ISSN 0030-400X, Optics and Spectroscopy, 2015, Vol. 119, No. 4, pp. 577–588. © Pleiades Publishing, Ltd., 2015. Original Russian Text © M.K. Lebedev, Yu.A. Tolmachev, 2015, published in Optika i Spektroskopiya, 2015, Vol. 119, No. 4, pp. 576–587.
THE INTERNATIONAL YEAR OF LIGHT 2015 No one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them… When there are only a few sources, say two, interfering, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used. Richard Feynman
Diffraction of Light: the Young and Huygens Models at a New Stage of Development of Optics M. K. Lebedev and Yu. A. Tolmachev St. Petersburg State University, Peterhof, St. Petersburg, 198504 Russia e-mail:
[email protected] Received January 19, 2015
Abstract—Solutions to classical problems of the theory of diffraction have been considered in two mutually complementary approximations: a monochromatic wave and a pulsed wave. An exact one-to-one correspondence between solution results in the Huygens–Fresnel and Young approximations has been found. The dynamics of the development of the spatial structure of a diffraction field of an ultrashort pulse taking into account edge waves and formation of X-shaped singularities moving at a superluminal velocity has been presented. DOI: 10.1134/S0030400X15100161
INTRODUCTION In the second half of the 17th century, Huygens [1], when studying the propagation of light, formulated the principle in accordance with which each point of the front of a light wave was considered as a source of secondary perturbation. A century later, Young [2], when analyzing particular features of scattering of light by the edge of an opaque screen, put forward the hypothesis that the field behind the screen is a combination of two components, the first of which is a part of the initial wave that passed through the hole, while the second component is the radiation of the hole edge caused by the perturbation of the screen surface by the incident wave. However, in 1816, Huygens’s notions have been supported by a physically clear mathematical description proposed by Fresnel [3]. Three years later, Young agreed with Fresnel’s arguments, accepting the inconsistency of his model in his private correspondence with Fresnel, and this model remained in the history of science as a certain curious scheme. At the same time, further development of the theory of diffraction [4] showed that the field behind the screen can indeed be represented as a superposition of a part of the initial wave that propagates in accordance with
laws of geometrical optics and a “boundary” (edge) wave. The main objective of this work is to show that there are no contradictions between the Huygens– Fresnel and Young models in the notion of the existence of an edge wave. Either of them proves to be useful for the description of diffraction and interference effects under certain conditions. Whereas the former (Huygens–Fresnel) is more universal, especially, in studies of stationary systems, the latter makes it possible to analyze the dynamics of the process of formation of the wave field, which is particularly important in the investigation of propagation of ultrashort pulses. We will not complicate the problem by considering special features related to the vector character of the electromagnetic field of the light wave, but, rather, we will use the scalar approximation [5]. Addressing the Young model is related to radical qualitative changes that have occurred in optics over the last two to three decades. Whereas, previously, a monochromatic plane or spherical wave was a typical object of traditional optics, now, the technique of generation of ultrashort light pulses, each of which contains only one to two oscillations, has been developed.
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The width of the spectrum and the duration of these ultimately short pulses obey the relation Δν Δ t ~ 1; correspondingly, the ordinary notion of “carrying frequency” becomes indeterminate for them. From the point of view of experimental science and technology, this means that it became possible for the first time to design an experiment in the light frequency range with a classically deterministic ordered wave packet. In addition, we would like to call attention to an object that is known not too well at the present, namely, a plane (or spherical) δ(t)-shaped wave. The use of this form is based on the possibility of direct application of the formalism of the theory of linear systems to the investigation of diffraction and interference processes of pulsed waves. As a rule, in order to describe the interaction of an ultrashort light pulse with an optical system, the field of the wave is represented as a superposition of fields of all monochromatic components of the spectrum of the pulse. The propagation of each spectral component is studied separately, and then calculated fields of all components are summed. The use of this approach for ultrashort pulses is connected with great difficulties, at least, because, upon solving diffraction problems, phase corrections of high orders are usually neglected. However, being small for some frequencies, they may prove to be large for other frequencies. In addition, with this method of solving a particular problem, the illustrative character of the description of the formation process itself of a resultant field is lost, as well as of the possibility of rather simple generalizing obtained results to other cases. The method that uses the notion of the impulse response, i.e., the reaction of the system to a signal in the shape of a δ-function, is opposite to the method described above. However, we emphasize that the use of the model of a δ(t)-wave is quite justified only if linear systems are investigated (or as a first step in analysis of the field for nonlinear systems).1 Below, we pres1 We
note that the δ(t)-shaped wave is none the greater idealization than the point light source or the monochromatic wave. Mathematically, each of these objects is described in terms of a δ-function of corresponding variables—time, spatial coordinates, or frequency. The passage from this ideal physical object or process to a real one is related to similar transformations in the corresponding spaces. The formalism of general functions, which should be used in the investigation of the impulse response, is rather well developed, and the clarity of appearing images is the question of the acquiring of new skills, new habit. One of the definitions of the δ-function relates it to the derivative of the Heaviside θ-function, which is used in analysis of transient processes occurring upon switching-on of a signal. Under this understanding, the δ(t)-shaped wave may be considered as the propagation of the front of a switched-on oscillation; then, the question of the energy of the δ-wave is lifted. Mathematically, this means that there is no definition for the square of the δ-function. Of particular note is the problem of the physical realizability of corresponding signals. It is studied in depth as applied to radiotechnical linear systems, whereas the case of spatial electromagnetic waves still remains to be investigated. Similar problems also arise for monochromatic waves, wavelets, and the Gaussian wave packet so much-loved by physicists.
ent several results of solutions that we obtained for a series of well-known diffraction problems. Mathematical simplicity, visual clearness, and exact coincidence with results of classical methods of calculation show the efficiency of the approach based on the notion of impulse response of the optical system. Relations presented below are based on the application of the Kirchhoff approximation, which is the development of the Huygens–Fresnel method. As distinct from the monochromatic wave, the δ(t)-wave is a localized perturbation, which makes it possible to separate all components of the diffraction field in space and time and makes its qualitative properties absolutely clear. However, the use of the notion of the δ(t)signal violates one of the basic assumptions of the Kirchhoff approximation: the infinite spectrum of this signal contains long wavelengths, which fall out of the applicability range of this theory. The notion of the impulse response may be used to solve physical problems only if the following two conditions are fulfilled: (i) The found mathematical form of the description of the wave propagation—the impulse response of the system under investigation—is considered only as a solution of the mathematical problem on the transformation of the δ(t)-wave. (ii) In order to determine the diffraction field of any real wave process, it is necessary to calculate the convolution of the impulse response found at the first stage with a function that describes the dependence of the signal on time. Only the result of the second stage may be considered as a solution of the physical diffraction problem. The theory of the diffraction process of the δ(t)shaped wave developed in [6, 7] based on the Kirchhoff approximation showed that solutions of most widespread classical problems (diffraction by slit, circle, diaphragm with a Gaussian transmission, and other objects closely related to them) can be represented in an algebraically simple form, which contains only two components of the field: transmitted and edge waves. This simple description of the field structure is completely consistent with the Young model and opens the way to an elementary interpretation of results of experiments on diffraction of ultrashort pulses in media without dispersion. In particular, it yields a simple explanation to the formation of Хshaped structures of the field [8–11], which move with a superluminal velocity in the direction of propagation of the incident wave and in the opposite direction. The transformation of the signal into its first derivative with respect to time at long distances from the obstacle and the formation of the Arago–Poisson spot, as well as many other diffraction and interference effects, become physically clear. The correctness of the application of the model of the δ(t)-wave was verified in several ways. The first, and the simplest and natural way is to perform comparison with results of analytical calculations for well-
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nected with the corresponding quantities in the incident wave as follows:
V = T ( x, y)V , (i )
P
r
φ
z
(i ) (2) ∂ V = T ( x, y) ∂ V . ∂n ∂n This definition of the boundary conditions implies that diffraction effects do not cause changes in the field distribution in the plane of the screen. In the case of a δ-shaped perturbation, this assumption is justified to an extent to which one can neglect rescattering effects of waves that are diffracted by elements of the screen and that retard with respect to the initial pulse.2 By introducing notation
r = S
( x − x 0 ) 2 + ( y − y0 ) 2 + z 02 ,
φ = arccos
x − x0
( x − x 0 ) 2 + ( y − y0 ) 2
(Fig. 1), the field at P can be represented [7] as
Fig. 1.
studied diffraction problems of monochromatic waves, including waves obtained by direct calculation of the Kirchhoff integral. The second way is to solve numerically the wave equation in the presence of an obstacle without involving the Kirchhoff theory. Finally, the third way is to give direct impulse interpretation of results of experiment on diffraction of an ultimately short pulse.
V (P, t ) = 1 4π
∫∫ T ( x, y)
S (3) z0 z0 ⎤ 1 ⎡ × 3 δ(t − r / c) + δ t (t − r / c) dS, 1+ cr ⎢⎣ r ⎥⎦ r where δ t (t − r / c) = ∂δ(t − r / c) ∂ t . The expression in the braces in (3) is the impulse response of the space; it has the meaning of the scattering function of an infinitely small spatial element of the δ-wave,
{
}
z0 δ(t − r / c) + 1 [1 + z 0 / r ] δ t (t − r / c) . (4) 3 cr r Taking into account that z 0 = r cos ψ , where ψ is the angle at which the perturbation propagates from the given element of the screen to the point of observation, we can see that the field of the diffracted wave is formed by two terms, one of which is referred to as the “near” field, the amplitude of which falls with distance as r −2 , while the other term is the ordinary field, which decreases as r −1 . If in (3) the term corresponding to the near field is neglected and the Fourier transform is performed (i.e., the passage from the impulse signal to the monochromatic wave), the Fresnel– Kirchhoff formula will be obtained [13, 14]. To the differentiation of the δ-function for the monochromatic wave, the shift of the oscillation phase by 90° corresponds; in the mathematical description, this corresponds to the appearance of multiplier i in front of the integral sign. Relation (4) can be considered as a kernel of an integral transformation that describes the propagation of a δ-wave with a distribution of amplitudes over the front proportional to T ( x, y). We note h0(r, t ) =
THEORETICAL In accordance with the Kirchhoff theorem, the field of a light wave at some point of observation P ( x 0, y0, z 0 ) that is in a region of space bounded by closed surface S is defined by values of the field and its normal derivative on the given surface [5],
= 1 4π
V (P, t ) [V ] ∂ 1 − 1 ∂ r ⎡⎢∂V ⎤⎥ − 1 ⎡⎢∂V ⎤⎥ dS. ∂ n r cr ∂ n ⎣ ∂ t ⎦ r ⎣ ∂ n ⎦
∫∫ { S
}
()
(1)
Let there be an absorbing screen (Fig. 1) that is located in plane ( x, y,0) and that contains a hole with amplitude transmission T ( x, y ) . Let wave
V (i )(0,0, z, t ) = δ(t − z / c) that propagates in the positive direction of the z axis be incident on the screen. Assume that the Kirchhoff boundary conditions are fulfilled, in accordance with which the field and its normal derivative in the plane of the screen are conOPTICS AND SPECTROSCOPY
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2 Our investigation [12] showed that the amplitudes of rescattering
do not exceed a few percent of the initial amplitude.
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that, only if the two terms in relation (3) are taken into account, does it allow one to correctly describe the propagation of the plane δ-wave in free space; to do this, it suffices to set T ( x, y) = 1 on the entire plane and to perform integration in the infinite range. This circumstance also manifestly shows that any inhomogeneity of the wave, e.g., a change in its amplitude at some point or region, a local delay in time (for the monochromatic wave, a local change in phase), etc., leads to the occurrence of a new additional wave, i.e., to diffraction. For the field formed behind the screen, relation (3) for the impulse wave can be written in the following form:
z V (P, t ) = c ⎡1 + 0 ⎤ ⎢ 4π ⎣ ct ⎥⎦ ∞ × {−Φ(ct )δ(r − ct ) + Φ '(ct )θ(r − ct )} r = z0 ,
z
Fig. 2. Qualitative pattern of the time dependence of the field of a wave diffracted by a circular hole at different points of space behind the screen. Dashes denote the light/shadow boundary in the geometrical optics approximation.
where 2π
Φ(r ) ≡
∫
T ( r 2 − z 02 cos φ, r 2 − z 02 sin φ)d φ
0
and θ is the Heaviside function. A detailed analysis [15] of the physical meaning of the obtained relation for the case of diffraction by hole with a unit transmission showed that the first term in the braces, which contains the δ-function, may be considered as an element of the incident wave that passed through the hole, while the second term, which contains θ(r − ct ), may be considered as a wave that was scattered by the edge of the hole. Solutions to some diffraction problems in the form of a superposition of a passed wave and a wave that is emitted from the edge were obtained already by Sommerfeld [5] and correspond to the description of the phenomenon of diffraction proposed by Young.3 Below, we will show results of the application of the developed model of the diffraction process for several most known examples, such as diffraction by circular hole, diffraction by half-plane, and diffraction by diaphragm with a Gaussian transmission. DIFFRACTION OF A δ-WAVE BY A CIRCULAR HOLE In [7], upon solving the problem of diffraction by a circular hole of radius a in an absorbing screen, it has been shown that, inside a cylinder that is based on the hole (i.e., within the area illuminated by the incident the region in which T ( x, y) ≠ 0 is bounded, the supports of functions Φ(r ) and Φ '(r ) also prove to be bounded. In this situation, it is always possible to indicate the “nearest” and the “farthest” points of the hole edge with respect to the point of observation. Consequently, in the Kirchhoff approximation, an impulse response of a hole of a finite dimension to a δ-shaped perturbation proves always to be bounded in time.
3 If
wave in the geometrical optics approximation), the impulse response at the point of observation consists of two components, passed wave V pass and diffracted wave V d , while, in the shadow area, only a diffracted wave exists, V ill (P, t ) = V pass + V d = cδ(z 0 − ct ) + V d in the illuminated area, V shad (P, t ) = V d in the shadow a rea.
(5)
In this case, in both areas, (except for the immediate surface of the boundary cylinder), the shape of the diffracted wave is described by the relation
V d (P, t ) ⎡ z ⎤ (ct )[(ct ) 2 + a 2 − x 02 − z 02 ] = − c ⎢1 + 0 ⎥ 2 2 2 2 2π ⎣ ct ⎦ [(ct )2 − z 02 ] −[rmin − (ct ) ][rmax − (ct ) ] × [θ ( rmax − (ct )) − θ ( rmin − (ct ))] . (6) To simplify the form of expression (6), minimum and maximum distances from the point of observation 2 to the hole edge are introduced, rmin = z 02 + (a − x 0 ) 2 2 and rmax = z 02 + (a + x 0 ) 2 . The difference of θ-functions in (6) determines the time interval within which the impulse response of the diffracted wave differs from zero. In this case, the structure itself of expression (6) shows that the moment of the appearance of the signal at the observation point is determined exactly by the distance from the hole edge; consequently, relation (6) describes precisely the shape of the edge wave. The amplitude of the edge wave has discontinuities of the second kind at the interval ends ( rmin c , rmax c ) and changes its sign immediately at the light/shadow boundary, with the wave of the positive polarity (with respect to the polarity of the inci-
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+ +
+
− − +
−
−
−
−
+ +
(a)
(b)
(c)
Fig. 3. Development of the diffraction field of a δ (t ) -wave upon its diffraction by a circular hole of unit radius. Black vertical segment indicates a positive passed wave; dashed lines show the section of a cylindrical light/shadow boundary. Signs of the amplitude of the edge wave are presented in each case. (а) Onset of an edge wave in the form of an expanding torus. (b) Formation of an Х-structure near the cylinder axis and formation of a negative δ-pulse on the symmetry axis. (c) Pulling of the amplitude of an edge wave to boundaries of the toroidal surface and approaching of a negative Х-node to the passed wave.
dent wave) being scattered toward the shadow area, while the wave of the negative polarity is scattered toward the illuminated area. Qualitatively, the shape of the impulse response from the circular hole is shown in Fig. 2, while Fig. 3 presents a more detailed structure of the edge wave at different moments of time. General properties of the edge wave were discussed in detail in [16]; among them, the most important for us is that the effective width of pulses near the ends of the existence time interval monotonically decreases with an increase in the distance from the screen, with the area of each of pulses remaining constant. This indicates that, as z increases, the properties of the positive and negative pulses of the edge wave approach those of the δ-function. The spatial shape of the wave tends to a torus with an infinitely thin shell, the outer radius of which increases as ρ = ct , while the axial line coincides with the edge of the hole. Expansion of the torus is accompanied by its selfintersection and formation of a field distribution with an Х-shaped cross section and an amplitude singularity on the symmetry axis of the system, which is clearly seen in Figs. 3b, 3c. To this singularity, which is in the illuminated region of the space, a δ-function of a negative sign corresponds.4 As a result, immediately on the axis, the impulse response has the form
V ( x 0 = 0, y0 = 0, z 0 )
()
2⎤ ⎧⎪ ⎡ ⎪⎫ = c ⎨δ ( z 0 − ct ) − 1 ⎢1 + 1 − a ⎥ δ( a 2 + z 02 − ct )⎬ . 2⎣ ct ⎦ ⎪⎭ ⎩⎪ (7)
4 Formation of a component of the edge δ(t)-wave on the symme-
try axis of the problem was also noted in [17]. OPTICS AND SPECTROSCOPY
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It is obvious that, at ct @ a , the expression in the square brackets tends to 2, and relation (7) acquires very simple form
V ( x 0 = 0, y0 = 0, z 0 @ a) 2 (8) ⎧ ⎛ ⎞⎫ = c ⎨δ ( z 0 − ct ) − δ ⎜ z 0 + a − ct ⎟⎬ . 2z 0 ⎩ ⎝ ⎠⎭ The second δ-function lags behind the first one by interval Δ t = a 2 2cz 0 , which tends to zero at z 0 → ∞ (but never turns to it); i.e., this pulse moves at a velocity that exceeds the speed of light in vacuum. Upon calculation of a convolution of impulse response (8) with a signal described by an ordinary smooth funci tion V ( ) = f (t − z / c ), at very long distances from the screen, we obtain
z ∂ f ⎛⎜ 0 − t ⎞⎟ ⎝c ⎠ . (9) f ( x 0 = 0, y0 = 0, z 0 @ a, t ) ≈ − a 2cz 0 ∂t 2
Transformation of the signal into its first derivative with respect to time due to the diffraction is an effect that is well known as early as the 1940s [18]. It is also observed upon diffraction by slit [6] and in the focus of a spherical converging wave (in this case, a decrease in its amplitude with increasing z 0 is completely compensated by a decrease in the radius of curvature of the wave upon propagation of the wave toward the center of the sphere) [19, 20]. In the diffraction of a converging monochromatic spherical wave, a phase shift by π/2 with respect to that calculated in the ray optics approximation corresponds—the Gouy phase effect. Relation (8) shows that it is the edge wave that is responsible for the formation of a signal proportional to the fist derivative of the input signal.
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v 4π
0.001
0.002
0.01
0.005
0.005
0.001 0.01 0.02 0.03
0.015
0.002 0.001
0.005
2π 0.005
0.001 0.05
0.9
0
0.7
0.3
0.2 0.1
0.02 0.005 0.01
0.015
0.01
0.03
0.001
2π
0.005
0.01
4π
0.001
6π
8π
u
Fig. 4. Comparison of the field intensity distributions near the focal point in the first quadrant calculated (black curves) by the impulse method and (gray curves) by Lommel functions.
The described structure of the impulse response on the symmetry axis makes it possible to draw several qualitative conclusions that are important for experiment. If a circular hole is illuminated by a spatially coherent biharmonic radiation with a period of T = λ c , interference of the passed and edge waves cause the formation of a director spot (or rings) at points on the z axis at which the delay time of the edge wave with respect to the passed one will be multiple of the period of oscillations, Δ t = nT , i.e., z n = a 2 2nλ . The first spot, which is the most distant from the screen, will be the most contrast, because the amplitude of the δ-pulse corresponding to the edge wave becomes close to the amplitude of the passed wave only at very small diffraction angles (i.e., at very long distances). Let the hole be replaced by an opaque disk. In accordance with the Babinet principle, in the space behind the disk, only an edge wave should be observed, which acquires the shape of a positive δpulse on the axis. Upon illumination of the disk by radiation with an arbitrary wavelength (including white light), a bright spot will be observed on the axis, which is termed the Arago–Poisson spot. The correctness of the developed model is also confirmed by comparing the results of its application to well-studied cases of diffraction of monochromatic waves with the data of their investigation obtained in terms of the Huygens–Fresnel model. Let us consider, e.g., the problem of diffraction of a plane monochromatic wave by a circular hole [12]. To this end, it suffices to calculate the Fourier transform of the obtained solution, and, while doing this, it should be taken into account that, at z 0 → ∞ , the angular dimension of the hole that is seen from the observation point becomes infinitely small, and the impulse response is determined only by edge wave (6). Corresponding calcula-
tions for the observation point with coordinates (ρ 0, z 0 ) , which were presented in [12], lead to the expression 2
V (ρ 0, z 0 ) = −i cka e 2z 0
2 ⎛ ρ ⎞ ik ⎜⎜ z 0 + 0 ⎟⎟ 2z ⎠ ⎝
⎡2J 1(kaρ 0 / z 0 )⎤ ⎢ kaρ / z ⎥ , ⎣ ⎦ 0 0
where k = 2π λ , which exactly coincides with that known in the Fraunhofer approximation [5]. In the same way as in the case of the diffraction of a plane wave by a circular hole in which two δ-functions were observed on the symmetry axis of the system, upon diffraction of a spherical convergent wave
V (i )(ρ, t ) = ρ −1δ (t − ( R − ρ ) c ) with initial radius R and current radius ρ , two δ-pulses also arise on the system axis. In a vicinity of the focal point, immediately on the z axis, we have [15]
(
) (
V (z, t ) = − 1 ⎡δ t − R − z − δ t − z ⎢⎣ c
)
R + z cos γ ⎤ (10) ⎥⎦ c
at z ! R , where z is the coordinate of the point of observation counted from the center of the sphere, and γ is the aperture angle counted from the axis to the edge of the hole. The passed wave corresponds to the first δ-function, while the edge wave, which lags behind the passed one by Δ t = (z cos γ)/с , corresponds to the second δ-function. The lag interval tends to zero at z → 0 (correspondingly, at the focal point, we have the derivative of the δ-function). By passing to monochromatic waves (by calculating the Fourier transform of expression (10)) and calculating the squared amplitude, we obtain the intensity distribution of the field
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Vb
2
4
6
8
the plane of the aperture, in accordance with (2), there is field V (ρ, 0, t ) = T (ρ, z = 0) δ (t ). Calculation of the Fresnel–Kirchhoff integral yields
t
1 −5
⎛ ρ2 ⎞ z V (P, t ) = exp ⎜ − 20 ⎟ δ ⎛⎜ t − 0 ⎞⎟ c⎠ ⎝ a ⎠ ⎝ ⎡ ⎛ 2ρ 0 ⎞ ⎛ 2ρ 0 ⎞⎤ ρ0 I1 ⎜ +⎢ − I0 ⎜ ⎟ ⎟⎥ (11) 2 2 2 2⎟ 2⎟ ⎜ 22 ⎜ 22 ⎣⎢ c t − z 0 ⎝ c t − z 0 ⎠ ⎝ c t − z 0 ⎠⎦⎥ ⎛ ρ 02 + c 2t 2 − z 02 ⎞ ⎛ z 0 ⎞ c(ct + z 0 ) × exp ⎜− ⎟ θ ⎜t − ⎟. c⎠ a2 a2 ⎝ ⎠ ⎝
2
−10
3
−15 4 Fig. 5. Time dependences of the amplitude of an edge wave pulse (Vb) upon its diffraction by a Gaussian diaphragm calculated for different distances from the screen: z = 1, 2, 5, and 8 for curves 1–4, respectively.
along the z axis in the vicinity of the focal point of the wave 2 2 V λ (z) ≈ 12 sin ⎡π z (1 − cos γ )⎤ ⎣⎢ λ ⎦⎥ z 2 2 = 12 sin ⎡2π z sin ( γ 2)⎤ . ⎢ ⎣ ⎦⎥ λ z Up to a constant multiplier, this expression coincides with that well known from the classical theory [5]. The first zeros of the field distribution along the z axis lie at points
z1 = ±
λ . 2 sin (γ 2) 2
In this calculation, we neglected corrections on the 2 order of ( z R ) in the amplitude of the wave; their consideration yields a small asymmetry of the distribution of field nodes along the z axis with respect to the focal point. The spatial distribution of the squared modulus of the Fourier transform of the total impulse response of the circular hole to a spherical convergent δ-wave [21, 22] in the vicinity of the focal point is shown in Fig. 4 in comparison with data obtained by the method described in [5]. We can see an excellent agreement between results of calculations performed by the absolutely different methods. DIFFRACTION OF A δ-WAVE BY A DIAPHRAGM WITH A GAUSSIAN TRANSMISSION In [15, 23], the impulse approach was compared with results of the “common” diffraction theory of monochromatic waves for an aperture with a Gaussian 2 transmission T (ρ, z = 0) = exp[− (ρ a ) ], where (ρ, z ) are coordinates in a cylindrical system, and a is parameter of the Gauss function. Immediately behind OPTICS AND SPECTROSCOPY
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Here, I 0 and I 1 are Bessel functions, and θ is, as before, the Heaviside function. The term that contains the δ-function is the initial wave that passed through the diaphragm, while the term in the square brackets is the wave that arose as a result of diffraction. One can draw an analogy between the first term and the Sommerfeld precursor, a wave of a small amplitude that arises upon propagation of a pulse with a sharp front in a medium with a dispersion [24]. The mechanisms by which these waves are formed are principally different; however, in both cases, they represent a part of the initial perturbation that does not interact with the corresponding optical system upon propagation through it. The shape of the signal that corresponds to the second term depends significantly both on time and on the distance to the z axis. At points of the front of the diffracted wave that are located at distance a from the axis, the field changes its sign: closer to the axis, the amplitude of this component of the field is negative, while, at ρ > a , it is positive [20]. Therefore, in the problem of diffraction of a plane wave by a diaphragm with a Gaussian transmission, the parameter a plays the same role as is played by the radius of the hole in the problem of diffraction of a wave by a circular diaphragm: the cylindrical surface ρ = a is similar to the boundary between the illuminated region and the region of the geometrical shadow. At long distances, the shape of the corresponding impulse approaches that of the derivative of the Gauss function. Immediately on the symmetry axis, the polarity of the diffracted wave is always opposite to that of the incident wave. The amplitude of this impulse increases linearly with increasing z (Fig. 5), and, simultaneously, the impulse narrows in time. Upon an increase in the distance from the screen, its area remains almost constant and tends rapidly to –1, with the whole impulse area being concentrated in a small vicinity of time moment t = z c . Therefore, this component of the scattered wave tends to the δ-wave at z → ∞ and, in combination with the passed wave forms the derivative of the δ-function on the symmetry axis. The problem of propagation of monochromatic radiation the transverse distribution of amplitudes of
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1
0 1
0
1
(а)
2
ρ
0
2
4
6
1
(b)
2
(c)
4
6
ρ
0
8
10
12
14 ρ
10
12
14
(d)
2
4
6
8
ρ
Fig. 6. Comparison of calculation results of diffraction by a Gaussian diaphragm obtained by different methods: (solid curve) δ -wave approximation, (dotted curve) quasi-optical approximation, and (dashed curve) space transfer function approximation. Parameters of calculations of vibronic interactions: c = 1, k = ω c = π 2 , (а) z = 0 , (b) z = 1, (c) z = 5 , and (d) z = 10 .
which is described by the Gauss function, i.e., of the Gaussian beam of the zeroth order, has been studied in detail; therefore, we compared the convolution of relation (11) with monochromatic wave with solutions obtained by the other two approaches. The first of them is based on the transfer function of free space [25]. This method yields an exact solution of the edge problem, but it is not the solution to the diffraction problem. Nevertheless, this method makes it possible to find exact relations for the entire space behind the screen if the distribution of amplitudes on some plane is specified. In comparison with the impulse calculation, calculations by this method require considerably greater computer burden because of the necessity to perform double integration. The second method is based on the Fresnel diffraction theory, which is a result of the application of the paraxial approximation in the Kirchhoff theory and which yields well-known relations for the zero-order Gaussian beam. At large values of the wave vector k = 2π λ , i.e., at λ ! a , all the three methods yield almost the same results at a rather long distance from the plane of the screen. Immediately on the plane, of the screen at z = 0, the exact solution and the field calculated in the Fresnel approximation coincide, whereas the calculation by the impulse method yields to distorted values of the field amplitude and phase, with this pattern being observed at any values of k . As the distance from the plane of the screen increases, these distortions decrease. If the distance to the plane of the screen is not too small, the absolute value of the amplitude of the field calculated by the impulse method at small k (Fig. 6) is much closer to the exact solution than the field calculated in the Fresnel approximation. Concerning the phase, the Fresnel approximation yields a satisfactory result only at rather large values of k in the paraxial region, whereas the phase of the field calculated by the impulse method almost coincides with the phase of the exact solution at all conditions, except for the case
in which the point of observation approaches the immediate vicinity of the screen. Therefore, at small values of the wavenumber, the impulse method yields a more exact description of the field at points of the space that are at not too small distances from the plane of the screen (≥ λ ), whereas, at large values of the wavenumber, the accuracy of the solution is, at least, not worse than that yielded by the Fresnel approximation. Near the plane of the screen, the result of the Fresnel solution is better compared to that of the impulse method because it is the solution of the approximate equation that satisfies to the exact boundary condition, whereas the impulse solution, being the result of the application of the Kirchhoff approximation, is the solution of the exact equation that satisfies approximate boundary conditions. In particular, upon using the Kirchhoff approximation, it was arbitrarily assumed that the normal derivative of the field behind the screen is equal to the product of the derivative of the field of the incident wave by the transmission coefficient of the screen. In reality, the value of the normal derivative of the field, which is an exact solution of the edge problem, differs from that calculated in this way. DIFFRACTION OF A δ-WAVE BY A SEMI-INFINITE SCREEN AND A SLIT Let us consider a problem that has an exact theoretical solution, namely, the diffraction of a plane monochromatic wave by a half-plane [5] and show results of numerical simulation of the diffraction process of the plane wave performed by different methods. A plane wave δ (t − z c ) is incident on a half-plane, interacts with the rectangular edge of the screen, and excites a cylindrical edge wave [26], the outer radius r of which increases at a rate с. In the illuminated region of space, a semi-infinite part of the plane δ-wave cut by the screen is observed, which continues its propagation in the initial direction, and a cylindrical edge wave. In the region of the geometrical shadow, only the edge
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V
V (а)
(b)
t
t Fig. 7. Impulse response of a screen in the shape of a half-plane: (а) in the region of the geometrical shadow and (b) in the illuminated region.
wave is observed. As well as in the case of a circular hole, the sign of the edge wave in the illuminated region is opposite to that of the incident wave, while, in the shadow region, its sign coincides with the sign of the initial wave. Let the coordinates of the point of observation be ( ζ, z 0 ), where ζ is its distance from the light/shadow boundary,
V ill (P, t ) = cδ(z 0 − ct ) + V d (P, t ) = cδ(z 0 − ct )θ ( ζ ) ζ + c θ ( ct − r ) 2π (z 0 − ct ) (ct ) 2 − r 2 in the illuminated region,
(12)
V shad (P, t ) = V d (P, t ) ζ = c θ ( ct − r ) 2π (z 0 − ct ) (ct ) 2 − r 2 in the shadow region, where r 2 = z 02 + ζ 2 . The shape of the impulse response is shown in Fig. 7. Using asymptotic relation for impulse response (12) at long distances in the range of small diffraction angles and applying the Fourier transformation, we obtain the following expression for the amplitude of the field of the monochromatic wave:
⎡ ⎛ ⎛ ⎞ ⎛ ⎞⎞ ⎤ Aν ( ζ, z 0 ) ≈ 1 ⎢(1 + i ) ⎜C ⎜ ζ 2ν ⎟ − iS ⎜ ζ 2ν ⎟⎟ + 1⎥ 2⎣ ⎝ ⎝ cz 0 ⎠ ⎝ cz 0 ⎠⎠ ⎦ 2 ⎡ ⎛ ζ ⎞⎤ × exp ⎢i 2πν ⎜ τ' + ⎟ . 2cz 0 ⎠⎦⎥ ⎣ ⎝ (13) Here, C and S are Fresnel integrals [5]. It may be easily verified that relation (13) is equivalent to formula (8.7.2) in monograph [5]. Figure 8 shows the plot of the squared modulus of the obtained function, which is identical to that well known from the literature. There is a qualitative difference between the impulse response of the rectangular edge of the halfplane and the circular hole: the perturbation at a certain point of observation caused by the edge wave difOPTICS AND SPECTROSCOPY
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fers from zero not in the limited time interval, but, rather, at all t beginning from the moment of arrival of light from the nearest point of the screen. The amplitude of the perturbation monotonically falls to zero at t → ∞ , since the contribution to the field amplitude at the observation point is made by all points of the screen edge, including arbitrarily distant ones, with a weight that decreases with increasing distance, which was noted even in [6]. As the distance of the point of observation from the screen increases, the field falls faster and faster with time, with the area of the impulse scattered in a certain direction being independent of the distance between the point of observation and edge of the screen. It can be easily verified that, at a moment of time that is apart from the arrival time r c of the impulse leading edge by an arbitrarily small fixed time interval Δ t , the field tends to zero with increasing r. Therefore, at long distances from the edge of the screen, the field manifests the properties of the δ-function.
This particular feature of the diffraction field of the δ-wave on the rectangular edge is consistent well with 1.4 1.2 1.0 0.8 0.6 0.4 0.2 −π
π
2π
3π
Fig. 8. Squared modulus of the Fourier transformation of impulse response (12), which yields the light intensity distribution in space behind a semi-infinite screen for a monochromatic wave. Calculation was performed for z0 = 10λ, c = 1, and ω = 1.
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1
2
3
4
5
6
7
8
9
10
Fig. 9. Development of the field of an ultrashort Gaussian pulse upon its diffraction by a conducting slit.
the result of our first work on diffraction by slit. Upon consideration of the impulse response of the slit with a width of 2a in the Fraunhofer region in the shadow [6], i.e., at z 0 @ a and z 0 @ ζ > a , it was shown that this response consists of a sum of two δ-functions,
⎡ ⎛ z − aζ / z 0 + ζ 2 /(2z 0 ) ⎞ A(t, ζ, z 0 ) = 1 ⎢δ ⎜ t − 0 ⎟ 2πζ ⎣ ⎝ c ⎠ (14) ⎛ z 0 + aζ / z 0 + ζ 2 /(2z 0 ) ⎞⎤ − δ ⎜t − ⎟⎥ . c ⎝ ⎠⎦ If we take into account that expression z 0 ± aζ / z 0 + ζ 2 /(2z 0 ) is (with an accuracy of up to small terms of higher orders) the distance from the point of observation to slit edges, we can see that these two δ-functions are two cylindrical δ-waves scattered by the slit edges. The sign of the part of the wave emitted toward the shadow coincides with the sign of the initial perturbation, while the sign of the part emitted toward the illuminated region is opposite. As well as upon diffraction of the δ-wave by the circular hole, the signal at ζ = 0 is described by the derivative
A(t, 0, z 0 ) =
a ∂ δ(t − z /c) 0 πz 0 c ∂t
(this can be easily shown by direct calculation of the Kirchhoff integral). Qualitatively, the correctness of the developed model of diffraction of a scalar δ-wave by a slit was obtained by numerical simulation of scattering of an ultrashort pulse by a “long” rectangular conducting diaphragm with a smaller side 2a and a larger side L @ 2a , which was placed inside of a long rectangular or cylindrical waveguide with a characteristic transverse dimension R @ L . As is known, the δ-function may be defined as a limit 2 2 δ( x) = lim[α exp(− α x )/ π]; therefore, the initial α→∞
pulse was considered to be plane of a Gaussian shape in time with parameter τ ! 2a/ c ; the amplitude distribution in the direction parallel to the plane of the slit was also assumed to be Gaussian, and the value of the parameter ensured an almost constant value of the pulse amplitude over the whole width of the aperture. At the initial moment of time, the pulse was near the
diaphragm plane. The configuration of the field as a function of coordinates of space and time was calculated. Upon calculation by the first method, the amplitude distribution was expanded into a series in terms of eigenfunctions of the waveguide and frequencies; then, obtained components were summated taking into account changes in phases of all components. Results of these calculations, which were performed by A.G. Antipov, for several moments of the development of the field (upon motion of the pulse from the right to the left) are shown in Fig. 9. The second calculation was performed by Sh. Bartch, and it was based on direct solving of the wave equation in the threedimensional space by the Euler method, the physical model of which is close to Huygens’s notions of propagation of waves [27]. We omit the corresponding figure, because obtained results are almost identical to those shown in Fig. 9. It can be easily seen that the application of classical methods of solving the mathematical problem of field formation to an ultrashort pulse, which models the δ-wave, yields the pattern that reflects particular features of the diffraction field described above, namely, the occurrence of a plane wave passed through the slit and of two cylindrical edge waves. A similar result was also obtained in [28] in terms of the theory of the edge diffraction wave. The diffraction of a pulse with a Gaussian transverse distribution of amplitudes upon its oblique incidence on a long slit was investigated. For the case in which the plane front of the incident pulse is parallel to slit edges, the obtained solution has the shape of a superposition of the passed “geometrical” wave and two waves that diverge from the slit edges. Let us consider in detail particular features of the field, which Fig. 9 shows. Frame 1 shows (by the light tone) the initial position of the positive pulse. Then the pulse enters the slit (frame 2) and excites an edge wave. The passed wave does not change its configuration as the distance from the diaphragm increases with time (frames 3, 4, 5,…). This element of the passed wave is adjoined by cylindrical edge waves, which are clearly seen in frames 3, 4, and so on. Their sign changes at the light/shadow boundary: it is opposite to the sign of the initial perturbation (negative) in the region illuminated by the incident wave and coincides
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with the initial sign in the shadow region. Frames 7– 10 reveal another detail of the diffraction, which is not taken into account by the Kirchhoff theory, namely, formation of secondary scattering waves of the edge wave.5 If the amplitude rather than the intensity is registered, the observation of a pulse scattered by the slit edges at a certain finite angle to the symmetry plane will yield a signal that consists of positive and negative pulses of the same shape, which can be overlapped if the delay time is shorter than the pulse duration. The delay time of one pulse relative to the other depends on the slit width and the angle of observation. The amplitudes of pulses decrease with increasing angle, as is seen from formula (13) and Fig. 9. At distances comparable with the slit width, the amplitudes of pulses will be different, and, with increasing distance, this difference levels off. Similar evolution of the shape of the signal was also shown in [29] upon mathematical simulation of the diffraction process of a pulse by a slit using methods of the theory of the transfer function of space. The authors of that work termed the effect of formation of two pulses the “splitting” of the initial signal. They also observed a violet shift of the frequency of the maximum of the spectrum of the pulse on the symmetry axis of the problem, which corresponds to the multiplication of the spectrum by frequency upon differentiation of the signal. Beginning from the fifth frame of Fig. 9, the geometry of the field is complicated: negative components of cylindrical edge waves intersect, forming two structures of an Х-shaped cross section, which propagate further in opposite directions. The region behind the screen (to the left of the diaphragm), in which both the passed and the diffracted waves coexist, is of main interest for the purposes of this work. The node of the Х-shaped intersection moves at a rate that is faster than c , gradually overtaking the passed wave. The sum of positive and negative Gaussians following one after another forms a derivative of the Gauss function at long distances from the slit. The formation of the first derivative indicates that, at long distances from the slit, the amplitude of the pulse at the node point becomes equal to the amplitude of the passed wave. This fact coincides with the well-known particular feature of the edge wave: as the light/shadow boundary is approached, its amplitude tends to a half of the amplitude of the incident wave. In the literature available to us, there are a good many theoretical publications on the diffraction of an ultimately short pulse, which contains a small number of field oscillations, and only sporadic experimental works in which the time dependence of observation conditions of the amplitude of such a pulse was investigated. Here, we dwell on one of them [30], with 5 Formation
of a negative wave in front of the diaphragm (to the right of it in frames) is determined by the zero boundary conditions taken in this problem. OPTICS AND SPECTROSCOPY
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detailed results of which we had the opportunity to become familiar. The authors of that work studied the diffraction of a terahertz pulse with a plane shape of its front by a long slit with a width of a few millimeters. The amplitude was registered in the far-field diffraction zone as a function of time at different displacements with respect to the symmetry plane of the slit. An analysis of the experimental data shows that, immediately on the symmetry axis, the diffracted pulse is described by the derivative of the pulse that illuminates the slit, whereas pulses that are scattered by an angle to the axis are satisfactorily described by a sum of pulses of the initial wave that are delayed in accordance with (14) and that are taken with different signs. CONCLUSIONS Theoretical and experimental data presented in this work showed that there are no contradictions between the Young and Huygens–Fresnel diffraction models, as well as there are no contradictions between the methods of investigation of linear systems using monochromatic and pulsed signals. The application of the notion of the impulse response to optical systems proves to be useful in studies of nonstationary and transient processes. A pictorial separation of the field into components that correspond to the passed and edge waves, which was proposed by Young, did not take into account peculiarities that appeared in our calculations, such as a change in sign of the perturbation at the light/shadow boundary, which, in principle, might be established even in those times with rather simple experiments. The development of the approach based on the notions of propagation of the δ-wave showed that the accuracy of calculations based on this method is as good as or even better than, in some cases, the accuracy of calculations based on traditional analytical methods of solution of diffraction problems of monochromatic waves. The high accuracy is achieved both in the Fraunhofer zone and in the Fresnel zone up to distances from the obstacle close to characteristic dimensions of the aperture. The experience of application of the impulse method showed its high informativity in analysis of the operation of optical systems. In conclusion, we note that Feynman’s words in the epigraph to the present article prove to be incorrect when one deals with a process that is localized in space and time. Interference of waves almost always accompanies the diffraction effect; however, for linear optical systems, it is always prove to be secondary. Only in the case of stationary wave processes does become the separation of diffraction and interference effects conventional.
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REFERENCES 1. Christiaan Huygens, TRAITÉ DE LA LUMIÈRE (ONTI NKTP, Moscow–Leningrad, 1935) [Russian translation]. 2. T. A. Young, Course of Lectures on Natural Philosophy and the Mechanical Arts (London, 1807), Vol. 1. 3. A. Fresnel, Ann. Chim. Phys. 1 (2), 239 (1816). 4. A. Rubinowicz, in Progress in Optics IV (North-Holland, Amsterdam, 1965), p. 199. 5. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1969; Nauka, Moscow 1973). 6. I. E. Suleimenov, M. K. Lebedev, and Yu. A. Tolmachev, Opt. Spektrosk. 88 (1), 104 (2000). 7. M. K. Lebedev and Yu. A. Tolmachev, Opt. Spectrosc. 90 (3), 398 (2001). 8. A. P. Kiselev, Opt. Spectrosc.102 (4), 603 (2007). 9. Localized Waves (Wiley, 2008). 10. P. Piksarv, P. Bowlan, M. Lõhmus, H. Valtna-Lukner, R. Trebino, and P. Saari, J. Opt. 14, 015701 (2012). 11. http://www.youtube.com/watch?v=4joXjiJcNWQ. 12. M. K. Lebedev, Yu. A. Tolmachev, A. G. Antipov, and M. V. Razmanova, Vestn. SPb. Univ. Ser. 4 (28), 44 (2002). 13. N. Kaliteevskii, Wave Optics (Vyssh. Shkola, Moscow, 1978) [in Russian]. 14. E. I. Butikov, Optics: A Textbook for Higher Education Institutes (Vyssh. Shkola, Moscow, 1986) [in Russian]. 15. M. K. Lebedev and Yu. A. Tolmachev, in Laser Investigations in the St. Petersburg State University (NII Ross. Tsentr Lazern. Fiz., St. Petersburg, 2004), Issue No. 3, p. 81. 16. Yu. A. Tolmachev, Opt. Zh. 72 (1), 3 (2005). 17. P. Hillion, Opt. Spectrosc. 94 (2), 268 (2003). 18. A. Sommerfeld, Vorlesungen über Theoretische Physik, Band IV, Optik (Wiesbaden, 1950).
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