Guiding of Waves Between Absorbing Walls

Journal of Modern Physics, 2012, 3, ***-*** Published Online July 2012 (http://www.SciRP.org/journal/jmp)

Guiding of Waves between Absorbing Walls

Dmitrii Kouznetsov, Makoto Morinaga Institute for Laser Science, University of Electro-Communications, Tokyo, Japan Email: {dima, morinaga}@ils.uec.ac.jp

Received ********* 2012

ABSTRACT Guiding of waves between parallel absorbing walls is considered. The principal mode is constructed; its absorption is estimated. The agreement with previous results about reflection of waves from absorbing walls is discussed. Roughly, the effective absorption of the principal mode is proportional the minus third power of the distance between walls, minus 1.5d power of the wavenumber and minus 0.5d power of the local absorption of the wave in the wall. This estimate is suggested as hint for the design of the atomic waveguides, and also as tool for optimization of attenuation of the amplified spontaneous emission (and suppression of parasitic oscillations) in high power lasers.

Keywords: Zeno Effect; Atom Optics; Waveguides; Suppression of Amplified Spontaneous Emission

1. Introduction The frequency of decay is denotes with  . The special system of units is used in such a way that  =1 and the The consideration of reflection of waves from absorbing energy of the particle in vacuum is assumed to be square walls had been stimulated by the experiments with ridged of its wavenumber. mirrors [1] and their interpretation in terms of the Zeno In Section 3, the special case of uniform absorption is effect [2,3]. The Zeno approximation [2] showed good considered; this gives relations between parameters of agreement with experiments in wide range of parameters the wave function and the physical quantities that can be [3,4]; it describes reflection of waves of any origin. In determined experimentally. Such physical quantities are particular, it applies to the atomic waves and to the energy  of the particle and its absorpfion s in the optical waves. In addition to the atom optics (discussed in [2]), the material of the wall; this quantities are assumed to be reflection and guiding of waves may affect the suppres- independent parameters. sion of the Amplified Spontaneous Emission (ASE) that In Section 4, the case of channeling is considered; is considered as serious problem [5-9]. At the scaling-up parameters of the principal mode (transversal wave- the power, the efficient suppression of ASE becomes number p , decay q and the damping  ) are defined more important, and the unwanted guiding of ASE by the and expressed through the holomorphic function 1 absorber (which is supposed to suppress it) might take cosc z = cos z z, its inverse function acosc = cosc place. The estimates of the conditions of such a guiding and acosq  zz = acosc exp iπ 4  ; properties of these is necessary tool for the design of powerful devices. functions are discussed and the efficient C++ imple- In this article, the wave function of a particle (atom, mentations are indicated. photon) between two absorbing walls is constructed. The In Section 5, the example of the principal mode is effective absorption of such a mode is estimated and considered; the real and imaginary parts of the wave compared to the previous results. The results are expect- functions are built through the trigonometric function of ed to have applications in both the atom optics (wanted complex argument and complex exponential for the guiding of neutral particles by the absorbing walls) and damping  =1 4 . The amplitude and phase of the the design of powerful lasers (estimates of conditions of principal mode are shown for  =1 4 and for the unwanted guiding of the ASE by the walls that are  =1 50. supposer to absorb the ASE, and optimization of suppres- In Section 6, the asymptotic behavior of parameters at sion of the ASE). small damping is considered; this is realistic case of good This article is organized in the following way: guiding conditions. In Section 2, the phenomenological absorbing Schrö- In Section 7, the estimates for the effective absorption denger equation is suggested. The case with absorbing of the principal mode are compared with the previous walls corresponds to the pure anti Hermitian potential. results; the agreement is interpreted as confirmation of

Copyright © 2012 SciRes. JMP 2 D. KOUZNETSOV, M. MORINAGA the validity of the analysis.  , consider the case of the uniform absorption of the In Section 8, the special case is suggested for phy- plane wave at U by (4); let  does not depend on the sically realistic parameters of the experimental condi- first argument, id est,   x,=zfz. Then, for tions of realization of the guiding of waves between f = fx  we get the equation absorbing walls. f i= f0 (6) In Section 9 (Conclusion), the asymptotic estimate of the effective absorption A of the guided mode through At positive values of  , the decaying in the direction the real and imaginary parts c , s of the wavenumber z solution has form in the wall is suggested. This main result is expected to f =expikz (7) be confirmed (or rejected) by the physical experiments with photons, atoms or any other wave of any origin. where kkk :  0 , 0 is solution of equation 2. Schrödinger Wave and the Absorption k 2 =i   (8) In this case the dimension less Schrödinger equation is Assuming   0 ,   0 , wavenumber k can be considered in the paraxial approximation. expressed as follows: For simplicity, in this section the special system of units is used such that  =1 and mass of the atom is k =i   (9) half. Then, the Schrödinger equation for the wave func- For the compactness of notations, let kcs=i , tion =,,x zt can be written as follows where c and s are real parameters. Then 2222 it =    x     zUx   (1) cs22 =  (10) where the potential UUx  depends only on the 2=cs  (11) transversal coordinate x (and does not depend on time Alternatively, we may consider the absorption of wave z in the direction of propagation). in the medium, id est,  k as initial parameter. Mak- In the simplest approximation, the entangling with   ing estimates for the ridged atomic mirror with distance numerous degrees of freedom of scattered (or relaxed) T between ridges, the absorption by intensity can be atom can be taken into account with non Hermitian approximated as 1 T , and the absorption by amplitude is potential. Such an approximation is considered, in of order of 12T particular, in the interpretation of the quantum reflection   in therms of the Zeno effect [2]. For the wave guide, the s ==12kT (12) absorption correspond to complex values of Ux at   Real part of k is determined by the mass and the x > X , where X is half-width of the channel between energy of the particle we intent to reflect or to guide, or the mirrors. just 2πn  , where  is vacuum wavelength of the Consider the quasi-monochromatic solution, assuming waves (perhaps, ASE) that could be guided, and n is the exponential dependence on t ; let the refraction index of the medium. x,,zt =exp i t xz ,  (2) In order to avoid additional guiding by the step of the Then, instead of (1) we get the stationary Schrödinger refraction index, the real part of the refraction index of equation the wall is supposed to be matched to that of the central region; but some reflection still may happen due to the = 2222 x    zUx   (3)  imaginary part. Then, the real and imaginary parts of the below, the two cases of the solution are considered: for swiare o wavenumber k can be expressed as follows: Ux=i=c  onst (4)  =kkkc22 = 22  = s2 (13) and for  ==kkk2 2)=  2sc (14) Ux=iUnitStep  x d (5)   Parameter  has sense of the energy of the particle, where UnitStep is the conventional unit step function and also sense of the square of wavenumber, while  implemented in various programming languages include- has sense of the decay rate at the time scale, if a wave ing Mathematica, and d  is constant, that has sense would be uniform in the space. For the estimates, the of half-width of the channel that confines the particle. term with k  in (13) can be neglected; however it is 3. Uniform Absorption kept in the deduction that may be applied not only to atoms, but also to other kinds of waves (optical, acous- In order to understand the physical sense to the constant tical, waves on the surface of a liquid, etc.) For optics

(both atom optics and conventional optics), the typical the “observation” (id est, x  d ), the better is the case is of low absorption; waves rather propagate than channeling. absorb, to, Substitution of (19) to (3) gives the equation for f  fx  in the following form: s k =1 (15) 2 ck fx i  UnitStep d f = 0 (20) In this case, for the estimate of the primary parameters, Search the solution of (20) as the combination of the we may use the approximation cosinusoidal and the exponential, let

2 2  kc  (16) cos px , x d fx=  (21) rqxdxdexp  ,  and treat ck=  and s = k as initial parameters   of the model describing the absorption of the wave inside the walls. where r  ,   , p  , q  are constant para- On the other hand, one may consider as “given” the meters. From the physical reasons (almost free propa- energy  of the particle and the absorption k  . gation inside the channel), parameter  is expected to Then, be of order of wavenumber k from the previous section. ck==  k2  (17) The substitution of (21) into (20) determines that 22 p= = k2 (22)  =2kk 2  (18)  22q =i (23) In the following consideration, parameters    and    and k  are supposed to be known. From the physical reasons, it is expected that q is These parameters determine behavior of the wave inside small compared to k =  and, therefore, q   . the absorbing wall. The continuity of f  x at x = d and the conti- For the atom optics, id est, for the atom wave, the nuity of f  give the relations absorption s = k should be positive. For the optical cos pd = r (24) wave, in principle, the absorption may be negative (gain medium), but the amplified spontaneous emission (un-  ppdqsin    r (25) avoidable in the gain medium) limits the application of The four Equations (22), (23), (24), (25) allows to ex- the formalism to very short distance of propagation. For press new parameters kpqr,,, in terms of the already this reason, the channeling in the pumped region may defined parameters  and  . have more applications. In such a way, this section gives Subtraction of (23) from (22) gives the sense to the parameters  and  that appear in the Equations (3) and (5) , that describe the channeling of pq22 =i  (26) a particle by the potential U by (5). This channeling is considered in the next section. Then, the effective pro- Dividing of (25) by (24) gives pagation constant   is estimated in terms of k 22 2 and half-width d of the channel. ppdtan = q (27) The combination of (26) and (27) gives 4. Channeling 1tan pd2 p2 =i  (28) For the case of potential U by (5), search the solution    in the following form: Using the relation 1 tan 2 = 1 cos  2, Equation  xz,= f  x ei z (19) (28) can be written as pd 2  2 where   is constant. 2 =i  d (29) For the experimental realization, B  is expect- cospd ed to be of order of ck, determined in the then previous section. As for the imaginary part, A   2 is expected to be small compared to s k  , and cospd i = (30) B  is expected to decrease as s k  in- pd 2 d 2 creases, allowing the interpretation in terms of the Zeno effect [2]: the stronger is the absorption in the region of This equation can be written as follows:

1 2 i where acosc = cosc is inverse function of cosc. Dam- coscpd = (31)  d 2 ping   is dimensionless parameter determining the efficiency of channeling, where for all complex z  0 , 1 cosz Damping = = coscz = (32) 1/2 z  d (36) Complex map of function cosc is shown in the left 11 ==1/2 hand side of figure 0; properties of this function and its 2kkd  2dsc inverse function acosc = cosc1 are described in TORI [10]. The name of function cosc is chosen in analogy with Properties of function acosc are described at TORI; the well established name of function sinc [11], defined with efficient implementation in C++ is suggested [10]. The sin z complex map of acosc is show in the right hand side of sincz = (33) z Figure 1 with levels of constant real part and levels of constant imaginary part. Behavior of real and imaginary As usually, the name of the inverse function is created adding prefix “a” or “arc”. parts of function acosq of real argument is shown in fthe Equation for one of solutions with acosc pd  > 0 left hand side of Figure 2. can be written as follows: The decay rate of mode in the region with absorption is determined with parameter q . Once p is deter- i coscpd = (34) mined, from Equation (27),  d 2 qd= pd tan pd = acosc ei/4  tan acosc ei/4  Equation (34) can be “inverted”, giving (37) iexp(iπ 4) = acosq tan acosq  = acosqq pd = acosc = acosc dd21/2   (35) Function acosqq is shown in the right hand part of =acoscexpiπ 4=acosq Figure 2.

Figure 1. Complex map of function f coscxiy by (32) , left, and that of f acosc xiy by [10], right, in the x , y plane. Levels uf  const and levels vf  const are shown.

Figure 2. y =acosq   and y =acosq  by (35), left; y =acosqq  and y =acosqq   by (35), right.

5. Assembling of Mode, Example =  p2 (45) The damping  by (36) determines the properties of the mode with given distance 2d between walls and The real and imaginary parts of  determine the given real and imaginary parts of the wavenumber k effective wavenumber and absorption of the guided mode that determines the propagation of wave in the material “exactly” in the mathematical sense. As for the physical of the wall. With tools defined above, the principal mode applications, the case of strong guiding (and low absorp- of wave guided between absorbing walls is expressed tion) is of interest. This case corresponds to the small with Equation (21). Parameters p , q and r are de- values of the damping parameter  1 , and the asym- fined with Equations (35), (37), (24). ptotic behavior of the absorption of the guided mode is As an example of the assembling of such a mode, the considered in this section. case  =1 4 is presented in Figure 3; the real and For the strong guiding, the propagation constant  imaginary parts of the components of function f are can be expanded as follows: plotted versus dimensionless product xd . In this case, 22 2 pp p 1  (46) pd 1.30652013112871 0.20108562381528i (38) 2 2  qd 2.63604614403057 2.93518290714528i (39) then, the absorption A of the mode can be expressed as r 0.26650956316919 0.19541505906974i (40) follows: The amplitude and phase of the mode f for  =1 4 App=      (47) and  =1 50 are shown in Figure 4. for  =1 50, the In order to provide the flux of probability from the parameters are center of mode to the absorbing walls, the imaginary part pd 1.54859380700524 0.02159861746934i (41) of the transversal wavenumber should be negative, p <0. The expansion of funciton acocq at zero qd 35.33791574606151 35.37182445894684i (42)   gives: r 0.02220587422227 0.02159497306720i (43) ππ acosqzz = eiπ/4Oz 2 (48) With functions ArcCosq and ArcCosqq implemented 22 in TORI through [10], one can easy assemble the prin- This gives the approximation for the transversal wave- cipal mode for other values of the damping parameter  number p in the following form: with minimal modification of the codes supplied there. ππ pd  eiπ/4  (49) 6. Application to Atomic Waves and the 22 Asymptotic and the estimate for the effective absorption The effective absorption of a guided mode is one of most ππ 1 important parameters of any waveguide. This section A    (50) consider the case of low damping and, correspondently, 2d d 2  strong channeling. 1 Then,   by (36) and 22s sc should According to (19) the effective absorption is deter- d  mined by parameter  , 1 Effective Absorption =vA = =   (44) be used, giving   . Then, the effective absorp-  dsc2 From Equation (22), tion of the mode

Figure 3. Combination of mode (21) from the cosinusoidal and the exponents for α = 1/4.

ridged mirror,  = pK plays role of the grazing angle. Substitution of (52) and (55) into (54) gives the following expression for the absorption  2 aKL 8 (56) 2d The grazing angle can be approximated with π   (57) 2dK giving the estimate for the efficient absorption Figure 4. Amplitude and phase of the mode f by (21) for α = π22L π 1/4 and for α = 1/50. aKL8 32 3 3 (58) 42dK K d π211 π2For the comparison to the previous result, L should A  = (51) 8 ddsc2 2 c 4 dc33/21/2 s be replaced to 12 s and K should be replaced to c , giving the absorption by probability where c has sense of wavenumber, and s is the π2 absorption in the wall. a  (59) In the similar way, the highest modes can be cons- 2 dc33/21/2 s tructed. For the mth transversal mode, the transversal However, the amplitude decays half slower than wavenumber scales proportionally to n ; and the absorp- probability, so, A  a 2 . In such a way, the asymptotic 2 tion of mode scales proportionally to m . estimates (51) and (59) agree. In the first approximation, the consideration of the 7. Comparison to Previous Results multiple reflection from absorbing walls and the con- The absorption of mode can be interpreted also in terms sideration of mode guided between the absorbing walls of the multiple reflection of guided wave from the walls. give the same prediction about effective absorption of this mode. The consideration of the multiple reflection The coefficient of reflection rzeno is estimated in the description of the ridged mirrors in terms of the Zeno from absorbing walls and the consideration of mode effect [2], guided between the absorbing walls give the same prediction about effective absorption of this mode. 1/ 4  1 1 2 rr==zeno  exp8  (52) 8. Numerical Example 1/ 4  1 1 2 Consider the application of the estimate (59) for the where guiding of the realistic laser beam. Assume, the absorp-  = KL (53) tion in the walls 0.5 K is wavenumber and can be replaced to c ; while s = 0.143 cm1 (60) L is distance between idealized absorbers that can be 35 cm approximated as 12s , and  is the grazing angle.  Following the ideology of the Zeno interpretation of (Notation p of [2] is not used here, to keep letter p absorbing walls [2], such an absorption may be appro- denoting the transversal wavenumber; so, in (52) and ximated with series of slits separated by distance 35 mm. (53) , notation  is used instead.) Let the wavenumber is For the good channeling conditions, the effective 2π absorption can be approximated with c =5.9 μ1 (61) 1.064 μ 1 r a = zeno (54) 2d  Let the halfwidth of the channel d =500μ (62) where d is half-width of the channel and  = pK is ratio of the transversal wavenumber This gives the estimate for the absorption of guided π modes, p = (55) 2d am 0.73 2m1 (63) to the wavenumber K . At the reflection of wave from a For the principal mode ( m =1), after to propagate

Copyright © 2012 SciRes. JMP D. KOUZNETSOV, M. MORINAGA 7 distance Z =35 cm, the attenuation factor is of order of suppression of the amplified spontaneous emission (ASE) in the high power lasers. The guiding of modes by the AF = expaZ exp   0.73 0.35   78% (64) absorption walls happens whenever the engineers want that means, that the most of the initial power of the this effect or not. Similar estimate is valid for the highest guided mode is still delivered. As for the second mode, modes. For the mth transversal mode, the asymptotic its attenuation estimate is suggested for the absorption AF = exp4aZ exp  1.02 36% (65) π22m A = 33/21/2 (66) that means significant dicrimination of the second mode. 4 dc s Using the approximation of the set of absorbers as a through the half-width d of the channel, wavenumber continuous medium [2], the example above may corres- c and absorption s of wave in the walls. pond to transfer of near infra-red light through the set of Similar estimate (with slightly higher absorption) corre- 10 slits separated with distance 35 mm. Roughly, the spond to the case with circular symmetry, that can be amplitude of field after the set of slits can be approxi- treated in the similar way; the mode is expressed with the mates with the cosinusoidal profile. However, the pre- Bessel function, parameter d plays role of the radius of sence of the highest modes, as well as the diffraction of the channel. At small value of damping  by (36), the the tails of the mode on the edges should make the transversal wavenumber p is almost real. similarity qualitative. Similar result one may expect to The result should be useful in both, wanted guiding of observe at propagation of light through the set of cold neutral particles by their detection (absorption) and pinholes of radius d . The similarity with the idealized the efficient suppression of the unwanted guiding of cosinusoidal or Besseliean mode should improve at the waves, for example, ASE in powerful optical amplifiers, increase of number of slits or pinholes; a hundred of silts and optimization of the ASE absorbers. or pinholes may be sufficient to get the quantitative agreement with the idealized cosinusoidal or Besselian 10. Acknowledgements profile. Authors are grateful to Dr. Hilmar Oberst and Prof. Fujio The accurate consideration of the discrete character of Shimizu and Prof. Kazuko Shimizu for the collaboration. the absorbing walls, as well as construction of the mode for the case with circular symmetry may be continuation of this work. For the paraxial case, the estimates are REFERENCES universal and are not sensitive to the origin of waves. In [1] F. Shimizu and J. Fujita, “Giant Quantum Reflection of particular, the results are expected to apply to the Neon Atoms from a Ridged Silicon Surface,” Journal of electromagnetic waves as well as to the cold atoms, the Physical Society of Japan, Vol. 71, No. 1, 2002, pp. 5-8. doi:10.1143/JPSJ.71.5 exhibiting the wave properties. [2] D. Kouznetsov and H. Obrest, “Reflection of Waves from 9. Conclusions a Ridged Surface and the Zeno Effect,” Optical Review, Vol. 12, No. 5, 2005, pp. 363-366. 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