aspects of the induced speckle motion due to rigid-body movements of a model using ... for both opaque and thin transparent diffusing models and for different ...
Speckle motions induced by rigid-body movements in freespace geometry: an explicit investigation and extension to new cases Pierre Jacquot and Pramod K. Rastogi
In line with the current interest in speckle motion in free-space propagation, it is proposed to investigate new aspects of the induced speckle motion due to rigid-body movements of a model using as a starting point the well-known concept of homology, first stated in holographic interferometry, the importance of which has
only recently been realized in speckle metrology. At the outset, stating the homology conditions leads to a general expression relating the speckle shift to the geometrical parameters at the recording and to the six degrees of movement of the object surface. The expression presented in this explicit form is found to be true
for both opaque and thin transparent diffusing models and for different conditions of illumination and observation. The study is then extended to (1) when the object is at rest, subjecting the illuminating source to a small displacement and (2) when the wavelength of the illuminating beam is changed between the two
exposures. The case of the rigid-body displacement of the object is experimentally verified and the results are found to be in good agreement with theory. The viability of the method to make measurements sessed, several applications are envisaged, and some advantages are pointed out.
1. Introduction
The phenomenon of speckle has been extensively studied and discussed by many authors both in the near- and far-fields, as well in free-space propagation as in imaging geometry (by means of a lens or a hologram) for different goals and purposes. In metrology one of these goals has been to describe the movements of speckle from diffusively reflecting surfaces.'- 3 An important result acquired in holographic interferometry is that, when an optically rough surface illuminated by a coherent beam undergoes a small rigid-body displacement, the amplitude distribution in the diffracted field remains identical3 -' 4 in the two successive positions of the object except for being locally shifted in space. The conditions in which this identity holds have
been established and are the well-known "homology conditions".
6 9
2 - ,11,"
The development of this condition, though initially only for holographic interferometry, should not be surprising if one considers that in this case the fringes are formed only in that region of the space where the two diffracted amplitudes have an identical structure
and the lateral and longitudinal displacement between them remains less than the volume of correlation. However, in speckle photography the interest of this condition was recognized later when it was realized that
the geometrical optics assumption, according to which the speckle moves as if attached to the object surface, was no longer adequate. In fact, ambiguity could arise if the object is not in focus, and serious errors may result
from the lens aberration effects.1 4-' 6
In this paper we shall invoke the homology conditions
to derive an explicit general relationship between the six degrees of freedom of movement of the object surface
and the lateral speckle shift in terms of the geometrical parameters in a free-space configuration. The expression is found to be valid for both opaque and thin transparent objects. The transparent object is believed
to be sufficiently thin so that, to a first approximation, the refraction effects may be neglected. The study is next extended to the case of the movement of the illuminating source and the object at rest. This is followed by a study of speckle movement when the object remains at rest and the wavelength of the illuminating beam is changed between the exposures. These cases have been of particular interest in the measurement of roughness,1 7 -' 9 contouring,
wedge angles,2 3 The authors are with SwissFederal Institute of Technology,Laboratory of Stress Analysis, CH-1015 Lausanne, Switzerland. Received 23 January 1979. 0003-6935/79/122022-11$00.50/0. ©1979 Optical Society of America. 2022
APPLIEDOPTICS/ Vol. 18, No. 12 / 15 June 1979
is as-
2 0 21
aberration
effects,2 2
etc. A discussion then outlines some general properties of the shift between the two fields and endeavors to indicate the influence the transparent model thickness has on the measurement of speckle shift. Finally, we present the experimental verification of the formulas developed theoretically.
II.
Recording and Reconstructioi
To test the validity of the expressionsderived in later sections, we twice expose the photographic plate, without any intermediate optics, to the object wave diffracted by a diffusing object illuminated by a laser beam (Fig. 1). The object undergoes a small rigid-body photographic transparency is placed in a classic Fourier
filtering setup shown in Fig. 2. Parallel equispaced fringes modulating the autocorrelation halo appear in
A.
or photographed. It can then be easily shown that the in-plane components Au and Av of the speckle shift in the observation plane are given by Xe
p
Expression for Speckle Shift as a Function of Six
Degrees
the Fourier plane, which can either be observed directly
Av =-f smy,
p
where p denotes the period of the correlation fringes and y is the angle these fringes make with the t axis in the Fourier plane. Ill.
displacement between the two exposures. Next the
Au =-fcosT,
of Freedom
of Movement
Notations Used
In Fig. 3 the notations used below are defined. The six components of displacement are designated by (T1,T 2 ,T3 , R1,R2 ,R3 ). The Cartesian coordinate system [O,xyz] is related to the object, the plane [O,xy] defining
the object plane. The respective direction cosines of the illuminating and the observation directions are (lemene) and (1omon o), the coordinates of unit vectors
S.
9e and go.
The coordinate system [S0,uvw] is related to the observation direction, such that the plane [Suvl] is perpendicular to OSo and the axis v lies in the [O,yw] plane in the Oy direction. The [SO,uv]plane defines the plane
of the photographic plate in which the shift (Au,Av) is measured between the two diffracted waves.
so
Lastly, we shall define an auxiliary coordinate system [O',XYZ], deducible from [O,xyz] by the displacement
undergone by the object, in which a part of the calcu-
Fig. 1. Schematic of speckle recording in free-space propagation. The object undergoes a small rigid-body movement between the two exposures.
11 _f
Laser
Aperture Photographic plate
Collimator
Fig. 2.
f
Fourier plane
Fourier filtering setup for visualizing the correlation fringes.
(a)
I (b) (C)
X
S.I0.:1: I0e.e
~_-I
I
So s ,'S. I v.:1: m
-w
U. ..
U. no
T3
Fig. 3.
Definition of the notations used. (a) The components of displacement; (b) coordinates of the illuminating source; (c) coordinates of the observation point SO in the observation system [S 0,uvw]. 15 June 1979 / Vol. 18, No. 12 / APPLIEDOPTICS
2023
lations will be carried out. The coordinates of the points (O,S,S, . . .) when expressed in this system take
the form (O',S',S, . .)
The object displacement is characterized by the rotation matrix [R] and the translation matrix [T] [RI =
[1
R3
-R3
1
-R,
-R
R2
(a)
R,;
(b)
Fig. 4. Recording geometry of speckle related to two different states of the object: (a) in the initial state in the coordinate system [O,xyzj;
1
(b) in the final state in the coordinate system [O',XYZ]. [R=-1
1
-R3
R3
1
-R 2
T11
R2 -Ri;
R,
[T]
1
(2)
T2]
T3
Although these conditions for homology can be es-
where the coordinate systems [O,xyz] and [O',XYZ] can
be deduced from each other as (without taking into account the change of origin)
[]
[]
=R]
[
[Y
[R
(3)
tablished from the first principles using any of the existing approachesl 6-9 (diffraction formula, grating model, plane wave development, etc.), in all that in-
terests us, we shall content ourselves only with writing these conditions directly: the identity [Eq. (6)] is verified if and only if the initial and final path lengths are equal:
S|P +PSO°P'+P's
(7)
The observation system [S,uvw] is related to [O,xyz] as Xi
for any point P(x,y) lying in the illuminated region of the object surface. It can easily be shown that in paraxial approximations and for plane objects, Eq. (7) reduces to
U
[r]
y= Z
v
[v = [r]Il yI [w
[z
Pe + Po = Pe + P,
where [r] is a rotation matrix given by
F0*o--1
.~~' °
- °
10 _°o° C,
lo
no
.0
_ l
0
no ao
_o
MO
no
(5)
where 0 =
['nI
o+ n
le +
(8)
= 1e + lo,
(9)
me + mO= me + o.
(10)
.This statement is the equivalent of saying that there will be no first-order change in the time required to travel either the initial (SePSo) or the final (S'eP'S,) path lengths for any points P and P' lying in the neighborhood of 0 and O' on the object surface before and after it has undergone a small rigid-body displacement. In other words, the partial derivative of the path difference (SePS - P'SO) with respect to x and y is zero to a first order. 2 4
B. Homology Conditions
For the sake of clarity, we consider independently the position of the object before and after it has undergone a small rigid-body displacement (Fig. 4). The concept of homology postulates that a point S' exists (in the vicinity of SO) where the amplitude of the diffracted field after the displacement of the object is identical to that at SO, the initial point, before the object displacement, i.e.,
The necessary and sufficient conditions for the homology, defined by Eqs. (8)-(10), allow us to calculate the coordinates of point S. The calculation is carried
out in three steps and assumes a systematic linearization of the formulas. The products like TiTj, TiRj, and RiRj are negligible compared with Tk and Rk. C. Calculation of the Apparent Movement of the Illumination Source
U(SO)
U(S.),
The coordinates of S'e can be inferred from that of Se
(6)
while imposing certain conditions to the coordinates of
by changing the axis [O,xyz] to [',XYZ].
S0,.
Pe = Pe
-
T *Se = e -
( T ile + T 2 Me + T3ne),
(11)
andthe coordinates of Se are given by ;n
I = [RI
2024
APPLIEDOPTICS/ Vol. 18, No. 12 / 15 June 1979
[Pele
Pele+ emeR3 - PeneR2 T1 ,I
PemeI - [T =
Pe-e
LPene
pene + eleR 2
-
PeleR + PeneRi -
PeMeRi
-
T2 T3
(12)
Solving for (le,m'e,n') we obtain le = me
-
-eneR2 + meR3 + [(1
=me + neRl -
ne = e
ieRl M
+ leMeT2 + leneT3]pe,
-1)T,
(13)
leR 3 + [lemeTi + (ml - 1)T2 + mneeT3]/Pe, (14)
1)T 3]IPe.
+ leR 2 + [leneTl + MeneT2 + (-
(15) T po = po + ( ile + T2ie + T3ne).
We verify l + m/2 +Tn,2= 1
(16)
Seand S, can be inferred from each other by introducing a small correction factor fe into each component For small displacements, the unit vectors
(17)
le = le + eel, me =
me + ee2,
(18)
ne =
ne + Ee3,
(19)
lefel + MeEe2+ neEe3 = 0.
(20)
D. Expression for the Coordinates of SO in [',XYZ]
From Eqs. (9) and (10) it followsthat the coordinates of s' and 9,^can be deduced from each other by applying a simple correction o (21)
l0 = lo + eon
m
(22)
= m + Eo2,
no = no +
(23)
Eo3.
Here, in order that the unitary condition for the vector s' be satisfied, the following condition manifests: lofol
+
ionEo2
+
noeo
3
(24)
= 0.
In accordance with Eqs. (9) and (10), Eqs. (17) and (18), and Eqs. (21) and (22), we have
Thus the relations [Eqs. (28)-(31)] give in an un-
equivocal manner the coordinates of point S' where the amplitude of the diffracted field is identical to that at S,. The coordinates of S' are a function of the displacement parameters, the coordinates of the illuminating source, and the point of observation. The results are applicable to all positions of Se and SO and to opaque and thin transparent diffusing objects as well: they will be verified experimentally. The various models used for establishing the conditions for homology are unbiased as to-the type of object used, whether opaque or transparent, provided the latter is thin enough for the refraction effects to be negligible. E. Expression for Speckle Shift in the Observation Plane [S0 ,uv]
One of the features of the present approach lies in the fact that we record directly, unlike holographic interferometry, the intensity of the speckle, as described in Sec. II. In practice it is not possible to observe at the same time the intensities at points S,, and S. In reality one records the intensity at S, the point of intersection of the line O'S', and the plane of the photographic plate [S0 ,uv].
The coordinates of SR can be obtained by
multiplying the direction cosines (,m',n') not by p', where = po - T - go = po - (Tl
P = jOS'
fol = -Eel,
(25)
Eo2= -Ee2-
(26)
The relations [Eqs. (9) and (10)] do not impose any condition on n', as we consider, a priori, the case of
plane objects. If we make a hypothesis to the contrary, no must satisfy the relation E03 =Ee 3 and Eq. (24); these, in general, give rise to an impossibility, the two conditions being incompatible. In the case of plane
(31)
by PRand
+ mT2 + nOT3). (32)
The fact that we observe at SR rather than at So, U(SO) and subsequently where the identity U(S,) lU(S,)l
2
I U(SO)12 strictly holds, presents no restric-
tions, as the depth of correlation is not zero. The two recorded fields remain practically identical when the longitudinal, shift given by AW= Po - P'-j (le + 1)TI + (me + mo)T2 + (ne + no)T, (33) = (Se+ go) T
objects, the question of incompatibility does not arise,
remains less than the depth of correlation2 5 PC
and it is sufficient to choose n,, so as not to disturb the unitary conditions for the vector s. From Eq. (24) we
where
have
-
EC3 =
(27)
(lEool + Moo 2 )Ino]
Using Eqs. (13)-(27), the direction cosines of O'S in [O',XYZ] are given by
The expression for the coordinates of SR in [O',XYZJ thus follows as
= 1, + nR 2 - meR3 - [(le- 1)T, + ei.T 2 + leneT3]/pe, +(ie -1)T2 + mnefleT ,o=io - nR + leR 3l/Pe, 3 - [lemeT
1-
fleno no
nelo2 R--Rno
(lemo - melo) no
(34)
21 being the diameter of the illuminated region.
(28)
1,
no n+
(2Xp)/12, p
(29)
R3
+ [leie+ + ( [(le- 1)lo+ leimenmo]Tl
-)moT2
+ menemolT3 + [lenelo
.
(30)
Peno
Finally, combining Eqs. (8) and (11), we obtain 15 June 1979 / Vol. 18, No. 12 / APPLIEDOPTICS
2025
1)T + lmeT2 + lnT 3] P
polo+ po(neR2 - meR 3) - [(-
['S
p~m0 + po(f-neRl + lR 3)
[0',XYZ]
-
l(loT, + moT2 + n0 T3)
-
Pe
[leMeTi+ (2
-)T
2
+ nemeT3 ] Po- mo(loT1 + moT2 + n.T3)
(35)
Pe
pono + P [nemoR, no
neloR 2 - (leMo
-
+ P [loleMe + (e noPe
-1)inT
lome)R3 ] + P
-
noPe
[l0(le- 1) + MioleM]T
+ P [lonele+ nememo]T noPe
2
nO(IoT, + moT2 + no T3 ).
3-
In the coordinate axis [O,xyz], we have [OS,] = [RI-' [OSR] + T. [O,xyz] [O,XYzI
(36)
and the shift in the observation plane [So,uv]is given by [Au
Fpolo
= [SOSIR = [r]-I [OS'] [S,,,uv] [O,xyz]
Av 0
i
- 1Pm.i -
(37)
pOn
Reintroducing /Žw as obtained in Eq. (33), we write the
three components of the speckle shift in [S,,,uv]
LAu1
1R2I IR3
AwJ
where
(38)
S.~~~~~~~
s.-' z /
1T21 LTa
v
[C] is a 3 X 6 matrix.
To simplify, it seems preferable to decompose Eq. (38) into two separate
expressions,
one giving the
w
speckle shift due to the object rotation and the other the shift due to the object translation: Au] AV
[Au] =
U
.
[Au]
[Ri]
IAV,
R3
I .wrot.
.Itrans.
+ . rot. J
[Au]
[C~r,,t. 1R 2
AU
[Au]
Av
Aw
Aw
Fig. 5. Schematic of speckle recording for the object at rest and the illuminating source undergoing a small displacement between the two exposures.
Av
(39)
AW .
trans.
[T11
=[ltran..
AVi
T2
IAWi
(40)
1
IT
where the matrices [Clrot.and [Cltrans. follow simply from Eqs. (37)-(40) as -mo0l, ( + ne) 2(1 + [C]rot.
O 0
-
o
k (Iomle- lome)- no(me+ Mi)
- (no + n) o
nO nO
o
Pe 'To
Pe Po
lm,,,,(l- Me,)-
leMe - 10mO Pe
1oleme
no (1 -
O'o
2) +
Po
both [Crot. and [C]trans. being 3 X 3 matrices. APPLIEDOPTICS / Vol. 18, No. 12 / 15 June 1979
(mO + me)
-- e(,le +lomime)_ lonO
2Pe Po
Po
(.+1)Pe (lo,+ le)-
2026
(41)
o
no [C]trans.=
e lo + le
-
Po
Po
Pe
-neme -
P
Po
(no + ne) Pe o Po
,no (42)
undergoes a small displacement-between the two exposures. Figure 5 schematizes the symbolsused. The
Equations (40)-(42) summarize the first result in the form of an expression for the speckle shift relating
coordinates of S'e in [Se, UVW] are given by
to the displacement between the two diffracted waves as a function of the six degrees of freedom of movement
of the object and the geometrical parameters, whatsoever, at the recording. One should note that in the case of spherically convergent illumination, either for opaque or thin transparent models, Pe simply transforms to -Pe as the convergent wave is the complex conjugate of the spherically divergent wave-their arguments having opposite signs.
SPeRH
[Se,UVWI
minating source in the U and V directions, respectively,
and TL is the longitudinal displacement in the W direction. In the coordinate system [O,xyz] the coordinates of
In practice, the illumination direction, the normal to the object, and the observation direction generally lie
S'e can be inferred by a simple transformation
in the same plane, for example, [O,xz]. In this case mo = me = 0; EJo= Ce = 1, and the matrices given by Eqs.
Se Pele+ -PeRH [OXyz]
+ ne Po
[(no +
no 0
te)
L0 -+ noP
no
Po
0
[C]trans. =
0
lo + leI'
0
o
Pe
no
1 +Po
Pe
(lo+le)Pe
0
Po
Po
0
(46)
e
R
-
Mene PeRk+ eTL Ure
Thus from Eqs. (17) and (18) we obtain
-
Rv +-TL,
Eel=ne RH.
He
(44) Ee2=
Ce
eRv + -
(47)
Pe (48)
TL.
Pe
(no+ne)Pe
Recalling the identity conditions given by Eqs.
Po
(8)-(10), the coordinates of SO in [O,xyzj are seen to be
Similarly, if the illumination and the observation direction lie in the [O,yz]plane, the matrices [C]rot.and [C]trans. follow by substituting
1
-
Se
le~~~1 Pe
-ne--lo
0
PeRv + eTL
-
Se re PeMe + CePeRv + meTL
Pe
(43)
of axis
lin
Se Pelene
(41) and (42), respectively, simplify to
=
(45)
TL
where RH and RV represent the tilt angles of the illu-
F. Simplifying Case
[Cirot.
[e PeRv
le = lo = 0, Elo= no, and
po(lo- el) po(Mo- e2)
Se = ne in Eqs. (41) and (42). The special case treated in this paragraph in no way
[Oxyz] p(no
undermines the usefulness of the general relations expressed by Eqs. (40)-(42), since the former case is valid
observation plane [,uv] follow by a straightforward calculation using Eq. (5): _ nope
is, for all lo,mo,no on the plate, provided that for those
II
Ee3
Co eel + lo
Aul
points far enough from the center one corrects the formula for the discrepancies introduced, ipso facto, since the orthogonality of the recording plane no longer strictly holds for those points.
-_
I
e
,in,,p
' el
0-,,
n
-0-oPoEe2
p
I
(50)
0-,,
The three shift components can be rewritten in the form
IV. Special Cases A. Case of the Object at Rest and the Displacement of the Illuminating Source
where [C'] is a 3 X 3 matrix. with Aw = TL, we get F
ne Co nO,,fe
0
o
1 no
eime0,o
loino
0-
0
e
-
ro
e
By stating eel and
-
11 e
o+
Ee2
and
oelo
Peno0,
Co
Pe
0e
1
e
(51)
= [C'] [TL
AV
In this section we extend the approach given in Sec. III to calculate the components of speckle shift for the case of the object at rest, and the illuminating source
=
+t E
whereby the components of speckle shift (Au, /Žv) in the
only for the central region of the photographic plate thereby preventing extraction of all the recorded information. The latter gives a representation of the displacement over the whole surface of the plate, that
WI
(49)
- Ee3)
15 June 1979 / Vol. 18, No. 12 / APPLIEDOPTICS
Io
* (52)
2027
As in Sec. III, the matrix [C'] simplifies considerably
in the case where the illumination direction, the normal to the object, and the observation direction lie in the same plane, for example, [O,xz]:
[
ne
[C'] = Po
_
nO
le Pefo *
0 0
-1 '
(53)
0 1
and the longitudinal shift Aw followsdirectly using Eqs. (32), (33), and (54): Aw = (Pe + PO) A(63
In the case, found most often in practice, where the illumination direction, the normal to the object, and the observation direction lie in the same plane, say, [O,xz],
the speckle shift components simplify to
In this simple form, a case most employed, the above result corresponds well with the formulas given by the different authors. 7 - 9 25
Au = ( + lo) P AX no X
AV = 0
B. Case of the Object at Rest and the Change of the Wavelength of the Illuminating Beam
The techniques of either the change of wavelength of the illuminating source or the change of index of refraction of the medium surrounding the recording system between the two exposures have found practical use
in roughness-measurement and depth-contouring applications. With this in mind, we shall extend our approach to calculate the components of speckle shift for the case of the object at rest and the object illuminated by two different wavelengths in successive exposures. In this case, in paraxial approximations, the homology conditions take the form: Ki(pe + po) = K2 (Pe + P' 0 ),
KI(le + l) = K 2 (le + lo) KI(me + m)
(54)
(55)
= K 2 (Me + M'o),
(56)
where K, = (2r)/X\,, K2 = (27r)/X2 ; and X 1 and A2 are the wavelengths of the illuminating source during the two exposures. To calculate the coordinates of SO,we first state the correction factors Eo,, o2, and C,3 using Eqs. (21)-(23), (27), (55), and (56), whereupon AX
E,,1 = (le+ lo) A
(57)
E,2 = (ie + m) + '
(58)
0
=
A
,(l2+inO+lole+inoie)AX °
no
A
(59)
where X is the wavelength used for the first exposure and AX= 1X1 -X21. The coordinates of SR' in the [O,xyz] system are given
by
A
= (
Polo + PoEol Pom + Pofo2
pon + Poo3
from which, using the same transformation as in Eq. (37) and on simplifying, the lateral speckle shift components in the observation plane are seen to be Au = [le + 1 + M(lome - lemo)]PO A
(61)
Po A X
Av = (e + m)
(62) co
2028
APPLIEDOPTICS / Vol. 18, No. 12 / 15 June 1979
+ PO) A X
V. Discussion A. Approximations
Involved
The above calculations have been carried out with the
restriction of paraxial approximation and for the case of plane objects. In what concerns the paraxial approximation, neglecting the terms of type (X/e)2 and (x/p,,)2 as compared to X/Pe and x/po is equivalent to limiting the aperture angle of the illuminating source and the point of observation to a value of pe
(60)
(64)
For the case where the object remains at rest and the index of refraction is changed between the two exposures, the results follow as a simple corollary to the above developments on replacing (X)/X by (\n)/n. One important point to note is that rapid decorrelation between the two recorded speckle patterns occurs with the increase in the spectral difference A\Xas is evident from Eq. (63). In fact, the two recorded speckle patterns remain homologsonly in the condition that the longitudinal shift remains less than the depth of correlation. This is precisely the reason why, in roughness-measurement applications, one takes the precaution of giving the photographic plate an additional longitudinal shift' 8 to compensate for the decorrelation effect caused by Aw, thus attributing the decorrelation taking place only to the microstructure of the surface. In other words, on analyzing such a transparency in the Fourier-filtering setup, the visibility of the fringes characterizing the correlation between the two speckle patterns is now related only to the standard deviation of the surface roughness.
I-1; s OST [O,xyz]
(63)
