Theory of lossless propagation of simultaneous different-wavelength optical pulses

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OPTICS COMMUNICATIONS

15 February 1981

INCOHERENT OPTICAL PROCESSING: SAMPLING THE INCOHERENT SPECTRUM B. GUILLAME and J. DUVERNOY Laboratoire de Physique Gdngrale et Optique, assoei# au CNRS no 214, Facultd des Sciences et des Techniques, Universitd de Franche-Comtg, 25030 Besancon Cedex, France

Received 23 October 1980 The incoherent spectrum of the object is linearly filtered by the modulation transfer function (MTF) of the incoherent imaging system. When coding a convenient filter upon the MTF, the system becomes an information processor. Statistical filters are computed from the objects to be classified, whose incoherent spectra must be measured. This paper presentsthe principle of the sampling of incoherent spectra, a computer-operated sampling system, and the first experimental results.

1. Introduction: incoherent spatial filtering

However the incoherent spectrum ~, defined by:

Coherent optical systems are used as fully parallel, linear processors [1 ]. They make it possible to perform any kind of spatial filtering, because of the direct access to the complex Fourier transform, 6~(u, v), of the processed object, o~(x,y), in their Fourier plane. Incoherent optical systems deal with the object intensity, o (x, y), their modulation transfer functions (MTF's) acting as linear filters. The information-processing scheme is based on the imaging relation: i(x,y) =o(x,y) * t(x, y)

(3)

is not accessible in any physical plane when using an incoherent system. Eq. (2) shows that 3 should be attainable through a proper MTF, which must take the form of a sampling function. This paper introduces the sampling of incoherent spectra by means of convenient MTF's. The principle, the basic system, and the first illustrative results are presented.

(1)

where i ( x , y ) denotes the image intensity, t ( x , y ) the impulse response of the system, and * the convolution. Fourier transforming this relation yields:

~u, o) = ~(u, o). r(u, o),

(u, o) = FT [o (x, y)] = ~ (u, o) * o~* (u, o)

(2)

where 7" and ~ denote the Fourier transform of i and o, and T the MTF of the system. The object information, ~(u, u), is filtered by T. It is therefore possible to assign a given shape to the MTF by using specified pupils [2,3] ; deterministic [4,5] or statistical [6] filters can be implemented in this way. Optical objects can be classified by using an incoherent optical processor insofar as the suitable statistical filter is available. According to the classical pattern recognition principles [7], this filter can be computed from a training set, i.e. a collection of informations { ~(u, o)} representative of the classes one has to deal with.

2. Sampling the incoherent spectrum In the following the problem will be taken in one dimension, for sake of simplicity. The incoherent spec. trum, 3(u), of the object is to be measured at N differ. ent sampling points, {~(kAu)}, k = 1 .... ,N, with a spatial frequency sampling space equal to Au. A possible implementation scheme consists of imaging the object through two different systems whose pupils are shown in fig. 1. For sampling the current value, ~(kAu), two successive pupils are needed: PtkI) = rect (u/a) * (,5 (u - k A u / 2 ) + 8 (u + kAu/2)), //k2) = rect (u/a),

(4)

where a denotes the width of the elementary pupil.

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PUPIL

incoherent source object

MTF

nl"nA

15 February 1981

A

/x

O<1

A

-k~-

I

object spectrum

SAMPLING FUNCTION

Subtracting twice the MTF of the second system from that of the f'trst one yields the following sampiing function: a A (u/2a) * (8 (u - kAu) + 8 (u + kZXu))

(6)

The sampling function, Dk, for the spatial frequency kAu is therefore defined by the recurrence relation:

o k = rk - 2 r k _ l + r k - 2 . 274

[

Fig. 2. Expedrnental set-up for sampling the object incoherent spectrum. The object is imaged by an optical system whose pupil is modified as a function of time, according to either schemes in figs. 1 or 3.

Pk

Tk -2 Tk_I .Tk. 2

Tk

(5)

where A denotes the triangular function [9] with width 2a. Using a collection of pupils, Pk(1), with an increasing distance kAu allows the sampling of the incoherent spectrum. Some difficulties in carrying out the automatic sampling by two moving slits led us to the alternative device shown in fig. 2. The incoherent imaging system has a pupil, Pk, whose width is progressively increased using a step-motor controlled by computer. Successive measurements are recorded in the image plane, and stored in the computer. Their nature and limitations will be discussed in the next section. The corresponding sampling function is generated according to fig. 3, in a recursive way that is well suited to the computerized data measurement. The width of the pupil Pk at the step # k is a multiple of a given increment Au (i.e.: (k+l)&u). Assuming that the pupil is described by a rectangle function, the transfer function is:

Tk=(k+l)Au A ( 2 ( k + ul ) A u )

imaging

I.,..o,o, H °o..,., H ' . ."" , , , . . . . ,I,

.k'~u

Fig. 1. Generating the sampling function in incoherent light. A two-channel system is used. The pupils "P~t)and "P)~) used for sampling the spatial frequency kAu are shown, as well as the corresponding MTF's. The sampling function is obtained by subtracting the two MTF's.

~

L,

(7)

2)AU/2

•

I I I I

'l

I I I !

I i I

1'

C:

I

A

(k-llAu/2 kAu

1

I i i

A -kAu

.kAu

k Au/2 PUPIL

MTF

SAMPLING FUNCTION

Fig. 3. Alternative scheme for generating the same sampling function as that in fig. 1. A single pupil is progressively opened by increment Au. The successive MTF's are processed according to the given recurrence relation.

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OPTICS COMMUNICATIONS

This function, which takes the form of two A functions (fig. 3), weights the incoherent spectrum of the object with a maximum transmittance for u = -+kAu. Its narrowness depends on the increment Au, which in turn is fixed by the sensitivity of the photodiode. The principle of a real-time, computer-operated incoherent spectrum sampler is based on eq. (7). Its implementation, is depicted in fig. 2.

3. Measurements-limitations In this section we restrict ourselves to the presentation of the headings of computations which will be discussed in a further publication.

3.1. Measuring the magnitude of one spatial frequency Let Tk(U ) be the MTF of the imaging system (fig.2). Let Ax be the diameter of the photodiode. The luminous intensity, ¢k(X0), measured in the image plane, around the point of coordinate x 0 is given by: xo + Ax/2

,k(Xo)= f

i(x)dx.

(8)

xo - Ax/2

Using the classical relations (1) and (2) yields: +0~

Ok(Xo)--f

x O+ Ax/2

f

rk(U)

-*~ xo - Ax/2

X exp (j2rmx) dxdu

=

f

Tk(U ) sin0ruAx) exp(j2rrx0} du.

(9a)

(9b)

This expression simplifies for x 0 = 0, i.e. when measuring the intensity on the optical axis. Nevertheless the measurement does not yield the expected value m k defined by:

mk = f

"o(u) rk(u ) flu

(10)

_ o o

which processed accordingto eq. (7), gives the sampled incoherent spectrum ~(kAu):

~(kAu) = m k - 2mk_ 1 + mk_ 2.

15 February 1981

A correcting procedure compensating for the weighting factor introduced by the finite size of the detector is digitally implemented.

3.2. Noise considerations The two possible measurement schemes do not deal with the same amount of light. The first one (fig. 1) works with a transfer function that remains at the same level, as the second one processes linearly increasing MTF's. In this case, according to eq. (11), we are facing measurements with a very small average value, but with a variance of the order of 4k. Therefore modulation techniques in the detection of the intensity ~k are to be used. In a first step we did not use them, taking advantage of the computerized set-up for performing multiple measurements, an applying classical estimation rules [10] to the determination of the most probable value of the measurements.

3.3. Number of independent samples The effect of the weighting function, introduced by the finite width, Ax, of the photo-diode (sect. 3.1), is in the limitation of the effective bandwidth of the measurable incoherent spectrum. The values of 3 (u) that fie in the spectral domains where sin OruAx)[lru is close to zero are not well determined. Any source of additive noise impairs their measurement. Let us assume that the effective bandwith is of the order of 1lax. The number of independent samples taken out from these band depend on the minimum value of the increment, Au, of the pupil width. In our experimental set-up (fig. 2), the pupil is placed in the plane of the image of the source. Even though this image is modulated by the object information it can be assumed in a first-step that the minimum value of the increment, Aumin, leading to a detectable increment of intensity at the output, is constant over the image of the source. Therefore the number of independent sampies, that can be measured in the incoherent spectrum of the object, is bounded by: N~< 1/Axz~Umin .

(12)

In this first approximation this number depends on the characteristics of the photodiode.

(11) 275

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O(k~u)

O(x)

15 February 1981

the finite size and sensitivity of the photoiode used, the number of independent samples is equal to 10, taken out from a spatial frequencies band of 4 mm -1.

4. Conclusion

(a)

experimental

(b~

Fig. 4. Result of the sampling of the incoherent spectrum of two simple 1-D objects, i.e. a slit (a) an a grating (b). Theoretical and experimental results are shown.

3.4. Experimental results Two simple 1-D objects were selected: a slit with width a = 0.5 mm, and a grating with period p = 0.5 mm. Their theoretical spectra are easily computed. The corresponding sampled spectra are in a good agreement with the former ones, as shown in fig. 4, either for the location of zeroes and the magnitude o f maximums. The result shown here were obtained by a single measurement (for the grating) and the superimposition o f four scannings (for the slit). Because o f

276

This paper has presented the principle and the first experimental results of the sampling of the incoherent spectra. The 1-D examples shown here evidence the possibilities of the method. A 2-D sampling is readily obtained by rotating the object in the input plane. A quantitative study of the sampling limitations, the S/N ratio, and the effects of the non-uniformity of the source will appear in a further publication. New trends in optical processing emphasize the advantages of incoherent systems. Introducing such 2-D, incoherent spectra samplers will probably restrict the laser, and the associated Fourier transformable properties, to specific uses.

References [1] J.W. Goodman, Proc. IEEE 65 (1977) 23. [2] A.W. Lohmann and W.T. Rhodes, Appl. Optics 17 (1978) 1141. [3] D. Gorlitz and F. Lanzl, Optics Comm. 20 (1977) 68. [4] B. Braunecker and R. Hauck, Optics Comm. 20 (1977) 234. [5] S. Lowenthal and P. Chavel, J. Opt. Soc. Am. 6 (1976) 14. [6] J. Duvernoy, Appl. Optics 18 (1979) 2737. [ 7 ] K. Fukunaga, Introduction to statistical pattern recognition, (Academic Press, New York 1972). [8] S. Lowenthal and P. Chavel, J. Opt. Soe. Am. 68 (1978) 721. [9] R. Bracewell, The Fourier transform and its applications (Mac Graw Hill, New York 1965). [10] H.L. Van Trees, Detection, estimation and modulation theory, Part. I, Chap. 2 (John Wiley, New York, 1968).