- Email: [email protected]
Position monitoring technique using double diffraction phenomenon
Volume 27, number 3
OPTICS COMMUNICATIONS
December 1978
POSITION MONITORING TECHNIQUE USING DOUBLE DIFFRACTION PHENOMENON K. PATORSKI ICarsaw Technical University~ Precision Mechanics Department, Institute of Design of Precise and Optical Instruments, 02-525 Warsaw, Poland
Received 7 September 1978
The application of far field double diffraction pattern to absolute position monitoring is analysed. Detection of intensity changes in two symmetrical double diffraction orders of two optical gratings separated by the specified air gap distance provides full information for the system performance.
1. I n t r o d u c t i o n
Monitoring of the absolute position, e.g., the amount of and direction of the movement, is frequently required in various technological fields. Before the last decade the widespread application of optical diffraction gratings for this purpose is observed. Diverse methods, utilize the moir6 effect, double diffraction phenomenon or interference phase detection [1-5]. The well known technique for providing directional information requires two signals being 90 ° out of phase at the output of the measuring system. This quadrature relationship is usually realized by special sector design of one of the gratings, appropriate disposition of photocells or polarisation effects. The purpose of this report is to describe an absolute position monitoring system based on the grating double diffraction phenomenon, in which the quadrature relationship is realised by providing the appropriate air gap separation between two optical gratings.
2. Principle The proposed configuration is basically the spectroscopic moir6 fringe detecting system analysed by Guild [6] and employed by Post for moir6 fringe multiplication [7]. It is shown schematically in fig. 1. The spatially coherent plane wave illuminates two identical diffraction gratings G1 and G2 of period d and separated by an air gap distance z. The lines of the two gratings are assumed mutually parallel. Double diffraction far field diffraction pattern is observed at the back focal plane of lens L2. Itcon~sts of discrete diffraction spots to which the resultant doubly diffractid beams are brought to focus. When one of the gratings is laterally displaced in the direction perpendicular to the grating lines, the intensity of double diffraction spots changes periodically 04 !
O2
L2
°q-N
I °--N
Fig. 1. Double diffraction spectroscopic arrangement for absolute position monitoring. S - point source, L1 - collimator lens, G1 and G2 - diffraction gratings, L2 - transforming objective, P - detection plane. 303
Volume 27, number 3
OPTICS COMMUNICATIONS
December 1978
with period d of the grating. The phase relationship of these intensity modulation changes in various spots depends on the separation distance z of the gratings. By properly adjusting the grid separation distance and detecting two symmetrical diffraction orders +N and - N it is possible to obtain two optical signals with quadrature phase relationship.
3. Analysis The amplitude transmittance of grating G1 is represented in the form TI(X ) = ~ an exp (i2~rnx/d},
(1)
n
where an designates the amplitude factor of the nth Fourier series harmonic, and d is the grating period. Assuming the plane wave illumination, the complex amplitude at a distance z from grating G1 is [8]
A'(x, z) ~ ~ a n exp(-iTm2Xz/d2)exp {i27mx/d}.
(2)
n
The second grating G2 is represented as T2(x ) = ~ am exp {i2zrm(x + Ax)Id},
(3)
m
where Ax relates to the value of mutual transverse displacement of grating G2 with respect to G1. The diffraction field just behind grating 132 equals
A"(x, z) ~ A'(x, z)T2(x ) = ~[~[~ arean exp(-irrXn2z/d2)exp {i2rr [(m + n)x + mAxl ) m
n
=a 2 +a 0 m~=l= am exp {i2n'm(x + Ax)/d}+a 0 m~=_1 am exp{i27rm(x + Ax)/d} oo
+ a0 ~
_
oo
an exp(-irr~n2z/d)exp {i21rnx/d} + a0 n=-I ~ an exp(-iTrkzn2/d)exp {i2rmx/d}
+ ~ a2n exp(-iTr~.n2/d2)exp (i2rm(2x + Ax)/d} m=n~O +
~ area n exp(-irrMn2/d2)exp (i2rt[(m + n)x + mAx]/d}. m -~n --/=O
(4)
Eq. (4) is written in the form enabling simple recognition and interpretation of subsequent stages of double diffraction at gratings G1 and G2. Let us analyse the double diffracted beam of spatial frequency N/d. Its amplitude is derived from eq. (4) as
A N = aoa N exp {i2zrN(x + Ax)/d} + aNa 0 exp(-ircMN2/d2)exp {i2rrNx/d} + a2Ni2 exp {ircM(N/2)2/d 2 } exp {i2zr[Nx + (N/2) Ax] Id} +anaN_ n exp(iTrkzn2/d2)exp (i21r [Nx + ( N - n)Ax]/d}. The resulting intensity modulation function [AN ]2 in the Nth double diffraction spot takes in a general case a
304
(5)
Volume 27, number 3
OPTICS COMMUNICATIONS
December 1978
rather complicated form. It strongly depends on the fourier coefficients a m and an defined by the grating profiles. Here, we limit our analysis to the case of frequently employed binary type amplitude gratings. In practice, the odd number double diffraction orders N are usually employed for observation and detection [7] with these gratings. The best situation occurs if Ronchi type gratings with opening ratio 0.5 are used. In this case the even diffraction orders m and n of each of the gratings are missing. Therefore, eq. (5) becomes A N = aoaN exp {i2zrN(x + Ax)/d} + aNa 0 exp {i27r(Nx/d - zN2/2d2)).
(6)
It is seen from eq. (6) that only the two strongest diffraction beams remain, namely, (0,N) and (N, 0), where the numbers in parenthesis designate the diffraction order number at the first and second grating, respectively. Therefore, two beam interference is encountered. The observed intensity is I N = 2a~a2[1 + cos {27r(Nx/a + zN2/2d2))].
(7)
Similarly, it might be shown that the intensity of the symmetrical - N double diffraction order is expressed as L N = 2a2a2[ l + cos {2;r(Nx/a - zN2/2d)}].
(8)
The following properties can be deduced from eqs. (7) and (8). - The frequency of the intensity modulation imposed by the transverse grating shift Ax is directly proportional to the order number N, in which the observation is conducted. - The phase shift XzN2[2d 2 is directly proportional to the air gap separation distance z between the gratings and to the square of double diffraction order number N. - The contrast of the intensity modulation curves does not depend on the separation distance z of the gratings. From eqs. (7) and (8) the phase difference between the intensity modulation curves detected in orders -iV and +Nis q~(N,z) = 27rXzN2/d2.
(9)
Now, the special cases can be discussed. Ifz equals the Talbot distance d2/~ or its multiplicity, the phase difference becomes a multiplicity of 2~r. For the planes lying in the middle between the self-images formed at Talbot distances, the phase difference becomes 7r. These two cases were already discussed in a previous paper [9]. Here, we aim at a quadrature phase relationship between the orders -At and +N. It is satisfied at distances z z = (M+ 1/4N2)d2/X,
(10)
where M = 0, 1, 2 . . . . . It is seen, that the higher double diffraction order N is used (for increasin] the position monitoring sensitivity); the number of quadrature relationship planes over the Talbot distance dZ/~. increases proportionally to N 2. 4. E x p e r i m e n t a l w o r k
For the experiments Ronchi type amplitude gratings of period 0.1 and 0.025 mm were used. The optical system shown schematically in fig. I was realized; signals from two photodetectors set at orders -At and +N (first, third and fifth diffraction orders were investigated) were recorded by an X Y plotter as a function of air gap distance z. The obtained curves of intensity modulation as function of the lateral shift Ax as well as plotted Lissajou figures provided an excellent verification of the principle derived above. 5. Conclusion The characteristics of a double diffraction moir6 detecting system were studied analytically and experimentally 305
Volume 27, number 3
OPTICS COMMUNICATIONS
December 1978
from the point o f view of realisation of a phase quadrature relationship required for absolute position monitoring. Detection of two photo-electric signals in double diffraction orders symmetrically located with respect to the optical axis was proposed. The phase delay between the signals is easily adjustable by the control of the air gap distance between the gratings. The derived phase relationships between the diffraction orders as function of mutual gratings displacement can serve as a general contribution to the theory of the double diffraction phenomenon.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L.A. Sayce, J. Phys. E 5 (1972) 193. K. Kodate, T. Makiya and M. Kamiyama, Jap. J. Appl. Phys. 10 (1971) 104. G. Makosz, Appl. Opt. 12 (1973) 2054. D.P. Jablonowski and J. Taamot, Appl. Opt. 15 (1976) 1437. Y. Torii and Y. Mizushima, Opt. Commun. 23 (1977) 135. J. Guild, The interference systems of crossed diffraction gratings: theory of moire fringes (Clarendon Press, Oxofrd, 1956). D. Post, AppL Opt. 6 (1967) 1939. R.F. Edgar, Opt. Acta 16 (1969) 281. K. Patorski, S. Yokozeki and T. Suzuki, Appl. Opt. 15 (1976) 1234.
306
OPTICS COMMUNICATIONS
December 1978
POSITION MONITORING TECHNIQUE USING DOUBLE DIFFRACTION PHENOMENON K. PATORSKI ICarsaw Technical University~ Precision Mechanics Department, Institute of Design of Precise and Optical Instruments, 02-525 Warsaw, Poland
Received 7 September 1978
The application of far field double diffraction pattern to absolute position monitoring is analysed. Detection of intensity changes in two symmetrical double diffraction orders of two optical gratings separated by the specified air gap distance provides full information for the system performance.
1. I n t r o d u c t i o n
Monitoring of the absolute position, e.g., the amount of and direction of the movement, is frequently required in various technological fields. Before the last decade the widespread application of optical diffraction gratings for this purpose is observed. Diverse methods, utilize the moir6 effect, double diffraction phenomenon or interference phase detection [1-5]. The well known technique for providing directional information requires two signals being 90 ° out of phase at the output of the measuring system. This quadrature relationship is usually realized by special sector design of one of the gratings, appropriate disposition of photocells or polarisation effects. The purpose of this report is to describe an absolute position monitoring system based on the grating double diffraction phenomenon, in which the quadrature relationship is realised by providing the appropriate air gap separation between two optical gratings.
2. Principle The proposed configuration is basically the spectroscopic moir6 fringe detecting system analysed by Guild [6] and employed by Post for moir6 fringe multiplication [7]. It is shown schematically in fig. 1. The spatially coherent plane wave illuminates two identical diffraction gratings G1 and G2 of period d and separated by an air gap distance z. The lines of the two gratings are assumed mutually parallel. Double diffraction far field diffraction pattern is observed at the back focal plane of lens L2. Itcon~sts of discrete diffraction spots to which the resultant doubly diffractid beams are brought to focus. When one of the gratings is laterally displaced in the direction perpendicular to the grating lines, the intensity of double diffraction spots changes periodically 04 !
O2
L2
°q-N
I °--N
Fig. 1. Double diffraction spectroscopic arrangement for absolute position monitoring. S - point source, L1 - collimator lens, G1 and G2 - diffraction gratings, L2 - transforming objective, P - detection plane. 303
Volume 27, number 3
OPTICS COMMUNICATIONS
December 1978
with period d of the grating. The phase relationship of these intensity modulation changes in various spots depends on the separation distance z of the gratings. By properly adjusting the grid separation distance and detecting two symmetrical diffraction orders +N and - N it is possible to obtain two optical signals with quadrature phase relationship.
3. Analysis The amplitude transmittance of grating G1 is represented in the form TI(X ) = ~ an exp (i2~rnx/d},
(1)
n
where an designates the amplitude factor of the nth Fourier series harmonic, and d is the grating period. Assuming the plane wave illumination, the complex amplitude at a distance z from grating G1 is [8]
A'(x, z) ~ ~ a n exp(-iTm2Xz/d2)exp {i27mx/d}.
(2)
n
The second grating G2 is represented as T2(x ) = ~ am exp {i2zrm(x + Ax)Id},
(3)
m
where Ax relates to the value of mutual transverse displacement of grating G2 with respect to G1. The diffraction field just behind grating 132 equals
A"(x, z) ~ A'(x, z)T2(x ) = ~[~[~ arean exp(-irrXn2z/d2)exp {i2rr [(m + n)x + mAxl ) m
n
=a 2 +a 0 m~=l= am exp {i2n'm(x + Ax)/d}+a 0 m~=_1 am exp{i27rm(x + Ax)/d} oo
+ a0 ~
_
oo
an exp(-irr~n2z/d)exp {i21rnx/d} + a0 n=-I ~ an exp(-iTrkzn2/d)exp {i2rmx/d}
+ ~ a2n exp(-iTr~.n2/d2)exp (i2rm(2x + Ax)/d} m=n~O +
~ area n exp(-irrMn2/d2)exp (i2rt[(m + n)x + mAx]/d}. m -~n --/=O
(4)
Eq. (4) is written in the form enabling simple recognition and interpretation of subsequent stages of double diffraction at gratings G1 and G2. Let us analyse the double diffracted beam of spatial frequency N/d. Its amplitude is derived from eq. (4) as
A N = aoa N exp {i2zrN(x + Ax)/d} + aNa 0 exp(-ircMN2/d2)exp {i2rrNx/d} + a2Ni2 exp {ircM(N/2)2/d 2 } exp {i2zr[Nx + (N/2) Ax] Id} +anaN_ n exp(iTrkzn2/d2)exp (i21r [Nx + ( N - n)Ax]/d}. The resulting intensity modulation function [AN ]2 in the Nth double diffraction spot takes in a general case a
304
(5)
Volume 27, number 3
OPTICS COMMUNICATIONS
December 1978
rather complicated form. It strongly depends on the fourier coefficients a m and an defined by the grating profiles. Here, we limit our analysis to the case of frequently employed binary type amplitude gratings. In practice, the odd number double diffraction orders N are usually employed for observation and detection [7] with these gratings. The best situation occurs if Ronchi type gratings with opening ratio 0.5 are used. In this case the even diffraction orders m and n of each of the gratings are missing. Therefore, eq. (5) becomes A N = aoaN exp {i2zrN(x + Ax)/d} + aNa 0 exp {i27r(Nx/d - zN2/2d2)).
(6)
It is seen from eq. (6) that only the two strongest diffraction beams remain, namely, (0,N) and (N, 0), where the numbers in parenthesis designate the diffraction order number at the first and second grating, respectively. Therefore, two beam interference is encountered. The observed intensity is I N = 2a~a2[1 + cos {27r(Nx/a + zN2/2d2))].
(7)
Similarly, it might be shown that the intensity of the symmetrical - N double diffraction order is expressed as L N = 2a2a2[ l + cos {2;r(Nx/a - zN2/2d)}].
(8)
The following properties can be deduced from eqs. (7) and (8). - The frequency of the intensity modulation imposed by the transverse grating shift Ax is directly proportional to the order number N, in which the observation is conducted. - The phase shift XzN2[2d 2 is directly proportional to the air gap separation distance z between the gratings and to the square of double diffraction order number N. - The contrast of the intensity modulation curves does not depend on the separation distance z of the gratings. From eqs. (7) and (8) the phase difference between the intensity modulation curves detected in orders -iV and +Nis q~(N,z) = 27rXzN2/d2.
(9)
Now, the special cases can be discussed. Ifz equals the Talbot distance d2/~ or its multiplicity, the phase difference becomes a multiplicity of 2~r. For the planes lying in the middle between the self-images formed at Talbot distances, the phase difference becomes 7r. These two cases were already discussed in a previous paper [9]. Here, we aim at a quadrature phase relationship between the orders -At and +N. It is satisfied at distances z z = (M+ 1/4N2)d2/X,
(10)
where M = 0, 1, 2 . . . . . It is seen, that the higher double diffraction order N is used (for increasin] the position monitoring sensitivity); the number of quadrature relationship planes over the Talbot distance dZ/~. increases proportionally to N 2. 4. E x p e r i m e n t a l w o r k
For the experiments Ronchi type amplitude gratings of period 0.1 and 0.025 mm were used. The optical system shown schematically in fig. I was realized; signals from two photodetectors set at orders -At and +N (first, third and fifth diffraction orders were investigated) were recorded by an X Y plotter as a function of air gap distance z. The obtained curves of intensity modulation as function of the lateral shift Ax as well as plotted Lissajou figures provided an excellent verification of the principle derived above. 5. Conclusion The characteristics of a double diffraction moir6 detecting system were studied analytically and experimentally 305
Volume 27, number 3
OPTICS COMMUNICATIONS
December 1978
from the point o f view of realisation of a phase quadrature relationship required for absolute position monitoring. Detection of two photo-electric signals in double diffraction orders symmetrically located with respect to the optical axis was proposed. The phase delay between the signals is easily adjustable by the control of the air gap distance between the gratings. The derived phase relationships between the diffraction orders as function of mutual gratings displacement can serve as a general contribution to the theory of the double diffraction phenomenon.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L.A. Sayce, J. Phys. E 5 (1972) 193. K. Kodate, T. Makiya and M. Kamiyama, Jap. J. Appl. Phys. 10 (1971) 104. G. Makosz, Appl. Opt. 12 (1973) 2054. D.P. Jablonowski and J. Taamot, Appl. Opt. 15 (1976) 1437. Y. Torii and Y. Mizushima, Opt. Commun. 23 (1977) 135. J. Guild, The interference systems of crossed diffraction gratings: theory of moire fringes (Clarendon Press, Oxofrd, 1956). D. Post, AppL Opt. 6 (1967) 1939. R.F. Edgar, Opt. Acta 16 (1969) 281. K. Patorski, S. Yokozeki and T. Suzuki, Appl. Opt. 15 (1976) 1234.
306