ACTA MECHANICA SINICA ,Vol.5, No.2, May, 1989 Science Press, Beijing, China Allerton Press, INC., New ,York ,U. S ,A.

ISSN 0567- 7718

P O L A R I Z E D SHEARING HOLOGRAPHIC-MOIRE INTERFEROMETRY*

Fang ling Dai Fulong

( Tsinghua University, Beijing)

ABSTRACT: A new holographic-moir6 method is presented to obtain the in-plane strain fringe patterns. During the recording, double object beams and dual reference beams of orthogonally polarized state are used to illuminate the object and the holographic plate, respectively. Two carriers modulated by deformation are obtained through double-exposure or in real time. Pure in-plane displacement derivative patterns are then obtained through twice filtering.

KEY WORDS: orthogonal polarization, in-plane strain patterns, optical f'dtering.

I. INTRODUCTION Among the interferometric methods in optical mechanics, some shearing techniques were

developed to obtain the displacement derivative patterns of deformed object. For example, the

speckle-shearing camera, developed by Hung ill, is a useful tool to acquire the derivative patterns of

displacements on rough surface. With moir6 interferometry, Weissman I21 obtained the strain fields

by mechanical shifting of the recording plate. Among our newly developed methods of the polarized

interferometry/3J, the polarization of laser beams was used by authors/4] in moir6 interferometry and

the in-plane strain fringe patterns were achieved in real time. However, up to no w, the application of

Shearing techniques to holographic interferometry has not been successful though the formation of

holographic-moir6 made the shearing of the separated displacement fields valuable. For the purpose

of differentiation, Gilbert lSJ proposed two shifting methods. In one method, multiplexing

techniques were used to incorporate an initial pattern into the shifting processing, but the results of

the method were a coupling of thein-plane and the out-of-plane displacement derivatives. In another

method, a holographic-moir6 pattern was shifted on itself to produce the derivative pattern directly.

In this case, however, the strain fringes were not distinct because the frequency of the in-plane

displacement fringes was usually not high enough and it is nearly impossible to separate the

subtractive second-order moir~ from the additive ones.

In this paper, a polarized holographic-moir6 method is proposed to acquire the in-plane

displacement derivative fringe patterns. Illuminating the object with double beams of orthogonally

polarized axes, and by changing the directions of dual reference beams to obtain carriers, the

necessary information can be recorded by double exposures or in real time as in common

holographic recording. After the recording plate is filtered twice, not only the out-of-plane

displacements are canceled, but also pure and clear strain patterns can be observed directly:

H. PRINCIPLES To eliminate the influence of the out-of-plane deformation, double beams E 1 and E 2 are used to

illuminate symmetrically the object (Fig.l). They are linearly polarized beams with perpendicular

Received 25 September 1987.

* The project is supported by the National Natural Science Foundation of China.

192 ACTA MECHANICA SINICA 1989

axes, and at an angle of at with the normal to the surface of object. Assuming the rough surface is of

the property of non-depolarization, reflective diffused rays O1 and 0 2 will have the same

polarization state as the incident light. When they reach the recording plane, the complex

amplitudes can be expressed in Jones vectors as

I ei~l(x'Y) I I 0 ] 0 1 = 0 0 2 = eiq~2(x,y ) (1)

where tpl(x, y), q~2(x, y) are random phases.

YO XO

, o , + §

2 t '

Fig.1 Schematic of holographic recording with orthogonally polarized beams

Moreover, collimated reference beams R1 and R e are also in an orthogonal form of linear

polarization. Before the object is loaded, they are symmetrical to the normal to the holographic plate

at an angle of 0x, and can be written as

[ :"~ E 0 R~ = 0 R2 = e_n,,:o (2)

where fo is spatial frequency of two plane wave~ and equal to sinl0xl/2. Therefore, in the first

exposure, these four beams are recorded:

11 = (0~ + 0 2 + R t + R2)'e(Ot + 0 2+R1 + R2) (3) where ' ~ ' means the conjugation and transposition of the matrix.

When the object is loaded, the phases q~l and ~0 2 will become ~0' a and 60 2 respectively, and object

beams will be 0 1 and 0 2. At the same time, we change the directions of R1 in plane X-z by A0~, and

R2 both in plane x-z by A0x and in its perpendicular plane by A0 r. Hence, the new reference beams

are:

[:""1 [ ] Ri = 0 Ri = 0 e_i2~lf lx+ f 2y ) (4)

where f~ = sin(0~ + A0~) / 2 and f2 = sin(A0r) / 2.

The second exposure records the intensity distribution of

12 = (O'1 + 02 + R] + R'2)*(O'~ + 02 + R', + R'2) (5) and the total intensity recorded by hologram is I = I 1 -4-12.

After the holographic plate is developed, it is reconstructed under the illumination of both R1

and R 2. In the condition of linear recording, the amplitude transmissivity is proportional to the

intensity. Neglecting the constants, the transmissive light will be given by

Vol. 5 No. 2 Fang'& Dai : Holographic- Moire Interferomet{~ !93

u, = Go + I2o)R, + Go + I2o)n'2 , i l , n t o , + l i d i ~ o ~ + n,il~*o', + il>n~*o7 (6) -t- RiOt1 R , -i- R 2 O t R 2 q- RIO'I4"R'I q- l ~ 2 0 ~ R 2

The terms of the first line in Eq.(6) are the zero-order diffractive waves and those of the last lirie

indicate the real image of the object. What we are interested in is the terms in the niiddle lhie because they form the virtual images of the objeet surface. Substituting the Jones vectors of these wavefl'onts

into this equation, we can express the four virtual images in the form of

ei~l(x,r) + e,[,#,(,,)) + ~(.,.~,) + 2~/~o* 1

where ak(x, y) = CPk(x, y) -- 9"k(X, y) (k = 1, 2)

fzo = f z - - f o = cos 0.~(aO,) / 2 (a0~, is sraall) f l = sin(A0r) / ~1 - (A0y) / 2 (A0 r is ~,'nall)

and the intensity of the virtual images is

I~ = U~ Uo=2 + e;(~l +2"fleX) + e - I ( ' l+:" f lO 'O

q_ e lEa2 + 2st(i/o x +.12Y)] _]_ e - i[a, + 2~(fi o,r +fl~')] (8)

This result can al.~o be obtained in real time when the hologram is processed in. si tu after the

first exposure and then ilhlminated by R~ and R 2. After the intensity of the virtual object is recorded by a plate, for example, the plate 1, it i.:

processed in filtering system. Under the ilhimination of a collimated light, die complex amplitude in

the Fourier transfornl plane is represented by

L~G,A.) = .~{/o(x, j-)} = 46(fx, f , . )+ Pm,(f.<-f,o.f,.) + D ~ , [ - ( f . , + / , o ) , - f , . ]

+ C'}2(f.,~-f~o,/~-l~)+ U}2[-(f.<, +Ao), -G+Y~)] (9)

where 6~,, f~) is the Dirac delta function centered at (fx, f~.) and

G , G, f,) = .~{e'=,~x"'} (93 b{f2(f.,-, f,.) = ~{ e'~2``''>}

This is a spectrmn of five diffracted points as shown in Fig.2. Apart from the zero-order

diffraction , ~ , fr) at the origi.nal of the axes, two first-order wavefrouts E-j-~ and /~'~t are symmetrical about the vertical axis with frequency o/'./'1o =f~ - r e . Another couple of the first-order diffraction, Lrrz and/;~2, have the spatial freqcncics b.th in tlae directionsf~, andj~, that is, f~ o and f2. Since the information of the in-plane displacement depends on the phase change ~1 (x, y) and

/

J -S,

Fig.2 Spectrum in the first filtering

194 ACTA MECHANICA SINICA 1989

a2(x, Y) as 4.rcsinc~.

~ ( x , .,-) ~,(, . y) = x " ( " ' ~') (10)

where u(x, y) is the compon~,nts of tile displacement vec tor i l l .a: y]irection, we are interested only in

tl~e diffractions Url and Uf2. \Vhell a field stop that lets only + 1 order diffraction wavefronts pass through is placed at the Fom-icr transform plane of the lens, the intensity distribution at image plane

i(x', y') is the Frcsncl diffl'action of Eq.(9')

y)-7oq(x, y') + 2rcf2y' ( l (x , y') = K ' I + cos 2 ( I I )

Differing from the filtered result of the conventional holographic-moir6, the image is a cosine grating of basic frequency.f2, that is, as the modulation to spatial frequency, the relative phase

change ~2(.r, y)--~l(:V, 3) remains. The newly generated carrier makes it possible to shear the dense moire frim~gcs on themselves to

produce the second-order sublractive moirb. Recording the intensity of Eq.(11) by another plate

plate 2, plate 1 is then shifted laterally by kx in x direction. In this two-dimensional linear shiftinvariant system, no posinon change occurs at the spectrum plan& Together with the shifted- image intensity//2' (x', ?'} = lit' (x' TAX', 3"), the total intensity distribution at plate 2 is

U,x', Y) = f;,(t', y')+ &(x', y)

[ c~176 k~=-acq] (12) - 2 K ' 1+cos 2 cos 2

where r - ~k(x' + zXx', y) + c~k(.r', y') (t,'- 1, 2) kc~ k = a~(x' + Ax', y') -- ak(x' , y') (k = I, 2)

Beca~,,,e of the existence of the frequency-difference f2, the frequency of the first cosine term is higher than that of the second one, whi(:h can be obtained by the filtering processing of plate 2. Replacing plate 1 with plate 2 and letting only one of tile diffi'actcd wavefl'onts pass through the back

focal plane, we can observe the intensity- of image of

" " " = K cos 2 I i (X , ) ") " a ~ 2 - - AI~I (] 3)

When Acx 2 -Acq = 2nrr, I'~' (x", 3") =/,,a~' that is, at points where the differentiation of Eq.(10)

satisfies 0u n l

c~ = 0x - 2si,,c~(Ax") (n=0, +1, +2 "")

bright fringes of isopleth will be observed. Similarly, if plate 1 is shifted Ay in y direction during the first filtering, the displacement

derivative of t?u/0y can be obtained. In the same way, the other in-plane components, t?v/8x or t?v/~y, may be acquired when we illuminate the object in-plane y-z and recording, filtering as

mentioned before.

I l L E X P E R I M E N T A L V E R I F I C A T I O N

Tile optical arrangement for the holographic recording is illustrated in Fig.3. After the laser beam is split into two beams by beam spliter BS, the polarized axis of one beam is turned by 90 ~ by a quartz crystal rotator R so that tile dual beams are:with perpendicular axes. Two plane mirrors M~

Vol.5 No. 2 Fang & Oai : Holograpilic -[,,'iuire Imerie, ometry 195

and M 2 are utilized to change the direction of tile ref~'reuce beams R~ and R 2 so that the carriers can

be introduced by double exposures or in real time. To localize the object surface on the image plane,

a lens L is used to lealize image holography.

~- _..A___-f/> v x

Fig.3 Optical arrangement of recording system. R, quartz crystal rolator O,

object M 1 and M2, mirrors to change the directions of R l and R 2

The devdoped hologram is reconstructed by reference beams R, and R z (or by R't and R'2),

and the carrier fringe patterns are then recorded and processed in filtering system as mentioned

above. When the second filtering is made, pure patterns ofdisplace!j~ent derivatives are observed o~,

image plane.

Fig.4 shows the displacement derivative patterns of a disk under diametrical compression. A

curve of strain c.~ along the verticalradius lineof the disk, which shows a very close correlation with

the theoretical values plotted together, is illustrated in Fig.5.

Fig.4 The displacement derivative patterns of a disk under diametrical compression

(~) o. / ox, (b) o,~/0I

--0.8 --0.4 "'> 0.4 0.8 i J 8 ~ x / R y

40

~ , ( x 10 -5 ~ )

F!g.5 A comparison of strain eomp0nent ~, obtained from experimental data with theoretical valueg

196 ACTA MECHANICA SINICA 1989

IV. CONCLUSION

Similar to the holographic recording for displacement mcasurement, the record of polarized

shearing holographic-moir6 for strain patterns is also realized by double exposures or in real time. In

this method, the light's polarization is used to avoid the interaction between the beams, and the

introduction of two carriers is used to realize the shearing of holographic-molt6. As the modulation

to the carrier frequencies, pure patterns of the in-plane displacement derivatives are obtained by a

series of optical information processing. The new technique of holographic interferometry can be applied to measuring the strain

distribution of the curve surface'. Most important, it can be used in the solution O f dynamic problems

becatrse it retains the advantage of real time in recording.

REFERENCES ['1] Hung~ u u snd Liang~ C. u Image-shearing camera f~r direct measurement ~f surface strains. Applied Opttr176

7(1979), 1046. [2] Weissman, E. M. and Post, D. Full-field displacement and strain rossettes by Moir$ Interferometry, Exp.Mech., 22,

9(1982), 324. [3] Fang Jing, Polarized Interferometry for Strain and Stress Analyses. Ph.D. thesis, Tsinghua University, 1987, (in

Chinese). [4] Dai Fulong and Fang Jing, Polarized shearing Moir~ interferometry, Optics Communication, (in press).

[5] Gilbert, J. A., Differentiation of Holographic-Moir$ Patterns, Exp.Mech., 18, 11(1978), 436.