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Effects of surface roughness of a moving test body on laser-interferometric displacement measurements
Volume 14, number 1
OPTICS COMMUNICATIONS
May 1975
EFFECTS OF SURFACE ROUGHNESS OF A MOVING TEST BODY ON LASER-INTERFEROMETRIC Keiji TANAKA Department
and Yoshihiro
of Engineering
DISPLACEMENT MEASUREMENTS
OHTSUKA
Science, Faculty of Engineering,
Hokkaido
University, Sapporo, Japan
Received 27 January 1975
Displacement measurements by laser-homodyne or laser-heterodyne methods are discussed analytically with respect to the relevant surface roughness. Expressions of the output signal and its signal-to-noise ratio are derived on the basis of gaussian statistics of the roughness. The roughness does not impose restrictions on the measurable minimum displacement but degrades the signal-to-noise ratio of the output signal.
1. Introduction
Recently, considerable work on the optical measurement of small displacements of a moving body has been reported. As far as optical homodyne or heterodyne methods are concerned, the displacement of an object in the microscopic range has been measured [l-6] . In these methods, a light beam from a laser is split into two components; one is reflected from a fixed reference surface and the other one from the moving object under study. These two reflected beams are recombined onto a photodetector surface, and the optical phase change induced by the motion of the object is electrically analyzed from the photocurrent output. Several authors have reported [2,3,5] measurements of displacement amplitudes down to 1O-2 A. Generally speaking, this value is much smaller than that of the surface roughness of the object. However, so far the analyses have been restricted to the motion of an idealized flat surface, and effects due to the roughness have not been taken into consideration. The purpose of the present work is to clarify the effects of the surface roughness of a moving object on the displacement measurement. In the following sections, expressions for the photocurrent and its signal-to-noise ratio are derived theoretically, then the limitation to the surface condition is discussed.
2. Analysis and discussion
The interferometer considered in this paper may be a Michelson type or other novel ones, and its conceptual basis is schematically illustrated in fig. 1. A plane, monochromatic light beam is split into the component E,, reflected from a reference surface R, and the component Es, scattered by a moving surface S, and then these components are optically mixed in the photodetector D. The frequency of either component may be shifted by a frequency modulator such as an ultrasonic cell. The direction of motion of the object S is assumed to be parallel to the optical axis. In this situation the distance of the object surface from a light source may be expressed in terms of the separate temporal and spatial parts: (1)
d (x’, t) = f(t) + 77@‘), where q,(x’) is the surface profile function 110
of the object andf(t)
its temporally
varying displacement,
i.e., the
May 1975
OPTICS COMMUNICATIONS
Volume 14, number 1
Fig. 1, Schematic diagram showing the optical system analyzed. The moving object is denoted by S, the reference plane by R, and the detector by D. The reference and scattered light beams diffused by R and S are represented by Er and Es, respectively.
D Fig. 1.
displacement of a plane determined by spatially averaging the surface profile function. As a consequence, the amplitude distribution of the light over the object surface S, perpendicularly illuminated by the plane monochromatic light, can be written as E&C’, t) = Au(x’) exp [i {G; t + kf(r) + key)]
,
(2)
where Au, w and k are the amplitude, the angular frequency and the wave number of the light, respectively. The scattered field Es on the detector is calculated on the basis of the Fresnel-Huygens principle in the Fraunhofer approximation (cf, ref. [7] ) as E,(x, t) = exp [i {WC t 2kf(t) - kL,}] s Ao(x’) exp (ikxx’/L,)
exp (2 ikq,(x’)} dx’,
(3)
where L, is the distance from the surface S of the moving object to a detecting plane at t = 0. Similar to the above calculation, the reference wave with an angular frequency of (o + o+,) and wave r -rlber k’ on the detector can be written as E,(x, t) = exp {i(wt + &$t - k’L,)}
J Ao(x’)exp (ik’xx’/L,)
exp { 2ik’rl,(x’)}
dx’,
(4)
where I is the surface profile function of the reference plane and L, is the distance from the reference plane R to the detecting surface. The homodyne method is the case in which 6+, = 0 and k =k’. When the phase-modulated signal beam Es is photomixed with the reference beam E,, the obtained photocurrent i(t) takes the form i(t)/Q = s [Es +Er I2 dx, where Q is the quantum
W/Q = /-dx &; ’
efficiency
of the detector. Therefore,
j-d+4,tx;M&,
eqs. (3) and (4) give
{G,( x,x;,x;)R,(x;)R:(x;)+G,(x,x;,x;)R,(x;)R,*(x;
@F(f)Grs(X, Xi ,X;)Rs(X;)R:(xi)’ #*F*(t)G:s(Xp Xi ,Xi)Rr(X;)Rz(X;)II,
(5)
where @J= exp ( -i(kLs
- k’L,)},
F(t)=exp[i{obt+2kf(t)}l,
6, b) 111
Volume 14, number 1
G,(x,x\ , xk)
OPTICS COMMUNICATIONS
= exp { ik’x(x; - xi)/,!,,}
G,,(x, xi, xi) = exp { ix(kx&
R&x;) = exp {2ikns(xj)),
,
- k’x$,)},
May 1975
G&x, xi, xi) = exp { ikx(x;
R,(xJ
- x;)/L,},
(5c, d)
j=1,2,
(se, f)
= exp { 2 ik’q,(xJ},
j=l,2.
(5g)
To calculate eq. (5) we assume that the surface profile functions or and q&x’) are both random variables obeying gaussian statistics and that the illuminating beam width is much larger than the correlation length of the profile functions. If this is the case we may replace the space averages by ensemble averages and neglect the contribu tion from the correlation area of the profile functions (see ref. [S]), eq. (5) becomes Ci(t)/Q) = [dx
Jdx;
J” dx;Ao(x;)Ao(x;)
+ G&x, ii, x;)R2(o,) where
R(ur) = exp (-2k’2
+ 2 Re {V’(0G,,(x, u,2),
and
[G,(x, x;,x;)R2(o,) xi ,x;))
R(or)R(a,)l ,
R(u,) = exp (-2k2u2) s .
(6) (6a, b)
The average value on the rough surface is expressed by L). ur and us are the standard deviation of 7),(x’) and V&X’), respectively, i.e., the root mean square surface roughness of the reference and object surface. In eq. (6) the first term is a dc component produced by the reference beam, the second one a dc component by the signal beam and the third one a phase-modulated beat component which will be considered here. The beat current depends on the roughness of the surfaces, as is expressed quantitatively by the product of eqs. (6a) and (6b), and decreases as or and us increase. It is obvious that if or = us = 0, &,(x’) = constant c-4, and diffraction effects are neglected, then eq. (6) reduces to the well-known result [4] : (i(?)/Q)=Jdx2,4;[1
tRe{@F(t)}].
(7)
If a shot-noise-limited detector such as a photomultiplier is employed, ratio on the roughness is calculated when L, = L, and k = k’ as SNR(ur,
u,)/SNR(O,
0) =R2(ur)R2(u,)/[R2(u,)
+R2(u,)].
the dependence
of the signal-to-noise
(8)
For the case where or = us s u, eq. (8) reduces to SNR(u)/SNR(O)
=; R2(u),
@a)
and in the case where 0 x or Q us 2 u to SNR(u)/SNR(O)
=R2(u)/[1
+R2(u)],
@b)
where R(u) = exp(-2k2u2).
(8~)
These relations are depicted in fig. 2 in units of SNR(0). The case of eq. (8b) is better than that of eq. @a), but in any case the normalized SNR plotted on a log scale (N.B., this corresponds to dB) decreases drastically around ku = 1. Consider a typical experimental situation under the following conditions: Q = 0.1, the light intensity received by the detector is 10e8 W, the wavelength of the light is 6328 A and the bandwidth of the circuit following the detector is lo5 Hz. In this case the actual SNR nearly equals unity when kux 2. From these observations it may be concluded that the criterion ku 2 1 should be satisfied in order to detect the signal in usual experimental circumstances. Note that the increase of the roughness which causes the degradation of the SNR’s has nothing with the magni-
112
OPTICS COMMUNICATIONS
Volume 14, number I
Fig. 2.
Fig. 2. Shot-noise-limited roughness.
May 1975
SNR dependence on the surface
tude of the small mean displacement f(r). That is, since the effects of the diffused surface are not essential for the measurement of the displacement if ka 5 1, we can measure the arbitrary displacement f(t) even if it is much smaller than the rms surface roughness us. The decrease in SNR(a) with the increase of ko may be due to the fact that the phase of the scattered light is spatially random and causes a deterioration of the coherent detection. Straightforward extension of the present calculation to the case where the light is obliquely incident upon the rough surface might yield similar results, if shadowing effects [9] can be neglected. In addition to the effects of roughness, there is a problem of temporal random vibrations of the surface. These may be due to mechanical noise or to thermal vibrations of surface atoms which fluctuate at an amplitude comparable to the measurable minimum displacement of the system. If the noise frequency is much higher than the detectable maximum frequency of the equipment used, then this random motion can be eliminated. The situation of thermal vibrations certainly corresponds to this case, since the frequencies involved are generally higher than lOlo Hz. On the other hand, it would be difficult to remove mechanical motions in the angstrom region, so that the actual detectable displacement may be limited by this noise component.
3. Conclusion As a concluding remark, we have clarified that the presence of roughness on the object surface does not impose any restrictions on the rn-asurable minimum displacement, but that it degrades the SNR of the output signal. If the beam width is not much larger than the correlation length of the profile functions, then, contrary to the assumption made in the derivation of eq. (6), the space averages cannot be replaced by the ensemble averages. In this case the analytic forms of the profile functions may be needed to carry out the analysis, and this will be the subject of a future study.
Acknowledgements The authors wish to thank Mr. N. Takai for valuable discussions and suggestions. Thanks are also due to Dr. M. Kitamura for critical reading of the manuscript. 113
Volume 14, number 1
OPTICS COMMUNICATIONS
References [l] [2] [3] [4] [S] [6] [7] [8] [9]
H.A. Deferrari, R.A. Darby and F.A. Am-hews, J. Acoust. Sot. Am. 42 (1967) 982. S.M. Khamra, J. Tonndorf a.td W.W. Walcott, J. Acoust. Sot. Am. 44 (1968) 1555. S. Sizgoric and A.A. Gundjian, Proc. IEEE 57 (1969) 1313. F.J. Eberhardt and F.A. Andrews, J. Acoust. Sot. Am. 48 (1970) 603. W. Puschert, Opt. Commun. 10 (1974) 357. Y. Ohtsuka and I. Sasaki, Opt. Commun. 10 (1974) 362. J.W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) p. 57. W.B. Ribbens, Appl. Opt. 8 (1969) 2173. P. Beckmann, Progress in Optics, ed. E. Wolf (North-Holland, Amsterdam, 1967) Vol. VI, Chap. II, p. 53.
114
May 1975
OPTICS COMMUNICATIONS
May 1975
EFFECTS OF SURFACE ROUGHNESS OF A MOVING TEST BODY ON LASER-INTERFEROMETRIC Keiji TANAKA Department
and Yoshihiro
of Engineering
DISPLACEMENT MEASUREMENTS
OHTSUKA
Science, Faculty of Engineering,
Hokkaido
University, Sapporo, Japan
Received 27 January 1975
Displacement measurements by laser-homodyne or laser-heterodyne methods are discussed analytically with respect to the relevant surface roughness. Expressions of the output signal and its signal-to-noise ratio are derived on the basis of gaussian statistics of the roughness. The roughness does not impose restrictions on the measurable minimum displacement but degrades the signal-to-noise ratio of the output signal.
1. Introduction
Recently, considerable work on the optical measurement of small displacements of a moving body has been reported. As far as optical homodyne or heterodyne methods are concerned, the displacement of an object in the microscopic range has been measured [l-6] . In these methods, a light beam from a laser is split into two components; one is reflected from a fixed reference surface and the other one from the moving object under study. These two reflected beams are recombined onto a photodetector surface, and the optical phase change induced by the motion of the object is electrically analyzed from the photocurrent output. Several authors have reported [2,3,5] measurements of displacement amplitudes down to 1O-2 A. Generally speaking, this value is much smaller than that of the surface roughness of the object. However, so far the analyses have been restricted to the motion of an idealized flat surface, and effects due to the roughness have not been taken into consideration. The purpose of the present work is to clarify the effects of the surface roughness of a moving object on the displacement measurement. In the following sections, expressions for the photocurrent and its signal-to-noise ratio are derived theoretically, then the limitation to the surface condition is discussed.
2. Analysis and discussion
The interferometer considered in this paper may be a Michelson type or other novel ones, and its conceptual basis is schematically illustrated in fig. 1. A plane, monochromatic light beam is split into the component E,, reflected from a reference surface R, and the component Es, scattered by a moving surface S, and then these components are optically mixed in the photodetector D. The frequency of either component may be shifted by a frequency modulator such as an ultrasonic cell. The direction of motion of the object S is assumed to be parallel to the optical axis. In this situation the distance of the object surface from a light source may be expressed in terms of the separate temporal and spatial parts: (1)
d (x’, t) = f(t) + 77@‘), where q,(x’) is the surface profile function 110
of the object andf(t)
its temporally
varying displacement,
i.e., the
May 1975
OPTICS COMMUNICATIONS
Volume 14, number 1
Fig. 1, Schematic diagram showing the optical system analyzed. The moving object is denoted by S, the reference plane by R, and the detector by D. The reference and scattered light beams diffused by R and S are represented by Er and Es, respectively.
D Fig. 1.
displacement of a plane determined by spatially averaging the surface profile function. As a consequence, the amplitude distribution of the light over the object surface S, perpendicularly illuminated by the plane monochromatic light, can be written as E&C’, t) = Au(x’) exp [i {G; t + kf(r) + key)]
,
(2)
where Au, w and k are the amplitude, the angular frequency and the wave number of the light, respectively. The scattered field Es on the detector is calculated on the basis of the Fresnel-Huygens principle in the Fraunhofer approximation (cf, ref. [7] ) as E,(x, t) = exp [i {WC t 2kf(t) - kL,}] s Ao(x’) exp (ikxx’/L,)
exp (2 ikq,(x’)} dx’,
(3)
where L, is the distance from the surface S of the moving object to a detecting plane at t = 0. Similar to the above calculation, the reference wave with an angular frequency of (o + o+,) and wave r -rlber k’ on the detector can be written as E,(x, t) = exp {i(wt + &$t - k’L,)}
J Ao(x’)exp (ik’xx’/L,)
exp { 2ik’rl,(x’)}
dx’,
(4)
where I is the surface profile function of the reference plane and L, is the distance from the reference plane R to the detecting surface. The homodyne method is the case in which 6+, = 0 and k =k’. When the phase-modulated signal beam Es is photomixed with the reference beam E,, the obtained photocurrent i(t) takes the form i(t)/Q = s [Es +Er I2 dx, where Q is the quantum
W/Q = /-dx &; ’
efficiency
of the detector. Therefore,
j-d+4,tx;M&,
eqs. (3) and (4) give
{G,( x,x;,x;)R,(x;)R:(x;)+G,(x,x;,x;)R,(x;)R,*(x;
@F(f)Grs(X, Xi ,X;)Rs(X;)R:(xi)’ #*F*(t)G:s(Xp Xi ,Xi)Rr(X;)Rz(X;)II,
(5)
where @J= exp ( -i(kLs
- k’L,)},
F(t)=exp[i{obt+2kf(t)}l,
6, b) 111
Volume 14, number 1
G,(x,x\ , xk)
OPTICS COMMUNICATIONS
= exp { ik’x(x; - xi)/,!,,}
G,,(x, xi, xi) = exp { ix(kx&
R&x;) = exp {2ikns(xj)),
,
- k’x$,)},
May 1975
G&x, xi, xi) = exp { ikx(x;
R,(xJ
- x;)/L,},
(5c, d)
j=1,2,
(se, f)
= exp { 2 ik’q,(xJ},
j=l,2.
(5g)
To calculate eq. (5) we assume that the surface profile functions or and q&x’) are both random variables obeying gaussian statistics and that the illuminating beam width is much larger than the correlation length of the profile functions. If this is the case we may replace the space averages by ensemble averages and neglect the contribu tion from the correlation area of the profile functions (see ref. [S]), eq. (5) becomes Ci(t)/Q) = [dx
Jdx;
J” dx;Ao(x;)Ao(x;)
+ G&x, ii, x;)R2(o,) where
R(ur) = exp (-2k’2
+ 2 Re {V’(0G,,(x, u,2),
and
[G,(x, x;,x;)R2(o,) xi ,x;))
R(or)R(a,)l ,
R(u,) = exp (-2k2u2) s .
(6) (6a, b)
The average value on the rough surface is expressed by L). ur and us are the standard deviation of 7),(x’) and V&X’), respectively, i.e., the root mean square surface roughness of the reference and object surface. In eq. (6) the first term is a dc component produced by the reference beam, the second one a dc component by the signal beam and the third one a phase-modulated beat component which will be considered here. The beat current depends on the roughness of the surfaces, as is expressed quantitatively by the product of eqs. (6a) and (6b), and decreases as or and us increase. It is obvious that if or = us = 0, &,(x’) = constant c-4, and diffraction effects are neglected, then eq. (6) reduces to the well-known result [4] : (i(?)/Q)=Jdx2,4;[1
tRe{@F(t)}].
(7)
If a shot-noise-limited detector such as a photomultiplier is employed, ratio on the roughness is calculated when L, = L, and k = k’ as SNR(ur,
u,)/SNR(O,
0) =R2(ur)R2(u,)/[R2(u,)
+R2(u,)].
the dependence
of the signal-to-noise
(8)
For the case where or = us s u, eq. (8) reduces to SNR(u)/SNR(O)
=; R2(u),
@a)
and in the case where 0 x or Q us 2 u to SNR(u)/SNR(O)
=R2(u)/[1
+R2(u)],
@b)
where R(u) = exp(-2k2u2).
(8~)
These relations are depicted in fig. 2 in units of SNR(0). The case of eq. (8b) is better than that of eq. @a), but in any case the normalized SNR plotted on a log scale (N.B., this corresponds to dB) decreases drastically around ku = 1. Consider a typical experimental situation under the following conditions: Q = 0.1, the light intensity received by the detector is 10e8 W, the wavelength of the light is 6328 A and the bandwidth of the circuit following the detector is lo5 Hz. In this case the actual SNR nearly equals unity when kux 2. From these observations it may be concluded that the criterion ku 2 1 should be satisfied in order to detect the signal in usual experimental circumstances. Note that the increase of the roughness which causes the degradation of the SNR’s has nothing with the magni-
112
OPTICS COMMUNICATIONS
Volume 14, number I
Fig. 2.
Fig. 2. Shot-noise-limited roughness.
May 1975
SNR dependence on the surface
tude of the small mean displacement f(r). That is, since the effects of the diffused surface are not essential for the measurement of the displacement if ka 5 1, we can measure the arbitrary displacement f(t) even if it is much smaller than the rms surface roughness us. The decrease in SNR(a) with the increase of ko may be due to the fact that the phase of the scattered light is spatially random and causes a deterioration of the coherent detection. Straightforward extension of the present calculation to the case where the light is obliquely incident upon the rough surface might yield similar results, if shadowing effects [9] can be neglected. In addition to the effects of roughness, there is a problem of temporal random vibrations of the surface. These may be due to mechanical noise or to thermal vibrations of surface atoms which fluctuate at an amplitude comparable to the measurable minimum displacement of the system. If the noise frequency is much higher than the detectable maximum frequency of the equipment used, then this random motion can be eliminated. The situation of thermal vibrations certainly corresponds to this case, since the frequencies involved are generally higher than lOlo Hz. On the other hand, it would be difficult to remove mechanical motions in the angstrom region, so that the actual detectable displacement may be limited by this noise component.
3. Conclusion As a concluding remark, we have clarified that the presence of roughness on the object surface does not impose any restrictions on the rn-asurable minimum displacement, but that it degrades the SNR of the output signal. If the beam width is not much larger than the correlation length of the profile functions, then, contrary to the assumption made in the derivation of eq. (6), the space averages cannot be replaced by the ensemble averages. In this case the analytic forms of the profile functions may be needed to carry out the analysis, and this will be the subject of a future study.
Acknowledgements The authors wish to thank Mr. N. Takai for valuable discussions and suggestions. Thanks are also due to Dr. M. Kitamura for critical reading of the manuscript. 113
Volume 14, number 1
OPTICS COMMUNICATIONS
References [l] [2] [3] [4] [S] [6] [7] [8] [9]
H.A. Deferrari, R.A. Darby and F.A. Am-hews, J. Acoust. Sot. Am. 42 (1967) 982. S.M. Khamra, J. Tonndorf a.td W.W. Walcott, J. Acoust. Sot. Am. 44 (1968) 1555. S. Sizgoric and A.A. Gundjian, Proc. IEEE 57 (1969) 1313. F.J. Eberhardt and F.A. Andrews, J. Acoust. Sot. Am. 48 (1970) 603. W. Puschert, Opt. Commun. 10 (1974) 357. Y. Ohtsuka and I. Sasaki, Opt. Commun. 10 (1974) 362. J.W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) p. 57. W.B. Ribbens, Appl. Opt. 8 (1969) 2173. P. Beckmann, Progress in Optics, ed. E. Wolf (North-Holland, Amsterdam, 1967) Vol. VI, Chap. II, p. 53.
114
May 1975