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Study of Frozen Waves’ theory through a continuous superposition of Bessel beams
ARTICLE IN PRESS
Optics & Laser Technology 39 (2007) 1370–1373 www.elsevier.com/locate/optlastec
Study of Frozen Waves’ theory through a continuous superposition of Bessel beams C.A. Dartoraa, K.Z. Nobregab,, Alexandre Dartorac, Gustavo A. Vianad, Horacio Tertuliano S. Filhoa a Electrical Engineering Department - UFPR, Brazil Electrical Engineering Department, CEFET-MA, Brazil c CEFET-PR, Pato Branco, Brazil d Institute of Physics Gleb Wataghin, State University of Campinas, Brazil b
Received 21 November 2005; received in revised form 7 October 2006; accepted 20 October 2006 Available online 18 December 2006
Abstract This paper presents a study of a recent solution to Maxwell’s equation, the so-called ‘‘Frozen Waves’’, whose main characteristics are to remain static in space, and to keep an arbitrary longitudinal field pattern previously chosen. These waves could be obtained by an adequate, but discrete, superposition of monochromatic Bessel beams. Contrary to that, we have here proposed a new way to get these waves through a continuous superposition of Bessel beams, and discussed some physical aspects and then exemplified for both loss and lossless media. r 2006 Elsevier Ltd. All rights reserved. Keywords: Non-diffracting waves; Spatial localized waves; Scalar diffraction theory
1. Introduction Since they were first discovered both theoretically and experimentally by Durnin [1], many investigations have been carried out to realize diffraction-free beams. These waves have been verified in many fields like optics, microwaves and acoustics, and their different applications [2,3]. As a fact, one can identify two classes of these localized waves: pulses and beams. Regarding the beams, Bessel [1–3], Bessel–Gauss [4] and Mathieu [3,5] are the most popular. Only a few papers have been addressed to the use of superpositions of Bessel beams with the same frequency, but with different longitudinal wave numbers to get confinement in both transverse or longitudinal direction. The possibility of transverse shape modeling using Bessel beams was previously examined by Bouchal et al. [6], while the control of longitudinal intensity Corresponding author.
E-mail addresses: [email protected] (C.A. Dartora), [email protected] (K.Z. Nobrega). 0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.10.013
pattern has been analyzed in Refs. [7–9] using numerical optimization techniques. Very recently, it has been reported that an analytical and interesting solution to obtain longitudinal localized waves through an adequate relationship between transverse and longitudinal wave vectors of a monochromatic Bessel beam, the so-called Frozen Waves [10]. That solution has the main characteristics to remain static in space and to keep their arbitrary field pattern. However, to find such a stationary pattern, it was shown that one should have a discrete sum of wave vectors of Bessel beams, expressed in terms of a truncated Fourier series and their coefficients. Relaxing that condition, in this paper it is presented a generalization of that discrete condition to the continuous case, it means, an analytical way that considers continuous superposition of monochromatic Bessel beams to get a Frozen Wave. Besides that, from our study one can clearly identify limits of resolution of these waves and, finally, two examples are shown to illustrate their potentiality in both lossy and lossless media.
ARTICLE IN PRESS C.A. Dartora et al. / Optics & Laser Technology 39 (2007) 1370–1373
2. Frozen Waves and their integral version 2.1. Theoretical framework The monochromatic Frozen Waves obtained by the method developed in Ref. [10] are written as follows: CFW ðr; z; tÞ ¼ expðio0 tÞ expðiQzÞ
N X
the longitudinal wave vector, kz ¼ Q þ b, must satisfy the condition 0:5k0 nR pQpk0 nR . As one can see, the functions AðbÞ and F ðzÞ are pairs of Fourier transforms. This last conclusion is clearer by considering the on-axis r ¼ 0 case of Eq. (3): Z bmax dbAðbÞ expðibzÞ, (5) CFW ðzÞ ¼ expðiQzÞ bmax
Am
m¼N
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2pm 2 A 2 2 @ k0 n Q þ r J 0 L 2pm exp i z , L
1371
ð1Þ
with bmax ¼ k0 nR Q. Another interesting case of study is the solution when z ¼ 0, corresponding to the aperture transmittance function (TF) of an arbitrary apparatus used to generate the pattern F ðzÞ: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z bmax CFW ðrÞ ¼ k20 n2R ðQ þ bÞ2 r . (6) dbAðbÞJ 0 bmax
being k0 ¼ o0 =c ¼ 2p=l0 the vacuum wave number and n ¼ nR þ inI the riffraction index of the medium. With the values of Q and L previously chosen the coefficients Am in Eq. (1) are obtained from the following definition: Z 1 L 2pm z , (2) Am ¼ dzF ðzÞ exp i L 0 L where F ðzÞ is the desired longitudinal pattern on the axis r ¼ 0. The condition 0pQ 2pN=Lpk0 nR must be satisfied in order to have only waves propagating forward along the z direction. Actually, the function F ðzÞ is a periodic function with spatial period L in the z-axis. However, practical realizations of F ðzÞ are possible only over a finite multiple of the spatial period L. In order to improve the attainable pattern, it is convenient to consider the spatial period tending to infinity (L ! 1), with a function F ðzÞ that is non-vanishing only over a finite range going from z ¼ 0 to z ¼ zmax , being zmax the depth of the field [3] of the Bessel beams composition [10]. In general it is determined by the experimental apparatus [9,10]. Making L ! 1 and also the number of composing Bessel beams going to infinity, N ! 1, we rewrite Eq. (1) as follows: Z ðk0 nR QÞ CFW ðr; z; tÞ ¼ expðio0 tÞ expðiQzÞ db ðk0 nR QÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðbÞJ 0 k20 n2R ðQ þ bÞ2 r expðibzÞ,
Considering the well-known Fourier transform properties, it is clear the limits of spatial resolution of any function F ðzÞ. Such properties impose an uncertainty relation, shown below: Db DLz X2p,
(7)
Db being the spectral bandwidth of AðbÞ and DLz the spatial resolution of the function F ðzÞ. The maximum value of Db is 2ðk0 nR QÞ, and considering Q ¼ k0 nR cos yQ (k0 ¼ o0 =c ¼ 2p=l) we have DLz X
l p ¼ . 2nR ð1 cos yQ Þ bmax
(8)
The previous equation has important characteristics because it quantifies true spatial resolution of F ðzÞ, DLz . First, the lower the Q, the better the spatial resolution of F ðzÞ. For example, if nR ¼ 1 and Q ¼ 0:9998k0 or 0:99996k0 , one has DLz 2500l and DLz 12560l, while when approaching the best case, Q ¼ 0:5k0 , the limit is about l considering forward propagation only. Qualitatively, one can also infer that the higher the Q, the less longitudinal localized the F ðzÞ is and the less localized the
ð3Þ where AðbÞ is obtained from the desired pattern, F ðzÞ, as follows: Z 1 1 AðbÞ ¼ dzF ðzÞ expðibzÞ. (4) 2p 1 Before moving on, looking at Eq. (3), one could notice that the longitudinal wave vector is centered around Q and the upper limit of such integral corresponds to the extreme case when kz ¼ k0 nR , but the lower limit is considered the negative value of the upper only to assure the symmetric limits of Fourier’s transform. Here, for the sake of comparison with Zamboni-Rached [10], if we restrict the solution to forward propagation only (0okz ok0 nR ), then
Fig. 1. Axial intensity pattern of field to illustrate a step function, F ðzÞ, near the point of transition with different Q values. l0 ¼ 632:8 nm.
ARTICLE IN PRESS C.A. Dartora et al. / Optics & Laser Technology 39 (2007) 1370–1373
1372
F ðzÞ is along the transverse direction, consequently (Q ! k0 implies kr ! 0). In the last paragraph, it was shown the compromise between parameter Q and spatial limits of the Frozen Waves. Now, let us continue to use properties of Fourier’s transform to clearly explain that Q is also very important to determine how abrupt F ðzÞ can vary (see Fig. 1). In Fig. 1 it is analyzed that the most critical possibility for an arbitrary F ðzÞ supposing three different values of Q, e.g., it represents a step function centered in z ¼ 20 cm. From that figure, it is shown that higher Q is, worse the step is. Thus, the condition of choosing Q satisfying spatial resolution is necessary but not sufficient to get the desired field profile F ðzÞ. Considering our extensive previous discussion, this fact can be easily understood taking into account that low Q values get the available spectrum, AðbÞ, large enough to rebuild field profiles with large frequencies, i.e., high discontinuities. P It is important to notice that if AðbÞ ¼ N m¼N Am dðb 2pm=LÞ, with dð:Þ being the Dirac delta function and Am calculated from Eq. (2), our integral theory of Frozen Waves is reduced to the discrete case presented in Ref. [10], where the later is a particular case from the former.
Fig. 2. The 3D intensity pattern of the Frozen Wave given pffiffiffiffiffiffi by Eq. (9) in a lossless medium, with l ¼ 632:8 nm, nR ¼ 1 and a ¼ 0:5.
2.2. Examples As a first example, let us consider l ¼ 632:8 nm, nR ¼ 1 (vacuum), Q ¼ 0:999k0 and 8 pffiffiffi 9 < 4 aðz 0:1Þðz 0:25Þ; 0:1pzp0:25 = F ðzÞ ¼ . 0:152 : ; 1; 0:25ozp0:35 (9) The grid considered has 500 200 points, along z and r directions, respectively. To find F ðzÞ, we have first calculated AðbÞ, using Eq. (4), and substituted it into Eq. (3). Fig. 2 shows three-dimensional field intensity pattern. To this case, we have calculated the percentual mean-squared error (s2 ) between the original and the reconstructed intensity patterns and found s2 ¼ 0:0074%. As a second example we address the problem of compensating attenuation in a lossy medium. In a weakly conductive lossy media (s5oe) with attenuation constant a (skin depth d ¼ 1=a), we have k ¼ k0 nR þ ia. If Qbbmax we can express Eq. (3) as follows: Z bmax CFW ðr; zÞ ¼ expðaz þ iQzÞ dbAðbÞ bmax qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a J 0 1 þ i k20 n2R ðQ þ bÞ2 r nR k 0 expðibzÞ.
ð10Þ
In order to compensate the attenuation and to obtain any arbitrary longitudinal pattern for r ¼ 0 and long distances (z4d) the spectrum AðbÞ must be modified to
Fig. 3. Axial intensity pattern of field in a lossy medium when F ðzÞ ¼ fuðz 0:2Þ uðz 0:3Þg, l0 ¼ 632:8 nm, nR ¼ 1:5 and a ¼ 0:023 cm1 .
the following: Aloss ðbÞ ¼
1 2p
Z
1
dzF ðzÞ exp½az expðibzÞ.
(11)
1
Effectively, we raise exponentially the desired pattern to compensate the exponential damping expðazÞ of the field in the lossy medium. In spite of the fact that we can compensate losses for the axis r ¼ 0 and obtain any desired pattern, we must remember that the energy absorption by the medium continues to occur normally; the difference is that the fields generated with spectrum (11) are capable to reconstruct their on-axis (r ¼ 0) intensity even for a lossy medium. Similar effects in trying to compensate loss
ARTICLE IN PRESS C.A. Dartora et al. / Optics & Laser Technology 39 (2007) 1370–1373
effects, but in nonlinear directional couplers, have been already reported in some previous works [11]. For a lossy medium with nR ¼ 1:5 and a ¼ 0:023 cm1 at l0 ¼ 632:8 nm, we chose Q ¼ 0:999k0 nR and F ðzÞ ¼ fuðz 0:2Þ uðz 0:3Þg, where uðzÞ is the Heaviside function. The grid considered has 300 200 points, along z and r direction, respectively. We have calculated AðbÞ through Eq. (11) (with compensation) and substituted it into Eq. (10) to obtain the desired pattern F ðzÞ in the lossy medium. Fig. 3 shows the two-dimensional on-axis intensity pattern for a better view and comparison between the reconstructed case and the one without compensation. 3. Conclusion and remarks In this paper the theory of Frozen Waves, first appeared in Ref. [10], has been generalized. In fact we have here proposed an analytical way that considers a continuous superposition of monochromatic Bessel beams. Besides that, we have mathematically identified the limit of resolution of the longitudinal pattern F ðzÞ, and we have considered two examples to illustrate the potentiality of our method, in both lossy and lossless media. For future works, practical realizations of these waves through finite apertures, as well as the use of higher order Bessel beams as basis functions will be analyzed.
1373
Acknowledgment We would like to thank CNPq for the financial support. References [1] Durnin J, Miceli Jr JJ, Eberly JH. Phys Rev Lett 1987;58:1499–501; Durnin J. J Opt Soc Am A 1987;4:651–4. [2] Recami E, Zamboni-Rached M, Nobrega KZ, Dartora CA, Hernandez-Figueroa HE. IEEE J Sel Top Quantum Electron 2003;9:59–73 and references therein; McGloin D, Dholakia K. Contemp Phys 2005;46:15–28. [3] Dartora CA, Zamboni-Rached M, Nobrega KZ, Recami E, Herna´ndez-Figueroa HE. Opt Commun 2003;222:75–80. [4] Greene PL, Hall DG. J Opt Soc Am A 1998;15:3020; Overleft PL, Kenney CS. J Opt Soc Am A 1991;8:732. [5] Roger-Salazar J, New GHC, Chavez-Cerda S. Opt Commun 2001;190:117–22; Gutierrez-Vega JC, Iturbe-Castillo MD, Chavez-Cerda S. Opt Lett 2000;25:1493–5. [6] Bouchal Z. Opt Lett 2002;27:1376–8; Bouchal Z, Wagner J. Opt Commun 2000;176:299–307. [7] Rosen J, Yariv A. Opt Lett 1994;19:843–5. [8] Piestun R, Spektor B, Shamir J. J Mod Opt 1996;43:1495–507. [9] Dartora CA, Nobrega KZ, Dartora A, Hernandez-Figueroa HE. Opt Commun 2005;249:407–13. [10] Zamboni-Rached M. Opt Express 2004;12:4001–6. [11] Nobrega KZ, Sombra ASB. Opt Commun 1998;151:31–4; Nobrega KZ, da Silva MG, Sombra ASB. Nonlinear Opt 2001;26:337–51.
Optics & Laser Technology 39 (2007) 1370–1373 www.elsevier.com/locate/optlastec
Study of Frozen Waves’ theory through a continuous superposition of Bessel beams C.A. Dartoraa, K.Z. Nobregab,, Alexandre Dartorac, Gustavo A. Vianad, Horacio Tertuliano S. Filhoa a Electrical Engineering Department - UFPR, Brazil Electrical Engineering Department, CEFET-MA, Brazil c CEFET-PR, Pato Branco, Brazil d Institute of Physics Gleb Wataghin, State University of Campinas, Brazil b
Received 21 November 2005; received in revised form 7 October 2006; accepted 20 October 2006 Available online 18 December 2006
Abstract This paper presents a study of a recent solution to Maxwell’s equation, the so-called ‘‘Frozen Waves’’, whose main characteristics are to remain static in space, and to keep an arbitrary longitudinal field pattern previously chosen. These waves could be obtained by an adequate, but discrete, superposition of monochromatic Bessel beams. Contrary to that, we have here proposed a new way to get these waves through a continuous superposition of Bessel beams, and discussed some physical aspects and then exemplified for both loss and lossless media. r 2006 Elsevier Ltd. All rights reserved. Keywords: Non-diffracting waves; Spatial localized waves; Scalar diffraction theory
1. Introduction Since they were first discovered both theoretically and experimentally by Durnin [1], many investigations have been carried out to realize diffraction-free beams. These waves have been verified in many fields like optics, microwaves and acoustics, and their different applications [2,3]. As a fact, one can identify two classes of these localized waves: pulses and beams. Regarding the beams, Bessel [1–3], Bessel–Gauss [4] and Mathieu [3,5] are the most popular. Only a few papers have been addressed to the use of superpositions of Bessel beams with the same frequency, but with different longitudinal wave numbers to get confinement in both transverse or longitudinal direction. The possibility of transverse shape modeling using Bessel beams was previously examined by Bouchal et al. [6], while the control of longitudinal intensity Corresponding author.
E-mail addresses: [email protected] (C.A. Dartora), [email protected] (K.Z. Nobrega). 0030-3992/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2006.10.013
pattern has been analyzed in Refs. [7–9] using numerical optimization techniques. Very recently, it has been reported that an analytical and interesting solution to obtain longitudinal localized waves through an adequate relationship between transverse and longitudinal wave vectors of a monochromatic Bessel beam, the so-called Frozen Waves [10]. That solution has the main characteristics to remain static in space and to keep their arbitrary field pattern. However, to find such a stationary pattern, it was shown that one should have a discrete sum of wave vectors of Bessel beams, expressed in terms of a truncated Fourier series and their coefficients. Relaxing that condition, in this paper it is presented a generalization of that discrete condition to the continuous case, it means, an analytical way that considers continuous superposition of monochromatic Bessel beams to get a Frozen Wave. Besides that, from our study one can clearly identify limits of resolution of these waves and, finally, two examples are shown to illustrate their potentiality in both lossy and lossless media.
ARTICLE IN PRESS C.A. Dartora et al. / Optics & Laser Technology 39 (2007) 1370–1373
2. Frozen Waves and their integral version 2.1. Theoretical framework The monochromatic Frozen Waves obtained by the method developed in Ref. [10] are written as follows: CFW ðr; z; tÞ ¼ expðio0 tÞ expðiQzÞ
N X
the longitudinal wave vector, kz ¼ Q þ b, must satisfy the condition 0:5k0 nR pQpk0 nR . As one can see, the functions AðbÞ and F ðzÞ are pairs of Fourier transforms. This last conclusion is clearer by considering the on-axis r ¼ 0 case of Eq. (3): Z bmax dbAðbÞ expðibzÞ, (5) CFW ðzÞ ¼ expðiQzÞ bmax
Am
m¼N
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2pm 2 A 2 2 @ k0 n Q þ r J 0 L 2pm exp i z , L
1371
ð1Þ
with bmax ¼ k0 nR Q. Another interesting case of study is the solution when z ¼ 0, corresponding to the aperture transmittance function (TF) of an arbitrary apparatus used to generate the pattern F ðzÞ: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z bmax CFW ðrÞ ¼ k20 n2R ðQ þ bÞ2 r . (6) dbAðbÞJ 0 bmax
being k0 ¼ o0 =c ¼ 2p=l0 the vacuum wave number and n ¼ nR þ inI the riffraction index of the medium. With the values of Q and L previously chosen the coefficients Am in Eq. (1) are obtained from the following definition: Z 1 L 2pm z , (2) Am ¼ dzF ðzÞ exp i L 0 L where F ðzÞ is the desired longitudinal pattern on the axis r ¼ 0. The condition 0pQ 2pN=Lpk0 nR must be satisfied in order to have only waves propagating forward along the z direction. Actually, the function F ðzÞ is a periodic function with spatial period L in the z-axis. However, practical realizations of F ðzÞ are possible only over a finite multiple of the spatial period L. In order to improve the attainable pattern, it is convenient to consider the spatial period tending to infinity (L ! 1), with a function F ðzÞ that is non-vanishing only over a finite range going from z ¼ 0 to z ¼ zmax , being zmax the depth of the field [3] of the Bessel beams composition [10]. In general it is determined by the experimental apparatus [9,10]. Making L ! 1 and also the number of composing Bessel beams going to infinity, N ! 1, we rewrite Eq. (1) as follows: Z ðk0 nR QÞ CFW ðr; z; tÞ ¼ expðio0 tÞ expðiQzÞ db ðk0 nR QÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AðbÞJ 0 k20 n2R ðQ þ bÞ2 r expðibzÞ,
Considering the well-known Fourier transform properties, it is clear the limits of spatial resolution of any function F ðzÞ. Such properties impose an uncertainty relation, shown below: Db DLz X2p,
(7)
Db being the spectral bandwidth of AðbÞ and DLz the spatial resolution of the function F ðzÞ. The maximum value of Db is 2ðk0 nR QÞ, and considering Q ¼ k0 nR cos yQ (k0 ¼ o0 =c ¼ 2p=l) we have DLz X
l p ¼ . 2nR ð1 cos yQ Þ bmax
(8)
The previous equation has important characteristics because it quantifies true spatial resolution of F ðzÞ, DLz . First, the lower the Q, the better the spatial resolution of F ðzÞ. For example, if nR ¼ 1 and Q ¼ 0:9998k0 or 0:99996k0 , one has DLz 2500l and DLz 12560l, while when approaching the best case, Q ¼ 0:5k0 , the limit is about l considering forward propagation only. Qualitatively, one can also infer that the higher the Q, the less longitudinal localized the F ðzÞ is and the less localized the
ð3Þ where AðbÞ is obtained from the desired pattern, F ðzÞ, as follows: Z 1 1 AðbÞ ¼ dzF ðzÞ expðibzÞ. (4) 2p 1 Before moving on, looking at Eq. (3), one could notice that the longitudinal wave vector is centered around Q and the upper limit of such integral corresponds to the extreme case when kz ¼ k0 nR , but the lower limit is considered the negative value of the upper only to assure the symmetric limits of Fourier’s transform. Here, for the sake of comparison with Zamboni-Rached [10], if we restrict the solution to forward propagation only (0okz ok0 nR ), then
Fig. 1. Axial intensity pattern of field to illustrate a step function, F ðzÞ, near the point of transition with different Q values. l0 ¼ 632:8 nm.
ARTICLE IN PRESS C.A. Dartora et al. / Optics & Laser Technology 39 (2007) 1370–1373
1372
F ðzÞ is along the transverse direction, consequently (Q ! k0 implies kr ! 0). In the last paragraph, it was shown the compromise between parameter Q and spatial limits of the Frozen Waves. Now, let us continue to use properties of Fourier’s transform to clearly explain that Q is also very important to determine how abrupt F ðzÞ can vary (see Fig. 1). In Fig. 1 it is analyzed that the most critical possibility for an arbitrary F ðzÞ supposing three different values of Q, e.g., it represents a step function centered in z ¼ 20 cm. From that figure, it is shown that higher Q is, worse the step is. Thus, the condition of choosing Q satisfying spatial resolution is necessary but not sufficient to get the desired field profile F ðzÞ. Considering our extensive previous discussion, this fact can be easily understood taking into account that low Q values get the available spectrum, AðbÞ, large enough to rebuild field profiles with large frequencies, i.e., high discontinuities. P It is important to notice that if AðbÞ ¼ N m¼N Am dðb 2pm=LÞ, with dð:Þ being the Dirac delta function and Am calculated from Eq. (2), our integral theory of Frozen Waves is reduced to the discrete case presented in Ref. [10], where the later is a particular case from the former.
Fig. 2. The 3D intensity pattern of the Frozen Wave given pffiffiffiffiffiffi by Eq. (9) in a lossless medium, with l ¼ 632:8 nm, nR ¼ 1 and a ¼ 0:5.
2.2. Examples As a first example, let us consider l ¼ 632:8 nm, nR ¼ 1 (vacuum), Q ¼ 0:999k0 and 8 pffiffiffi 9 < 4 aðz 0:1Þðz 0:25Þ; 0:1pzp0:25 = F ðzÞ ¼ . 0:152 : ; 1; 0:25ozp0:35 (9) The grid considered has 500 200 points, along z and r directions, respectively. To find F ðzÞ, we have first calculated AðbÞ, using Eq. (4), and substituted it into Eq. (3). Fig. 2 shows three-dimensional field intensity pattern. To this case, we have calculated the percentual mean-squared error (s2 ) between the original and the reconstructed intensity patterns and found s2 ¼ 0:0074%. As a second example we address the problem of compensating attenuation in a lossy medium. In a weakly conductive lossy media (s5oe) with attenuation constant a (skin depth d ¼ 1=a), we have k ¼ k0 nR þ ia. If Qbbmax we can express Eq. (3) as follows: Z bmax CFW ðr; zÞ ¼ expðaz þ iQzÞ dbAðbÞ bmax qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a J 0 1 þ i k20 n2R ðQ þ bÞ2 r nR k 0 expðibzÞ.
ð10Þ
In order to compensate the attenuation and to obtain any arbitrary longitudinal pattern for r ¼ 0 and long distances (z4d) the spectrum AðbÞ must be modified to
Fig. 3. Axial intensity pattern of field in a lossy medium when F ðzÞ ¼ fuðz 0:2Þ uðz 0:3Þg, l0 ¼ 632:8 nm, nR ¼ 1:5 and a ¼ 0:023 cm1 .
the following: Aloss ðbÞ ¼
1 2p
Z
1
dzF ðzÞ exp½az expðibzÞ.
(11)
1
Effectively, we raise exponentially the desired pattern to compensate the exponential damping expðazÞ of the field in the lossy medium. In spite of the fact that we can compensate losses for the axis r ¼ 0 and obtain any desired pattern, we must remember that the energy absorption by the medium continues to occur normally; the difference is that the fields generated with spectrum (11) are capable to reconstruct their on-axis (r ¼ 0) intensity even for a lossy medium. Similar effects in trying to compensate loss
ARTICLE IN PRESS C.A. Dartora et al. / Optics & Laser Technology 39 (2007) 1370–1373
effects, but in nonlinear directional couplers, have been already reported in some previous works [11]. For a lossy medium with nR ¼ 1:5 and a ¼ 0:023 cm1 at l0 ¼ 632:8 nm, we chose Q ¼ 0:999k0 nR and F ðzÞ ¼ fuðz 0:2Þ uðz 0:3Þg, where uðzÞ is the Heaviside function. The grid considered has 300 200 points, along z and r direction, respectively. We have calculated AðbÞ through Eq. (11) (with compensation) and substituted it into Eq. (10) to obtain the desired pattern F ðzÞ in the lossy medium. Fig. 3 shows the two-dimensional on-axis intensity pattern for a better view and comparison between the reconstructed case and the one without compensation. 3. Conclusion and remarks In this paper the theory of Frozen Waves, first appeared in Ref. [10], has been generalized. In fact we have here proposed an analytical way that considers a continuous superposition of monochromatic Bessel beams. Besides that, we have mathematically identified the limit of resolution of the longitudinal pattern F ðzÞ, and we have considered two examples to illustrate the potentiality of our method, in both lossy and lossless media. For future works, practical realizations of these waves through finite apertures, as well as the use of higher order Bessel beams as basis functions will be analyzed.
1373
Acknowledgment We would like to thank CNPq for the financial support. References [1] Durnin J, Miceli Jr JJ, Eberly JH. Phys Rev Lett 1987;58:1499–501; Durnin J. J Opt Soc Am A 1987;4:651–4. [2] Recami E, Zamboni-Rached M, Nobrega KZ, Dartora CA, Hernandez-Figueroa HE. IEEE J Sel Top Quantum Electron 2003;9:59–73 and references therein; McGloin D, Dholakia K. Contemp Phys 2005;46:15–28. [3] Dartora CA, Zamboni-Rached M, Nobrega KZ, Recami E, Herna´ndez-Figueroa HE. Opt Commun 2003;222:75–80. [4] Greene PL, Hall DG. J Opt Soc Am A 1998;15:3020; Overleft PL, Kenney CS. J Opt Soc Am A 1991;8:732. [5] Roger-Salazar J, New GHC, Chavez-Cerda S. Opt Commun 2001;190:117–22; Gutierrez-Vega JC, Iturbe-Castillo MD, Chavez-Cerda S. Opt Lett 2000;25:1493–5. [6] Bouchal Z. Opt Lett 2002;27:1376–8; Bouchal Z, Wagner J. Opt Commun 2000;176:299–307. [7] Rosen J, Yariv A. Opt Lett 1994;19:843–5. [8] Piestun R, Spektor B, Shamir J. J Mod Opt 1996;43:1495–507. [9] Dartora CA, Nobrega KZ, Dartora A, Hernandez-Figueroa HE. Opt Commun 2005;249:407–13. [10] Zamboni-Rached M. Opt Express 2004;12:4001–6. [11] Nobrega KZ, Sombra ASB. Opt Commun 1998;151:31–4; Nobrega KZ, da Silva MG, Sombra ASB. Nonlinear Opt 2001;26:337–51.