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Transmission of a surface profile through a single optical fiber
Volume 67, number 3
OPTICS C O M M U N I C A T I O N S
1 July 1988
T R A N S M I S S I O N OF A SURFACE P R O F I L E T H R O U G H A S I N G L E OPTICAL FIBER A.A. TAGLIAFERRI ", J. CALATRONI b and C. FROEHLY c Universidade Federal Fluminense, Inst. de Fisica, C.P. 100296, 24210 Niteroi, R.J. Brazil b Universidad Simdn Bolivar, Dep. Fisica, A.P. 89000, Caracas I080-A, Venezuela c Universitd de Limoges, Institut de Recherches en Communications Optiques et Microondes, U.A.C.N.R.S. no 356, 123 R. Albert Thomas, 87060 Limoges Cedex, France a
Received 24 February 1988
An entirely optical method which allows the transmission of a 2-D surface profile through a single optical fiber is presented. The system performs a double encoding of the surface: in a first step a white light interferogram is used to store the relief. In a second step a chromatic encoding is used to introduce, in real time, the 2-D interferogram in a single multimode fiber. The number of transmitted pixels is limited by the luminosity of the image.
1. Introduction Here we present a purely optical method for transmission of the relief z(x, y) of a 2-dimensional surface through a single optical fiber in real time. Taking into account that the object needs the three spatial coordinates to be completely described, the optical fiber being a one dimensional channel, it is necessary to encode the object in order to transmit a two-dimensional information through a one-dimensional channel (the fiber). Here, we present a system which involves a double encoding in a first step, the surface profile is stored in a white light interferogram (channeled spectrum). In a second step this interferogram is sampled by a double spectroscopic device which to each sampled point object associates a characteristic wavelength in Zke visible spectrum. This last process is known as chromatic or frequencial encoding, and it was used to 'transmit images of 1-, or 2-D objects through one or several optical fibers [ 1-8 ]. Lacourt and Boni [ 9 ] reported the thickness transmission of a transparent film through an array of parallel multimode fibers. Quite recently, the colour of the object was also transmitted [ 10,11 ] by using a related encoding process. In our system, the chromatic encoding transforms the 2-D white light interferogram into an intensity 180
spectral distribution I ( k ) which is introduced into a multimode fiber. At the output of the fiber an identical double-spectroscopic system performs the inverse operation, giving rise to an image of the interferogram.
2. Principle of the method In fig. 1 we show the first step of the surface encoding. The object is the surface z(x, y) which plays the role of one of the mirrors of a Michelson interferometer. The other mirror gives the reference plane. The interferometer is illuminated with a white light
/'i \
Mr
BS
~
Lo
z(x;y)
~x'
Fig. 1. Interferometric encoding of the surface profile z(x; y). The light source is a tungsten filament.
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 67, number 3
OPTICS COMMUNICATIONS
source. At the output plane (x', y' ), the light distribution is
(where Nl is the frequency o f G , 0b~ is the blaze angle and kij the diffraction order), for each value 0i of the diffraction angle. The mean difference between wavelengths which satisfies eq. (2) is the free spectral range A2,
I(x', y' ) = ~ B ( 2 ) [ ( l + r2) + 2r cos2~z 2z/2 ] d2 ,
(1) where B(2) is the spectral energy distribution of the source and r(2) is the light amplitude reflection coefficient. (It will be shown later that the method requires that r(2) ~ constant. ) The second step of the encoding (fig. 2), which is in series with the first one, acts on the intensity distribution given by eq. ( 1 ), in the (x', y' ) plane. It consists of a double spectroscopic system which contains two diffraction gratings (GI, G2). One of the gratings (G~) is of the "echelette" type blazed in the infrared (blaze wavelength 8 ~tm), and therefore when it is used in the visible spectral region it works in high diffraction orders. (G~) has its grooves vertically so it diffracts light horizontally. The other grating (G2) is used in the first diffraction order. (G2) has its grooves horizontal, performing a vertical dispersion. To show the action of the system of fig. 2, let us suppose that a fictitious white light points source is located in O (in the place of the entrance face of the fiber). Taking into account that G~ is used at high diffraction orders, a sequence of wavelengths 2~j satisfies the grating equation: sin 0b~+ sin 0i = kiAiN 1 - ( k ~ - 1 ))]./_+IN] =
(ki-2)2i_+2N, .... (k,--j)2i_+jN, ,
(2)
1 July 1988
=•2/2bl
( 3)
,
with 2hi, the blaze wavelength of G~ (2hi = 2 sin Ob~/N~ ) and 2=0.5 Ixm, the mean wavelength of the visible spectrum. If we restrict our attention to the visible spectrum (2max=0.6 ~tm, 2min=0.4 ~m) diffracted at the blaze angle, the number of spectral components for each diffraction angle 0~ will be 2jmax + 1 = (~-max - - ) ~ m i n ) / ~ -
-
The second grating G2 diffracts - along the y-axis the sequence of wavelengths 2~j given by eq. (2),
sin ~ t +sin Oj =N2&j,
(4)
where 82 is the grating frequency of N2, and 0 the angular coordinate associated to y-axis. Both expressions (2) and (4) establish the dispersion equation of the overall system. Fig. 3 shows the spatial distribution of the continuous spectrum emitted by the fictitious source. The spectrum looks like a zig-zag in which the number of segments is given by n = (2max --,~,min) / z~k~ •
(5)
Each segment has a slope given by the ratio of the dispersions of G~ and G2. The coordinates ((9, ~) of
y' i ?t rnin = O'4/u m
".
G2
~ N-I tg/~. cos(Jbt WNI .l. bl cosObl N2 .~-
G1
,
X.~_
61~x '
3.t o
NT1 Fig. 2. Doublespectroscopicsystemwith two crosseddiffraction gratings (lensesare not shown).
i
Fig. 3. Scheme of the spatial distribution of the spectrum of a white lightpoint sourcewhenanalyzedthroughthe spectroscopic system of fig. 2. 181
Volume 67, number 3
OPTICS COMMUNICATIONS
each wavelength are determined by eqs. (2) and (4). It must be pointed out that the system of fig. 2 sets a correspondence between the wavelengths of the visible spectrum and those points of the (x', y ' ) plane which lie on the segments of the zig-zag indeed, that is the reason why the object profile must be stored in a white light interferogram. Now, let us replace the fictitious luminous source by the entrance face of an optical fiber. Letalso suppose by the moment that the fiber diameter is infinitely small. If the (x' ,y' ) plane is illuminated with an intensity distribution given by ( 1 ), those points which satisfy eqs. (2) and (4) (zig-zag points), will be imaged on the fiber by the system of fig. 2. In this way the double spectroscopic system performs a chromatic sampling of the intensity distribution I(x', y' ) selecting some points which will be transmitted through the optical fiber. The intensity distribution along certain lines of the object (zig-zag lines) is thus transformed in a spectral distribution Is(2). Each point of the object is characterized without any ambiguity by a particular wavelength. An identical system at the exit of the fiber performs the decoding operation and reproduces the original intensity distribution, sampled and filtered with a new spectral factor B[2(x's; Y's) ] , I~ = B [2(X's; Y's) ]
×[l+r2+2rcos2n2z(x's;y's)/2(x's;y's)]
,
ever, in order to calculate the number of points which are transmitted by the system we have to consider the finite diameter of the fiber. This means that the lines of the zig-zag have a finite width. Then more points will enter into the fiber, giving rise to a more intense image but, of course, of lower resolution. In the next section we consider the problem of resolution versus luminance in order to select the optical components of the system and to calculate the transmission performance.
3. Experimental design In order to design the optical system a fundamental property of chromatic encoding must be taken into account: the luminosity of each image pixel is equal, in mean value, to the luminosity of the corresponding object pixel divided by the total number of transmitted pixels. This fact imposes a compromis between the number of transmitted points and the luminous flux per pixel necessary to its detection. In other words: the product of luminance by resolution is a constant. In our experiment we have choosen the number of lines of the zig-zag spectrum (n) equal to 6. This determines the free spectral range of Gl and the blaze wavelength
(6) n = (2max - - ~ m i n ) / / ~ .
where (x's; Y's) are the coordinates of a point which belongs to the set of solutions of eqs. (2) and (4). This means that Is represents a "rainbow" image of the surface with one comer in violet and the other in red. Due to the characteristics of the chromatic encoding process B(2) and r(2) must be approximately constant. Otherwise, if we are dealing with a coloured object (r(2) ~ constant), it may occur that the colour of some region of the object does not match with the wavelengths with which this region will be imaged into the fiber. In that case black areas without information will appear in the image. It is important to note that the double spectroscopic system of fig. 2 performs a sampling of the intensity distribution I(x'; y' ). There are points of the objects - those which do not belong to the zigzag - which will not be imaged into the fiber. How182
1 July 1988
(7)
This gives 2 2 ~ 3 × 10 -2 ~tm, and from (3) 2b1~7.5 txm, taking 2=0.5 ~tm. Actually Gl is an infrared grating with 2bt = 8 ~tm, N = 150 m m - l , k = 16 for 2=261, and M (the total number of grooves) is 7500. Fig. 4 shows the whole experimental set-up. The total number of resolved pixels is
e=np,
(8)
where p is the number of pixels in each line of the zig-zag: p=A2/d2 where d2 is the resolved spectral element. If we call V 2 the number of modes of the light beam (equal to the normalized spatial frequency of the multimode fiber) we get d2= V~52,
(9)
where 52 is the intrinsic resolved spectral element of
Volume 67, number 3
OPTICS COMMUNICATIONS
Fig. 4. Complete experimental set-up which encodes the profile function z(x; y) to be transmitted through the optical fiber. Gt: ~2=2/Ri, R i being the theoretical resolution of GI. Then we have for p
p= z~/ ( V,~/Ri) = M / V .
(10)
Eq. (10) establishes the quantitative relationship connecting luminosity and resolution. The number of resolved points in each zig-zag line determines the maximum depth z that the system can detect. Effectively, one fringe of the channeled spectrum satisfies the relation 2z/2 = integer where 2z is the path difference and 2 the wavelength for that fringe. This means that between adjacent fringes we have
dz/z=2/2z+d2/2.
( 11 )
When considering path differences z>>2/2 (it is always possible to introduce an additional path difference by changing the position of the reference mirror M) we obtain that the longitudinal resolution z/dz is equal to the effective resolution of G1. Taking dz/z~0.2%, we get p ~ 2 0 , and from (10) V~375. This last number V allows to estimate the diameter of the fiber's core. Using the classical relation V~ na sin 0/2,t, (where a is the core diameter and
1 July 1988
sin 0 the numerical a p e r t u r e s 0 . 2 for a silica fiber coated with plastic cladding) we obtain a ~ 600 ~tm. On the other hand G2 has to resolve A2=0.03 lim which lets a great freedom of choice. In this work we have used a grating blazed at 0.5 ~tm with 1200 lines/mm, 5 cm width. The lens L~ must have a diameter larger than GI and also a numerical aperture higher than that of the fiber. These conditions set for L~ an f n u m b e r smaller than 2.5. Lenses L'~ and L2 are introduced to modify the linear dispersion of each grating independently. Because of the finite diameter of the fiber's core the lines of the zig-zag spetrum have an angular width: dq~=f'~ a/f2fl. In order to prevent the superposition of the zig-zag lines we need to have the angular width d0~N2A2/cos ~1. This last condition determines the ratio fq/f2. At the fiber output, an identical double spectroscopic system performs the decoding step. Actually, in our experiment we have used GI twice reinjecting the light from the fiber in a "round trip" mounting, because of the lack of a second grating identical to G1. With the system described before we were able to transmit a channeled spectrum through a single fiber, as shown in fig. 5. The relationship 2z/2 = integer for each fringe allows to calculate the depth increment dz corresponding to the neighboring fringe. In this way it is possible to calculate z in any point when Z(Xo; Yo) is known a priori in a selected point (Xo; Yo) of the surface.
4. Conclusions We have presented a method which makes it possible to transmit the optical information about the shape of a 2-D surface through a single optical fiber. The basic idea involved is the extension of chromatic encoding for 2-D phase objects. This means that each point object is characterized by a wavelength which defines its (x, y) coordinates. In this sense, the use of white light interferograms to store the surface profile is well adapted to the encoding method. One limitation of the procedure may be estimated from photometric considerations: any attempt to increase luminosity produces a reduction of the image 183
Volume 67, number 3
OPTICS COMMUNICATIONS
1 July 1988
of the object: it must be sufficient low in order to give a constant phase over the entrance face of the fiber. Otherwise, the fringe frequency in the interferogram will be higher than the i n s t r u m e n t a l resolution limit. In spite of the rather low geometric resolution the system allows the transmission of a 2-D phase object ( z = z(x, y) ) through a single optical fiber, by purely passive optical means, at very high temporal rates (up to 100 G b s-~, over tens of meters of fiber). By c o m b i n i n g the described system to an optical data aquisition device, an automatic single fiber endoscope may be built-up for technical applications.
Acknowledgements The authors wish to t h a n k the C N P q a n d F I N E P (Brazil), C O N I C I T a n d U n i v e r s i d a d Sim6n Bolivar (Venezuela), a n d the Service de C6operation Technique de l'Ambassade de France au V r n r z u r l a for its financial support in the form of grants a n d traveling aids.
Fig. 5. Received image of a channeled spectrum after fiber transmission. This zig-zagchanneled spectrum corresponds to a plane surface of 2 X 3 cm2 which lies 60 gm behind the reference mirror. The interferogram consists of periodic fringes along the wavenumber (a) coordinate. Fringes shown here does not exhibit exact periodicity because grating dispersions are not linear with a. resolution. Our experiment was limited to the transmission of an image of 6 × 20 = 120 pixels. Each resolved pixel will correspond to the image of the entrance face of the fiber formed by the double spectroscopic device on the object surface. This means that the object phase profile is sampled by the image of the entrance face of the fiber in 120 points. Another l i m i t a t i o n is concerned to the local slope
184
References [ 1] C.J. Koester, J. Opt. Soc. Am. 58 (1968) 63. [2] P. Cielo and C. Delisle, Can. J. Phys. 54 (1976) 2332. [3 ] H.O. Bartelt, Optics Comm. 27 ( 1978) 365. [4] A.A. Friesem and U. Levy, Optics Lett. 2 (1978) 133. [ 5 ] H.O. Bartelt, Optics Comm. 28 (1979) 45. [6] L.N. Deryugin, A.V. Chekan and V. Demchenkov, Opt. Spectrosc. 43-48 (1980). [7] G. Liang, P. Facq andJ. Arnaud, Electron. Lett. 18 (1982) 660. [ 8 ] T.C. Yang and C. Froehly, Optical waveguide science, eds. H.C. Huang and A.W. Snyder (Martinus Nijhoff, La Hague, 1983) pp. 333-334. [ 9 ] A. Lacourt and P. Boni, Optics Comm. 27 ( 1978) 57. [ 10] J. Calatroni, C. Froehly and T.C. Yang, Appl. Optics 26 (1987) 2202. [ 11 ] J. Calatroni, C. Froehly and H. A1Mawie, Appl. Optics 26 (1987) 2206.
OPTICS C O M M U N I C A T I O N S
1 July 1988
T R A N S M I S S I O N OF A SURFACE P R O F I L E T H R O U G H A S I N G L E OPTICAL FIBER A.A. TAGLIAFERRI ", J. CALATRONI b and C. FROEHLY c Universidade Federal Fluminense, Inst. de Fisica, C.P. 100296, 24210 Niteroi, R.J. Brazil b Universidad Simdn Bolivar, Dep. Fisica, A.P. 89000, Caracas I080-A, Venezuela c Universitd de Limoges, Institut de Recherches en Communications Optiques et Microondes, U.A.C.N.R.S. no 356, 123 R. Albert Thomas, 87060 Limoges Cedex, France a
Received 24 February 1988
An entirely optical method which allows the transmission of a 2-D surface profile through a single optical fiber is presented. The system performs a double encoding of the surface: in a first step a white light interferogram is used to store the relief. In a second step a chromatic encoding is used to introduce, in real time, the 2-D interferogram in a single multimode fiber. The number of transmitted pixels is limited by the luminosity of the image.
1. Introduction Here we present a purely optical method for transmission of the relief z(x, y) of a 2-dimensional surface through a single optical fiber in real time. Taking into account that the object needs the three spatial coordinates to be completely described, the optical fiber being a one dimensional channel, it is necessary to encode the object in order to transmit a two-dimensional information through a one-dimensional channel (the fiber). Here, we present a system which involves a double encoding in a first step, the surface profile is stored in a white light interferogram (channeled spectrum). In a second step this interferogram is sampled by a double spectroscopic device which to each sampled point object associates a characteristic wavelength in Zke visible spectrum. This last process is known as chromatic or frequencial encoding, and it was used to 'transmit images of 1-, or 2-D objects through one or several optical fibers [ 1-8 ]. Lacourt and Boni [ 9 ] reported the thickness transmission of a transparent film through an array of parallel multimode fibers. Quite recently, the colour of the object was also transmitted [ 10,11 ] by using a related encoding process. In our system, the chromatic encoding transforms the 2-D white light interferogram into an intensity 180
spectral distribution I ( k ) which is introduced into a multimode fiber. At the output of the fiber an identical double-spectroscopic system performs the inverse operation, giving rise to an image of the interferogram.
2. Principle of the method In fig. 1 we show the first step of the surface encoding. The object is the surface z(x, y) which plays the role of one of the mirrors of a Michelson interferometer. The other mirror gives the reference plane. The interferometer is illuminated with a white light
/'i \
Mr
BS
~
Lo
z(x;y)
~x'
Fig. 1. Interferometric encoding of the surface profile z(x; y). The light source is a tungsten filament.
0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 67, number 3
OPTICS COMMUNICATIONS
source. At the output plane (x', y' ), the light distribution is
(where Nl is the frequency o f G , 0b~ is the blaze angle and kij the diffraction order), for each value 0i of the diffraction angle. The mean difference between wavelengths which satisfies eq. (2) is the free spectral range A2,
I(x', y' ) = ~ B ( 2 ) [ ( l + r2) + 2r cos2~z 2z/2 ] d2 ,
(1) where B(2) is the spectral energy distribution of the source and r(2) is the light amplitude reflection coefficient. (It will be shown later that the method requires that r(2) ~ constant. ) The second step of the encoding (fig. 2), which is in series with the first one, acts on the intensity distribution given by eq. ( 1 ), in the (x', y' ) plane. It consists of a double spectroscopic system which contains two diffraction gratings (GI, G2). One of the gratings (G~) is of the "echelette" type blazed in the infrared (blaze wavelength 8 ~tm), and therefore when it is used in the visible spectral region it works in high diffraction orders. (G~) has its grooves vertically so it diffracts light horizontally. The other grating (G2) is used in the first diffraction order. (G2) has its grooves horizontal, performing a vertical dispersion. To show the action of the system of fig. 2, let us suppose that a fictitious white light points source is located in O (in the place of the entrance face of the fiber). Taking into account that G~ is used at high diffraction orders, a sequence of wavelengths 2~j satisfies the grating equation: sin 0b~+ sin 0i = kiAiN 1 - ( k ~ - 1 ))]./_+IN] =
(ki-2)2i_+2N, .... (k,--j)2i_+jN, ,
(2)
1 July 1988
=•2/2bl
( 3)
,
with 2hi, the blaze wavelength of G~ (2hi = 2 sin Ob~/N~ ) and 2=0.5 Ixm, the mean wavelength of the visible spectrum. If we restrict our attention to the visible spectrum (2max=0.6 ~tm, 2min=0.4 ~m) diffracted at the blaze angle, the number of spectral components for each diffraction angle 0~ will be 2jmax + 1 = (~-max - - ) ~ m i n ) / ~ -
-
The second grating G2 diffracts - along the y-axis the sequence of wavelengths 2~j given by eq. (2),
sin ~ t +sin Oj =N2&j,
(4)
where 82 is the grating frequency of N2, and 0 the angular coordinate associated to y-axis. Both expressions (2) and (4) establish the dispersion equation of the overall system. Fig. 3 shows the spatial distribution of the continuous spectrum emitted by the fictitious source. The spectrum looks like a zig-zag in which the number of segments is given by n = (2max --,~,min) / z~k~ •
(5)
Each segment has a slope given by the ratio of the dispersions of G~ and G2. The coordinates ((9, ~) of
y' i ?t rnin = O'4/u m
".
G2
~ N-I tg/~. cos(Jbt WNI .l. bl cosObl N2 .~-
G1
,
X.~_
61~x '
3.t o
NT1 Fig. 2. Doublespectroscopicsystemwith two crosseddiffraction gratings (lensesare not shown).
i
Fig. 3. Scheme of the spatial distribution of the spectrum of a white lightpoint sourcewhenanalyzedthroughthe spectroscopic system of fig. 2. 181
Volume 67, number 3
OPTICS COMMUNICATIONS
each wavelength are determined by eqs. (2) and (4). It must be pointed out that the system of fig. 2 sets a correspondence between the wavelengths of the visible spectrum and those points of the (x', y ' ) plane which lie on the segments of the zig-zag indeed, that is the reason why the object profile must be stored in a white light interferogram. Now, let us replace the fictitious luminous source by the entrance face of an optical fiber. Letalso suppose by the moment that the fiber diameter is infinitely small. If the (x' ,y' ) plane is illuminated with an intensity distribution given by ( 1 ), those points which satisfy eqs. (2) and (4) (zig-zag points), will be imaged on the fiber by the system of fig. 2. In this way the double spectroscopic system performs a chromatic sampling of the intensity distribution I(x', y' ) selecting some points which will be transmitted through the optical fiber. The intensity distribution along certain lines of the object (zig-zag lines) is thus transformed in a spectral distribution Is(2). Each point of the object is characterized without any ambiguity by a particular wavelength. An identical system at the exit of the fiber performs the decoding operation and reproduces the original intensity distribution, sampled and filtered with a new spectral factor B[2(x's; Y's) ] , I~ = B [2(X's; Y's) ]
×[l+r2+2rcos2n2z(x's;y's)/2(x's;y's)]
,
ever, in order to calculate the number of points which are transmitted by the system we have to consider the finite diameter of the fiber. This means that the lines of the zig-zag have a finite width. Then more points will enter into the fiber, giving rise to a more intense image but, of course, of lower resolution. In the next section we consider the problem of resolution versus luminance in order to select the optical components of the system and to calculate the transmission performance.
3. Experimental design In order to design the optical system a fundamental property of chromatic encoding must be taken into account: the luminosity of each image pixel is equal, in mean value, to the luminosity of the corresponding object pixel divided by the total number of transmitted pixels. This fact imposes a compromis between the number of transmitted points and the luminous flux per pixel necessary to its detection. In other words: the product of luminance by resolution is a constant. In our experiment we have choosen the number of lines of the zig-zag spectrum (n) equal to 6. This determines the free spectral range of Gl and the blaze wavelength
(6) n = (2max - - ~ m i n ) / / ~ .
where (x's; Y's) are the coordinates of a point which belongs to the set of solutions of eqs. (2) and (4). This means that Is represents a "rainbow" image of the surface with one comer in violet and the other in red. Due to the characteristics of the chromatic encoding process B(2) and r(2) must be approximately constant. Otherwise, if we are dealing with a coloured object (r(2) ~ constant), it may occur that the colour of some region of the object does not match with the wavelengths with which this region will be imaged into the fiber. In that case black areas without information will appear in the image. It is important to note that the double spectroscopic system of fig. 2 performs a sampling of the intensity distribution I(x'; y' ). There are points of the objects - those which do not belong to the zigzag - which will not be imaged into the fiber. How182
1 July 1988
(7)
This gives 2 2 ~ 3 × 10 -2 ~tm, and from (3) 2b1~7.5 txm, taking 2=0.5 ~tm. Actually Gl is an infrared grating with 2bt = 8 ~tm, N = 150 m m - l , k = 16 for 2=261, and M (the total number of grooves) is 7500. Fig. 4 shows the whole experimental set-up. The total number of resolved pixels is
e=np,
(8)
where p is the number of pixels in each line of the zig-zag: p=A2/d2 where d2 is the resolved spectral element. If we call V 2 the number of modes of the light beam (equal to the normalized spatial frequency of the multimode fiber) we get d2= V~52,
(9)
where 52 is the intrinsic resolved spectral element of
Volume 67, number 3
OPTICS COMMUNICATIONS
Fig. 4. Complete experimental set-up which encodes the profile function z(x; y) to be transmitted through the optical fiber. Gt: ~2=2/Ri, R i being the theoretical resolution of GI. Then we have for p
p= z~/ ( V,~/Ri) = M / V .
(10)
Eq. (10) establishes the quantitative relationship connecting luminosity and resolution. The number of resolved points in each zig-zag line determines the maximum depth z that the system can detect. Effectively, one fringe of the channeled spectrum satisfies the relation 2z/2 = integer where 2z is the path difference and 2 the wavelength for that fringe. This means that between adjacent fringes we have
dz/z=2/2z+d2/2.
( 11 )
When considering path differences z>>2/2 (it is always possible to introduce an additional path difference by changing the position of the reference mirror M) we obtain that the longitudinal resolution z/dz is equal to the effective resolution of G1. Taking dz/z~0.2%, we get p ~ 2 0 , and from (10) V~375. This last number V allows to estimate the diameter of the fiber's core. Using the classical relation V~ na sin 0/2,t, (where a is the core diameter and
1 July 1988
sin 0 the numerical a p e r t u r e s 0 . 2 for a silica fiber coated with plastic cladding) we obtain a ~ 600 ~tm. On the other hand G2 has to resolve A2=0.03 lim which lets a great freedom of choice. In this work we have used a grating blazed at 0.5 ~tm with 1200 lines/mm, 5 cm width. The lens L~ must have a diameter larger than GI and also a numerical aperture higher than that of the fiber. These conditions set for L~ an f n u m b e r smaller than 2.5. Lenses L'~ and L2 are introduced to modify the linear dispersion of each grating independently. Because of the finite diameter of the fiber's core the lines of the zig-zag spetrum have an angular width: dq~=f'~ a/f2fl. In order to prevent the superposition of the zig-zag lines we need to have the angular width d0~N2A2/cos ~1. This last condition determines the ratio fq/f2. At the fiber output, an identical double spectroscopic system performs the decoding step. Actually, in our experiment we have used GI twice reinjecting the light from the fiber in a "round trip" mounting, because of the lack of a second grating identical to G1. With the system described before we were able to transmit a channeled spectrum through a single fiber, as shown in fig. 5. The relationship 2z/2 = integer for each fringe allows to calculate the depth increment dz corresponding to the neighboring fringe. In this way it is possible to calculate z in any point when Z(Xo; Yo) is known a priori in a selected point (Xo; Yo) of the surface.
4. Conclusions We have presented a method which makes it possible to transmit the optical information about the shape of a 2-D surface through a single optical fiber. The basic idea involved is the extension of chromatic encoding for 2-D phase objects. This means that each point object is characterized by a wavelength which defines its (x, y) coordinates. In this sense, the use of white light interferograms to store the surface profile is well adapted to the encoding method. One limitation of the procedure may be estimated from photometric considerations: any attempt to increase luminosity produces a reduction of the image 183
Volume 67, number 3
OPTICS COMMUNICATIONS
1 July 1988
of the object: it must be sufficient low in order to give a constant phase over the entrance face of the fiber. Otherwise, the fringe frequency in the interferogram will be higher than the i n s t r u m e n t a l resolution limit. In spite of the rather low geometric resolution the system allows the transmission of a 2-D phase object ( z = z(x, y) ) through a single optical fiber, by purely passive optical means, at very high temporal rates (up to 100 G b s-~, over tens of meters of fiber). By c o m b i n i n g the described system to an optical data aquisition device, an automatic single fiber endoscope may be built-up for technical applications.
Acknowledgements The authors wish to t h a n k the C N P q a n d F I N E P (Brazil), C O N I C I T a n d U n i v e r s i d a d Sim6n Bolivar (Venezuela), a n d the Service de C6operation Technique de l'Ambassade de France au V r n r z u r l a for its financial support in the form of grants a n d traveling aids.
Fig. 5. Received image of a channeled spectrum after fiber transmission. This zig-zagchanneled spectrum corresponds to a plane surface of 2 X 3 cm2 which lies 60 gm behind the reference mirror. The interferogram consists of periodic fringes along the wavenumber (a) coordinate. Fringes shown here does not exhibit exact periodicity because grating dispersions are not linear with a. resolution. Our experiment was limited to the transmission of an image of 6 × 20 = 120 pixels. Each resolved pixel will correspond to the image of the entrance face of the fiber formed by the double spectroscopic device on the object surface. This means that the object phase profile is sampled by the image of the entrance face of the fiber in 120 points. Another l i m i t a t i o n is concerned to the local slope
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