Talbot effect in cylindrical waveguides

Optics Communications 268 (2006) 215–225 www.elsevier.com/locate/optcom

Talbot effect in cylindrical waveguides L. Praxmeyer

a,b,*

, K. Wo´dkiewicz

a,c

a

Institute of Theoretical Physics, Warsaw University, ul. Hoza 69, 00–681 Warsaw, Poland Theoretical Physics Division, Sofia University, James Bourchier 5 blvd, 1164 Sofia, Bulgaria Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA b

c

Received 10 April 2006; received in revised form 5 July 2006; accepted 6 July 2006

Abstract We extend the theory of Talbot revivals for planar or rectangular geometry to the case of cylindrical waveguides. We derive a list of conditions that are necessary to obtain revivals in cylindrical geometry. A phase space approach based on the Wigner and the Kirkwood– Rihaczek functions provides a pictorial representation of interference phenomena that lead to the Talbot effect. Ó 2006 Elsevier B.V. All rights reserved. PACS: 42.30.Va; 42.25.Hz Keywords: Talbot effect; Self-imaging; Waveguides; Interference

1. Introduction Despite of the fact that the Talbot effect was discovered in the 19th century (1836) and then rediscovered many times in different systems and concepts, it has never become a widely known phenomenon in the physics community. In problems involving time evolution of quantum systems this effect was often called ‘‘self-imaging’’ or ‘‘quantum revivals’’. Even in the optical domain a reference to Talbot name is not mentioned very often when this phenomenon is described and discussed. In fact it was W.H.F. Talbot, who was the first to observe that monochromatic light passing through a periodic grating forms its ideal image at a certain distance from the grating. In addition, consecutively at integer multiples of this distance, similar images are reproduced. Because of this, we believe that all ‘‘selfimaging’’ effects in wave optics and atomic optics should be referred as Talbot effects. * Corresponding author. Address: Institute of Theoretical Physics, Warsaw University, ul. Hoza 69, 00–681 Warsaw, Poland. Tel.: +48 22 55 32 313; fax: +48 22 621 94 75. E-mail address: [email protected] (L. Praxmeyer).

0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.07.016

After the original paper by Talbot [1], followed by the work of Lord Rayleigh [2], and then by a series of papers of Wolfke [3], many papers have been written on this subject. A comprehensive description of the Talbot effect and its rediscoveries in classical optics can be found in Patorski [4]. Similar historical review and a detailed description of ‘‘quantum revivals’’ can be found in [5]. In the last years of the 20th century fractional aspects of the Talbot effect attracted considerable interest [6–11] and they were studied both in the optical and the quantum mechanical domains. As regards Talbot effect in optical waveguides most significant are works of Ulrich [12], who investigated waveguides of planar and ribbon geometries. The case of planar waveguides was experimentally studied also in Ref. [13], where interesting possible application in high precision interferometry have been suggested. Similar suggestions for applications of the Talbot effect for precise determining of irrational distances were made in [14]. Even if the literature on the Talbot effect in planar or rectangular waveguide geometries is quite rich (including some US patents for applications of the Talbot effect in those systems), the much more practical cylindrical geometry was not taken into account in optical studies. In this

216

L. Praxmeyer, K. Wo´dkiewicz / Optics Communications 268 (2006) 215–225

paper, a comprehensive theoretical study of the Talbot effect in cylindrical waveguides is presented. This should fill the gap in the available descriptions of self-imagining phenomena in various waveguides. There is a limited number of papers mentioning cylindrical geometry in quantum revivals [15], but the results presented in our contribution go beyond the ones obtained so far. Quite recently an interesting experiment on the Talbot effect in arrays of waveguides was reported [16], which certainly adds new interest to this problem. The paper is organized as follows: general assumptions from which our study starts are presented in Section 2. Then, a formal similarity between field propagation in waveguides and the dynamics of wave packets in potential wells in a given geometry is emphasized. The phase space Wigner and Kirkwood–Rihaczek functions are applied to give an intuitive description of interference effects, that are the basis of the Talbot revivals. In Section 3, the case of dielectric fibers is studied and the possibility of revivals in this most practical for possible application system is described. In Section 4, solutions of the wave equation in cylindrical mirror waveguides are analyzed and conditions essential to obtain revivals are derived and imposed. In the same section also a detailed study of approximations used is included. Finally, a summary of our results is given and a short paragraph devoted to a presentation of numerical method applied is presented. 2. Talbot effect in phase space 2.1. General assumptions The reason why the Talbot effect appears in optical and quantum mechanical systems, mathematically can be summarized very briefly: The Helmholtz equation is common for classical electrodynamics and quantum mechanics. The form of the dependence of the electromagnetic field in the direction of the field propagation can be regarded as an analogue of the quantum mechanical time dependence of a wave packet. In the regime where paraxial approximation is justified this analogy is especially clear. This paper is focused on Talbot revivals in different types of cylindrical waveguides. It is a purely optical phenomenon but, as we shall see in the next paragraphs, its most simple explanation can be given referring to quantum mechanical concept of phase space distributions. The key question we shall pose and answer is whether the Talbot effect in cylindrical waveguides exists and whether it can be use in practice. We assume that harmonic, monochromatic plane waves propagate through a waveguide which symmetry axis was chosen as the z direction of the system. Inserting the fields Eðx; y; z; tÞ ¼ Eðx; yÞeikzixt ;

Bðx; y; z; tÞ ¼ Bðx; yÞeikzixt ; ð1Þ

into the Maxwell equations, we obtain the two-dimensional Helmholtz equation:

   2  E ¼ 0; r ? þ c2 B

where

c2 ¼ le

x2  k2; c2

o2 : oz2 The propagating constants ki, that correspond to the specific modes are to be derived from appropriate boundary conditions (see, e.g., [17]). r2? ¼ r2 

2.2. Phase space distributions As we have already mentioned the phase space description emphasizes the basis of Talbot effect, i.e., constructive and distractive interference. Typically, the quasi-distribution functions used for the study of electromagnetic fields, especially pulses, are in the time and frequency domain. For the description of the Talbot effect we shall use as a phase space the position and the wave vector (position and momentum representation) of the electric field. Some complications result from the fact that the electromagnetic field requires a vector description. However, in many cases (e.g., mirror waveguides) solutions of the Maxwell equations are divided into TM and TE modes, i.e., are entirely determined by the Ez or the Bz field components, respectively. Then, one can treat the Ez (Bz) component, that fully describes the field, as a single scalar function. This fact can be used as a formal correspondence of the scalar field to the quantum mechanical wave function in a potential well problem. To present problems of revivals in the phase space we shall use the following two quasi-distribution functions: the Wigner function [18,19], Z 1 ~ W W ð~ WH ð~ x;~ pÞ ¼ x þ~ n=2Þei~pn Wð~ x ~ n=2Þd d n; ð2Þ d ð2pÞ which is the most commonly known phase space distribution, and the Kirkwood–Rihaczek (K–R) function [20,21]: Z 1 ~ d d nWH ð~ Kð~ x;~ pÞ ¼ nÞeiðn~xÞ~p Wð~ xÞ: ð3Þ d ð2pÞ that is very convenient for calculations. In the definitions above the subscript d denotes the number of spatial dimensions of the system, whereas W is a function that characterizes the system (e.g., a wave function in quantum mechanical applications, or the electric field component Ez for TM modes in waveguides, etc.) For three-dimensional systems the phase space distributions are six-dimensional, thus graphically only selected cross-sections can be presented. The choice of suitable graphical cross-sections is made easier for the solutions given by Eq. (1), where the z-dependence of the field separates from the transversal components. Fig. 1 shows examples of the ‘‘transversal’’ Wigner and K–R functions calculated for Ez = J0(j1r)  J0(j2r), which corresponds to the superposition of TM01 and TM02 modes in cylindrical mirror waveguide: j1, j2 denote theqfirst ffiffiffiffiffiffiffiffiffi and the second zero of Bessel function J0(q), r ¼

x2 þy 2 , a2

and a is a waveguide radius.

L. Praxmeyer, K. Wo´dkiewicz / Optics Communications 268 (2006) 215–225

0.25 1 0.75 0.5

0

1

0.1

0.2

0.05

0.1

0

0.75 0.5 0.25

0

0 20 6 4 10 8

10 3 2 5 4

Fig. 1. The y = 0 and py = 0 cross-sections of ‘‘transversal’’ quasidistributions corresponding to Ez = J0(j1r)  J0(j2r): (a) cross-section of the Wigner function; (b) cross-section of the real part of the K–R function.

As mentioned above in the case of plane wave solutions of Maxwell equations we consider here, the z-dependence of the field separates from its transversal components. Thus, when describing the Talbot effect, we can simply ignore the transversal dependence of the field and concentrate on the z–pz cross-section of the phase space distribution which is essential for the Talbot revivals. Integrals corresponding to the exp(ikiz) factors are quite elementary and the cross-sections of the quasi-distribution functions for given x, y, px, py, i.e., with fixed transversal position and momentum components, are easy to obtain. The Wigner function for a superposition of two plane waves exp(ik1z) + exp(ik2z) is given by W 2 ðz; pz Þ  dðpz  k 1 Þ þ dðpz  k 2 Þ   k1 þ k2 þ 2d pz  cosðzðk 2  k 1 ÞÞ: 2

When there are N superposed waves, the Wigner function becomes:  X X  ki þ kj W N ðz; pz Þ  dðpz  k i Þ þ 2 d pz  2 i¼1 1¼i
Re½K N ðz; pz Þ 

N X

ð6Þ

j¼1

dðpz  k j Þ 1 þ

N X

of ci’s is a sufficient condition. That is why we shall often limit ourselves to modes with the lowest propagating constants. A definite advantage of working within the range where the linear approximation of square root holds for systems having ci’s proportional to each other is the fact that for a given wavelength the Talbot distance is settled, it does not depend on superposed modes. Otherwise, different modes superpositions shall revive at different distances. All the features characteristic for the Talbot effect can be easily explained by looking at the behavior of the interference of cos [z(ki  kj)] terms, as all depends on whether for a given range of z the oscillating terms interfere in a constructive or destructive way. The phase space description brings us directly this simple idea and indicates how fundamental this concept is. 3. Talbot effect in dielectric waveguides

Re½K 2 ðz; pz Þ  dðpz  k 1 Þ þ dðpz  k 2 Þ þ ½dðpz  k 1 Þ þ dðpz  k 2 Þ cos½zðk 2  k 1 Þ: ð5Þ

and, the real part of the K–R function is given by:

Obviously, all superpositions of just two different modes, like in Eqs. (4), (5), will revive perfectly at multiples , no matter what kind of waveguide we consider. of k22p k 1 But, when there is more superposed modes, commensurability of all possible (ki  kj) factors is needed to obtain perfect revivals. As the propagating constants ki vividly depend on the type of waveguide and its parameters, such a commensurability is rather an exceptional then a typical case. Even dealing with such highly symmetric problems as mirror waveguides with planar or square cross-sections we qffiffiffiffiffiffiffiffiffiffiffiffiffiffi have to keep in mind that k i ¼ k 20  c2i , which means that commensurability of ci’s is not automatically followed by commensurability of propagating constants ki. Only when qffiffiffiffiffiffiffiffiffiffiffiffiffiffi linear approximation of k 20  c2i holds, commensurability

ð4Þ

The real part of the K–R distribution for such a superposition takes the form

 cos½zðk i  k j Þ;

217

!

cos½zðk i  k j Þ ;

i¼1;i6¼j

ð7Þ with appropriate coefficients. In both cases the whole z dependence is inserted into interference cos [z(ki  kj)] terms. Initially, at z = 0, all this cosines are equal to 1. The further z dependence is guided by (ki  kj) factors. When all these (ki  kj) factors are commensurable with each other, perfect regular revivals are obtained at such zrev for which all cos [zrev(ki  kj)] are simultaneously equal to 1 again.

Cylindrical dielectric waveguides are called optical fibers. We shall analyze here only the step–index fibers, i.e., the fibers with constant refractive indexes in the core and the cladding. Such fibers are entirely characterized by cores’ and cladding radii and reflective indexes n1 ¼ pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi l1 e1 , n2 ¼ l2 e2 . If we assume that the cladding and the core differ only by dielectric constants (magnetic permeabilities l1 = l2), the standard boundary conditions lead to the following equation [22,23]:  2  0   1 J m ðCÞ 1 K 0m ðjÞ n21 J 0m ðCÞ n22 K 0m ðjÞ m2 x2 n2 n2 þ þ ¼ 2 2 12 þ 22 ; C J m ðCÞ j K m ðjÞ C J m ðCÞ j K m ðjÞ k c C j ð8Þ where a denotes the core radius, C = ac, j = ab, c2 ¼ 2 2 l1 e1 xc2  k 2 and b2 ¼ k 2  l2 e2 xc2 . In this equation Jm and Km are Bessel and Macdonald functions, respectively. Although Eq. (8) has a quite regular form, there is no way to solve it analytically. In the simplest case when the field has no azimuthal dependence, i.e., m = 0, the solutions can be divided into TE and TM types. Then a clear presentation of the solutions can be obtained graphically in full analogy to the finite cylindrical potential well problem from quantum mechanics.

L. Praxmeyer, K. Wo´dkiewicz / Optics Communications 268 (2006) 215–225

218

However, we have to keep in mind that the u  independence, corresponding to m = 0, is not a typical case. General solutions of Eq. (8) are u dependent and, actually, the lowest propagating mode in step-index dielectric fibers are obtained for m = 1.

faithful revivals are obtained. The situation of appropriate cosines are getting in and out of phase repeats almost cyclicly. Plots of initial light intensity corresponding to superposition of TM01  TE01  HE11  2HE21 modes, its first Talbot revival and a characteristic inverted image at the half of the Talbot distance are presented in Fig. 2. A standard measure

3.1. General solutions in step index fibers The lowest propagating mode in a cylindrical dielectric waveguide is always an HE11 mode (in this notation HE means that field Hz dominates over Ez field, while for EH modes the Ez field dominates). Single mode fibers are of a great practical importance, but in this study we are not interested in them because a single mode propagates without a change in its transverse distribution. Next modes are TE01, TM01, and HE21. They appear almost simultaneously as their propagating constants are nearly the same. Let us firstly consider a fiber in which there are only these four propagating modes (e.g., k = 1550 nm, n1 = 1.46, n2 = 1.45, a = 4.5 lm). We shall refer to this situation as to the ‘‘limit of small number of modes’’. Numerical solutions of Eq. (8) obtained for these parameters are: acHE11 ¼ 1:79268, acTE01 ¼ 2:75973, acTM01 ¼ 2:76234, acHE21 ¼ 2:76342. While analyzing the problem of revivals of superpositions of HE11, TE01, TM01, and HE21 modes, one has to realize that although four modes are superposed, it is a special case when three propagating constants (corresponding to TE01, TM01, and HE21 modes) are really close to each other. As we have already mentioned, superpositions of arbitrary two modes with propagating constants k1, k2 shall revive at all integer multiples of zt ¼ k12p . It is easy to calculate that initial k 2

kIð0Þ  IðzÞk kIð0Þk !1 R1 R 2p 2 2 du 0 dqqjIðq; u; 0Þ  Iðq; u; zÞj 0 ¼ ; R1 R 2p 2 du 0 dqqjIðq; u; 0Þj 0

f0 ðzÞ ¼

ð9Þ

can be used to compare fidelities of subsequent revivals. We shall call this measure infidelity because its value increases with increasing deviation of I(z) from the original I(0) and only when f0 = 0 the copy is perfect. For superposition of HE11  TE01  TM01  2HE21 modes the infidelities of successive revivals are quite low (f0 = 0.0000276, f0 = 0.0029825, f0 = 0.0260536, f0 = 0.0945819 correspond to the 1st, 10th, 30th, and 60th revival, respectively) and in such a four-mode fiber the infidelities of revivals for any initial field distribution would be of this order. It is seen that in the limit of small number of modes Talbot revivals in dielectric fibers can be obtained. Changing the fiber parameters will result in the change of Talbot distance. It is quite obvious that for a small number of propagating modes Talbot revivals can be obtained. Just as we have demonstrated using phase space representations – appropriate numbers of cosines have to get in phase. This simple and intuitive method of looking for the Talbot distance starts to be more complicated when the number of propagating modes increases. Then, different methods have to be applied to calculate revival distance as we shall see in the next section on the example of TE0i modes.

images constructed from a superposition of pairs (HE11, TyE01), or (HE11, TM01) or (HE11, HE21) shall revive at multiples of zt1 ¼ 3:40787 mm, zt2 ¼ 3:3967 mm, zt3 ¼ 3:39214 mm, respectively. Those values are so close to each other that images constructed from all four modes shall revive at a mean value which is zt ¼ 3:39891 mm. In numerical simulations of propagation the first, second and even 20th or 50th Talbot revival can be observed. Obviously, higher revivals are less accurate, and after several tenths of faithful revivals they finally get out of phase. Then, after some distance of totally dephased propagation they get in phase again, and again some

3.2. The u-independent case (m = 0) The u-independent solutions of Eq. (8) can be divided into TE and TM modes. For TE modes the following relation is obtained: f 0 =0.0000276

1.5

1.5

1

1 2

0.5 0 –2

a

1

1 2

–2

b

0 –2

0 –1

–1

0 1 2

–2

2

0.5

1

0 –2

–1

0

1 2

0.5

0 –1

1.5

c

1 0 –1

–1

0 1 2

–2

Fig. 2. Superposition of TM01  TE01  HE11  2HE21 modes: (a) the original intensity; (b) intensity at the half of the Talbot distance; (c) 1st Talbot revival.

L. Praxmeyer, K. Wo´dkiewicz / Optics Communications 268 (2006) 215–225

1 J 1 ðCÞ 1 K 1 ðjÞ þ ¼ 0: C J 0 ðCÞ j K 0 ðjÞ

ð10Þ

Because the parameters C and j are (by definition) correlated, C2 þ j2 ¼

a2 x2 2 ðn1  n22 Þ :¼ V 2 ; c2

ð11Þ

we obtain a closed set of equations that can be solved graphically or numerically. It is convenient to introduce a normalized frequency parameter V, Eq. (11), and depict and j1 KK 10 ðjÞ as a function of C. for example  C1 JJ 10 ðCÞ ðCÞ ðjÞ For TM modes instead of Eq. (10) we would have 1 J 1 ðCÞ e1 1 K 1 ðjÞ þ ¼ 0: C J 0 ðCÞ e2 j K 0 ðjÞ

ð12Þ

Graphical solutions of Eq. (12) are similar to those of Eq. (10). The only difference is that the Macdonald part of the K 1 ðjÞ plot jK is modified by a fixed factor ee12 , which is usually 0 ðjÞ close to 1. In practice, this means that TE0n and TM0n modes have nearly the same propagation constants and they will tend to appear simultaneously. In the previous subsection, we were dealing with the small number of modes, so now let us focus on a limit of large number of propagating modes (large frequency parameter V). In the limit V ! 1, solutions acn are given by zeros of the Bessel J1(q) function, and propagating constants correspond exactly to those obtained for TE0n modes in the mirror waveguides. This case is widely discussed in the next Section, thus, now we shall take into consideration only the finite values of V. Analyzing the graphical representation of Eq. (10) for different values of frequency parameter V one finds that with the increase of V the MacK 1 ðjÞ donald part of the plot, jK , starts to be parallel to C axis 0 ðjÞ for the increasing range of Cn’s. Moreover, it is also getting K 1 ðjÞ ’ V1 . For large closer to this axis as for C ! 0 value of jK 0 ðjÞ

219

V and low mode numbers solutions would be of very regular form: the first one corresponding to TE01 mode will be given by, say C0, and the approximate formula for the next solutions would be Cn = C0 + anp. Obviously the larger V, the more accurate this formula is, and it works well only for modes having numbers low in comparison to the total number of modes. Sometimes, however, it is easier to omit such analytical approximations and to simply calculate numerically infidelities of intensity distribution as a function of z and look for the minima of this function. 3.2.1. Examples of revivals In this Section, we consider only the u-independent fields that are fully characterized by their cross-section along the radius. Thus, the figures presented show crosssections of the light intensity versus the normalized distance from a fiber center q/a. All of the examples were calculated numerically for a quite thick fiber (a = 1 mm) with refractive indexes of the core and cladding equal to n1 = 1.47, n2 = 1.45, respectively, and a wavelength of k = 850 nm. For these values of k, n1, and n2 the frequency parameter V equals 1786.35, which corresponds to more then 550 propagating TE modes, and C0 = 3.82956. 3.2.1.1. Symmetric superposition of TE01 + TE02 + TE03 + TE04 modes. Fig. 3 presents the light intensity of superposition TE01 + TE02 + TE03 + TE04 at z = 0, z = zt = 27.7036 m, z = 10zt, z = 30zt, z = 60zt, and z = 100zt. The original intensity is plotted in black, the revivals in blue/ gray. Above the plots of revivals their infidelities are depicted. The value of the Talbot distance zt was determined numerically by finding minimum of the infidelity function, Eq. (9). It is seen that although revivals are not perfect they are certainly faithful enough even at distances of 1 km. For comparison, Fig. 4 presents examples of the light intensities at distances between revivals: it is clear that

Fig. 3. Light intensity for the symmetric superposition of TE0n for n 2 {1, . . ., 4}. The original intensity distribution are plotted in black, the intensity at the Talbot distance zt = 27.7038 m and its multiples in blue/gray. (a) The original intensity; (b) 1st Talbot revival; (c) 10th Talbot revival; (d) 30th Talbot revival; (e) 60th Talbot revival; (f) 100th Talbot revival. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

220

L. Praxmeyer, K. Wo´dkiewicz / Optics Communications 268 (2006) 215–225

the intensities at multiples of the Talbot distance differ significantly from typical intensity distribution during propagation.

uting modes effects not only the fidelities of revivals, but also the optimal Talbot distance, which we shall clearly see comparing this and the next example.

3.2.1.2. Gaussian ‘‘25’’. Fig. 5 presents revivals of an initial intensity with a Gaussian radial distribution. The Gaussian function was obtained from a superposition of the first 25 TE0. modes. The numerically calculated Talbot distance is zt = 27.7035 m and the infidelities of revivals (depicted above every plot) are quite low. This example shows that revivals of intensities of a given shape can be observed. Obviously, this result holds true only on the assumption that other modes (e.g., modes depending on u) do not contribute to the initial image. Besides, it should be noted that the fidelities of revivals depend significantly on the effective number of terms of the Bessel–Fourier (BF) series contributing to the initial image. Although in the example presented here we have taken first 25 terms of the BF series, only first 9 coefficients were larger then 1/1000 and only first 4 were larger then 1/100. The coefficients from 11th to 25th were of the order of 5/10000. If we propagate, say, a more slim Gaussian these proportions would be different. This effective number of contrib-

3.2.1.3. Symmetric superposition of TE01 + TE02 +    + TE025 modes. This example of revivals in a thick fiber illustrates how the effective number of contributing modes might modify a Talbot distance. As the initial intensity we take the one corresponding to the symmetric superposition of TE01 + TE02 +    + TE025 modes. Fig. 6 presents the initial intensity, its 1st Talbot revival and then 10th, 30th, 60th, and 100th Talbot revival. The numerically calculated ‘‘optimal’’ Talbot distance is equal in this case zt = 27.7013 m, which is slightly smaller then in the case presented in examples 3.2.1.1 and 3.2.1.2. We have already stressed the fact that different initial images have different Talbot distances can be explained by the effective number of modes that are superposed. It is, again, a consequence of importance of the quality of approximations used. In our simplified analysis we have assumed that cn ’ c0 + np. On one hand this is true only K 1 ðjÞ for n small enough for the jK plot to be parallel to 0 ðjÞ the ac axis, on the other hand the distance between the

Fig. 4. Examples of intensity distribution between revivals calculated for the same modes superposition as in Fig. 3. The original intensity is plotted in black, in orange/light gray are plotted its ‘‘counterparts’’ at (a) z = 1 m, (b) z = 5 m, (c) z = 10 m. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Light intensity for the superposition of TE0n, n 2 {1, . . ., 25}, providing a Gaussian distribution. The original intensity distribution are plotted in black, the intensity at the Talbot distance zt = 27.7035 and its multiples in blue/gray. (a) Original intensity; (b) 1st Talbot revival; (c) 10th Talbot revival; (d) 30th Talbot revival; (e) 60th Talbot revival; (f) 100th Talbot revival. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

L. Praxmeyer, K. Wo´dkiewicz / Optics Communications 268 (2006) 215–225

221

Fig. 6. Light intensity for the symmetric superposition of TE0n modes for n 2 {1, . . ., 25}. The original intensity distribution are plotted in black, the intensity at the Talbot distance zt = 27.7013 m and its multiples in blue/gray. (a) The original intensity; (b) 1st Talbot revival; (c) 10th Talbot revival; (d) 30th Talbot revival; (e) 60th Talbot revival; (f) 100th Talbot revival. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

neighboring zeros of Bessel functions is p only for higher modes. Thus, it is obvious that when only first few modes contribute to the initial image (or their contribution dominates) the Talbot distance is slightly different then in the case of larger number of contributing modes. Differences in optimal Talbot distance zt for superpositions presented in examples (3.2.1.1–3.2.1.3) can be found by calculating the infidelities of revivals. They start to be important at 20th, or 50th revival when an initial 2 mm difference between optimal Talbot distances results in 5 or 10 cm divergence from the distance of optimal revival. As long as only the first Talbot revival is considered an average zt can be taken and the infidelities of revivals of different initial images at zt should not be larger then 1/100 which is sufficient for most applications. When revivals at larger distances are to be of interest, all initial images can be divided in classes having the same ‘‘effective number of contributing modes’’ and the revivals at multiples of the corresponding ‘‘optimal Talbot distances’’ would be obtained for all the images within the class. Numerical simulations presented indicate that the Talbot revivals of images constructed from TE0n modes can be obtained in optical fibers. Examples (a–c) were limited to superpositions obtained from TE modes, but for superpositions of TM0n modes revivals should be even more faithful, as ee12 < 1 and thus Cn are even closer to zeros of J1(q) function then it is for TE modes. This observation brings us to the case of mirror waveguides. 4. Talbot effect in mirror waveguides In this section, we present a model of ideal mirror waveguides. This model is very useful to illustrate the basic properties of the Talbot effect. It is analytically soluble for systems of standard geometries because boundary conditions are quite simple: normal component of B and tangential component of E have to vanish at the boundary mirror surface.

4.1. Planar mirror waveguides In the elementary case of a planar mirror waveguide a propagation constant for nth mode is given by k 2n ¼ 2 2 k 20  ndp2 , where d denotes a separation distance between mirrors plates [24,22]. It is clear that, in general, the field changes its transverse distribution as it travels through the waveguide because different modes travel with different propagation constants and different group velocities. The following expansion of the propagation constant kn, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! n2 p2 n2 p2 1 n2 p2 2 kn ¼ k0  2 ¼ k0 1  2 2 ’ k0 1  ; ð13Þ 2 d 2 k 20 d d k0 2

shows, however, that within this approximation for z ¼ 4kp0 d the initial field is obtained. Obviously, requirements for above linear approximation are not met for an arbitrary kn. Higher modes have to be prevented from contributing to initial images, because only then approximation of the square root with accuracy to the linear term is sufficient. The reason why revivals appear is that ci factors are all of the form constant (characterizing the system) times integer. The question arises whether in cylindrical mirror waveguides similar analytical formula for the Talbot distance can be obtained. Let us stress that the necessity of taking care of the paraxial approximation is the main difference between optical and quantum mechanical Talbot revivals. In the case of an infinite potential well in quantum mechanics, eigenvalues 2 2 of the system are given simply by ndp2 and states evolve like 2 2 expði ndp2 tÞ. Because n are integers, the revivals at multiples of trev = 2d2/p are obtained for all possible wave functions superpositions. In planar waveguides, however, the ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n2 p2 z-dependance of the field is governed by expði k 0  d 2 zÞ factor and revivals are obtained only for these modes for which the linearization of the square root given by Eq. (13), is a sufficient approximation.

L. Praxmeyer, K. Wo´dkiewicz / Optics Communications 268 (2006) 215–225

222

4.2. Cylindrical mirror waveguides Solutions of the wave equation in cylindrical mirror waveguides are classified accordingly to their angular dependance e±imu, where m is an integer corresponding to a Bessel function Jm(q) characterizing the radial solution. qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

Propagating constants are k 20  jamn2 for ffi of the form of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 TM modes and k 20  jamn2 for TE modes, where jmn denotes

the nth root of Bessel function Jm(q) and j0mn the nth root of its derivative. In this case not only a linear approximation of square root is needed but also approximation for zeros of Bessel function jmn or jmn. Standard asymptotical expansions for jmn, j0mn are given by jmn ’ np þ ðm  12Þ p2 and j0mn ’ np þ ðm  32Þ p2. They are believed to be good enough for n > m (or in more rigorous manner for n > 2m). Using these formulas we can repeat procedure from Eq. (13) and obtain: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 2 2 j j 1 j mn k 20  mn2 ¼ k 0 1  2mn2  k 0 1  2 k 20 a2 a k0a 2

 k0  Thus, qffiffiffiffiffiffiffiffiffi e

iz

j2

k 20  mn 2 a

 eik0 z e

p2 ð4n þ 2m  1Þ : 32a2 k 0

i2p

k 128a2



ð4nþ2m1Þ2 z

ð14Þ

;

and

  mðm  1Þ 2 ð4n þ 2m  1Þ ¼ 8 2n2 þ 2nm  n þ þ 1: 2 2pk Omitting a common phase factor expðik 0 z þ i 128a 2 zÞ we find that for a given wavelength k and the waveguide’s radius a at distance zt = 16a2/k and at its integer multiples Talbot revivals will be obtained. Similar calculation made for TM modes using approximation for j0mn leads to the same Talbot distance zt = 16a2/k.

4.2.1. Examples of revivals in cylindrical mirror waveguides To illustrate Talbot revivals in cylindrical mirror waveguides we have chosen an initial intensity function, Iorg(0) of the form:  sin 2pq for q 2 ½0; 0:5; u 2 ½0; 2p I org ð0Þ ¼ 1 ð15Þ sin 2pq for q 2 ½0:5; 1; u 2 ½0; 2p: 2 Figs. 7 and 8 show cross-sections of this initial light intensity Iorg(0) along ‘‘normalized’’ radius and its Talbot revivals obtained for ak = 103 and ak = 104, respectively. Analytical function from Eq. (15) and its approximation by Bessel functions corresponding to TM0i modes are plotted together in Figs. 7a and 8a. A numerical procedure allowing to approximate a given intensity function employed here is described in Section 6 at the end of the paper. Quality of this approximation, f0a , is calculated using measure from Eq. (9). To indicate that the infidelity of approximation is measured, we shall write f0a , while the infidelities of revivals are denoted without this ‘‘a’’ superscript. Figs. 7b and 8b present the first Talbot revival of the initial field, Figs. 7c, 8c; 7d, 8d; 7e and 8e show 2nd, 5th, and 10th Talbot revivals, respectively. Above every plot the corresponding infidelities are depicted. It is very instructive to compare plots and infidelities obtained for different values of ak. It is clearly seen that infidelities are much lower for smaller k to a ratio. This effect is connected with the quality of linear approximation of the square root from Eq. (14). The smaller percentage of all modes is used to construct initial images the lower infidelities of revivals are to be expected. 4.2.2. Some comments on used approximations Revivals studied in previous paragraph were quite faithful, which means that approximations used to predict their existence were justified. However, please note, that all the examples presented so far had the following property: superposed modes were of the same angular dependence (they corresponded to the Bessel functions of fixed

Fig. 7. I(q) = sin(2pq) for q 2 [0, 0.5] and IðqÞ ¼ 12 sinð2pqÞ for q 2 [0.5, 1]. The intensity of light is plotted: (a) shows initial I(q) (an origin and an approximation given by the first 50 terms of BF series, plotted together); (b) shows the first Talbot revival at z = zt and the initial intensity shown together; (c–e) show 2nd, 5th, and 10th Talbot revival, respectively.

L. Praxmeyer, K. Wo´dkiewicz / Optics Communications 268 (2006) 215–225

223

Fig. 8. I(q) = sin(2pq) for q 2 [0, 0.5] and IðqÞ ¼ 12 sinð2pqÞ for q 2 [0.5, 1]. The intensity of light is plotted: (a) shows initial I(q) (an origin and an approximation given by the first 50 terms of BF series, plotted together); (b) shows the first Talbot revival at z = zt and the initial intensity shown together; (c–e) show 2nd, 5th, and 10th Talbot revival, respectively.

number m). Studying more complicated combinations one finds out that superpositions of modes with different azimuthal mode numbers do not revive at the Talbot distance zt = 16a2/k [25]. Numerical simulations of field propagation draw attention to a really interesting fact: superposition of TM or TE modes having the same angular dependance revive quite faithfully at the Talbot distance and only for the superpositions of ‘‘mismatched’’ modes revivals are not obtained. Explanation of this unexpected phenomenon comes from the higher terms of the asymptotic formulae for the roots of Bessel functions, and its derivatives [26]: 



2



ð16Þ

and   3 p 4m2 þ 3



j0mn ¼ np þ m   2 2 8 np þ m  32 p2 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} B0

112m4 þ 328m2  9 

3  . . . ; 384 np þ m  32 p2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð18Þ

B02

C

A0

2

ð4m2 þ 3Þ

2 : 64 np þ m  32 p2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

02

B

ð4m2  1Þð28m2  31Þ

3  . . . ; 384 np þ m  12 p2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

2A B

A02

þ

1 p 4m  1



jmn ¼ np þ m   2 2 8 np þ m  12 p2 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} A

distance for arbitrary m and n fails, because different powers of p would be included. So how we can observe any revivals at all? Let us take a closer look at approximation (17), i.e., the one considering TE modes – keeping in mind that similar analysis can be made for TM modes as well. Taking two first terms of this approximation the following expression for j02 mn is obtained:    2 3 p 4m2 þ 3 j02 np þ m   mn ¼ 2 2 4 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflffl{zfflfflffl} 0 0

ð17Þ

C0

which are definitely more important then we have assumed so far. In many papers and textbooks only first terms (A, A 0 ) of above approximations are used. We have also limited ourselves to them in preliminary calculations but to obtain a faithful approximation at least two first terms should be taken into account. From formulas (16), (17) it is, however, clear that when higher terms are taken into consideration, finding the Talbot

02

Ratios B /A and 2A 0 B 0 /A 0 2 for a wide range of parameters m and n are shown in Fig. 9. It is seen that although the ratio B 0 2/A 0 2 for n > m is close to zero and can be neglected, the percentage value of 2B 0 /A 0 can be quite significant. Obviously, this is the effect we were looking for as the terms 2A0 B0 ¼ m2 þ 34 depend only on the azimuthal mode number m and not on the radial one n. That is the reason why modes having the same angular depen2 dence revive at the Talbot distance zt ¼ 16a , while for a k superposition of modes with different angular dependence we do not obtain such revivals. In the first case factor exp(i(4m2 + 3)zt/4) is merely a phase common for all the terms of the superposition. While, when the initial Hz is the superposition of a form H z ð0Þ ¼ J m ðj0mn qÞeimu þ J l ðj0lm qÞeilu , at z = zt phase factors corresponding to Jm and Jl differ for m5l and, consequently, they do not cancel. This relative phase destroys the Talbot revivals promised by less accurate approximation. Similar analysis shows that superpositions of TM and TE modes will not revive at the same distance because of the relative phase. Only for TM1i and TE0j modes this relative phase disappears and Talbot effect can be observed. However, due to possibility of using polarizers, restrictions

224

L. Praxmeyer, K. Wo´dkiewicz / Optics Communications 268 (2006) 215–225

Fig. 9. Errors of ‘‘asymptotic’’ approximation of ðj0mn Þ2 : notation is taken from Eqs. (17), (18). Plots (a) and (b) are shown in two different scales percentage ratio of 2B 0 /A 0 for m 2 {0, . . ., 12}, n 2 {1, . . ., 25}. Unicolor dotted lines correspond to one value of azimuthal mode number m. Plots (c) and (d) show present percentage ratio of B 0 2/A 0 2. It is seen that for n > m this term can be neglected.

imposed by necessity of choosing polarization of modes is experimentally less demanding then that concerning u-dependance.

For an arbitrary spherically symmetric TM field with an initial Ez of a form 50 X xi J 0 ðj0i qÞ; ð19Þ Ez ð0Þ ¼

5. Summary We have discussed in details different approximations that appear in the study of the Talbot effect in the cylindrical mirror waveguides dielectric waveguides. We have shown that in many cases almost perfect Talbot revivals can be obtained. It’s also interesting that a dephased propagation can be used as a method of coding. This is related to the fact that even if one knows the field distribution obtained for arbitrary z, without knowing the exact distance from the starting point of the waveguide, it is impossible to guess what image is being send. We have stressed that a phase space description (sometimes regarded as an unnecessary complicated representation) extracts an essence of the constructive and destructive interference phenomena. The phase space approach provides in a natural way the conditions needed to be fulfilled to obtain Talbot revivals. The main aim of this paper was to study the possibility of applying the Talbot effect in optical waveguides to transmit information codded in images. It is, however, worth noting that potential applications of this phenomenon include the possibility of experimental determination of rational and irrational Talbot distances [14] and can be used in high precision interferometry [13].

6. Methods The following numerical procedure was used for decomposing given initial light intensity Iorg into TM0i modes:

i¼1

we have evaluated the light intensity I(0) and this I(0) was expanded again in a Bessel–Fourier (BF) series corresponding to J0. Numerical evaluation of some integrals was required at this point. In such a way general ‘‘basis’’ (e1, e2, . . ., e50) was obtained (every ei being a sum of all possible pairs xi Æ xj with numerically calculated coefficients). Then, an arbitrary u-independent intensity Iorg that we propagate through the waveguide was decomposed in BF series I org ð0Þ 

50 X

ci J 0 ðj0i qÞ;

ð20Þ

i¼1

and a set of quadratic equations c1 = e1, c2 = e2, . . ., c50 = e50 was solved numerically to find ðxi Þ50 i¼1 . Figs. 7a and 8a can be treated as a test of faithfulness of the solutions found in the procedure described above – the infidelities of approximation f0a are of the order of 102. Acknowledgements We acknowledge useful discussions with Professor W.P. Schleich. This research was partially supported by Polish MEN Grant No. 1 P03B 137 30 and European Union’s Transfer of Knowledge project CAMEL (Grant No. MTKD-CT-2004-014427). References [1] W.H. Fox Talbot, Philos. Mag. 9 (IV) (1836) 401. [2] Lord Rayleigh, Philos. Mag. 11 (1881) 196.

L. Praxmeyer, K. Wo´dkiewicz / Optics Communications 268 (2006) 215–225 [3] M. Wolfke, Ann. Phys. (Germany) 40 (1913) 194, and references therein. [4] K. Patorski, The Self-Imaging Phenomenon and Its Applications, in: E. Wolf (Ed.), Progress in Optics, vol. 27, North Holland, Amsterdam, 1989, p. 3. [5] R.W. Robinett, Phys. Rep. 392 (2004) 1. [6] M.V. Berry, S. Klein, J. Mod. Opt. 43 (1996) 2139. [7] K. Banaszek, K. Wo´dkiewicz, W. Schleich, Opt. Express 2 (1998) 169. [8] M.V. Berry, J. Phys. A 29 (1996) 6617. [9] I. Marzoli et al., Acta Physica Slov. 48 (3) (1998) 323. [10] M. Berry, I. Marzoli, W. Schleich, Phys. World 14 (6) (2001) 39. [11] M.T. Flores-Arias et al., J. Opt. Soc. Am. A 17 (2000) 1007. [12] R. Ulrich, Opt. Commun. 13 (3) (1975) 259. [13] Y.B. Ovchinnikov, T. Pfau, Phys. Rev. Lett. 87 (2001) 123901. [14] H.C. Rosu, J.P. Trevino, H. Cabrera, J.S. Murguia, Int. J. Mod. Phys. B 20 (11–13) (2006) 1860. [15] B. Rohwedder, Phys. Rev. A 63 (2001) 053604. [16] R. Iwanow et al., Phys. Rev. Lett. 95 (2005) 053902.

225

[17] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975. [18] E. Wigner, Phys. Rev. 40 (1932) 749. [19] W.P. Schleich, Quantum Optics in Phase Space, Wiley-vch, New York, 2001. [20] J.G. Kirkwood, Phys. Rev. 44 (1933) 31; A.N. Rihaczek, IEEE Trans. Inf. Theor. 14 (1968) 369. [21] L. Praxmeyer, K. Wo´dkiewicz, Opt. Commun. 223 (2003) 349. [22] Peter K. Cheo, Fiber Optics and Optoelectronics, Prentice-Hall International Editions, Englewood Cliffs, NJ, 1990. [23] C.D. Cantrell, Dawn M. Hollenbeck, Fiberoptic mode functions: a tutorial. Available from: . [24] B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, Wiley, New York, 1991. [25] L. Praxmeyer, Classical and quantum interference in phase space, PhD thesis, 2005. [26] F.W.J. Olver, Introduction to Asymptotics and Special Functions, Academic Press, London, 1974.