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Real time contour line visualization of an object
Volume 12, number 2
OPTICS COMMUNICATIONS
REAL TIME CONTOUR
LINE VISUALIZATION
October 1974
OF AN OBJECT *
P. BENOIT and E. MATHIEU Laboratoire d'Optique Coh~rente, Centre d'Etudes et de Recherches de la Compagnie Electro-M~canique, 55, Avenue Jean Jaurds, 93350 Le Bourget, France
Received 16 May 1974 Revised version received 2 August 1974
Two methods of real time observation of contour lines of an object are discussed and experimental results are given. An analysis of the visibilty of these contour fringes is presented and some experimental limitations are considered.
Introduction Several methods of contour line generation have been described during the last years. Two essential approaches may be found in the literature: the first one uses holographic techniques [ 1 ], the second one uses the Moir6 phenomenon with incoherent light [ 2 - 4 ] . Nevertheless, those methods, either need gratings o f the dimensions of the object to be tested, or do not work in real time. In this paper we present experimental results on real time contouring using two different grid projection techniques. A short analysis o f the methods and their limitations is given. The technique of fabricating appropriate transmission masks is also mentioned. In section 1 a method is discussed where the observation o f the Moir6 fringes has a fixed direction: one has to observe through a mask. The second m e t h o d leads to contour lines which are directly visible on the object.
structure on the object and a transmission mask which defines the reference plane (fig. 1). The object is illuminated by incoherent, spatially structured light of spatial period p. The angle between the observation direction and the illumination direction is given by/3. The object is observed by imaging the illuminated object through the lens L on a mask M containing the information about the reference plane. The contour lines of the object are directly visible by placing a diffuser D upon the transmission mask M. For the most simple case of illumination the spatial distribution of the light intensity I ( X , Y) is given by
1. Observation of contour lines through a mask The experimental setup for contour mapping consists of an appropriate light projector which images a grid * Work supported by Comit6 de la M6canique de la D.G.R.S.T. in collaboration with Centre Technique des Industries MScaniques (C.E.T.I.M.).
Fig. 1. Arrangement for observation of contour lines of the object O through the mask M (Z = direction of illumination, z = direction of observation,/3 = angle between these two directions). 175
Volume 12, number 2
OPTICS COMMUNICATIONS
) ](X, II) =10 ~n an e x p /27rin ~X
an = a_n ,
;
(1)
where the two axes X, Y are orthogonal to the illumination direction Z. I 0 is a normalization factor. By rotating the X, Y, Z coordinate system around the Y axis of an angle/3 the new z coordinate corresponds to the observation direction. In this x, y, z system the object form is represented by the function z =f(x, y) =fxy (with z = 0 as reference plane) and the spatial distribution of the illuminating light intensity I(x, y, fxy) on the surface of the object is given by
I(x'Y'fxy)=Io ~n an
/2ran (x exp~--~-
cos/3+fxy
sin/3))
.
(2) This intensity distribution is imaged on the transmission mask M containing the spatial transmission distribution T(x', y ' ) with
T(x',y')=Ko~m amexpt2n~mx'cos @ •
October 1974
adjacent lines Afis given by A f = p/sin/3. Since for each term Im, n, where m + n = 0, the carrier frequency vanishes, formulas similar to eq. (5) are found by replacing cos ((27r/p) fx')/sin/3 } by cos (n (2rr/p) fx'y' × sin/3} and zSf equals p/(n sin/3) (n integer). Usually it is the fundamental term I_+1,+1 given by eq. (5) which is investigated. All other terms can be considered as terms of noise which reduce the visibility v of the contour lines. To eliminate the terms Ira, n with m + n :~ 0 (i.e. containing the carrier) one has to fix the spatial resolution behind the mask M in such a manner that the carrier frequency is no more resolved. (Naturally the Moir6 fringes forming the contour lines have to be resolved). With this low resolution all terms with m + n 4= 0 are equal to zero. Neglecting terms of higher order (i.e. m = 2, n = --2, etc.) which perturb the fundamental contour line system, the visibility v can reach theoretically a maximum value of
u = 2a~/a 2 .
(6)
(3)
We have assumed a 1 : 1 imaging system (K 0 is a constant). The spatial intensity distribution I(x', y') in the diffuser is therefore equal to the product ofI(x,Y, fxy) and T R (x', y'), viz.
I(x" Y')= IoKo G aman exp ( 21ri(pm+n) x' 2rrin sin/3 ) . X exp ,'l~p--fx,y,
(4)
The last exponential factor contains the information of the form of the object and is modulated by the spatial carrier frequency (rn + n)(cos/3)[p. The first term of the summation (m = n = O) is the average intensity (I) on the image plane. For m = O, n = + 1, I01 is proportional to
fxy
cos ((2rr/p)(x' cos/3 + fx'y' sin/3) } , whereas for m = +- 1, and n = ~- 1 the corresponding intensity I+l, zl contains no carrier frequency: I-+1,~I
=2IoKoa~c°s(-~fxySin/3) "
(5)
It is this term 1±1, :~1 which gives rise to contour lines in the image plane. The height difference between two 176
Fig. 2. Contour lines of a turbine blade (65 X 12 cm). Using an angle/3 of 25° and a period of 1.3 mm on the plane of reference a height difference 2xfof two adjacent lines of 3.3 mm is achieved.
Volume 12, number 2
OPTICS COMMUNICATIONS
For the projection of a Ronchi ruling (a 0 = 1/2, a 1 = l/n, a 2 = 0, a 3 = - 1 / 3 rr) the first harmonic contribution (n = 3) is negligible behind the fundamental one (V 1 = 8/7r2, V 3 "~ 1/7r2). An experimental result is shown in fig. 2. It has been obtained by projecting an appropriate transmission mask M 1 on the blade. The projection has the following special properties: - The spatial illumination structure is a regular grid on the reference plane (z = 0) with a period p' in this plane. - The reference plane is exactly the image plane of the mask M 1 . With this technique it is possible to observe contour lines with a reference plane which is orthogonal to the observation direction. The transmission mask M 1 has to be carefully made so as to eliminate a part of the divergence of the imaging system. By divergence we mean the fact that the magnification of the optical system varies across the field due to the tilted object and image planes. Hence the grating period p would be nonuniform, unless special precautions are taken. By simply projecting e.g. a Ronchi ruling on the object (with an angle fl with respect to the observation direction) and using an appropriate Ronchi ruling behind the lens L the reference of this system would no more be a plane, but a cylindrically curved surface. A way of obtaining the appropriate mask M 1 is shown in fig. 3. A regular mask M (usually a Ronchi ruling) is projected in the direction z of observation on the reference plane (x, y). The illuminated refer-
October 1974
ence plane is imaged in the direction Z on a photo-
graphic plate. The developed plate is used as a transmission mask M 1 . By illuminating M 1 and projecting this transmission mask on the reference plane the initial light distribution on the reference plane is achieved. The mask M is just the well-aligned mask of observation which is needed for this method.
2. Direct observation o f contour lines on the object
In this second method two illuminating light beams are used. The method has been treated in a geometrical way by Wasowski [5]. Since this method generates the contour lines directly on the object, this technique seems quite promising for several applications of object form control. The experimental set up is shown in fig. 4. The two illumination directions are symmetrically arranged with respect to the observation axis z. The two angles between illumination axes and z axis are therefore given by fl and -fl, respectively. Supposing equal illumination I 0 of the object from both sides, the intensity I(x,Y, fxy ) on the object is given by
I(x,Y, fxy)=Io { ~n an exp
(2~
(x
c°sfl+fxy sin fl))
using the same notation as for eq. (2). Since eq. (7) can be written as
a n =an,
+ /
Y\
, i
/
/
? Fig. 3. Arrangement for obtaining the appropriate mask M1 (M = regular grating, which is used - after the production of MI - as well adapted mask of observation). M1 is tilted such that the reference plane (z = 0) is imaged onto it.
Fig. 4. Contour line generation using two directions (Z1, Z2) of illumination. On the object O two different fringe systems are superposed. This leads to Moirg fringes which depict contours lines upon this object. 177
Volume 12, number 2
OPTICS COMMUNICATIONS
I(x'Y"Ixy)= 210 { aO+N>O~--'2aNc°s ~ ( 2rrxPc°s/3 × cos
(N 2rr(xysin/3}t! _ \
(8)
p
In this method the Moir6 structure is visible without any mask, but every Moir6 term is multiplied by a carrier frequency, i.e. it is not possible to see the contour lines without the spatial modulation of a carrier. If a J Ronchi-type illumination structure is chosen, the resolution of the observing system has to integrate spatially the carrier frequency (3 cos/3)/p to suppress the contour line system given by cos sin/3} which perturbs the fundamental Moir6 fringes. Using a spatial resolution of &x and Ay, respectively, one observes a spatially averaged intensity showing a fringe system with a visibility depending on the angles and 6 (x, y ) of the object defined by
{(6n/P)fxy
e(x,y)
e(x,y) = arctg of and 6 ( x , y ) = arctgV," .at ay"
October 1974
It is also important in this experiment to use two appropriate masks M 1 and M 2 for the object illumination. Using this technique we investigated the possibilities of visualizing a contour line system with distances between two adjacent contour lines of the order of some hundred microns. This scheme is of interest for surface control on small and well defined parts of a workpiece. A result is shown in fig. 5 using the experimental arrangement of fig. 4. It has been obtained by simply photographing the doubly illuminated medal which has a diameter of 10 cm. The angle t3 was chosen equal to 60 °. The period p' on the plane of reference was taken equal to 490/J. This leads to a A f = 140 #. It should be mentioned that the quality and the visibilty of the contour line system can be improved by classical optical filtering techniques using coherent or even incoherent light. With these techniques one can eliminate the carrier frequency, but this optical filtering cannot be done in real time. To maintain real time contouring of an object it is in certain cases depending primarily on the carrier frequency involved - possible to use an appropriate electro-optical arrangement [6].
(9) 3. Limitations of the methods A restriction of contour line generation is imposed by the object angles and [cf. (9)]. Using for a zone tx -x0[, ] y - y 0 1 < &r the Taylor development off(x, y), the intensity I on the surface of the object [eq. (3)] can be written as
e(x,y)
6(x,y)
I(x,Y, fxy ) =I 0 (a 0 + a I (exp [i(K" r ~- ~)1 + e x p [ - i ( K - r ÷ ~)l)}
(10)
by taking only the fundamental terms n = 0, + 1 into account. The components of K are given by
Kx=p_2~(cos/3 + ~x3¢"sin/3)
and
Ky -2zr3f-p3ysin/3 .
The effects of e and 6 on the observed fringe system are: - locally the resulting period is given by
p(x,y)
Fig. 5. Experimental result of this method. The medal has a diameter of 8 cm. The height difference between two adjacent lines 2x.fhas the value Af= 140 U.
p/cosl]
P(x'Y)-v/i[l+tge(x,y)tg/3]2+
[tg8 (x,y) tg/3] 2 (lla)
178
Volume 12, number 2
OPTICS COMMUNICATIONS
October 1974
2~/p_
S
....... ,
,osp.j'
~
I
D-
J
Fig. 6. Representation of the angular limitations of the object angles e(x, y) and a (x, y) in a normalized K-space. a) Case of the mask method: only the points Q(e, 6) inside the circle with radius p/a can be used for correct object contouring; b) case of the second method: the resolution circle (hatched) is centered around the origin. The circle around the point (cos ~3,0) describes the limitations due to the fact that also the Moire frequency K M has to be resolved by the resolution cell with diameter a. - the fringe system is locally rotated by an amount a(x,y) with respect to the Y-axis: tg a(x, y ) =
tg 6 (x, y ) ctg/3 + t g e ( x , y ) "
(1 l b )
presented in K-space. All points Q(e, 6) lying inside of the circle of resolution can be used for contour mapping. For 6 ~ 0 and e > 0 the experiment gave a value of 0.35 for the product tg e tg/3 of eq. (12a).
This description corresponds to a first fringe system with a vector K 1 . The second vector K 2 depends on the contour mapping method which is used.
3.2. Double projection method
3.1. Mask method
K2x = ( 2n /p)(cos /3 - tg e sin/3)
Here K 2 is given by
and The components of K 2 are given by K2x = (2n/p) cos/3 and K2y = 0. The Moir~ vector K M is given by K M = K 1 - K 2. The corresponding Moir~ period PM is therefore:
K2y = - (27r/p) tg 6 sin/3. The two Moir~ components are
KMx = (27r/p) 2 tg 6 sin/3
P
sin /3 x/tg2 e (x, y ) + tg 2 6 (x, y ) "
and
This period PM has to be resolved by the observation system with spatial resolution o. The condition
KMy = (2rt/p) 2 tg 6 sin/3,
PM > o
(12)
is therefore necessary and limits the object angles e and 6 which can be used for contour mapping. It has to be noted that for the particular case 6 ~ 0 ° the condition has to be modified and becomes more severe, because K M has the same direction as K 1 and K2, respectively. To separate in this case correctly K M from K 1 or K2, the condition (for/3 > 0)
- 1 < tge(x,y) tg/3 < ½
(12a)
has to be fulfilled. In fig. 6a these restrictions are re-
respectively. The Moird period PM is given by (P/2) sin/3 ~/tg2e + tg26. However, as was shown in eq. (8), it is necessary in this method to resolve the carrier frequencies K 1 and K 2. A resolution cell o f diameter a corresponds in Kspace to a circle (fig. 6b), but here the circle is centered around the origin (hatched curve). Moreover, the Moir4 frequency K M has to be resolved, too. This leads to an additional limiting circle of radius p/2o around the point (cos/3, 0). Since all Moird vectors pass through this point (cos/3) on the Kx-axis, the 179
Volume 12, number 2
OPTICS COMMUNICATIONS
symmetrical curve to the hatched curve has to be traced, too (curve: . . . . . . ). The end points of the vectors K1,2 which are directly related to the object's angles e and 6, must therefore lie inside of the zone being traced by thick lines. For 6 ~ 0 ° the restriction on tg e is more important, too.
October 1974
Acknowledgements Acknowledgements are due to A. Thomas for his help with the experiments.
References 4. Conclusion With illumination by special masks it is possible to observe in real time the contour lines of an object. The spatial resolution of the detecting system influences fringe visibility and perturbations. The angles e and 6 of the object determine the geometry of the arrangement and limit the value Af.
180
[ 1] W. Schmidt, A. Vogel and D. Preussler, Appl. Phys. 1 (1973) 103. [2] D.M. Meadows, W.O. Johnson, J.B. Allen, Appl. Opt. 9 (1970) 94. [3] H. Takasaki, Appl. Opt. 12 (1973) 845. [4] J. der Hovanesian and Y.Y. Hung, Appl. Opt. 10 (1971) 2734. [5] J. Wasowski, Opt. Commun. 2 (1970) 321. [6] B. Dessus, J.P. Gerardin and P. Mousselet, to be published.
OPTICS COMMUNICATIONS
REAL TIME CONTOUR
LINE VISUALIZATION
October 1974
OF AN OBJECT *
P. BENOIT and E. MATHIEU Laboratoire d'Optique Coh~rente, Centre d'Etudes et de Recherches de la Compagnie Electro-M~canique, 55, Avenue Jean Jaurds, 93350 Le Bourget, France
Received 16 May 1974 Revised version received 2 August 1974
Two methods of real time observation of contour lines of an object are discussed and experimental results are given. An analysis of the visibilty of these contour fringes is presented and some experimental limitations are considered.
Introduction Several methods of contour line generation have been described during the last years. Two essential approaches may be found in the literature: the first one uses holographic techniques [ 1 ], the second one uses the Moir6 phenomenon with incoherent light [ 2 - 4 ] . Nevertheless, those methods, either need gratings o f the dimensions of the object to be tested, or do not work in real time. In this paper we present experimental results on real time contouring using two different grid projection techniques. A short analysis o f the methods and their limitations is given. The technique of fabricating appropriate transmission masks is also mentioned. In section 1 a method is discussed where the observation o f the Moir6 fringes has a fixed direction: one has to observe through a mask. The second m e t h o d leads to contour lines which are directly visible on the object.
structure on the object and a transmission mask which defines the reference plane (fig. 1). The object is illuminated by incoherent, spatially structured light of spatial period p. The angle between the observation direction and the illumination direction is given by/3. The object is observed by imaging the illuminated object through the lens L on a mask M containing the information about the reference plane. The contour lines of the object are directly visible by placing a diffuser D upon the transmission mask M. For the most simple case of illumination the spatial distribution of the light intensity I ( X , Y) is given by
1. Observation of contour lines through a mask The experimental setup for contour mapping consists of an appropriate light projector which images a grid * Work supported by Comit6 de la M6canique de la D.G.R.S.T. in collaboration with Centre Technique des Industries MScaniques (C.E.T.I.M.).
Fig. 1. Arrangement for observation of contour lines of the object O through the mask M (Z = direction of illumination, z = direction of observation,/3 = angle between these two directions). 175
Volume 12, number 2
OPTICS COMMUNICATIONS
) ](X, II) =10 ~n an e x p /27rin ~X
an = a_n ,
;
(1)
where the two axes X, Y are orthogonal to the illumination direction Z. I 0 is a normalization factor. By rotating the X, Y, Z coordinate system around the Y axis of an angle/3 the new z coordinate corresponds to the observation direction. In this x, y, z system the object form is represented by the function z =f(x, y) =fxy (with z = 0 as reference plane) and the spatial distribution of the illuminating light intensity I(x, y, fxy) on the surface of the object is given by
I(x'Y'fxy)=Io ~n an
/2ran (x exp~--~-
cos/3+fxy
sin/3))
.
(2) This intensity distribution is imaged on the transmission mask M containing the spatial transmission distribution T(x', y ' ) with
T(x',y')=Ko~m amexpt2n~mx'cos @ •
October 1974
adjacent lines Afis given by A f = p/sin/3. Since for each term Im, n, where m + n = 0, the carrier frequency vanishes, formulas similar to eq. (5) are found by replacing cos ((27r/p) fx')/sin/3 } by cos (n (2rr/p) fx'y' × sin/3} and zSf equals p/(n sin/3) (n integer). Usually it is the fundamental term I_+1,+1 given by eq. (5) which is investigated. All other terms can be considered as terms of noise which reduce the visibility v of the contour lines. To eliminate the terms Ira, n with m + n :~ 0 (i.e. containing the carrier) one has to fix the spatial resolution behind the mask M in such a manner that the carrier frequency is no more resolved. (Naturally the Moir6 fringes forming the contour lines have to be resolved). With this low resolution all terms with m + n 4= 0 are equal to zero. Neglecting terms of higher order (i.e. m = 2, n = --2, etc.) which perturb the fundamental contour line system, the visibility v can reach theoretically a maximum value of
u = 2a~/a 2 .
(6)
(3)
We have assumed a 1 : 1 imaging system (K 0 is a constant). The spatial intensity distribution I(x', y') in the diffuser is therefore equal to the product ofI(x,Y, fxy) and T R (x', y'), viz.
I(x" Y')= IoKo G aman exp ( 21ri(pm+n) x' 2rrin sin/3 ) . X exp ,'l~p--fx,y,
(4)
The last exponential factor contains the information of the form of the object and is modulated by the spatial carrier frequency (rn + n)(cos/3)[p. The first term of the summation (m = n = O) is the average intensity (I) on the image plane. For m = O, n = + 1, I01 is proportional to
fxy
cos ((2rr/p)(x' cos/3 + fx'y' sin/3) } , whereas for m = +- 1, and n = ~- 1 the corresponding intensity I+l, zl contains no carrier frequency: I-+1,~I
=2IoKoa~c°s(-~fxySin/3) "
(5)
It is this term 1±1, :~1 which gives rise to contour lines in the image plane. The height difference between two 176
Fig. 2. Contour lines of a turbine blade (65 X 12 cm). Using an angle/3 of 25° and a period of 1.3 mm on the plane of reference a height difference 2xfof two adjacent lines of 3.3 mm is achieved.
Volume 12, number 2
OPTICS COMMUNICATIONS
For the projection of a Ronchi ruling (a 0 = 1/2, a 1 = l/n, a 2 = 0, a 3 = - 1 / 3 rr) the first harmonic contribution (n = 3) is negligible behind the fundamental one (V 1 = 8/7r2, V 3 "~ 1/7r2). An experimental result is shown in fig. 2. It has been obtained by projecting an appropriate transmission mask M 1 on the blade. The projection has the following special properties: - The spatial illumination structure is a regular grid on the reference plane (z = 0) with a period p' in this plane. - The reference plane is exactly the image plane of the mask M 1 . With this technique it is possible to observe contour lines with a reference plane which is orthogonal to the observation direction. The transmission mask M 1 has to be carefully made so as to eliminate a part of the divergence of the imaging system. By divergence we mean the fact that the magnification of the optical system varies across the field due to the tilted object and image planes. Hence the grating period p would be nonuniform, unless special precautions are taken. By simply projecting e.g. a Ronchi ruling on the object (with an angle fl with respect to the observation direction) and using an appropriate Ronchi ruling behind the lens L the reference of this system would no more be a plane, but a cylindrically curved surface. A way of obtaining the appropriate mask M 1 is shown in fig. 3. A regular mask M (usually a Ronchi ruling) is projected in the direction z of observation on the reference plane (x, y). The illuminated refer-
October 1974
ence plane is imaged in the direction Z on a photo-
graphic plate. The developed plate is used as a transmission mask M 1 . By illuminating M 1 and projecting this transmission mask on the reference plane the initial light distribution on the reference plane is achieved. The mask M is just the well-aligned mask of observation which is needed for this method.
2. Direct observation o f contour lines on the object
In this second method two illuminating light beams are used. The method has been treated in a geometrical way by Wasowski [5]. Since this method generates the contour lines directly on the object, this technique seems quite promising for several applications of object form control. The experimental set up is shown in fig. 4. The two illumination directions are symmetrically arranged with respect to the observation axis z. The two angles between illumination axes and z axis are therefore given by fl and -fl, respectively. Supposing equal illumination I 0 of the object from both sides, the intensity I(x,Y, fxy ) on the object is given by
I(x,Y, fxy)=Io { ~n an exp
(2~
(x
c°sfl+fxy sin fl))
using the same notation as for eq. (2). Since eq. (7) can be written as
a n =an,
+ /
Y\
, i
/
/
? Fig. 3. Arrangement for obtaining the appropriate mask M1 (M = regular grating, which is used - after the production of MI - as well adapted mask of observation). M1 is tilted such that the reference plane (z = 0) is imaged onto it.
Fig. 4. Contour line generation using two directions (Z1, Z2) of illumination. On the object O two different fringe systems are superposed. This leads to Moirg fringes which depict contours lines upon this object. 177
Volume 12, number 2
OPTICS COMMUNICATIONS
I(x'Y"Ixy)= 210 { aO+N>O~--'2aNc°s ~ ( 2rrxPc°s/3 × cos
(N 2rr(xysin/3}t! _ \
(8)
p
In this method the Moir6 structure is visible without any mask, but every Moir6 term is multiplied by a carrier frequency, i.e. it is not possible to see the contour lines without the spatial modulation of a carrier. If a J Ronchi-type illumination structure is chosen, the resolution of the observing system has to integrate spatially the carrier frequency (3 cos/3)/p to suppress the contour line system given by cos sin/3} which perturbs the fundamental Moir6 fringes. Using a spatial resolution of &x and Ay, respectively, one observes a spatially averaged intensity showing a fringe system with a visibility depending on the angles and 6 (x, y ) of the object defined by
{(6n/P)fxy
e(x,y)
e(x,y) = arctg of and 6 ( x , y ) = arctgV," .at ay"
October 1974
It is also important in this experiment to use two appropriate masks M 1 and M 2 for the object illumination. Using this technique we investigated the possibilities of visualizing a contour line system with distances between two adjacent contour lines of the order of some hundred microns. This scheme is of interest for surface control on small and well defined parts of a workpiece. A result is shown in fig. 5 using the experimental arrangement of fig. 4. It has been obtained by simply photographing the doubly illuminated medal which has a diameter of 10 cm. The angle t3 was chosen equal to 60 °. The period p' on the plane of reference was taken equal to 490/J. This leads to a A f = 140 #. It should be mentioned that the quality and the visibilty of the contour line system can be improved by classical optical filtering techniques using coherent or even incoherent light. With these techniques one can eliminate the carrier frequency, but this optical filtering cannot be done in real time. To maintain real time contouring of an object it is in certain cases depending primarily on the carrier frequency involved - possible to use an appropriate electro-optical arrangement [6].
(9) 3. Limitations of the methods A restriction of contour line generation is imposed by the object angles and [cf. (9)]. Using for a zone tx -x0[, ] y - y 0 1 < &r the Taylor development off(x, y), the intensity I on the surface of the object [eq. (3)] can be written as
e(x,y)
6(x,y)
I(x,Y, fxy ) =I 0 (a 0 + a I (exp [i(K" r ~- ~)1 + e x p [ - i ( K - r ÷ ~)l)}
(10)
by taking only the fundamental terms n = 0, + 1 into account. The components of K are given by
Kx=p_2~(cos/3 + ~x3¢"sin/3)
and
Ky -2zr3f-p3ysin/3 .
The effects of e and 6 on the observed fringe system are: - locally the resulting period is given by
p(x,y)
Fig. 5. Experimental result of this method. The medal has a diameter of 8 cm. The height difference between two adjacent lines 2x.fhas the value Af= 140 U.
p/cosl]
P(x'Y)-v/i[l+tge(x,y)tg/3]2+
[tg8 (x,y) tg/3] 2 (lla)
178
Volume 12, number 2
OPTICS COMMUNICATIONS
October 1974
2~/p_
S
....... ,
,osp.j'
~
I
D-
J
Fig. 6. Representation of the angular limitations of the object angles e(x, y) and a (x, y) in a normalized K-space. a) Case of the mask method: only the points Q(e, 6) inside the circle with radius p/a can be used for correct object contouring; b) case of the second method: the resolution circle (hatched) is centered around the origin. The circle around the point (cos ~3,0) describes the limitations due to the fact that also the Moire frequency K M has to be resolved by the resolution cell with diameter a. - the fringe system is locally rotated by an amount a(x,y) with respect to the Y-axis: tg a(x, y ) =
tg 6 (x, y ) ctg/3 + t g e ( x , y ) "
(1 l b )
presented in K-space. All points Q(e, 6) lying inside of the circle of resolution can be used for contour mapping. For 6 ~ 0 and e > 0 the experiment gave a value of 0.35 for the product tg e tg/3 of eq. (12a).
This description corresponds to a first fringe system with a vector K 1 . The second vector K 2 depends on the contour mapping method which is used.
3.2. Double projection method
3.1. Mask method
K2x = ( 2n /p)(cos /3 - tg e sin/3)
Here K 2 is given by
and The components of K 2 are given by K2x = (2n/p) cos/3 and K2y = 0. The Moir~ vector K M is given by K M = K 1 - K 2. The corresponding Moir~ period PM is therefore:
K2y = - (27r/p) tg 6 sin/3. The two Moir~ components are
KMx = (27r/p) 2 tg 6 sin/3
P
sin /3 x/tg2 e (x, y ) + tg 2 6 (x, y ) "
and
This period PM has to be resolved by the observation system with spatial resolution o. The condition
KMy = (2rt/p) 2 tg 6 sin/3,
PM > o
(12)
is therefore necessary and limits the object angles e and 6 which can be used for contour mapping. It has to be noted that for the particular case 6 ~ 0 ° the condition has to be modified and becomes more severe, because K M has the same direction as K 1 and K2, respectively. To separate in this case correctly K M from K 1 or K2, the condition (for/3 > 0)
- 1 < tge(x,y) tg/3 < ½
(12a)
has to be fulfilled. In fig. 6a these restrictions are re-
respectively. The Moird period PM is given by (P/2) sin/3 ~/tg2e + tg26. However, as was shown in eq. (8), it is necessary in this method to resolve the carrier frequencies K 1 and K 2. A resolution cell o f diameter a corresponds in Kspace to a circle (fig. 6b), but here the circle is centered around the origin (hatched curve). Moreover, the Moir4 frequency K M has to be resolved, too. This leads to an additional limiting circle of radius p/2o around the point (cos/3, 0). Since all Moird vectors pass through this point (cos/3) on the Kx-axis, the 179
Volume 12, number 2
OPTICS COMMUNICATIONS
symmetrical curve to the hatched curve has to be traced, too (curve: . . . . . . ). The end points of the vectors K1,2 which are directly related to the object's angles e and 6, must therefore lie inside of the zone being traced by thick lines. For 6 ~ 0 ° the restriction on tg e is more important, too.
October 1974
Acknowledgements Acknowledgements are due to A. Thomas for his help with the experiments.
References 4. Conclusion With illumination by special masks it is possible to observe in real time the contour lines of an object. The spatial resolution of the detecting system influences fringe visibility and perturbations. The angles e and 6 of the object determine the geometry of the arrangement and limit the value Af.
180
[ 1] W. Schmidt, A. Vogel and D. Preussler, Appl. Phys. 1 (1973) 103. [2] D.M. Meadows, W.O. Johnson, J.B. Allen, Appl. Opt. 9 (1970) 94. [3] H. Takasaki, Appl. Opt. 12 (1973) 845. [4] J. der Hovanesian and Y.Y. Hung, Appl. Opt. 10 (1971) 2734. [5] J. Wasowski, Opt. Commun. 2 (1970) 321. [6] B. Dessus, J.P. Gerardin and P. Mousselet, to be published.