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Fiber-optic Moiré Interference Principle
OPTICAL FIBER TECHNOLOGY ARTICLE NO.
4, 224]232 Ž1998.
OF980247
Fiber-optic Moire ´ Interference Principle Libo Yuan Department of Physics, Harbin Engineering Uni¨ ersity, Harbin 150001, People’s Republic of China
and Limin Zhou Department of Mechanical Engineering, The Hong Kong Polytechnic Uni¨ ersity, Hong Kong Received October 24, 1997
A novel fiber-optic interference method called fiber-optic Moire ´ interferometry has been developed and demonstrated. Fiber-optic moire ´ interference is based on the special relative positions and the polarization directions of three HiBi fiber ends. A helium]neon laser and three HiBi fibers are used to configure a moire ´ interferometer in our experiments. The optical field intensity distribution function of the moire ´ interference pattern is analyzed and the simulation results as compared with experimental interference patterns are given. Q 1998 Academic Press
I. INTRODUCTION
The moire ´ phenomenon is based on the fact that when two or more gratings that lie in contact with a small angle between the grating lines, we see a fringe pattern of much lower frequency than for the individual gratings. This is an example of the moire ´ effect and the resulting fringes are called moire´ fringes or moire´ pattern w1x. The mathematical description of moire ´ pattern resulting from the superposition of sinusoidal gratings is the same as for interference patterns formed by two plane waves. The moire ´ effect is therefore often termed mechanical interference. The moire technique can be used to measure displacement by two angularity´ w x displaced gratings 2 or to measure in-plane deformation and strains w3x. It also can be used in the measurement of out-of-plane deformations such as the shadowmoire ´ technique w4x or the projected fringes method as well as the reflection moire´ technique w5x, the latter of which is called Lighternberg’s method. Moire ´ patterns can be formed by many types of gratings, for example, circular gratings, radial 224 1068-5200r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
´ INTERFEROMETRY FIBER-OPTIC MOIRE
225
gratings, circular zone-plates, spirals, etc. Lehmann and Wiemer w6x, as early as 1953, presented the theory of moire ´ patterns formed by different types of gratings based on the indicial representation method. Pirard w7x, in 1960, gave a complete discussion of moire ´ phenomena produced by line, radial, circular, and parabolic gratings, as well as their most interesting combinations which was also based on the method of indicial representation of sets of curves. In this paper we present a new method whereby the moire ´ patterns are obtained by three HiBi single-mode optical fibers other than gratings. The experiments and the theoretical analysis method of the fiber optic moire ´ interference technique are given in Sections II and III. Finally, the simulation results as compared with the experimental interference patterns are given.
´ INTERFERENCE PHENOMENA II. FIBER-OPTIC MOIRE For the case of two single-mode optical fiber, the experimental background of fiber-optic Mach]Zehnder interferometry was provided in Ref. w8x. The interference pattern is shown in Fig. 1a. Compared with two fiber optic interference fringes, the typical three fiber moire ´ interference patterns fringes are given by Figs. 1b and 1c. Figure 1b looks like a moire ´ pattern formed by a small angle and illuminated by a diffuser; Fig. 1c looks like moire ´ fringes formed by parallel gratings of slightly different pitch, and the fringes run parallel to the rulings of the gratings. Figure 2 shows the schematic diagram of a three HiBi fiber moire ´ interferometer optical setup. A stable helium]neon laser was chosen as the light source in the interferometer. The light from a 633-nm He]Ne laser is split by two prism splitters, undergoes three polarizers, and then is launched into the three single-mode HiBi fibers. At the output end, the three HiBi fibers are inserted into a glass tube and glued with epoxy. From the front view of the polished three fiber ends, its relative positions are shown in Fig. 3 and the three fiber end output optical field polarization directions are shown in Fig. 4. The CCD detecting target is mounted on a solid frame and perpendicular to the three fiber ends. The distance between the CCD detecting target surface and the three fiber end is D. The HiBi fiber parameters and the three fiber end coordinate positions and output optical field polarization
FIG. 1. Fiber-optic Mach]Zehnder and moire ´ interference patterns. Ža. Typical two single-mode optical fiber Mach]Zehnder interference pattern. Žb. Typical three HiBi single-mode fiber moire ´ interference patterns. Žc. Typical three HiBi single-mode fiber moire ´ interference patterns.
226
YUAN AND ZHOU
FIG. 2. Three fiber-optic moire ´ interference experimental setup.
direction angles from the experiments are given in Tables 1 and 2, respectively. For different relative positions and polarization direction states of the three fiber ends, different moire ´ interference patterns can be obtained as shown in Figs. 1b and 1c. III. THEORY ANALYSIS
Consider the general case of three HiBi fiber ends arranged as in Fig. 5. the three fiber end surfaces lie in the j Oz plane and the fiber 1, 2, and 3 coordinate positions of the core center point are F1Ž0, 0., F2 Ž a, b ., and F3 Ž c, d ., respectively Žsee Fig. 4.. The observation plane XOY ŽCCD target surface plane. is opposite from the fiber end plane and is at a distance D. For the three HiBi fibers, based on the superposition principle, the lightwave vector at arbitrary point QŽ x, y . in the observation plane XOY can be expressed as
3
Ý < Pk : ,
Ž 1.
ks1
where < Pk : s Uk eyjŽ v tq f k . cos u k < Px : q sin u k < Py : 4 ,
k s 1, 2, 3,
Ž 2.
represents the linearly polarized lightwave vector in direction u k of HiBi fibers 1, 2, and 3. Here, v and f k are the circularity frequency and the phase of the lightwave. Ui is the amplitude of lightwave and < Px : s 1 , 0
ž/
< Py : s 0 1
ž/
FIG. 3. Three fiber end coordinate system.
Ž 3.
´ INTERFEROMETRY FIBER-OPTIC MOIRE
227
FIG. 4. Three fiber polarization direction coordinate system.
TABLE 1 HiBi Single-Mode Optical Fiber Characteristics Quantity Type Cutoff wavelength Attenuation Measured beat length B Numerical aperture Optimum launch spot-size Fiber OD Coating OD
Values HB600 595 nm 10 dBrKm at 633 nm 1.15 mm at 633 nm 5.5 = 10y4 0.17 3 mm 125 m m 240 m m
TABLE 2 Three HiBi Fiber Interferometer Parameters Quantity Polarized angle Coordinate parameter Coordinate parameter Coordinate parameter Coordinate parameter Distance Wavelength
Symbol
Values wFig. 1bx
Values wFig. 1cx
u1 s u2 s u3 a b c d D l
908 82 m m 150 m m 150 m m 450 m m 1598 m m 0.633 m m
908 197 m m 0 405 m m 0 1598 m m 0.633 m m
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YUAN AND ZHOU
are base vectors representing waves, linearly polarized in the x and y directions, respectively. Then the intensity distribution can be calculated as 3
I s ²U < U : s
3
Ý Ý
Uk Um eyjŽ f ky f m . cos u k² Px < q sin u k² Py < 4
ks1 ms1
? cos um < Px : q sin um < Py : 4 3
s
3
Ý Ý
Uk Um eyjŽ f ky f m . cos u k cos um q sin u k sin um 4
Ž 4.
ks1 ms1 3
s
3
Ý Uk2 q ks1
Uk Um cos Ž f k y fm . cos Ž u k y um .
Ý ks1, ms1 k/m
since ² Px < Px : s ² Py < Py : s 1,
² Px < Py : s ² Py < Px : s 0.
Ž 5.
The phase f k of Eq. Ž4. can be written
fk s
2p
l
Ž n c l k q rk . q f k 0 ,
k s 1, 2, 3,
Ž 6.
where n c is the refractive index of the fiber core, l k Ž k s 1, 2, 3. is the length of the fiber k, r k is the distance between the fiber end center and the point QŽ x, y ., and f k 0 is the initial phase of the lightwave vector < Pk : formed by the end of fiber k. Thus we have
f k y fm s
2p
l
Ž r k y rm . q f k m ,
k, m s 1, 2, 3.
Ž 7.
Here,
fk m s
2p
l
n c Ž l k y l m . q Ž f k 0 y fm 0 . ,
k, m s 1, 2, 3.
Ž 8.
Assume that the three fibers have not been disturbed and the initial phases are determined by the polarizers. Then f k m can be considered as a constant phase. Under the condition D 4 x, y
Ž 9.
´ INTERFEROMETRY FIBER-OPTIC MOIRE
229
FIG. 5. Arrangement of fiber-optic moire ´ interference system.
approximately, in the coordinate system shown in Fig. 5, we have r1 s D 2 q x 2 q y 2 2
r 2 s D 2 q Ž x y a. q Ž y y b . 2
r3 s D 2 q Ž x y c . q Ž y y d .
1r2
2 1r2
2 1r2
fD 1q fD 1q fD 1q
x2 q y2 2 D2 2 2 Ž x y a. q Ž y y b .
2 D2
Ž 10 .
2 2 Ž x y c. q Ž y y d.
2 D2
and
f2 y f1 s f3 y f1 s f3 y f2 s
2p
lD
2p
lD 2p
lD
y Ž ax q by . q d 21 y Ž cx q dy . q d 31
Ž 11 .
y Ž c y a . x y Ž d y b . y q d 32 ,
where
d 21 s f 21 q d 31 s f 31 q d 32 s f 32 q
p lD
p lD p lD
Ž a2 q b 2 . Ž c2 q d2 .
Ž c 2 q d 2 y a2 y b 2 . .
Ž 12 .
230
YUAN AND ZHOU
Substituting Eq. Ž11. into Ž4., the intensity distribution function of the fiber optic moire ´ interference pattern is obtained as I Ž x, y . s I1 q I2 q I3 q 2 I1 I2 cos Ž u 1 y u 2 . cos
'
q 2 I1 I3 cos Ž u 1 y u 3 . cos
'
q 2 I2 I3 cos Ž u 2 y u 3 . cos
'
½ ½
2p
lD 2p
lD
½
2p
lD
Ž ax q by . y d 21
Ž cx q dy . y d 31
5
5
Ž 13 .
Ž c y a. x q Ž d y b . y y d 32 ,
5
where I1 , I2 , and I3 are the light intensity of fibers 1, 2, and 3 at point QŽ x, y .. From the intensity distribution function Ž13., the visibility of fiber moire ´ interference optical field can be defined as V12 s V13 s V23 s
2 I1 I2 cos Ž u 1 y u 2 .
'
I1 q I2 q I3 2 I1 I3 cos Ž u 1 y u 3 .
'
Ž 14 .
I1 q I2 q I3 2 I2 I3 cos Ž u 2 y u 3 .
'
I1 q I2 q I3
for the three interference fringe families, respectively. The linear equations of the three interference fringe line families are given by 2p
lD 2p
lD 2p
lD
Ž ax q by . y d 21 s 2hp Ž 15 . Ž cx q dy . y d 31 s 2 np
Ž c y a. x q Ž d y b . y y d 32 s 2 mp ,
where h , n , m are arbitrary integer numbers. From equation Ž15., the pitches of each family line can be calculated as Ph s Pn s Pm s
Dl 2 Dl 2
ž ž
1 a2 1 c2
Dl
1
2
Ž c y a.
2
q q q
1 b2 1 d2
1r2
/ /
Ž 16 .
1r2
1r2
1
Ž d y b.
2
.
´ INTERFEROMETRY FIBER-OPTIC MOIRE
231
IV. NUMERICAL SIMULATION RESULTS
The fiber-optic moire ´ interference patterns of three HiBi fiber ends arranged as different positions and for some special polarization directions were calculated numerically by the optical field distribution intensity function given above. For comparison of the calculated pattern with the experimental results, a simple mesh program was used to draw the pattern by filling the small blocks with different shades of gray, where the gray value corresponds to the intensity value at each small block center. For brevity, suppose I1 s I2 s I3 s 1r3, and drop the phase constant d i j Ž i, j s 1, 2, 3.. Then, the intensity is simplified as I Ž x, y . s 1 q V12 cos q V23 cos
½
½
2p
lD
2p
lD
5
Ž ax q by . q V13 cos
½
Ž c y a. x q Ž d y b . y
2p
lD
5
Ž cx q dy .
5
Ž 17 .
.
In our experiment, the parameters of the single-mode fiber are core radius s 4 m m, cladding radius r s 62.5 m m. The light wavelength l s 0.633 m m and the
FIG. 6. Experimental fiber-optic moire ´ interference patterns simulation. Ža. Oblique fringe pattern. Žb. Parallel fringe pattern.
232
YUAN AND ZHOU
distance between the fiber ends and the CCD target is D s 1598 m m. The simulation fiber-optic moire ´ patterns are shown in Fig. 6 with the experimental data given by Table 2. For arbitrary u , if u i y u j / p2 Ž i, j s 1, 2, 3; i / j ., when the condition ab / dc is satisfied, the oblique fiber-optic fringe pattern is obtained as shown in Fig. 1b. a When the fiber end relative positions have the relationship ab s dc s dc y y b , the three HiBi fiber interference fringe pattern is formed as in Fig. 1c. For cases given by our experiments, the simulation results are given in Fig. 6. V. CONCLUSIONS
In this paper we present a new method for forming a fiber-optic interference moire ´ pattern. Three HiBi single-mode fibers were used to configurate a fiber-optic moire ´ interferometer. We have analyzed the formation of the interference pattern and deduced a distribution intensity formula of the interference optical field at the end of the three HiBi fibers. By using a numerical simulation method, we present the simulation patterns for comparison with the experimental patterns and evaluate the theoretical results. Combined with moire ´ fringe detecting technique, this new method can be easily used to improve the detecting sensitivity of this kind of fiber-optic sensor. Research results will be presented in the future.
ACKNOWLEDGMENTS This work was supported by the Science Foundation for Youth of Heilongjiang Province, China.
REFERENCES w1x C. A. Sciammarella, ‘‘The moire ´ method}A review,’’ Exp. Mech., vol. 22, 418 Ž1982.. w2x G. T. Reid, ‘‘Moire ´ fringes in metrology,’’ Opt. Lasers Eng., vol. 5, 63 Ž1984.. w3x C. A. Sciammarella, ‘‘Use of gratings in strain analysis,’’ J. Phys. E., vol. 5, 833 Ž1972.. w4x H. Takasaki, ‘‘Moire ´ topography from its birth to practical application,’’ Opt. Lasers Eng., vol. 3, 3 Ž1982.. w5x K. J. Gasvik, ‘‘Moire ´ technique by means of digital image processing,’’ Appl. Opt., vol. 22, 3543 Ž1983.. w6x R. Lehmann and A. Wiemer, ‘‘Untersuchungen zur Teorie der Doppelraster als Mittel zur Messanzeigen,’’ Feingerate Technik, vol. 2, 199 Ž1953.. w7x A. Pirard, ‘‘Consideration sur la methode du moire ´ en photoelasticite,’’ Analyse des Contraintes, Mem. GAMAC, 5, pp. 1]24, 1960. w8x G. B. Hocker, ‘‘Fiber optic sensing of pressure and temperature,’’ Appl. Opt., vol. 18, 1445 Ž1978..
4, 224]232 Ž1998.
OF980247
Fiber-optic Moire ´ Interference Principle Libo Yuan Department of Physics, Harbin Engineering Uni¨ ersity, Harbin 150001, People’s Republic of China
and Limin Zhou Department of Mechanical Engineering, The Hong Kong Polytechnic Uni¨ ersity, Hong Kong Received October 24, 1997
A novel fiber-optic interference method called fiber-optic Moire ´ interferometry has been developed and demonstrated. Fiber-optic moire ´ interference is based on the special relative positions and the polarization directions of three HiBi fiber ends. A helium]neon laser and three HiBi fibers are used to configure a moire ´ interferometer in our experiments. The optical field intensity distribution function of the moire ´ interference pattern is analyzed and the simulation results as compared with experimental interference patterns are given. Q 1998 Academic Press
I. INTRODUCTION
The moire ´ phenomenon is based on the fact that when two or more gratings that lie in contact with a small angle between the grating lines, we see a fringe pattern of much lower frequency than for the individual gratings. This is an example of the moire ´ effect and the resulting fringes are called moire´ fringes or moire´ pattern w1x. The mathematical description of moire ´ pattern resulting from the superposition of sinusoidal gratings is the same as for interference patterns formed by two plane waves. The moire ´ effect is therefore often termed mechanical interference. The moire technique can be used to measure displacement by two angularity´ w x displaced gratings 2 or to measure in-plane deformation and strains w3x. It also can be used in the measurement of out-of-plane deformations such as the shadowmoire ´ technique w4x or the projected fringes method as well as the reflection moire´ technique w5x, the latter of which is called Lighternberg’s method. Moire ´ patterns can be formed by many types of gratings, for example, circular gratings, radial 224 1068-5200r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
´ INTERFEROMETRY FIBER-OPTIC MOIRE
225
gratings, circular zone-plates, spirals, etc. Lehmann and Wiemer w6x, as early as 1953, presented the theory of moire ´ patterns formed by different types of gratings based on the indicial representation method. Pirard w7x, in 1960, gave a complete discussion of moire ´ phenomena produced by line, radial, circular, and parabolic gratings, as well as their most interesting combinations which was also based on the method of indicial representation of sets of curves. In this paper we present a new method whereby the moire ´ patterns are obtained by three HiBi single-mode optical fibers other than gratings. The experiments and the theoretical analysis method of the fiber optic moire ´ interference technique are given in Sections II and III. Finally, the simulation results as compared with the experimental interference patterns are given.
´ INTERFERENCE PHENOMENA II. FIBER-OPTIC MOIRE For the case of two single-mode optical fiber, the experimental background of fiber-optic Mach]Zehnder interferometry was provided in Ref. w8x. The interference pattern is shown in Fig. 1a. Compared with two fiber optic interference fringes, the typical three fiber moire ´ interference patterns fringes are given by Figs. 1b and 1c. Figure 1b looks like a moire ´ pattern formed by a small angle and illuminated by a diffuser; Fig. 1c looks like moire ´ fringes formed by parallel gratings of slightly different pitch, and the fringes run parallel to the rulings of the gratings. Figure 2 shows the schematic diagram of a three HiBi fiber moire ´ interferometer optical setup. A stable helium]neon laser was chosen as the light source in the interferometer. The light from a 633-nm He]Ne laser is split by two prism splitters, undergoes three polarizers, and then is launched into the three single-mode HiBi fibers. At the output end, the three HiBi fibers are inserted into a glass tube and glued with epoxy. From the front view of the polished three fiber ends, its relative positions are shown in Fig. 3 and the three fiber end output optical field polarization directions are shown in Fig. 4. The CCD detecting target is mounted on a solid frame and perpendicular to the three fiber ends. The distance between the CCD detecting target surface and the three fiber end is D. The HiBi fiber parameters and the three fiber end coordinate positions and output optical field polarization
FIG. 1. Fiber-optic Mach]Zehnder and moire ´ interference patterns. Ža. Typical two single-mode optical fiber Mach]Zehnder interference pattern. Žb. Typical three HiBi single-mode fiber moire ´ interference patterns. Žc. Typical three HiBi single-mode fiber moire ´ interference patterns.
226
YUAN AND ZHOU
FIG. 2. Three fiber-optic moire ´ interference experimental setup.
direction angles from the experiments are given in Tables 1 and 2, respectively. For different relative positions and polarization direction states of the three fiber ends, different moire ´ interference patterns can be obtained as shown in Figs. 1b and 1c. III. THEORY ANALYSIS
Consider the general case of three HiBi fiber ends arranged as in Fig. 5. the three fiber end surfaces lie in the j Oz plane and the fiber 1, 2, and 3 coordinate positions of the core center point are F1Ž0, 0., F2 Ž a, b ., and F3 Ž c, d ., respectively Žsee Fig. 4.. The observation plane XOY ŽCCD target surface plane. is opposite from the fiber end plane and is at a distance D. For the three HiBi fibers, based on the superposition principle, the lightwave vector at arbitrary point QŽ x, y . in the observation plane XOY can be expressed as
3
Ý < Pk : ,
Ž 1.
ks1
where < Pk : s Uk eyjŽ v tq f k . cos u k < Px : q sin u k < Py : 4 ,
k s 1, 2, 3,
Ž 2.
represents the linearly polarized lightwave vector in direction u k of HiBi fibers 1, 2, and 3. Here, v and f k are the circularity frequency and the phase of the lightwave. Ui is the amplitude of lightwave and < Px : s 1 , 0
ž/
< Py : s 0 1
ž/
FIG. 3. Three fiber end coordinate system.
Ž 3.
´ INTERFEROMETRY FIBER-OPTIC MOIRE
227
FIG. 4. Three fiber polarization direction coordinate system.
TABLE 1 HiBi Single-Mode Optical Fiber Characteristics Quantity Type Cutoff wavelength Attenuation Measured beat length B Numerical aperture Optimum launch spot-size Fiber OD Coating OD
Values HB600 595 nm 10 dBrKm at 633 nm 1.15 mm at 633 nm 5.5 = 10y4 0.17 3 mm 125 m m 240 m m
TABLE 2 Three HiBi Fiber Interferometer Parameters Quantity Polarized angle Coordinate parameter Coordinate parameter Coordinate parameter Coordinate parameter Distance Wavelength
Symbol
Values wFig. 1bx
Values wFig. 1cx
u1 s u2 s u3 a b c d D l
908 82 m m 150 m m 150 m m 450 m m 1598 m m 0.633 m m
908 197 m m 0 405 m m 0 1598 m m 0.633 m m
228
YUAN AND ZHOU
are base vectors representing waves, linearly polarized in the x and y directions, respectively. Then the intensity distribution can be calculated as 3
I s ²U < U : s
3
Ý Ý
Uk Um eyjŽ f ky f m . cos u k² Px < q sin u k² Py < 4
ks1 ms1
? cos um < Px : q sin um < Py : 4 3
s
3
Ý Ý
Uk Um eyjŽ f ky f m . cos u k cos um q sin u k sin um 4
Ž 4.
ks1 ms1 3
s
3
Ý Uk2 q ks1
Uk Um cos Ž f k y fm . cos Ž u k y um .
Ý ks1, ms1 k/m
since ² Px < Px : s ² Py < Py : s 1,
² Px < Py : s ² Py < Px : s 0.
Ž 5.
The phase f k of Eq. Ž4. can be written
fk s
2p
l
Ž n c l k q rk . q f k 0 ,
k s 1, 2, 3,
Ž 6.
where n c is the refractive index of the fiber core, l k Ž k s 1, 2, 3. is the length of the fiber k, r k is the distance between the fiber end center and the point QŽ x, y ., and f k 0 is the initial phase of the lightwave vector < Pk : formed by the end of fiber k. Thus we have
f k y fm s
2p
l
Ž r k y rm . q f k m ,
k, m s 1, 2, 3.
Ž 7.
Here,
fk m s
2p
l
n c Ž l k y l m . q Ž f k 0 y fm 0 . ,
k, m s 1, 2, 3.
Ž 8.
Assume that the three fibers have not been disturbed and the initial phases are determined by the polarizers. Then f k m can be considered as a constant phase. Under the condition D 4 x, y
Ž 9.
´ INTERFEROMETRY FIBER-OPTIC MOIRE
229
FIG. 5. Arrangement of fiber-optic moire ´ interference system.
approximately, in the coordinate system shown in Fig. 5, we have r1 s D 2 q x 2 q y 2 2
r 2 s D 2 q Ž x y a. q Ž y y b . 2
r3 s D 2 q Ž x y c . q Ž y y d .
1r2
2 1r2
2 1r2
fD 1q fD 1q fD 1q
x2 q y2 2 D2 2 2 Ž x y a. q Ž y y b .
2 D2
Ž 10 .
2 2 Ž x y c. q Ž y y d.
2 D2
and
f2 y f1 s f3 y f1 s f3 y f2 s
2p
lD
2p
lD 2p
lD
y Ž ax q by . q d 21 y Ž cx q dy . q d 31
Ž 11 .
y Ž c y a . x y Ž d y b . y q d 32 ,
where
d 21 s f 21 q d 31 s f 31 q d 32 s f 32 q
p lD
p lD p lD
Ž a2 q b 2 . Ž c2 q d2 .
Ž c 2 q d 2 y a2 y b 2 . .
Ž 12 .
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YUAN AND ZHOU
Substituting Eq. Ž11. into Ž4., the intensity distribution function of the fiber optic moire ´ interference pattern is obtained as I Ž x, y . s I1 q I2 q I3 q 2 I1 I2 cos Ž u 1 y u 2 . cos
'
q 2 I1 I3 cos Ž u 1 y u 3 . cos
'
q 2 I2 I3 cos Ž u 2 y u 3 . cos
'
½ ½
2p
lD 2p
lD
½
2p
lD
Ž ax q by . y d 21
Ž cx q dy . y d 31
5
5
Ž 13 .
Ž c y a. x q Ž d y b . y y d 32 ,
5
where I1 , I2 , and I3 are the light intensity of fibers 1, 2, and 3 at point QŽ x, y .. From the intensity distribution function Ž13., the visibility of fiber moire ´ interference optical field can be defined as V12 s V13 s V23 s
2 I1 I2 cos Ž u 1 y u 2 .
'
I1 q I2 q I3 2 I1 I3 cos Ž u 1 y u 3 .
'
Ž 14 .
I1 q I2 q I3 2 I2 I3 cos Ž u 2 y u 3 .
'
I1 q I2 q I3
for the three interference fringe families, respectively. The linear equations of the three interference fringe line families are given by 2p
lD 2p
lD 2p
lD
Ž ax q by . y d 21 s 2hp Ž 15 . Ž cx q dy . y d 31 s 2 np
Ž c y a. x q Ž d y b . y y d 32 s 2 mp ,
where h , n , m are arbitrary integer numbers. From equation Ž15., the pitches of each family line can be calculated as Ph s Pn s Pm s
Dl 2 Dl 2
ž ž
1 a2 1 c2
Dl
1
2
Ž c y a.
2
q q q
1 b2 1 d2
1r2
/ /
Ž 16 .
1r2
1r2
1
Ž d y b.
2
.
´ INTERFEROMETRY FIBER-OPTIC MOIRE
231
IV. NUMERICAL SIMULATION RESULTS
The fiber-optic moire ´ interference patterns of three HiBi fiber ends arranged as different positions and for some special polarization directions were calculated numerically by the optical field distribution intensity function given above. For comparison of the calculated pattern with the experimental results, a simple mesh program was used to draw the pattern by filling the small blocks with different shades of gray, where the gray value corresponds to the intensity value at each small block center. For brevity, suppose I1 s I2 s I3 s 1r3, and drop the phase constant d i j Ž i, j s 1, 2, 3.. Then, the intensity is simplified as I Ž x, y . s 1 q V12 cos q V23 cos
½
½
2p
lD
2p
lD
5
Ž ax q by . q V13 cos
½
Ž c y a. x q Ž d y b . y
2p
lD
5
Ž cx q dy .
5
Ž 17 .
.
In our experiment, the parameters of the single-mode fiber are core radius s 4 m m, cladding radius r s 62.5 m m. The light wavelength l s 0.633 m m and the
FIG. 6. Experimental fiber-optic moire ´ interference patterns simulation. Ža. Oblique fringe pattern. Žb. Parallel fringe pattern.
232
YUAN AND ZHOU
distance between the fiber ends and the CCD target is D s 1598 m m. The simulation fiber-optic moire ´ patterns are shown in Fig. 6 with the experimental data given by Table 2. For arbitrary u , if u i y u j / p2 Ž i, j s 1, 2, 3; i / j ., when the condition ab / dc is satisfied, the oblique fiber-optic fringe pattern is obtained as shown in Fig. 1b. a When the fiber end relative positions have the relationship ab s dc s dc y y b , the three HiBi fiber interference fringe pattern is formed as in Fig. 1c. For cases given by our experiments, the simulation results are given in Fig. 6. V. CONCLUSIONS
In this paper we present a new method for forming a fiber-optic interference moire ´ pattern. Three HiBi single-mode fibers were used to configurate a fiber-optic moire ´ interferometer. We have analyzed the formation of the interference pattern and deduced a distribution intensity formula of the interference optical field at the end of the three HiBi fibers. By using a numerical simulation method, we present the simulation patterns for comparison with the experimental patterns and evaluate the theoretical results. Combined with moire ´ fringe detecting technique, this new method can be easily used to improve the detecting sensitivity of this kind of fiber-optic sensor. Research results will be presented in the future.
ACKNOWLEDGMENTS This work was supported by the Science Foundation for Youth of Heilongjiang Province, China.
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