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Radial and rotational slope contours in speckle shear interferometry
Volume 56, number 5
OPTICS COMMUNICATIONS
1 January 1986
RADIAL AND R O T A T I O N A L S L O P E C O N T O U R S IN S P E C K L E S H E A R I N T E R F E R O M E T R Y C J O E N A T H A N , C S N A R A Y A N A M U R T H Y and R S S I R O H I Engmeermg Deszgn Centre, lndtan lnstttute of Technology, Madras 600 036, lndla Received 9 September 1985
Radial and rotational slope contours are obtamed using speckle shear mterferometry and a dual refracttve mdex ~mmers~on method Unhke speckle shear mterferometry which compares the deformed state of any object with its normal state and thus brings out the structural defects, the method reported here displays prommently any departures from an otherv, lse regular profile
1 Introduction
2 Description of the method
An object is contoured either by mechanical, electromc or optical probes, or by mterferometry Generally Interferometry is more wzdely used because it is a whole-field technique, but suffers due to over-crowdmg of fringes for very small path variations that are introduced, thus making it chfficult to analyse Shear lnterferometry Is one of the many ways by which one obtains relatwely fewer number of fnnges as it responds to the derwatlve of the path difference Moreover, speckle shear mterferometry enjoys some advantages over other mterferometnc techniques since it can be used on an object in its unpohshed state Hung et al [ 1] have obtained slope contours by speckle shear mterferometry using the conventional dual index Immersion method Jalsmg et al [2], Butters et al [3] and Joenathan et al [4] have obtained true depth contours by defocus speckles, two wavelength and dual index immersion methods respectively Recently, two new types of shears were introduced by Knshnamurthy et al [5] and Mohanty et al [6] in speckle shear lnterferometry Using these techniques and the dual Index immersion method, we report two methods for generating radial and rotational slope contours When using these techniques for testing or spotting defect sites m rotatlonally symmetrical objects, rotational shear mterferometry is useful since it contours only the defect areas
The object to be contoured is immersed in a tank contmnmg liquid of refractive index n 1 The front surface of the tank has a glass window (optically flat) through which the object is lllummated A small opening is provided in the base of the tank to drain away the liquid The object is Illuminated by a colhmated laser beam at an angle 0 as shown in fig J and is imaged on to the photographic plate A device to mtroduce either the rotational or radial shear IS placed m front of the imaging lens The tmagmg lens is apertured by a screen havmg two holes A double exposure record with the liquid changed between the exposures is made The two lrradlance distributions recorded can be written down as
0 030-4018/86/$03 50 © Elsevier Science Pubhshers B V (North-Holland Physics Pubhshmg Division)
I I = A 2 +A 2 + 2A 1A2 cos $12,
/2 = A I2 +A2 + 2A1A2 c°s($12 + 612), where $12 = $1 - $2, +2zrGx, 612 = 61 - 62,A1,A2 tank
/ ~
,mag,ng system
image ptane
Y
Flg 1 General schematic of the expertmental set up 309
Volume 56, number 5
OPTICS COMMUNICATIONS
are the amplitude distributions, q~l, q~2 are the random phases of waves via the two apertures at any point on the Image plane and 61, (52 are the phase changes Introduced due to the change In the refractive Index G is the grating frequency which depends on the aperture separation and image to lens distance The phase change (51 of one of the waves can be expressed as (51
:
(27r/X) z ( x , y ) An,
where An = n2(1 + cos n ) - n l ( l + c o s t 2 ) Here z ( x , y ) is the contour of the surface, nl, n 2 are the refractive indices of the two hqulds and rl, r 2 are the angles of refraction in the hqulds respectively and X is the wavelength of light The total lrradlance distribution recorded can be written as I = 2[A 2 +A 2 + 2A!A 2 cos(q~12 + (512/2) cos(612/2)] When this specklegram is Fourier filtered, fringe patterns depicting slope contours are obtained The bright fringes are formed when (512 = 2n~
(n = 0, 1, 2,
)
The relative phase change (512 between a point O(x,y) and ItS nelghbouring point O(x + A x , y + Ay) IS (512 : (27r/X) An [z(x + Ax, y + Ay) - z(x, y)] Expanding the terms in the above equation in Taylor's series we obtain t
t
(512 = (27r/X) An [Zx(X,y)Lx + Zy(X,y)Ay],
1 January 1986
3 Radial shear The expertmental set up to introduce radial shear is given in fig 2 It makes use of a split lens arrangement reported earher [5] Two apertures are placed, one on each half of the split lens and one segment of the spilt lens is displaced In the z-direction by a small amount Therefore two images of the object but with different magnifications are superposed at the Image plane Two types of radial shear configurations can be obtained with this set up (a) Focussed-Defocussed configuration (FD) (b) Focussed-Focussed configuration (FF) The former configuration is achieved by first focussing the object by both the spht-lens segments Now by translating one segment in the z-direction, a defocussed field is superposed on the focussed field In the second configuration, the two lens segments are in their conjugate positions with the magnification M 1 a n d M 2 bearing inverse relationship The variable shear Ar in the object plane can be written as [5] Ar
=
-(1
-
M 2)r
M 1<
l
This equation is applicable to both FF and FD configurations The shear vanishes at the centre and is maximum at the periphery of the field It can be shown that the bright fringes are formed when az
nX
r ~ - = (1 - M Z ) [ n 2 ( 1 + cos r2) - nl(1 + cos rl)] This expression gives us contour fnnges o f rOz/Or, from which radial slope contour ~z/Or can be obtained
t
where z x = Oz/3x and Z'y = ~z/Oy The above equation can also be written in polar coordinates as (512 : (27r/X)An [(Oz/ar)Ar
4 Rotational shear
+ (3z/aO)AO ]
If Ar = 0, then pure rotational shear of magnitude A0 transfers a point at (r, 0) to (r, 0 + A0) on the surface Similarly when A0 = 0, the radial shear brings the contrlbutlons of two points (r, 0) and (r + Ar, 0) to a single point at the image plane Therefore, both radial and rotational shears can as well be employed independently if one o f them is made zero at a time
The schematic arrangement for Introducing rotacNbmafed laser light ~ Ir1 o~ I - ' - ' ~
oblecf ptane
£~
2 Schematic ~L~magement to o b t ~
splat lens 310
double aperture ~spt,tlens
radm] shear using
Volume 56, number 5
OPTICS COMMUNICATIONS
1 January 1986
co[hmafed {aser hght
.
~
,mageptane
object ptane
Ftg 3 Schematic arrangement for obtaining rotatmnal shear
tlonal shear is shown m fig 3 A parr of ldent]cal Dove prisms are mounted m front o f the apertures The prisms can be rotated about their axes and translated keeping the axes parallel to each other. Inmally both the Images are made to coincide and by rotating one of the Dove prisms by A0/2, a rotational shear o f magmtude A0 is introduced m the two images [6] To introduce shear about any arbitrary point, approprmte hnear shear can be incorporated along with the rotat]onal shear Agaan it can be shown that the bright fringes m case of rotational shear are formed when
a0
A0 [n2(1 + cos r2) - nl(1 + cos r l ) ]
This expression gives us rotational slope contour fringes It is interesting to note that for a circularly symmetric object Oz/~O = 0, prowded the 0 shear ]s mtroduced about the centre of the object But ff the object that is contoured carnes a discontinuity that spoils the circular symmetry, the fringes of ~z/Ox centred at the &scontmmty are obtamed
Fig 4 az/~r contours for spheneal surface with R = 97 mm 3% ethyl alcohol solution was used to contour this surface
duced about a point on the axis o f the cyhnder The equat]on for the variation of the elevation with respect to the angle ]s gwen by a z = r 2 sm 0 cos 0 _ xy ~0 R R' where R is the radms of the cyhnder and r cos 0 and r sm 0 are any point on a tangent plane (which is the project]on plane) from a hne wh]ch suffers the maximum path length from the surface to the recording
5 Experimental results Ethyl alcohol &ssolved m water was used to obtain the reqmred refraetwe index change The first exposure was made with water and the second exposure w]th a solution o f ethyl alcohol and water. Fig 4 shows the photograph o f radial slope contours for a section of a spherical surface (R = 97 mm, dmmeter 60 mm) The specklegram is recorded at a demagnlfica tlon of 0 9 approximately and 3% ethyl alcohol solution was used for the second exposure The refractwe index change is 0 0008 at room temperature In another experiment, a section of a cyhndncal surface was used as an object and a 0-shear was mtro-
Fig 5 Oz/Oo contour slopes for cylindncal surface 5% ethyl alcohol solution was used to contour ttus surface
311
Volume 56, number 5
OPTICS COMMUNICATIONS
1 January 1986
a Fig 6 Contour slopes for a spherical surface with a dlscontmmty, (a) ~z/ao contour slopes, (b) az/Or contour slopes
plane The equation o f fringes for such an object can be written as
xy = nXR/AO An Tlus is an equation of a hyperbole Fig 5 shows the photograph o f bz/~O flanges for this object, the rotational shear (A0) being 200 mrad Here 5% solution o f ethyl alcohol in water was used The variation o f refractive index is 0 0014 at room temperature The value o f R can be evaluated from a particular fringe order There is a good agreement between theory and experiment For centro-symmetrlc object, rotational shear about the centre wdl gwe rise to zero fnnge field If a deliberate dlscontmulty is introduced then fringes about the defect site are obtained Fxg 6(a) show the photographs o f a 0-slope contour for spherical surface with a discontinuity The radius o f curvature of the object IS - 1 0 0 m m and the discontlnmty is - 4 5 m m Fig 6(b) Is the photograph o f radial slope contours for the same object The shear and magnlficatxon are as given m fig 4 In the above experiment 4% o f ethyl alcohol solution was used
312
6 Conclusion In a broad sense, these techniques are similar to the speckle mterferometnc techmque in which an object is compared with its aerial tmage using the diffuse reference beam, whereas in shear geometry, a point on the object is compared to a nearby point on the object (the points may bear lateral, radial or rotational relahon with each other) These techniques require less constraints on vibration isolation and have a lesser demand on the resolution o f the photographic plate For detecting size defects m circularly symmetrxc objects, rotational shear can be used, since it contours only the defect
References [1] Y Y Hung, J L Turner, M Tafrahan, J D Hovanestan and C E Taylor, Appl OpUcs 17 (1978) 128 [2] G K Jatsmgh andF P Chlang, Appl Optics 20 (1981) 3385 [3] J N Butters and J A Leendertz, Proe Electro-optics '74, Inter Conf (Brighton, England 1972)p 43 [4] C Joenathan, R K Mohanty and R S Strohl, Optics Letters, to be pubhshed [5] R Krlshna Murthy, R K Mohanty, R S Slrohl and M_P Kothlyal, Opt~k 67 (1984) 85 [6] R K Mohanty, C Joenathan and R S Strohl, Optics Comm 47 (1983) 27
OPTICS COMMUNICATIONS
1 January 1986
RADIAL AND R O T A T I O N A L S L O P E C O N T O U R S IN S P E C K L E S H E A R I N T E R F E R O M E T R Y C J O E N A T H A N , C S N A R A Y A N A M U R T H Y and R S S I R O H I Engmeermg Deszgn Centre, lndtan lnstttute of Technology, Madras 600 036, lndla Received 9 September 1985
Radial and rotational slope contours are obtamed using speckle shear mterferometry and a dual refracttve mdex ~mmers~on method Unhke speckle shear mterferometry which compares the deformed state of any object with its normal state and thus brings out the structural defects, the method reported here displays prommently any departures from an otherv, lse regular profile
1 Introduction
2 Description of the method
An object is contoured either by mechanical, electromc or optical probes, or by mterferometry Generally Interferometry is more wzdely used because it is a whole-field technique, but suffers due to over-crowdmg of fringes for very small path variations that are introduced, thus making it chfficult to analyse Shear lnterferometry Is one of the many ways by which one obtains relatwely fewer number of fnnges as it responds to the derwatlve of the path difference Moreover, speckle shear mterferometry enjoys some advantages over other mterferometnc techniques since it can be used on an object in its unpohshed state Hung et al [ 1] have obtained slope contours by speckle shear mterferometry using the conventional dual index Immersion method Jalsmg et al [2], Butters et al [3] and Joenathan et al [4] have obtained true depth contours by defocus speckles, two wavelength and dual index immersion methods respectively Recently, two new types of shears were introduced by Knshnamurthy et al [5] and Mohanty et al [6] in speckle shear lnterferometry Using these techniques and the dual Index immersion method, we report two methods for generating radial and rotational slope contours When using these techniques for testing or spotting defect sites m rotatlonally symmetrical objects, rotational shear mterferometry is useful since it contours only the defect areas
The object to be contoured is immersed in a tank contmnmg liquid of refractive index n 1 The front surface of the tank has a glass window (optically flat) through which the object is lllummated A small opening is provided in the base of the tank to drain away the liquid The object is Illuminated by a colhmated laser beam at an angle 0 as shown in fig J and is imaged on to the photographic plate A device to mtroduce either the rotational or radial shear IS placed m front of the imaging lens The tmagmg lens is apertured by a screen havmg two holes A double exposure record with the liquid changed between the exposures is made The two lrradlance distributions recorded can be written down as
0 030-4018/86/$03 50 © Elsevier Science Pubhshers B V (North-Holland Physics Pubhshmg Division)
I I = A 2 +A 2 + 2A 1A2 cos $12,
/2 = A I2 +A2 + 2A1A2 c°s($12 + 612), where $12 = $1 - $2, +2zrGx, 612 = 61 - 62,A1,A2 tank
/ ~
,mag,ng system
image ptane
Y
Flg 1 General schematic of the expertmental set up 309
Volume 56, number 5
OPTICS COMMUNICATIONS
are the amplitude distributions, q~l, q~2 are the random phases of waves via the two apertures at any point on the Image plane and 61, (52 are the phase changes Introduced due to the change In the refractive Index G is the grating frequency which depends on the aperture separation and image to lens distance The phase change (51 of one of the waves can be expressed as (51
:
(27r/X) z ( x , y ) An,
where An = n2(1 + cos n ) - n l ( l + c o s t 2 ) Here z ( x , y ) is the contour of the surface, nl, n 2 are the refractive indices of the two hqulds and rl, r 2 are the angles of refraction in the hqulds respectively and X is the wavelength of light The total lrradlance distribution recorded can be written as I = 2[A 2 +A 2 + 2A!A 2 cos(q~12 + (512/2) cos(612/2)] When this specklegram is Fourier filtered, fringe patterns depicting slope contours are obtained The bright fringes are formed when (512 = 2n~
(n = 0, 1, 2,
)
The relative phase change (512 between a point O(x,y) and ItS nelghbouring point O(x + A x , y + Ay) IS (512 : (27r/X) An [z(x + Ax, y + Ay) - z(x, y)] Expanding the terms in the above equation in Taylor's series we obtain t
t
(512 = (27r/X) An [Zx(X,y)Lx + Zy(X,y)Ay],
1 January 1986
3 Radial shear The expertmental set up to introduce radial shear is given in fig 2 It makes use of a split lens arrangement reported earher [5] Two apertures are placed, one on each half of the split lens and one segment of the spilt lens is displaced In the z-direction by a small amount Therefore two images of the object but with different magnifications are superposed at the Image plane Two types of radial shear configurations can be obtained with this set up (a) Focussed-Defocussed configuration (FD) (b) Focussed-Focussed configuration (FF) The former configuration is achieved by first focussing the object by both the spht-lens segments Now by translating one segment in the z-direction, a defocussed field is superposed on the focussed field In the second configuration, the two lens segments are in their conjugate positions with the magnification M 1 a n d M 2 bearing inverse relationship The variable shear Ar in the object plane can be written as [5] Ar
=
-(1
-
M 2)r
M 1<
l
This equation is applicable to both FF and FD configurations The shear vanishes at the centre and is maximum at the periphery of the field It can be shown that the bright fringes are formed when az
nX
r ~ - = (1 - M Z ) [ n 2 ( 1 + cos r2) - nl(1 + cos rl)] This expression gives us contour fnnges o f rOz/Or, from which radial slope contour ~z/Or can be obtained
t
where z x = Oz/3x and Z'y = ~z/Oy The above equation can also be written in polar coordinates as (512 : (27r/X)An [(Oz/ar)Ar
4 Rotational shear
+ (3z/aO)AO ]
If Ar = 0, then pure rotational shear of magnitude A0 transfers a point at (r, 0) to (r, 0 + A0) on the surface Similarly when A0 = 0, the radial shear brings the contrlbutlons of two points (r, 0) and (r + Ar, 0) to a single point at the image plane Therefore, both radial and rotational shears can as well be employed independently if one o f them is made zero at a time
The schematic arrangement for Introducing rotacNbmafed laser light ~ Ir1 o~ I - ' - ' ~
oblecf ptane
£~
2 Schematic ~L~magement to o b t ~
splat lens 310
double aperture ~spt,tlens
radm] shear using
Volume 56, number 5
OPTICS COMMUNICATIONS
1 January 1986
co[hmafed {aser hght
.
~
,mageptane
object ptane
Ftg 3 Schematic arrangement for obtaining rotatmnal shear
tlonal shear is shown m fig 3 A parr of ldent]cal Dove prisms are mounted m front o f the apertures The prisms can be rotated about their axes and translated keeping the axes parallel to each other. Inmally both the Images are made to coincide and by rotating one of the Dove prisms by A0/2, a rotational shear o f magmtude A0 is introduced m the two images [6] To introduce shear about any arbitrary point, approprmte hnear shear can be incorporated along with the rotat]onal shear Agaan it can be shown that the bright fringes m case of rotational shear are formed when
a0
A0 [n2(1 + cos r2) - nl(1 + cos r l ) ]
This expression gives us rotational slope contour fringes It is interesting to note that for a circularly symmetric object Oz/~O = 0, prowded the 0 shear ]s mtroduced about the centre of the object But ff the object that is contoured carnes a discontinuity that spoils the circular symmetry, the fringes of ~z/Ox centred at the &scontmmty are obtamed
Fig 4 az/~r contours for spheneal surface with R = 97 mm 3% ethyl alcohol solution was used to contour this surface
duced about a point on the axis o f the cyhnder The equat]on for the variation of the elevation with respect to the angle ]s gwen by a z = r 2 sm 0 cos 0 _ xy ~0 R R' where R is the radms of the cyhnder and r cos 0 and r sm 0 are any point on a tangent plane (which is the project]on plane) from a hne wh]ch suffers the maximum path length from the surface to the recording
5 Experimental results Ethyl alcohol &ssolved m water was used to obtain the reqmred refraetwe index change The first exposure was made with water and the second exposure w]th a solution o f ethyl alcohol and water. Fig 4 shows the photograph o f radial slope contours for a section of a spherical surface (R = 97 mm, dmmeter 60 mm) The specklegram is recorded at a demagnlfica tlon of 0 9 approximately and 3% ethyl alcohol solution was used for the second exposure The refractwe index change is 0 0008 at room temperature In another experiment, a section of a cyhndncal surface was used as an object and a 0-shear was mtro-
Fig 5 Oz/Oo contour slopes for cylindncal surface 5% ethyl alcohol solution was used to contour ttus surface
311
Volume 56, number 5
OPTICS COMMUNICATIONS
1 January 1986
a Fig 6 Contour slopes for a spherical surface with a dlscontmmty, (a) ~z/ao contour slopes, (b) az/Or contour slopes
plane The equation o f fringes for such an object can be written as
xy = nXR/AO An Tlus is an equation of a hyperbole Fig 5 shows the photograph o f bz/~O flanges for this object, the rotational shear (A0) being 200 mrad Here 5% solution o f ethyl alcohol in water was used The variation o f refractive index is 0 0014 at room temperature The value o f R can be evaluated from a particular fringe order There is a good agreement between theory and experiment For centro-symmetrlc object, rotational shear about the centre wdl gwe rise to zero fnnge field If a deliberate dlscontmulty is introduced then fringes about the defect site are obtained Fxg 6(a) show the photographs o f a 0-slope contour for spherical surface with a discontinuity The radius o f curvature of the object IS - 1 0 0 m m and the discontlnmty is - 4 5 m m Fig 6(b) Is the photograph o f radial slope contours for the same object The shear and magnlficatxon are as given m fig 4 In the above experiment 4% o f ethyl alcohol solution was used
312
6 Conclusion In a broad sense, these techniques are similar to the speckle mterferometnc techmque in which an object is compared with its aerial tmage using the diffuse reference beam, whereas in shear geometry, a point on the object is compared to a nearby point on the object (the points may bear lateral, radial or rotational relahon with each other) These techniques require less constraints on vibration isolation and have a lesser demand on the resolution o f the photographic plate For detecting size defects m circularly symmetrxc objects, rotational shear can be used, since it contours only the defect
References [1] Y Y Hung, J L Turner, M Tafrahan, J D Hovanestan and C E Taylor, Appl OpUcs 17 (1978) 128 [2] G K Jatsmgh andF P Chlang, Appl Optics 20 (1981) 3385 [3] J N Butters and J A Leendertz, Proe Electro-optics '74, Inter Conf (Brighton, England 1972)p 43 [4] C Joenathan, R K Mohanty and R S Strohl, Optics Letters, to be pubhshed [5] R Krlshna Murthy, R K Mohanty, R S Slrohl and M_P Kothlyal, Opt~k 67 (1984) 85 [6] R K Mohanty, C Joenathan and R S Strohl, Optics Comm 47 (1983) 27