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Interferometric analysis for small rotation
Optics and Lasers in Engineering 4 (1983) 8 l-90
Interferometric
Jacob
Analysis for Small Rotation
Wen-Kuang
Physics Department,
Towson
Huang
and Song Yu-Dong?
State University, (Received:
Baltimore,
Maryland
21204,
USA
12 July 1982)
ABSTRACT Both holographic interferometry and speckle interferometry are used to analyse small rotation. The experimental set-up and results are presented. A brief comparison of the two methods is also given.
1.
INTRODUCTION
The application of holography and speckle technique to metrology is a major improvement in the field of interferometry. It makes possible the analysis of displacements of opaque objects.1*2 Using an unpolished metal plate as the opaque object, we analyse its small outof-plane rotations from 1.4 x lo-’ to 2.8 X lop4 rad holographically and measure the rotations from 1.4 x 10e4 to 3.4X lop3 rad by speckle interferometry. The experimental results agree well with the setting value. In both cases, Agfa 8E75 film and the same processing procedure are used as in a previous paper.3 2.
HOLOGRAPHIC
ANALYSIS
A sketch of the experimental set-up is shown in Fig. 1. The plate is a steel 6 in. (150 mm) ruler. For rotation the plate is pushed by a ? On leave from Fuxin Mining Institute, Fuxin, Lianing 123001, The People’s Republic
of China. 81
optics and Lasers in Engineering 0143-8166/83/0004-0081/$03.0~ Publishers
Ltd, England,
1983. Printed
in Northern
Ireland
Applied
Science
.Iacoh Wen-Kuang
82
Huang, Song Yu-Dong
MIRROR
\ROTATING
PLATE
IOmwXe-Ne
Fig. 1.
Experimental
set-up
for recording
the hologram.
micrometer, Fig. 2. The micrometer (MITUTOYO 250-189) has a graduation of 1 *rn and can be estimated to O-5 pm and that is also the largest experimental error for this experiment. A plastic (6in.) handle is clamped to the micrometer spindle to ease the control of its rotation. The total holographic exposure is divided into two sequential exposures. The first one is made with the plate (object) in a position shown in Fig. 1 and the second one is made after the plate is rotated by a selected amount from the initial position. To be more specific, let us consider a surface which is rotated by an angle 8 horizontally. One exposure is made when the object point is at P and another exposure is made while the same point on the object is moved to P’ (Fig. 3). Let us represent the reference wave by eiorand the scattered waves from P and P’ by eip and eiCp-‘), while 6 is the phase shift between the
MICRONIETER
Fig. 2.
Setting
up the small rotation
around
fulcrum
0 with a micrometer.
Interferometric analysis for small rotation
83
ORIGINAL
IO FILM Fig. 3.
Schematic diagram for fringe analysis.
two waves travelled through P and P’, or 6=F =F For the adjacent
(AP + PB) = F
PP’(sin Or+ sin 13,)
(xO)(sin 8, + sin
es>
points on the surface A6 = $
(Axe)(sin 8t + sin 0,)
(1)
The intensity of the first exposure is 11 = (cia + ciP (2= 2 + ei(a-P) + c+(a-p) Similarly the intensity distribution of the second exposure is 12= leiu +
ei(P--S)12=2+
ei(or-P+S)+
e-i(or-P+S)
The intensity of the total exposure is I = II +12. After processing, if the hologram is illuminated by the reference wave, the amplitude of the diffracted wave is
The first term is the reference wave. The second and third terms represent diffuse waves that do not form images. They can, however, contribute to a uniform background and reducing contrast. The last two terms are the reconstructed object waves scattered from P and P’. The resultant intensity of these two waves is I= I&@+ei(@-s)12= (1+e-is12=2(1+cos
6)
(2)
84
Jacob Wen-Kuang
Huang, Song Yu-Dong
a sinusoidal function of 6. One sees a virtual image with fringe separations varying with 6. From eqns. (2) and (1) the fringe spacing will be Ax such that A6 = 2~. In our experiment, 8, = 8s = 70” sin B1+ sin 8, = 1.88 Hence, Ax=-
A 1*88y
or
(3) 0*53A ‘Y=
Ax
For pure rotation Ax is the same for any two adjacent fringes. If the hologram is illuminated with the conjugate reference wave, we have p(II
+
12)
=
4p +
+ e-iP
e-i(2a-13) +
+
e-i(2a-P+S)
e-i(P-S)
The first term is the conjugate reference wave. The second and third terms are the conjugate diffused waves. The last two terms are the conjugate object waves. If a photographic film (or a screen) is placed at the object position, a real image with interference fringes varying with 6 will be formed. The intensity, of course, will also be I, = (c-i6 + e-i(P-S) (2 _ -2(1+cos
S)
To record the fringes no camera is needed in this case because the light travels backward towards the original object. Most of our pictures were taken in this way. (The results from the two methods, virtual or real, are identical for the line of viewing direction except by a scale factor.) In our experiment, for each displacement setting of the plate, three holograms were made. The experimentally generated straight and uniform fringe patterns compared with the computed ones for each setting. The consistency of the fringes was within the limit of the conservative estimate of the micrometer reading (Fig. 4).
Interferometric
Fig. 4.
Fringe
patterns
analysis for small rotation
by holographic interferometry. bottom, y = 2.43 x 10 4 rad.
Top,
85
y = 1.00 x 10 -4 rad;
86
Jacob Wen-Kuang
3.
SPECKLE
Huang, Song Yu-Dong
INTERFEROMETRIC
METHOD
The speckle method involves two steps. First, a double-exposure specklegram which records the small rotation is made with the optical arrangement illustrated in Fig. 5. Second, the specklegram is illuminated with a laser beam to form the Young’s fringe pattern.
PLANK fig. 5.
EILti
Optical arrangement for recording small rotation by speckle interferometry.
In the first step, a narrow beam from a 5 mW He-Ne laser illuminates the diffusely reflecting surface of a rectangular alluminium plank which we use as the rotable object instead of the steel ruler in Fig. 2. The scattering light from different parts of the diffuse surface will interfere with one another to form speckles in space. A piece of sandwiched film is put at the back focal plane of the lens and exposed twice. Between the two exposures the plank is rotated. Two speckle patterns which are shifted a little bit on the film will act like Young’s double-slit. When the developed film is illuminated with a beam from a 2 mW He-Ne laser directly, a pattern which consists of a bright central spot due to the transmitted light surrounded by a set of Young’s fringes can be seen on a screen. The experimental results are shown in Fig. 6. They agree well with the equation which we have derived for our experimental situation4 i.e. the rotation AD y = f(2 cos 0) d where D is the separation between the specklegram and the screen, f is the focal length of the lens and d is the fringe spacing on the screen.
Interferometric analysis for small rotation
87
Fig. 6. Fringe patterns by speckle interferometry. Top, y = 5.71 x 10m4rad, D = 100 cm; centre, y = 8.57 X 10m4rad, D = 100 cm; bottom, y = 18.57 x lo-“ rad, D = 100 cm. For the most of sixteen data points, the experimental errors in y are less than 2%.
88
Jacob
Huang,Song Yu-Dong
Wen-Kuang
4.
DISCUSSION
To analyse small rotation holographically, the measurable are restricted by the fringe spacing. In our experiment,
rotations the fringe
patterns which correspond to rotations beyond 2.8 x 10P4 rad are too close-packed to resolve, while the speckle interferometric method is workable
in the
range
1.4 x 1OF” to
3.4 x lo-” rad.
So
these
two
analyses are complementary in a sense. The measurable limitations in the speckle method are related to the speckle size which is proportional to the f number of the lens. Choosing the lens with the smaller f number is good for measuring a larger rotation. On the other hand, the lower limit to resolution rotation is restricted by the shift of speckle pattern which should be more than the speckle size.
Fig.
7.
Fringe
patterns
relating
to
Setting
the value
bending is 6 pm.
by
holographic
interferometry.
Interferometric
89
analysis for small rotation
In comparison with the set-up for holographic analysis (Fig. l), the optical system for speckle method, as shown in Fig. 5, is much simpler. No vibration-free table is needed. However, the set-up we used for measuring small rotation by holography can be applied to the bending analysis directly. In such a case, we set the steel ruler with right end 0 fixed to a rigid block and push the left end of the ruler to a setting position. The resultant fringe patterns relating to bending are illustrated in Fig. 7. Fringe spacing Ax varies continuously across the ruler. If we count the bright fringes from the fixed end, the nth bright fringe corresponds to a phase shift 6 = 27rN. From Fig. 3, the normal deformation of these points where the Nth bright fringe is located can be calculated as pp’=
The experimental
/
/
/
/
/
(5)
results are shown in Fig. 8.
0 1 THE SCALE ON THE RULER (INCH)
/
NA sin Bit-sin &
/
/
/
//
/
/
/’
/
/
/’
/
/
2
/
/
d
,/’
/
/ I’
/
, /'
/
/
4 _/r,';/ - cc cx 00 ,x //' .& / /'
,/’
I’
3
5
/i/'
-2
/4’
-4
-6
/ /’ SETTId VALUES I Q
NORMAL DEFORMATION Fig. 8.
I8
The abscissa is the scale on the ruler. The ordinate is the normal deformation (pm) calculated from experimental results.
90
Jacob Wen-Gang Huang, Song Yu-Dong
REFERENCES 1. C. M. Vest, Holographic interferometry, John Wiley & Sons, New York, 1979. 2. W. E. Tagliaferro and P. L. Lee, Am. J. Phys., 46, (1978) 46. 3. J. W.-K. Huang, Am. J. Phys. 46, (1978) 737. 4. Y.-D. Song and J. W.-K. Huang, Am. J. Phys. 50, (1982) 664.
Interferometric
Jacob
Analysis for Small Rotation
Wen-Kuang
Physics Department,
Towson
Huang
and Song Yu-Dong?
State University, (Received:
Baltimore,
Maryland
21204,
USA
12 July 1982)
ABSTRACT Both holographic interferometry and speckle interferometry are used to analyse small rotation. The experimental set-up and results are presented. A brief comparison of the two methods is also given.
1.
INTRODUCTION
The application of holography and speckle technique to metrology is a major improvement in the field of interferometry. It makes possible the analysis of displacements of opaque objects.1*2 Using an unpolished metal plate as the opaque object, we analyse its small outof-plane rotations from 1.4 x lo-’ to 2.8 X lop4 rad holographically and measure the rotations from 1.4 x 10e4 to 3.4X lop3 rad by speckle interferometry. The experimental results agree well with the setting value. In both cases, Agfa 8E75 film and the same processing procedure are used as in a previous paper.3 2.
HOLOGRAPHIC
ANALYSIS
A sketch of the experimental set-up is shown in Fig. 1. The plate is a steel 6 in. (150 mm) ruler. For rotation the plate is pushed by a ? On leave from Fuxin Mining Institute, Fuxin, Lianing 123001, The People’s Republic
of China. 81
optics and Lasers in Engineering 0143-8166/83/0004-0081/$03.0~ Publishers
Ltd, England,
1983. Printed
in Northern
Ireland
Applied
Science
.Iacoh Wen-Kuang
82
Huang, Song Yu-Dong
MIRROR
\ROTATING
PLATE
IOmwXe-Ne
Fig. 1.
Experimental
set-up
for recording
the hologram.
micrometer, Fig. 2. The micrometer (MITUTOYO 250-189) has a graduation of 1 *rn and can be estimated to O-5 pm and that is also the largest experimental error for this experiment. A plastic (6in.) handle is clamped to the micrometer spindle to ease the control of its rotation. The total holographic exposure is divided into two sequential exposures. The first one is made with the plate (object) in a position shown in Fig. 1 and the second one is made after the plate is rotated by a selected amount from the initial position. To be more specific, let us consider a surface which is rotated by an angle 8 horizontally. One exposure is made when the object point is at P and another exposure is made while the same point on the object is moved to P’ (Fig. 3). Let us represent the reference wave by eiorand the scattered waves from P and P’ by eip and eiCp-‘), while 6 is the phase shift between the
MICRONIETER
Fig. 2.
Setting
up the small rotation
around
fulcrum
0 with a micrometer.
Interferometric analysis for small rotation
83
ORIGINAL
IO FILM Fig. 3.
Schematic diagram for fringe analysis.
two waves travelled through P and P’, or 6=F =F For the adjacent
(AP + PB) = F
PP’(sin Or+ sin 13,)
(xO)(sin 8, + sin
es>
points on the surface A6 = $
(Axe)(sin 8t + sin 0,)
(1)
The intensity of the first exposure is 11 = (cia + ciP (2= 2 + ei(a-P) + c+(a-p) Similarly the intensity distribution of the second exposure is 12= leiu +
ei(P--S)12=2+
ei(or-P+S)+
e-i(or-P+S)
The intensity of the total exposure is I = II +12. After processing, if the hologram is illuminated by the reference wave, the amplitude of the diffracted wave is
The first term is the reference wave. The second and third terms represent diffuse waves that do not form images. They can, however, contribute to a uniform background and reducing contrast. The last two terms are the reconstructed object waves scattered from P and P’. The resultant intensity of these two waves is I= I&@+ei(@-s)12= (1+e-is12=2(1+cos
6)
(2)
84
Jacob Wen-Kuang
Huang, Song Yu-Dong
a sinusoidal function of 6. One sees a virtual image with fringe separations varying with 6. From eqns. (2) and (1) the fringe spacing will be Ax such that A6 = 2~. In our experiment, 8, = 8s = 70” sin B1+ sin 8, = 1.88 Hence, Ax=-
A 1*88y
or
(3) 0*53A ‘Y=
Ax
For pure rotation Ax is the same for any two adjacent fringes. If the hologram is illuminated with the conjugate reference wave, we have p(II
+
12)
=
4p +
+ e-iP
e-i(2a-13) +
+
e-i(2a-P+S)
e-i(P-S)
The first term is the conjugate reference wave. The second and third terms are the conjugate diffused waves. The last two terms are the conjugate object waves. If a photographic film (or a screen) is placed at the object position, a real image with interference fringes varying with 6 will be formed. The intensity, of course, will also be I, = (c-i6 + e-i(P-S) (2 _ -2(1+cos
S)
To record the fringes no camera is needed in this case because the light travels backward towards the original object. Most of our pictures were taken in this way. (The results from the two methods, virtual or real, are identical for the line of viewing direction except by a scale factor.) In our experiment, for each displacement setting of the plate, three holograms were made. The experimentally generated straight and uniform fringe patterns compared with the computed ones for each setting. The consistency of the fringes was within the limit of the conservative estimate of the micrometer reading (Fig. 4).
Interferometric
Fig. 4.
Fringe
patterns
analysis for small rotation
by holographic interferometry. bottom, y = 2.43 x 10 4 rad.
Top,
85
y = 1.00 x 10 -4 rad;
86
Jacob Wen-Kuang
3.
SPECKLE
Huang, Song Yu-Dong
INTERFEROMETRIC
METHOD
The speckle method involves two steps. First, a double-exposure specklegram which records the small rotation is made with the optical arrangement illustrated in Fig. 5. Second, the specklegram is illuminated with a laser beam to form the Young’s fringe pattern.
PLANK fig. 5.
EILti
Optical arrangement for recording small rotation by speckle interferometry.
In the first step, a narrow beam from a 5 mW He-Ne laser illuminates the diffusely reflecting surface of a rectangular alluminium plank which we use as the rotable object instead of the steel ruler in Fig. 2. The scattering light from different parts of the diffuse surface will interfere with one another to form speckles in space. A piece of sandwiched film is put at the back focal plane of the lens and exposed twice. Between the two exposures the plank is rotated. Two speckle patterns which are shifted a little bit on the film will act like Young’s double-slit. When the developed film is illuminated with a beam from a 2 mW He-Ne laser directly, a pattern which consists of a bright central spot due to the transmitted light surrounded by a set of Young’s fringes can be seen on a screen. The experimental results are shown in Fig. 6. They agree well with the equation which we have derived for our experimental situation4 i.e. the rotation AD y = f(2 cos 0) d where D is the separation between the specklegram and the screen, f is the focal length of the lens and d is the fringe spacing on the screen.
Interferometric analysis for small rotation
87
Fig. 6. Fringe patterns by speckle interferometry. Top, y = 5.71 x 10m4rad, D = 100 cm; centre, y = 8.57 X 10m4rad, D = 100 cm; bottom, y = 18.57 x lo-“ rad, D = 100 cm. For the most of sixteen data points, the experimental errors in y are less than 2%.
88
Jacob
Huang,Song Yu-Dong
Wen-Kuang
4.
DISCUSSION
To analyse small rotation holographically, the measurable are restricted by the fringe spacing. In our experiment,
rotations the fringe
patterns which correspond to rotations beyond 2.8 x 10P4 rad are too close-packed to resolve, while the speckle interferometric method is workable
in the
range
1.4 x 1OF” to
3.4 x lo-” rad.
So
these
two
analyses are complementary in a sense. The measurable limitations in the speckle method are related to the speckle size which is proportional to the f number of the lens. Choosing the lens with the smaller f number is good for measuring a larger rotation. On the other hand, the lower limit to resolution rotation is restricted by the shift of speckle pattern which should be more than the speckle size.
Fig.
7.
Fringe
patterns
relating
to
Setting
the value
bending is 6 pm.
by
holographic
interferometry.
Interferometric
89
analysis for small rotation
In comparison with the set-up for holographic analysis (Fig. l), the optical system for speckle method, as shown in Fig. 5, is much simpler. No vibration-free table is needed. However, the set-up we used for measuring small rotation by holography can be applied to the bending analysis directly. In such a case, we set the steel ruler with right end 0 fixed to a rigid block and push the left end of the ruler to a setting position. The resultant fringe patterns relating to bending are illustrated in Fig. 7. Fringe spacing Ax varies continuously across the ruler. If we count the bright fringes from the fixed end, the nth bright fringe corresponds to a phase shift 6 = 27rN. From Fig. 3, the normal deformation of these points where the Nth bright fringe is located can be calculated as pp’=
The experimental
/
/
/
/
/
(5)
results are shown in Fig. 8.
0 1 THE SCALE ON THE RULER (INCH)
/
NA sin Bit-sin &
/
/
/
//
/
/
/’
/
/
/’
/
/
2
/
/
d
,/’
/
/ I’
/
, /'
/
/
4 _/r,';/ - cc cx 00 ,x //' .& / /'
,/’
I’
3
5
/i/'
-2
/4’
-4
-6
/ /’ SETTId VALUES I Q
NORMAL DEFORMATION Fig. 8.
I8
The abscissa is the scale on the ruler. The ordinate is the normal deformation (pm) calculated from experimental results.
90
Jacob Wen-Gang Huang, Song Yu-Dong
REFERENCES 1. C. M. Vest, Holographic interferometry, John Wiley & Sons, New York, 1979. 2. W. E. Tagliaferro and P. L. Lee, Am. J. Phys., 46, (1978) 46. 3. J. W.-K. Huang, Am. J. Phys. 46, (1978) 737. 4. Y.-D. Song and J. W.-K. Huang, Am. J. Phys. 50, (1982) 664.