Time impulse response and time frequency response of optical pupils.:Experimental confirmations and applications (1973)

The concepts and experiments discussed in this article are scarcely used in physical optics although they can introduce new ways of processing the information content of any optical pupil. They are related to diffraction phenomena in polychromatic light. One aspect deals with the well-known channelled spectra. The other consists in the definition of the time response function of an optical system - that is the result given by any pupil receiving a very short impulse of light. The time response function allows to generalize the explanation of the channelled spectra. It is also deduced from Fourier techniques and brings an argument to the parallelism between space and time domains. If the entrance pupil of a spectrometer is set in the interference pattern of a two-beam device illuminated in white light, the coloured spectrum is crossed by dark bands that are closer and closer as the difference between the optical paths increases. Along a time frequency (or wave-number) scale, the dark bands are sinusoidal. In this manner, one can say that the time spectrum is the Fourier transform of the couple of the wave trains. The time spectrum of white light being broad, a beam of white light should be considered as juxtaposition of very short wave groups. So, one can consider two individual wave groups passing along each arm of the interferometer. The same reasoning holds as one deals with several well-arranged wave trains, issued for instance from a grating or a multiple beam interferometer, or even further, with a sequence of wave trains in complete disorder. The situation is summarized in the diagram (fig. i): an incident plane wave is diffracted when transmitted through various devices. The spectrometer slit is placed in the region where the parallel beams combine with each other, at infinity or not. One should note the Fourier relationships of the spectral modulation curves with each sequnce of wave trains drawn along a time axis. These considerations immediately open out into new possibilities of conveying and collecting information. While a familiar way of transmission is to let the signal modulate the carrier which is in optics a parallel beam of monochromatic light, here the carrier consists of a parallel beam of polychromatic light, the modulation law being of temporal type. In turn this law is determined by the configuration of the optical system. Therefore any given function can be transmitted to any receiver station having a spectrometer at disposal - the role of this spectrometer being the spectral analysis of the message (or of its Fourier Transform). As an example suppose the tansmission of a (sinX/X)2 function is desired. It is the intensity distribution in the far-field diffraction pattern of a rectangular aperture. Therefore one has to perform the diffraction of a beam of white light by a slit of suitable width, a. In a first approximation (fig. ii), in a direction θ, the sequence of wave trains is limited by a rectangle function of width a cosθ and the power spectrum is expected to be given by the square modulus of a sinc-function. Likewise another test signal for studying the transmission mechanism would be a pair of thin slits that yields a cos2 function, or else a pupil of gaussian transmittance would be nessary to build up a message described by the reciprocal gaussian law. The previous process suffers of limitations. It applies to real and positive function whose Fourier transforms are real and positive. For a wider class of functions, namely complex ones, one has to resort to an equivalent of holography. The transposition comes to send a reference wave-group before or after the information proper. The phase terms are thus kept, same as in conventional holography. Such a simple experiment was carried out by means of a narrow slit set in the same plane as that of the diffracting pupil. A temporal hologram was then recorded. It appears as an image of the time spectrum striped with thin dark fringes of sinusoidal profile, carrying the amplitude and phase temrs. The usual reconstruction process in monochromatic light applies to that temporal Fourier hologram. Information concerning the previous rectangular pupil has been transmitted in the form of a hologram recorded at the output of a spectrometer 10 meters away from the source. Incidentally a much more interesting and promising technique enables real-time operating. All one has to do is get the temporal autocorrelation function of the whole of the message with its reference waves group. The pair of antisymmetrical side band distribution represents the image and its conjugate. Practically such phenomena are observable as the amplitude and phase modulation of the visibility function of the fringes displayed at the output of a Michelson interferometer. An important requirement is the use of a spatially coherent source of high luminance, with a broad spectrum (its transform - the reference - would be narrow then). The input function of the Michelson interferometer is the sequence of wave trains. Therefore this represents the basic arrangement in Fourier transform spectroscopy, with a slight difference: instead of gathering a signal and making an interferogram whose spectrum would describe the source, one gets here the autocorrelation function of the information under test, that is of the time holographic signal coming from the pupil. As a matter of fact the time-hologram itself is never formed; no intermediate recording is needed as one directly reaches three terms: the central one being the superposition of the autocorrelation of the signal and the reference respectively, the other two the symetrical side-bands displaying, as said before, the cross correlations of the signal by the reference which was assumed as a narrow impulse along a time axis. A more rigourous treatment has proved necessary to give a full account of the mechanisms involved in the transmission of a time message. In particular a time impulse response and a time transfer function can be defined for any pupil. Looking at figure iii, one notes that the scale of the Fraunhofer spatial diffraction pattern of a rectangular aperture varies as the reciprocal of the time frequency ν. As ν of the incident plane wave is tuned continuously from 0 to ∞, an observer set in a fixed position at infinity will see the « breathing » of the pattern according to the law sin Kν/Kν To complete the determination of the amplitude H(ν) at the observation point, one must add an important fact: usually, in the calculation of the amplitude diffracted by any pupil by means of the Fresnel-Kirchhoff integral, one omits the multiplicative factor, - j/λ. Therefore: H(ν) = (j2πν sin Kν/Kν. The reciprocal of ν in the time domain is t. The time response of the system to a unit impulse h(u, t) in the u-direction is then the Fourier Transform of H(ν). After a well-known derivative theorem, the transform of H(ν) is the derivative of a rectangle function. The same procedure would show that if one considers any pupil, whatever its contour and amplitude distribution, the time impulse response is the derivative of the pupil function. A similar result is obtained with phase pupils. Generally, the time impulse response of a complex pupil in a given direction is described by the derivative of the complex pupil function projected on this direction. This is illustrated by various drawings of functions corresponding to one or several slit apertures. Applications are envisaged, namely in metrology - for instance in the assessment of the quality of surfaces and the measurement of thicknesses.