CC BY
43
УДК 535.4
Вестник СПбГУ. Сер. 4. 2013. Вып. 4
M. K. Lebedev, Yu. A. Tolmachev
REINCARNATION OF THOMAS YOUNG'S IDEAS AS THE DIDACTIC BASIS FOR TEACHING THE DIFFRACTION OF ULTRASHORT PULSES*
Introduction. It was the general primary idea of Thomas Young [1] that the field of the wave behind the screen containing aperture consists of two components:
1. the first one belongs to some part of the initial wave passing through the aperture;
2. the second one is emitted by the boundary of aperture under illumination by the initial wave.
The development of Huygens presentation of the wave propagation, in general, and the wave diffraction, in particular, together with bright mathematical description of those processes given by Fresnel led to situation when Young ideas were referred to as some interesting curiosity. Meanwhile, the theory given by Rubinowicz demonstrated that indeed the field of the wave behind the screen is the superposition of "geometric optics" part and the "boundary" waves. The main aim of the present paper is to demonstrate that the idea of the boundary wave is highly effective for the description of the wave propagation in non-dispersive media and for the analysis of the effect of diffraction.
The ô-wave approach. In optics, for the description of the diffraction field of a monochrome wave the Kirchhoff approximation is often used. The very first look at the corresponding integral transformation permits one to state that there is a great difference between the results of its application to the monochrome wave and to rather complicated wave signal that contains wide band of spectral components. This difference is determined by the multiplier i/X (or iv) for the main integral.
As the most general attitude demonstrating the properties of the boundary waves and the effects of their combination with the incident wave we consider the ô(i)-wave approach. Such a wave is the localized perturbation that allows to separate in time and space all the components of the diffraction field and makes its qualitative properties absolutely clear. This approach violates one of the main assumptions for Kirchhoff approximation, namely the infinite spectrum of ô-signal contains large wavelengths which are out of the range of this theory usage. Nevertheless, it can be used for the solution of physical problems when two conditions are met:
1. the mathematical description of the wave propagation is treated only as a solution of the mathematical problem of the ô-wave transformation; the result of calculations is "the pulse response" of the system under study;
2. in order to determine the diffraction field of any physically realizable wave signal one must calculate the convolution of the pulse response with the waveform of the signal.
Only this last result may be treated as the solution of physical problem and only upon the condition that the whole system possesses the property of linearity.
Mikhail Konstantionovich Lebedev — Saint Petersburg State University; e-mail: m.k.lebedev@gmail.com
Yurii Aleksandrovich Tolmachev — Professor, Saint Petersburg State University; e-mail: ytolmachev@gmail.com
* По материалам международного семинара «Collisional processes in plasmas and gas laser media», 22—24 апреля 2013 г., физический факультет СПбГУ.
Семинар был проведён при софинансировании фондом «Династия».
© M. K. Lebedev, Yu. A. Tolmachev, 2013
The theory developed in [2] for the process of diffraction of S-shaped wave based on the Kirchhoff approximation has shown that the solution of the most common diffraction problems (the slit, the circle, the Gaussian aperture), and others closely related to them, indeed can be presented in algebraic-simple form containing only two components mentioned above. Such simplicity opens a way to elementary interpretation of the results of experiments with ultrashort pulses in non-dispersive media. In particular, it permits to reveal the details of the formation of X-structures of the diffracted field moving at the superluminal velocity in the direction of the incident wave propagation, as well as in the opposite direction.
Testing of the approach. Verification of the correctness of the S-wave approach was performed in two ways; both of them are in the comparison of Fourier transform of the pulse response with the direct solution of the Kirchhoff integral. We omit here diffraction from the slit for the corresponding solution was received by the integration of well-known formula for the monochrome waves over [0, to) interval of frequencies. This integration results in the sum of two S-functions of different signs, each of them appears in the observation point P at the moment corresponding to the distance between P and the edge of the slit. Analysis of diffraction process of the plane S-wave from the rectilinear edge of half-space screen confirmed the data obtained for the slit: in the illuminated zone there exist two waves the first is the passing one and the second is the boundary wave. This last tends to a S-function for z ^ to. In the shadow, only the boundary wave exists. The boundary waves in the illuminated zone and in the shadow are of different signs.
The field of the S-wave diffracted from the circular aperture of radius a in the illuminated zone also contains the passing through aperture circular fragment of the incident wave S(t — z/c) and the boundary wave; in the shadow zone only the boundary wave remains. Fourier transform of this pulse response at large distances (z ^ a) has a form of a Bessel function Ji divided by its argument and precisely coincides with the solution given in [3] and classical textbooks. At the symmetry axis, two S-functions exist of the opposite signs and approximately equal amplitudes, one running after another. They form the first derivative of S-function on time at z ^to, producing the first derivative of ordinary functions in the convolution. The same behavior is observed in the focal point of converging spherical wave diffracted from the circular aperture.
The detailed comparison of the S-wave approach with the results of "conventional" theory for monochrome waves was fulfilled for the aperture characterized with Gaussian transparency function T(p) = exp [—(p/a)2] where p is the coordinate at the plane of aperture and a is the parameter of Gaussian distribution in the cylindrical coordinate system. In the Kirchhoff approximation, immediately behind the aperture the diffracted wave amplitude is V(p,0,t) = T(p)S(t). For z > 0, the Kirchhoff—Fresnel integral provides [4]:
Here (p,z) are the coordinate of the observation point P, I0 and Ii are the Bessel functions and © is the Heaviside function. This result may be interpreted as a superposition of the passing initial wave (the first item) and the scattered wave field (the second item);
at z ^ 0 and p = 0, the form of the second item becomes similar to the first derivative of Gaussian function.
In our approach, the solution of the diffraction problem from Gaussian aperture for the monochrome wave can be found as the convolution of V(p, z,t) with f (t) = exp(—iwt) that is nothing but the Fourier transform of the obtained formula. The result was compared with the data obtained using two other approaches. The first one is based on the transfer function of free space [5]. This method gives an exact solution to the boundary problem, but the problem itself is not that of diffraction from an aperture. For the purpose of the present study it is essential that if the field in some plane is described by Gaussian function, then using this approach we obtain an exact result for all the points behind that plane. For the problem under study, in comparison with the S-wave approach it lacks the computation efficiency because of the double integration. The second method uses the quasioptical approach [6]. The latter deals with the Fresnel diffraction, which is a further approximation in the Kirchhoff theory, and provides the well-known expressions for the Gaussian beam of the 0th order. The results of these two approaches for the amplitude of the field at different distances from the aperture are compared with our data in Figure.
Comparison of the results obtained using the S-wave approach with the data of conventional methods:
solid line — S-wave, dashed line — quasioptics, dash-dotted line — the free space transfer function (c =1, k = rn/c = n/2); a — z = 0; b — z = 1; c — z = 5; d — z = 10
Conclusion. The S-wave model has clearly demonstrated that the primary Thomas Young's idea of scattering wave by the boundary of the obstacle (or any other non-homogeneity) as the main reason of the phenomenon of diffraction was absolutely correct, the fine nicety of changing the sign of the scattered wave at the light/shadow boundary could be established even at his epoch with rather simple experiments but was not mentioned. The development of S-wave approach to demonstrate obviously this peculiarity and proves that the Fourier transform of pulse response gives the same analytical solution for the classical problems of monochrome wave diffraction in the Fraunhofer zone that the Kirchhoff—Fresnel theory gives. Moreover, the application of this method for much more complicated problem of the wave diffraction from Gaussian aperture has demonstrated that the results of pulse
method practically coincide with those of most used theories and provides better approach to the precise solution in the Fresnel zone of diffraction and even at the distances close to characteristic dimensions of the aperture.
References
1. Rubinowicz A. Thomas Young and theory of diffraction // Nature. 1957. Vol. 180. P. 160-162.
2. Lebedev M. K., Tolmachev Yu. A. Diffraction of an ultra-short pulse by an aperture // Optics and Spectroscopy. 2001. Vol. 90. P. 457-463.
3. BornM., WolfE. Principles of optics: 7th (expanded) edition. Cambridge, 2002. 952 p.
4. Лебедев M. K., Толмачёв Ю. A. Дифракция плоской дельта-волны на гауссовой диафрагме // Вестн. С.-Петерб. ун-та. Сер. 4: Физика, химия. 2003. Вып. 4. С. 15-22.
5. Литвиненко О. Н. Основы радиооптики. Киев, 1974. 208 c.
6. Svelto O. Principles of lasers: 5th edition. Berlin, 2010. 642 p.
Статья поступила в редакцию 22 апреля 2013 г.
