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Hologram interferometric study of zero frequency filtered images; application to observation of small phase defects
HOLOGRAM ZERO APPLICATION
TO
INTERFEROMETRIC
FREQUENCY
de Physique
G&n&rale et Optique,
STUDY
FILTERED
OBSERVATION
C. FROEHLY, Laboratoire
October
OPTICS COMMUNICATIONS
Volume 4, number 2
OF
1971
OF
IMAGES;
SMALL
PHASE
DEFECTS
E. EDEES and J. CH. VIENOT Facult6
Received
des
Sciences,
la Bouloie,
25 Besanqon,
France
21 July 1971
Zero frequency spatial filtering techniques (strioscopy) are known to be scarcely adequate for the observation of very small phase variations. However, attaining high phase sensitivity becomes possible by using a new method associating hologram - or classical - interferometry to strioscopy.
1. INTRODUCTION The present work was suggested by some photographic results previously obtained by holographic strioscopy and phase-contrast [1,2]. Sufficiently well localized phase defects of an optical wave surface can be conveniently revealed by using simple low frequency spatial filtering techniques rather than classical two beam interferometry. In the case where the path variations introduced by localized defects in the transmitting medium are small enough with respect to the illumination wavelength, the phase-contrast method is very sensitive. The situation is particularly simple if one uses complete low frequency occultation by means of an opaque screen (socalled “strioscopy”) instead of a h/4 phase plate. However, due to the linear relationship between the filtered image local luminance and the corresponding optical path difference, phase contrast keeps an advantage. Indeed, good accuracy is also obtained by combining classical or holographical interferometry and strioscopic techniques. Such a new method is shown to be highly precise; it yields detailed information about the sign and modulus of phase variations under test.
2. COMPLEX AMPLITUDE IN THE STRIOSCOPIC IMAGE OF A PHASE OBJECT The defect corresponds to a variation 6 of the optical thickness. The complex amplitude of the transmitted light in the corresponding region can $ Now at the Universite du Benin, Lome,
Togo.
Fig. 1. Fresnel diagram showing relations between the complex amplitudes at the strioscopic imAge (vector 6%‘) and corresponding phase object (vector OM). The circle centered at 0’ is equal %that centered at 0; (OO’( = ]OM/ =A,,,
be written as: A(6)
=A0 exp ((211i/X)6},
(1)
where A0 and h refer to the modulus and the wavelength of the incident light respectively. Classical calculations [4,5] show that, for suitable transverse size of the opaque screen acting as a spatial filter (e.g. 0.1 mm), the transmitted image amplitude is approximately described by the simple form: A’(6)
= A(6) - 1 = A0 [exp ((27~i/x)6}-
l]
.
It should be pointed out that the same result would occur for the image of this phase object projected through a two beam interferometer working at or about n-phase difference [6,2] (dark uniform background for the observation field). At any object point characterized by an optical thickness variation 6, the image illumination 103
October
Photo Fig. 2. Photographs (slig ;htly defocussed).
104
1971
2
Photo 1: normal observati of the same phase object (thickness defect in ljhotogrnphic emulsion). Photo 2: classical or holographical t\\o-bennl interferogram showing the local amplitude of the defect.
on
Volume 4, number
2
OPTICS
COMMUNICATIONS
October
1971
Photo 3
Photo 4 image (= two-beam interferograms, Fig. 2. I ‘hoto 3: strioscopic dark uniform background) Photo 4: ho100 Tlplnica !I strioscopy; the 7i phase shift inside the circle corresponds to phase dc:fecl ;s clsci Illal .ing (or clas ;sical .) ilIterferometric around the zero value (see photo 2). 105
Volume
4, number 2
OPTICS COMMUNICATIONS
October
1971
E(6) takes a value proportional to the product A(6)A*(6) where A* means complex conjugate quantity of A. E’(6) = IL4 ‘(6) A’*( 6) E(6) = 4KA2 sin2 {(n/X)6}
(3)
0
(the factor K depends on the geometrical eters of the device). The phase $ of the light wave at this point happens to fit the relation: tan+
= cot(n 6/X) = tan(&
+ nS/h)
=tan(-$n+nb/h) If 6/h
paramimage
if 6 > 0 if 6<0.
(4)
<< 1, the approximation,
tan + = X/776 shows that, when 6 reaches zero by positive or negative values, phase $ shifts to & $I respectively, with a discontinuity at the origin. This results obviously from Fresnel diagram (fig. 1): + the OM vector associated with the complex amplitude A (6) in the object is transformed by spatial filtering -cf. eq. (2) -into the OM vector such as M’G
= @d
= -A;
for
small
of the image-to-object
values
3. EXPERIMENTAL
phase ratio
of 6.
VERIFICATI9N
A slight defocussing of the image of the phase object (photo 1, fig. 2) underlines its contour. Optical thickness variations of this object range from h,‘2 to h,‘lO (see the interferogram photo 2, fig. 2). The strioscopic image (photo 3, fig. 2) is actually quite identical to the corresponding interferogram which would be obtained with uniform dark background of the observation field. In the image the oscillations of phase are then studied by classical or holographical interferometry (e.g. fig. 3). A first recording is made, the holographic plate being placed in the interference field due to superposition of waves coming from the source S and from the filtered image (9) of the object (0 fi. At the second exposure the object and opaque screen are removed and the reference source laterally displaced at S2. 106
The reconstructed image from this hologram (H) (photograph 4, fig. 2) appears to be streaked with well contrasted fringes; discontinuities of phase + clearly appear inside the circle, indicating changes of the path difference sign, whereas the corresponding fringes of the conventional interferogram (photo 2, fig. 2) exhibit small distortions only.
4. DISCUSSION
.
The complex amplitude in the image is proportional to the vector a’, whose phase actually jumps from - &n to+in as 6 passes through the zero-value; we may thus observe an infinite
umplificntion
Fig. 3. Experimental set-up for recording double exposure holograms of a strioscopic image. First exposure: (3): object with phase defects; the telescopic system tl, L2 gives a distortion-free image (9) of (0); low frequencies are filtered out in the image (screen F). The latter interferes with the reference beam coming from SI on the photographic plate (H). Second exposure. (0) and F are removed; S1 comes to S2 position.
The value of the optical path thus calculated by measurements of local visibility of the interference pattern shown fig. 2 (photo 4) is more accurate than that obtained by performing photometric recordings on the simple strioscopic image If 6/x << 1, after eq. (2), the amplitude A’(6 ) assumes the well-known approximated form:
.
A’(6) N A0 (2ni/h)6
If A1 is the amplitude of the reference beam interfering with A’(6) the visibility u of these fringes at the defect 6 may be written as the following function of A 1 and A ‘( 6): 2lAll lA’(6)l 2, = _._____
lAl/2+(A’(6)l
~ 2Ao 2n 6
2
A1X
’
(7)
v is a linear function of 6, whose slope increases as the reference amplitude decreases. For instance, in order to illustrate the sensitivity of the method, let us consider a typical situation where Al = 0.1 A0 (thus El = 0.01 EO); if one supposes that noise terms are negligikle, a defect with an optical thickness equal to 5A theoretically brings the fringe visibility to the measurable value z: = 0.1; such orders of mag-
Volume 4, number
2
OPTICS COMMC’NICATIONS
nitude are quite similar to those usually accepted is absorbing phase contrast methods, yet this new device comparatively increases the brightness of the image: the observation of a h/200 thick phase defect requires one to interpose (phase contrast method) an absorbing phase plate of optical density equal to 1. It would result in a 90% waste of the light flux, whereas the same defect would appear five times more luminous when observed by the interferometric strioscopy method; this gain rises strongly as the expected precision increases.
October
1971
REFERENCES [l] J. Ch. Vie’not and J. Monneret, Compt. Rend. Acad. Sci. (Paris) 262B (1966) 671. [2] E. Edee, These de 3eme cycle, Universite de BesanTon (1970). [3] M. Frangon, Le contraste de phase en optique et microscopic (Rev. Opt., Paris, 1950). [4] E. L. Gayhart and R. Prescott. J. Opt. Sot. Am. 39 (1949) 546. [5] E. B. Temple, J. Opt. Sot. Am. 41 (1957) 91. [6] R. Barer, in: Contraste de phase et contraste par interferences (Rev. Opt., Paris, 1952) p. 56.
107
TO
INTERFEROMETRIC
FREQUENCY
de Physique
G&n&rale et Optique,
STUDY
FILTERED
OBSERVATION
C. FROEHLY, Laboratoire
October
OPTICS COMMUNICATIONS
Volume 4, number 2
OF
1971
OF
IMAGES;
SMALL
PHASE
DEFECTS
E. EDEES and J. CH. VIENOT Facult6
Received
des
Sciences,
la Bouloie,
25 Besanqon,
France
21 July 1971
Zero frequency spatial filtering techniques (strioscopy) are known to be scarcely adequate for the observation of very small phase variations. However, attaining high phase sensitivity becomes possible by using a new method associating hologram - or classical - interferometry to strioscopy.
1. INTRODUCTION The present work was suggested by some photographic results previously obtained by holographic strioscopy and phase-contrast [1,2]. Sufficiently well localized phase defects of an optical wave surface can be conveniently revealed by using simple low frequency spatial filtering techniques rather than classical two beam interferometry. In the case where the path variations introduced by localized defects in the transmitting medium are small enough with respect to the illumination wavelength, the phase-contrast method is very sensitive. The situation is particularly simple if one uses complete low frequency occultation by means of an opaque screen (socalled “strioscopy”) instead of a h/4 phase plate. However, due to the linear relationship between the filtered image local luminance and the corresponding optical path difference, phase contrast keeps an advantage. Indeed, good accuracy is also obtained by combining classical or holographical interferometry and strioscopic techniques. Such a new method is shown to be highly precise; it yields detailed information about the sign and modulus of phase variations under test.
2. COMPLEX AMPLITUDE IN THE STRIOSCOPIC IMAGE OF A PHASE OBJECT The defect corresponds to a variation 6 of the optical thickness. The complex amplitude of the transmitted light in the corresponding region can $ Now at the Universite du Benin, Lome,
Togo.
Fig. 1. Fresnel diagram showing relations between the complex amplitudes at the strioscopic imAge (vector 6%‘) and corresponding phase object (vector OM). The circle centered at 0’ is equal %that centered at 0; (OO’( = ]OM/ =A,,,
be written as: A(6)
=A0 exp ((211i/X)6},
(1)
where A0 and h refer to the modulus and the wavelength of the incident light respectively. Classical calculations [4,5] show that, for suitable transverse size of the opaque screen acting as a spatial filter (e.g. 0.1 mm), the transmitted image amplitude is approximately described by the simple form: A’(6)
= A(6) - 1 = A0 [exp ((27~i/x)6}-
l]
.
It should be pointed out that the same result would occur for the image of this phase object projected through a two beam interferometer working at or about n-phase difference [6,2] (dark uniform background for the observation field). At any object point characterized by an optical thickness variation 6, the image illumination 103
October
Photo Fig. 2. Photographs (slig ;htly defocussed).
104
1971
2
Photo 1: normal observati of the same phase object (thickness defect in ljhotogrnphic emulsion). Photo 2: classical or holographical t\\o-bennl interferogram showing the local amplitude of the defect.
on
Volume 4, number
2
OPTICS
COMMUNICATIONS
October
1971
Photo 3
Photo 4 image (= two-beam interferograms, Fig. 2. I ‘hoto 3: strioscopic dark uniform background) Photo 4: ho100 Tlplnica !I strioscopy; the 7i phase shift inside the circle corresponds to phase dc:fecl ;s clsci Illal .ing (or clas ;sical .) ilIterferometric around the zero value (see photo 2). 105
Volume
4, number 2
OPTICS COMMUNICATIONS
October
1971
E(6) takes a value proportional to the product A(6)A*(6) where A* means complex conjugate quantity of A. E’(6) = IL4 ‘(6) A’*( 6) E(6) = 4KA2 sin2 {(n/X)6}
(3)
0
(the factor K depends on the geometrical eters of the device). The phase $ of the light wave at this point happens to fit the relation: tan+
= cot(n 6/X) = tan(&
+ nS/h)
=tan(-$n+nb/h) If 6/h
paramimage
if 6 > 0 if 6<0.
(4)
<< 1, the approximation,
tan + = X/776 shows that, when 6 reaches zero by positive or negative values, phase $ shifts to & $I respectively, with a discontinuity at the origin. This results obviously from Fresnel diagram (fig. 1): + the OM vector associated with the complex amplitude A (6) in the object is transformed by spatial filtering -cf. eq. (2) -into the OM vector such as M’G
= @d
= -A;
for
small
of the image-to-object
values
3. EXPERIMENTAL
phase ratio
of 6.
VERIFICATI9N
A slight defocussing of the image of the phase object (photo 1, fig. 2) underlines its contour. Optical thickness variations of this object range from h,‘2 to h,‘lO (see the interferogram photo 2, fig. 2). The strioscopic image (photo 3, fig. 2) is actually quite identical to the corresponding interferogram which would be obtained with uniform dark background of the observation field. In the image the oscillations of phase are then studied by classical or holographical interferometry (e.g. fig. 3). A first recording is made, the holographic plate being placed in the interference field due to superposition of waves coming from the source S and from the filtered image (9) of the object (0 fi. At the second exposure the object and opaque screen are removed and the reference source laterally displaced at S2. 106
The reconstructed image from this hologram (H) (photograph 4, fig. 2) appears to be streaked with well contrasted fringes; discontinuities of phase + clearly appear inside the circle, indicating changes of the path difference sign, whereas the corresponding fringes of the conventional interferogram (photo 2, fig. 2) exhibit small distortions only.
4. DISCUSSION
.
The complex amplitude in the image is proportional to the vector a’, whose phase actually jumps from - &n to+in as 6 passes through the zero-value; we may thus observe an infinite
umplificntion
Fig. 3. Experimental set-up for recording double exposure holograms of a strioscopic image. First exposure: (3): object with phase defects; the telescopic system tl, L2 gives a distortion-free image (9) of (0); low frequencies are filtered out in the image (screen F). The latter interferes with the reference beam coming from SI on the photographic plate (H). Second exposure. (0) and F are removed; S1 comes to S2 position.
The value of the optical path thus calculated by measurements of local visibility of the interference pattern shown fig. 2 (photo 4) is more accurate than that obtained by performing photometric recordings on the simple strioscopic image If 6/x << 1, after eq. (2), the amplitude A’(6 ) assumes the well-known approximated form:
.
A’(6) N A0 (2ni/h)6
If A1 is the amplitude of the reference beam interfering with A’(6) the visibility u of these fringes at the defect 6 may be written as the following function of A 1 and A ‘( 6): 2lAll lA’(6)l 2, = _._____
lAl/2+(A’(6)l
~ 2Ao 2n 6
2
A1X
’
(7)
v is a linear function of 6, whose slope increases as the reference amplitude decreases. For instance, in order to illustrate the sensitivity of the method, let us consider a typical situation where Al = 0.1 A0 (thus El = 0.01 EO); if one supposes that noise terms are negligikle, a defect with an optical thickness equal to 5A theoretically brings the fringe visibility to the measurable value z: = 0.1; such orders of mag-
Volume 4, number
2
OPTICS COMMC’NICATIONS
nitude are quite similar to those usually accepted is absorbing phase contrast methods, yet this new device comparatively increases the brightness of the image: the observation of a h/200 thick phase defect requires one to interpose (phase contrast method) an absorbing phase plate of optical density equal to 1. It would result in a 90% waste of the light flux, whereas the same defect would appear five times more luminous when observed by the interferometric strioscopy method; this gain rises strongly as the expected precision increases.
October
1971
REFERENCES [l] J. Ch. Vie’not and J. Monneret, Compt. Rend. Acad. Sci. (Paris) 262B (1966) 671. [2] E. Edee, These de 3eme cycle, Universite de BesanTon (1970). [3] M. Frangon, Le contraste de phase en optique et microscopic (Rev. Opt., Paris, 1950). [4] E. L. Gayhart and R. Prescott. J. Opt. Sot. Am. 39 (1949) 546. [5] E. B. Temple, J. Opt. Sot. Am. 41 (1957) 91. [6] R. Barer, in: Contraste de phase et contraste par interferences (Rev. Opt., Paris, 1952) p. 56.
107