Operation over an optical signal using generalized diffraction grating Текст научной статьи по специальности «Физика»

YflK 535.4

BecTHHK Cn6ry. Cep. 4. 2013. Bun. 1

Yu. A. Tolmachev

OPERATION OVER AN OPTICAL SIGNAL USING GENERALIZED DIFFRACTION GRATING

Introduction. There is nothing more well-known for any optical scientist or engineer who deals with optical spectrometers than the diffraction grating (DG). In the most convenient case, this Hi-Tech instrument consists of a great series of rectilinear grooves parallel one to another periodically covering a plane surface (reflecting or transparent). The length of these grooves is many times greater than the grooves' spatial period and usually is considered as infinite in one direction (let it be y direction). The scattering coefficient of the groove is constant along y and periodically varies in x-direction that is orthogonal to y. In this respect, one can speak of ID-grating.

Why is the DG efficient for the analysis of a wave spectrum? The answer to this question is in two fields. The first one is hidden in the word "periodically". When speaking of the expansion of some incoming process illuminated the grating into its spectrum, one usually assumes the expansion into a harmonic series, i. e. into Fourier spectrum. The process to be analyzed is supposed to consist of a rather simple set of frequencies vik corresponding to the Plank law

hvik = Ei — Ek.

The second point to be emphasized on especially is in the properties of the process of wave diffraction from the grating: when the distance between grooves is of the same order of magnitude as the wavelengths of incoming wave process, the angle of monochrome wave component diffraction is great enough to be easily measured. This is why one uses DG of 10 to 100 mm-1 grooves spatial frequency for the analysis of IR emission and more then 2000 mm-1 for UV.

There are existing well-developed theoretical descriptions of the DG mode of operation over the field of monochrome wave that are based on Huygens principle or on Kirchhoff equations but those descriptions are applied directly to stationary wave processes. Monochrome waves (being stationary by definition) transfer no information while the most urgent problem today is particularly in the information transfer and information processing.

The main purpose of this paper is to show that even 1D diffraction grating is able to realize physically, at the rate of velocity of light, at least two significant and well-known operations over the incoming signal that are of principal value for optical signal processing, namely cross-correlation and convolution of functions.

Method of analysis and terms.

Significant assumptions. From this moment on, we shall assume that the optical system to be studied is linear with respect to the wave amplitude. For linear system analysis, there exist two main approaches: the first one is in using monochrome incoming process and determination of the exit monochrome oscillation amplitude and phase as a function of frequency (in some way it is realized in optics by traditional spectroscopic instruments). The second approach is in using 6(t —t0) pulse (6 being the Dirac function) for the excitation of the system under examination and analysing its pulse response h(t) as the exit signal. In

Yurii A. Tolmachev — professor, faculty of physics, Saint-Petersburg state university; e-mail: yurii.tolmach@rambler.ru

© Yu. A. Tolmachev, 2013

the linear system theory they prove that two methods are equivalent one to another with respect to the system properties description.

The optical physics tradition refers usually to the first method but the changes in optics of ultra-fast processes analysis happened in two last decades make us to turn to the second one [1-5]. In parallel and independently, the same approach was developed by P. Saari and co-workers [6, 7] in Estonia (now in US).

In optical spectroscopy, the response of a spectrometer to the incident monochrome wave is usually called "apparatus function" that is indeed the response to the "pulse" 6(v — vo) or 6(X — X0) in the corresponding spaces. In radio-optics the term "impulse response" or "point spread function" is used for the description of amplitude distribution of the image of a point monochrome source. In our publications and hereafter in this paper the term "impulse (or simply pulse) response" will be used for the time-and-space response h(t, r) of an optical system to the plane wave of 6(t) time structure. In this paper, we concentrate on the time transformation of a signal.

Pulse response of the traditional diffraction grating.

For example, take the diffraction grating consisting of N +1 equal slits of 2a-width positioned at b period along x axis on (x,y) plane (2a < b). As it is shown in [3], being observed at the scattering angle P = 0, the pulse response of a single slit at the great distance

zo ^ a is

This form is valid for a = 0 angle of incident wave illumination.

In full accordance to Young's interpretation of diffraction phenomenon, two edges of the slit respond to the incident pulse with their own two delta-pulses. This general property of the slit pulse response was confirmed in the form not referring to Kirchhoff approximation in [8]. The significant difference of simple Young's treating of diffraction is in the fact that pulses emitted by one slit edge are of different signs depending on the scattering direction: that one propagating towards the "shadow" zone (to the geometrical optics approximation) is positive while the other emitted towards the "illuminated" zone is negative.

For the linear system, one may assume that the general pulse response of the DG can be presented as the sum of elementary slit responses the DG consists of, one delayed to another by (b sin P)/c time. Using the Dirac comb(x) function the pulse response of an infinite DG can be written down as

Symbol ® denotes the operation of convolution. Note that at this stage of our study we operate with infinitely short pulses and infinitely long grating.

One can easily turn from the infinite DG pulse response (2) to the traditional case, namely to the reaction of DG to the imaginary infinitely long monochrome wave. This operation is fulfilled by calculation of convolution of hx(t) with exp(«2nvt) function, i. e. by realization of the Fourier transformation of (2). As it was shown in [3], this operation leads to the infinite series of 6(v ± nv0) "spectral lines" where v0 = c/(bsinp0), and the integer n is named by physicists "order of diffraction".

The first step towards the formulation of the problem indicated in the introduction is in replacing the infinite DG with the finite one consisting of N + 1 slits. Instead of infinite sequence, it leads to formation of finite N +1 number of pulse pairs (1). Mathematically,

(1)

(2)

this operation may be described by multiplication of the infinite sequence hx(t) by rect(t/x) function

rect(t/x)J1' T/2

[0, |t| < t/2.

This function "extracts" the necessary number of pulses of the infinite sequence. As a result one obtains the pulse response of finite DG as

, , , , ct

h(t) = rect

Nb sin R

ct

ni(t) (g) comb

b sin R

(3)

The Fourier transformation of (3) in X-domain provides the following result normalized by unity in its maximum

• /NbsinR , H(k, (3) = sine ( -!- ) eg)

. a sin R\ ( Nb sin R

smc —-- comb '

X V X

here sinc z = (1/nz) sinnz. The square of this formula is that one given in textbooks for the apparatus function of the perfect DG spectrometer.

Note that in the frame of developed treatment, diffraction grating is considered not as a spatial filter of an incoming wave but as the active instrument that generates its own signal being irradiated by the pulse. One may speak of the transformation of the initial wave process by the DG. The form of the resulting signal depends on the grating spatial characteristics (the groove width a, period b and DG length Nb) as well as on the angle of observation R. In fact, the dependence on the angle value means the dependence on b size measured from the point of view of a remote observer.

General approach to signal transformation by 1D plane scattering system.

Generalized DG structure. When analyzing the mode of operation of a simple plane DG over the monochrome wave process we followed tradition and supposed that the grating consists of similar grooves infinite in one direction, those grooves are parallel one to another and placed at equal distances along x-direction. In this section of the paper, we shall consider the system which grooves position and the groove ability to scatter a wave may vary along x from one point to another. This system we name Generalized Diffraction Grating (GDG). The system under analysis may be semi-transparent or reflecting, here we analyze the reflecting one. For the sake of simplicity we suppose that the scattering efficiency of an elementary (x,x + dx) element of GDG does not depend on the scattering angle.

Let r(x) be the scattering coefficient of an (x, x + dx) element of GDG. The r(x) function may be real or complex, in any case for passive GDG (non-amplifying the wave amplitude) its modulus is |r(x)| ^ 1. Moreover, we suppose the grating matter to have instantaneous reaction to the incident excitation, i. e. in this matter exists no delay, aftershock or oscillations excited by the incident wave. The effect of matter resonances on the pulse response properties was discussed in [9].

It would be correct now to point out that in general the scattering device may have curved grooves on the plane, they can be plotted on the concave surface or even to be the 3D phase or amplitude grating that realizes scattering and focusing simultaneously for operation over non-plane wave. Such a grating needs much more complicated mathematical description and will not be considered here.

Illumination of GDG. Consider now the 6(t — l/c) plane wave that falls onto GDG at the negative angle a = 0 (see Figure) so that it illuminates some groove simultaneously along y coordinate. The signal scattered at the angle R (that is positive in the Figure) is observed at

the distant plane orthogonal to this direction and the scattered field amplitude is integrated over this plane (to realize this operation one can take the infinite perfect lens and record the amplitude in its main focal point). The reaction of the whole system to b(t) pulse is nothing but the pulse response h(t) of the GDG. On finding h(t), one can easily receive the output GDG reaction to any incoming ty)(t) signal as the convolution

+^

$(t - l/c) = J q(t)h [(t - l/c) - t\ = ^(t) ® h(t - l/c).

To find the response of (x, x + dx) element, one must calculate the time needed for b(t)-pulse to come from some initial reference plane (it is shown in the Figure with a bold line orthogonal to a direction) to the groove and further to the observation plane that is shown with a bold dashed line. Denote the distance from the initial reference plane to x as l0, and that from x to the observation plane as m0. Under scattering, the wave amplitude changes proportionally to r(x), hence the pulse response is

h(t)

r(x)b

t

l0 + m0 — x(sin a + sin ß)

dx.

(4)

In this formula, the term in brackets is time of propagation from one reference plane to another, including zero delay time at the groove itself. The infinite ranges for integration must not mislead one that relation (4) may be applied only for infinite GDG, for the finite dimensions can be taken into account by zeroing the scattering coefficient r(x) out of some interval.

It can be easily shown that the (l0+m0)/c value is the constant parameter of the problem, so one can change the origin of time and set it to be zero. As a result one receives

+tt

haß (t) = r(x)b

t +

cj (sin a + sin ß)

dx.

(5)

The physical sense of the denominator in brackets can be understood on considering two extreme cases. Let P = 0, then

haß(t) = J r(x)b

— oo

t +

x sin a

dx

r(x)b

t

— cj sin a

dx

r(x) b

x

t--

Va

dx.

The velocity va = -c/ sin a is the rate at which the crossing line of the initial b(t) wave propagates over the (x,y) plane. In the Figure this line shows itself as a crossing point at the x-axis, it moves from left to right when the incidence angle a is negative and in the

c

x

x

c

opposite direction when a is positive. Note that for any a = n/2, va exceeds the velocity of light in vacuum. This is a well-known fact in the theory of waves interference.

The same consideration can be made on the component vY = —c/ sin p when the incident wave illuminates all the GDG plane simultaneously i. e. when a = 0. The vp is now the velocity of variation of coordinate of the emitting GDG element from the point of view of the observer.

The combination of two velocities in brackets in (5) is some new velocity. It can be interpreted as the velocity of variation of the emitting "groove" coordinate under S-wave excitation as seen by the observer:

c

W =--. (6)

sin a + sin p

In contrast to va or vp values, this velocity can take a value \W\ ^ c/2, from -c/2 to +c/2 passing through its magnitude depends on the combination of angle values.

Some special cases.

1. To show that the developed approach can correctly describe the interaction of GDG with some oscillation process, consider first the most familiar variant of the monochrome wave transformation by the infinite DG which scattering coefficient fits the cosine law:

1 1 / x \

r(x) =-(1 + to cos ilx) = - (1 + to cos 2jt— ) . (7)

1 + to 1 + toV b /

In this relationship, parameter b is similar to the DG grooves period studied above.

Traditionally, in (7) the to value is supposed to be \to\ < 1 and describes the contrast of grating as in amplitude holographic grating, for example. We shall omit this demand and include \to\ > 1 as well as the infinite value that permits us to demonstrate the effect of periodic phase n-shifts of scattered waves on the diffracted field properties.

The initial wave ^(t, l) = cos(2nvt — kl) falls onto the GDG at the angle a and the grating response is to be calculated for p angle. For these calculations, S(t) function in (5) must be replaced with its convolution with ^(t, l) that is cos(2nvt — kl) itself. Recollect also that cosine wave may be presented as the sum of two exponential functions, then the integral (5) becomes the sum of two Fourier transforms of r(x).

From (5), (7) one gets immediately that the response consists of three S-functions. The first one originates from the item 1/(1 + to) in (7) and fits the requirement

sin a + sinp = 0. (8)

From (6) it follows that under this condition W ^ <x> and all the beams scattered by GDG come to the observer simultaneously as in the case of plane mirror. Really, from (8) one gets a = —p that is indeed the law of reflection. In optics this extremum is named "zero diffraction order". Note that the amplitude of this pike decreases as modulation index to is increased.

The other two S-functions correspond to

O _+23tv

They fit the well-known condition

b(sin a + sin p) = ±X.

These pikes are ±1 diffraction orders formed by a plane DG. The amplitudes of these b-functions are proportional to the factor m/2(m +1). The ratio of first-order pike amplitudes to zero one is m/2. One can see nothing new for m ^ 1 but for m > 1 some parts of the scattering coefficient (7) become negative, which results in the decrease of zero order amplitude (even to 0 for m ^ to) . This is the well-known effect of zeroing the amplitude of the wave reflected from the boundary of metal mirror deposited on the part of hypotenuse of total internal reflection prism.

The method we have developed permits one to take easily into account the finite dimensions of GDG. Suppose, for example, that just the previously described DG is the band of width L in the x direction. Multiplication of r(x) by the rectangle function means that the integral (5) is taken over the (-L/2,L/2) interval. On recollecting that in this particular case one deals with Fourier transformation of r(x), it can be easily shown that instead of two b(Q - Qi) functions we receive two sincQ whose widths are inverse proportional to L value. Those functions are typical for the perfect optical spectrometers, and sinc2 Q is the limiting form of the "apparatus contour" for ideal spectrometer. Variation of the DG grooves scattering coefficient r(x) or their length along y coordinate permits for correction of the form of apparatus contour i. e. for apodisation.

We shall not study here the obvious case when the period of GDG grooves is great enough that permits for the non-overlapping periodic repetition of the incident pulse that is finite in time or for the generation of an encoded sequence of pulses and its decoding. This possibility was demonstrated in [10] for the circular grating.

2. The most interesting result of time-domain analysis of the initial wave transformation instead of frequency analysis is the variant sin a + sin|3 = ±1 in (5):

By introducing new variable t = x/c for the incoming signal ^(t), one gets immediately the output signal in the form:

The upper sign corresponds to cross-correlation function of the incident process and the function describing the scattering coefficient. The lower sign in (9) is for the convolution of those two functions. Both variants are of great significance for the information processing and are realized at the velocity of light by the simple optical device described in paper.

Conclusion. The data presented in the paper demonstrates the surprising abilities of such a simple traditional optical instrument as the Diffraction Grating. Usual periodic grating provides the Fourier spectrum of an initial wave operating as a spatial system that permits or does not permit the wave to propagate in some pre-determined direction. This filter works as a resonator, possessing very peculiar properties that can be revealed only by the examination of scattered field amplitude dependence on time using methods described in this paper. Traditional grating groove's spatial periodicity itself is complimentary to this or that period of oscillations.

Being clearly understood, time approach automatically leads to the conclusion that some other structure of DG grooves organization may correspond to another type of transformation. For example, the cos(x2 ) GDG can expand the incoming signal into a series of Gaussian

— œ

(9)

— œ

wavelets. The author believes that some complicated processes which exist in nature can be presented in a much more simple and "compact" form as a sum of functions other then harmonic. This may sufficiently help in the solution of problems of signal recovering or pattern recognition. The Generalized Diffraction Grating considered in the paper can provide the instrument that will help in realization of multichannel analysis "at the speed of light".

References

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Статья поступила в редакцию 2 октября 2012 г.