phase reverser. These four square waves with 90 ° phase difference between each other then pass through four differentiators to generate pulses. The square waves and pulses are then sent to two AND gates in such a way that, when the interference fringes shift in some direction, only one AND gate gives an output (high voltage). When the direction of rotation changes the two AND gates change their outputs. Finally the two outputs of the two AND gates pass another AND gate to respond with pulses for the u p - d o w n counter. The outputs of the two AND gates are also used to drive a flip-flop to get the control voltage of the counter, counting either up or down according to the direction of the rotation.
4. Experimental verification The set-up of Fig. 2 was used to verify the principle. Two identical right-angle prisms, with sides a - - 3 0 mm, were placed on the rotating table. In order to get 10 ÷ 5 pulses per degree, the separation of the two prisms should be r = 113.31 mm. Two photodiodes were positioned as shown in Fig. 4 to transform the interference pattern in a response of two sinusoidal signals with a 90 ° phase difference. An oscilloscope was used to check the phase difference of the two signals received by D1 and D2 and adjust it properly by slightly tuning the position of the phase plate with respect to the incident beam. We checked the repeatability of the set-up of Fig. 2, by measuring the maximum angle of the rotating table 36 times. A relative standard deviation a / N between 0.04526% and 0.06109% was obtained. In this experiment, we did not calibrate the set-up, only the repeatability was checked. From [2] we know indeed that the linearity of the set-up is adequate, so that when the set-up response has good linearity and repeatability, it can be used for accurate measurements. The overall accuracy of the actual set-up depends only on the accuracy with which the calibration factor can be determined, using either a highly sensitive autocollimator or gauge blocks.
Fig. 4. Two-photodiodearrangement to transform the response to the interferencepattern in two sinusoidal signals with a 90 degree phase difference.
74
Pan Shi, E. Stijns / Highly sensitive angular measurement
5. Conclusion A Michelson interferometer can be used to measure small-angle rotations in a simple and accurate method. The sensibility can be increased up to 10-s degree, with a range of measurement from + 5 ° to - 5 ° by increasing the distance between the two prisms complemented with an electronic fringe dividing technique.
References [1] Chapman, G.D., Interferometric angular measurement, Appl. Opt. 13 (1974) 1646. [-2] Pan Shi and E. Stijns, New optical method for measuring small-angle rotations, Appl. Opt. 27 (1988) 4342.