- Email: [email protected]
Generation of light string and light capillary beams
15 June 2002
Optics Communications 207 (2002) 131–138 www.elsevier.com/locate/optcom
Generation of light string and light capillary beams Andrey S. Ostrovsky a,b,*, Gabriel Martınez-Niconoff b, Julio C. Ramırez-San-Juan b b
a Facultad de Ciencias Fısico Matem aticas, Universidad Aut onoma de Puebla, 72570 Puebla, Pue., Mexico Departamento de Optica, Instituto Nacional de Astrofısica, Optica y Electr onica, 72000 Puebla, Pue., Mexico
Received 20 February 2002; accepted 3 April 2002
Abstract Recently we reported a new modal theory of propagation-invariant optical fields [Opt. Commun. 195 (2001) 27–34]. Within the framework of this theory we predicted the existence of so-called light string and light capillary beams. Here, we propose an optical technique for generating these beams and demonstrate it through experimental results. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.25.F; 42.25.K Keywords: Propagation-invariant field; Coherent-mode structure; Bessel beam; Bessel-correlated beam; Light string beam; Light capillary beam
1. Introduction Recently, we studied the coherent-mode structure of a so-called propagation-invariant field characterized by the invariance of its transverse cross-spectral density function in the propagation direction [1]. We have found three fundamentally different coherent-mode representations of the cross-spectral density function which define three possible classes of the propagation-invariant fields termed by us as propagation-invariant fields of the first, the second and the third kind. The propagation-invariant fields of the first kind represent
*
Corresponding author. Fax: +52-222-247-29-40. E-mail address: [email protected] (A.S. Ostrovsky).
completely coherent so-called nondiffracting beams [2], including the fundamental Bessel beam [3]. Good approximations of such beams can be readily produced in the laboratory by means of a rather simple optical system with an annular aperture [4], an axicon [5] or computer-generated hologram [6]. The propagation-invariant fields of the second and the third kind are partially coherent fields with fundamentally different coherentmode structures. A typical representative of the propagation-invariant fields of the second kind is the so-called Bessel-correlated beam [7], which can be also readily produced in laboratory conditions [8]. Within the class of the propagation-invariant fields of the third kind, we have predicted in theory the existence of the peculiar beams termed by us as light string beam and light capillary beam, in view
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 5 3 1 - 6
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of their very specific extremely sharply localized energy distribution in the transverse section. However, up to now, the physical realization of such beams remained undone. In this paper, we propose two possible optical techniques for generating light string and light capillary beams and demonstrate these through experimental results.
where ðr; uÞ are the polar coordinates in the space– ðlÞ frequency domain. On substituting for WIII from Eq. (1) into Eq. (2) and realizing the integration, firstly in azimuthal and then in radial directions, with due regard for the relations [10] Z il 2p exp½iðx cos a þ laÞ da ¼ Jl ðxÞ; ð3Þ 2p 0
2. Light string and light capillary beams
Jl ðxÞ ¼ Jl ðxÞ ¼ ð1Þ Jl ðxÞ; Z 1 b Jl ðbxÞJl ðb0 xÞx dx ¼ dðb b0 Þ;
l
The light string and light capillary beams are characterized by the transverse cross-spectral density function given by [1] ðlÞ
WIII ðq1 ; h1 ; q2 ; h2 ; zÞ ¼ cos½lðh1 h2 Þ
1 X n¼1
q q Jl al;n 1 Jl al;n 2 ; R R
we find ð0Þ
AIII ðr1 ; u1 ; r2 ; u2 Þ 2 1 a X R a0;n 0;n ¼ d r1 d r2 a0;n 2pR 2pR n¼1
ð6Þ
and
where l ¼ 0 for the light string, l ¼ 1 for the light capillary, Jl denotes the Bessel function of the first kind and of order l, al;n is the nth zero of the function Jl , R is a real positive constant, and ðq; hÞ are the polar coordinates in the plane z ¼ const. The transverse intensity distributions of these ð0Þ ð1Þ beams, WIII ðq; h; q; hÞ and WIII ðq; h; q; hÞ, calculated with truncating the summation in Eq. (1) by the different values of the number N of zeros of the Bessel function are shown in Fig. 1. As one may conclude from this figure, when N tends to infinity, the field with an intensity distribution ð0Þ WIII ðq; h; q; hÞ represents the infinitely thin light beam and the field with an intensity distribution ð1Þ WIII ðq; h; q; hÞ represents the infinitely thin light tube, a fact that explains the origin of the names given to the corresponding fields. For further consideration it will be useful to know the angular correlation functions or the cross-angular spectra [9] of light string and light capillary beams defined as ðlÞ
AIII ðr1 ; u1 ; r2 ; u2 Þ Z Z 1 Z Z 2p ðlÞ q1 q2 WIII ðq1 ; h1 ; q2 ; h2 ; 0Þ ¼ 0
expfi2p½r2 q2 cosðu2 h2 Þ r1 q1 cosðu1 h1 Þg dq1 dq2 dh1 dh2 ;
ð5Þ
0
ð1Þ
0
ð4Þ
ð2Þ
ð1Þ AIII ðr1 ; u1 ; r2 ; u2 Þ
2 1 X R ¼ cosðu1 u2 Þ a1;n n¼1 a a 1;n 1;n d r1 d r2 : 2pR 2pR ð7Þ
As will be shown further, Eqs. (6) and (7) together with Eq. (2) can be taken as a principle of optical generating the light string and light capillary beams. 3. Optical synthesis of light string and light capillary beams As one may see from Eq. (2), the cross-angular ðlÞ spectrum AIII ðr1 ; u1 ; r2 ; u2 Þ represents a four-dimensional Fourier transform of the cross-spectral density function characterizing a propagation-invariant field of the third kind. On the other hand, it is well known that the cross-spectral densities of the wave fields in the front and the back focal planes of a thin spherical lens are also related by a four-dimensional Fourier transform. Both these facts can be used to realize an optical synthesis of the light string and light capillary beams by producing a secondary planar source with a cross-spectral ðlÞ density function WS ¼ AIII in the front focal plane of a Fourier-transforming lens. We will show now that such a secondary source can be produced by
A.S. Ostrovsky et al. / Optics Communications 207 (2002) 131–138
133
Fig. 1. Transverse intensity distributions of light string (a) and light capillary (b) beams calculated for different values of truncating parameter N.
means of an appropriate spatial-time modulation of the homogeneous plane monochromatic wave. To produce the secondary source with a crossð0Þ spectral density function WS ¼ AIII , one may use a spatial light modulator with a complex amplitude transmittance that can be approximated as follows: N X 1 T ðq ; h Þ ¼ dða0;n q0 q0 Þ expðiWn Þ; a 0;n n¼1 0
0
0
where ðq ; h Þ are the polar coordinates in the front focal plane of a Fourier-transforming lens, q0 is a constant and, for each n, Wn is a real random variable that is uniformly distributed in the interval ð0; 2pÞ and Wn and Wm are statistically independent when n 6¼ m. When illuminating this modulator by a homogeneous plane wave with amplitude V0 the cross-spectrum density function in the output of modulator is given by
WS ðq01 ; h01 ; q02 ; h02 Þ ¼ V02
N X 1 dða0;n q0 q01 Þ 2 a 0;n n¼1
dða0;n q0 q02 Þ:
ð11Þ
Eq. (11) represents the finite sum approximation of ð0Þ the cross-angular spectrum AIII (with due regard for corresponding substitution of arguments), that had to be shown. To produce the secondary source with a crossð1Þ spectral density function WS ¼ AIII , one may use a spatial light modulator with complex amplitude transmittance T 0 ðq0 ; h0 Þ ¼
ð9Þ
where the angle brackets denote the statistical average. Substituting for T ðq0 ; h0 Þ from Eq. (8) into Eq. (9) and taking into account that, with due regard for the accepted statistical properties of random variables Wn ,
ð10Þ
where dnm is the Kronecker symbol, we obtain
ð8Þ
0
WS ðq01 ; h01 ; q02 ; h02 Þ ¼ V02 hT ðq01 ; h01 ÞT ðq02 ; h02 Þi;
hexpðiWn Þ expðiWm Þi ¼ dnm ;
N X 1 dða1;n q0 q0 Þ exp½iUn ðh0 Þ; a 1;n n¼1
ð12Þ where ( 0
Un ðh Þ ¼
2p þ h0 Wn h 0 Wn
for 0 6 h0 6 Wn ; for Wn < h0 6 2p
ð13Þ
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and Wn has the same meaning as in Eq. (8), followed by a Mach–Zehnder interferometer with a Dove prism in one of its arms as is shown in Fig. 2. The Dove prism realizes the mirror mapping of the incident wave field, providing thereby the phase conjugation of the distribution given by Eq. (12). The whole of the system in Fig. 2 acts as a new spatial light modulator with equivalent amplitude transmittance T ðq0 ; h0 Þ ¼
N 1X 1 dða1;n q0 q0 Þ 2 n¼1 a1;n
fexp½iUn ðh0 Þ þ exp½iUn ðh0 Þg ¼
N X 1 dða1;n q0 q0 Þ cosðh0 Wn Þ: a 1;n n¼1
ð14Þ 0
Substituting for T ðq0 ; h Þ from Eq. (14) into Eq. (9) and taking into account that hcosðh01 þ Wn Þ cosðh02 þ Wm Þi ¼ dnm cosðh01 h02 Þ;
ð15Þ
we come to the result WS ðq01 ; h01 ; q02 ; h02 Þ ¼ V02 cosðh01 h02 Þ
N X 1 2 a 1;n n¼1
dða1;n q0 q01 Þdða1;n q0 q02 Þ; ð16Þ which represents the finite sum approximation of ð1Þ the cross-angular spectrum AIII : Now, we will discuss briefly the possibility of physical realization of the space light modulators
with the complex amplitude transmittances given by Eqs. (8) and (12) (for more details see the next section). It is obvious that each of this modulators may be realized as a combination of the static purely amplitude modulator and the dynamic phase modulator with appropriate transmittance functions. The static amplitude modulator may be realized in a good approximation as a binary mask in the form of the transparent rings with the required radii and the widths which are proportional to the weight coefficients attached to the delta functions in Eqs. (8) and (12) (Fig. 3). Such a mask may be easily manufactured by means of a standard photolithographic technique. The dynamic phase modulator must represent a transparent plane screen that introduces the required azimuthal phase delays in the annular zones corresponding to the transparent rings of the binary mask that are changed randomly and independently in discrete moments of time (Fig. 4). As a such random phase screen, the liquid crystal display (LCD) controlled by a computer may be successfully used [11]. Concluding this section, we note that the proposed optical techniques allow us to realize only an approximate synthesis of the propagation-invariant beams described by Eq. (1). There are three main reasons for this. Firstly, as it is well known (see, e.g., [8]), if the source is bounded within a circle of diameter DS and the Fourier transforming lens with the focal length f has a circle input pupil of diameter DF ; the maximum distance at which the propagation-invariant behavior of the generated field can be expected is limited by the value zmax ¼ fDF =DS :
ð17Þ
Secondly, approximating the weighted delta functions by transparent rings of finite width, also
(a) Fig. 2. Secondary source for generating the light capillary beam: SLM – spatial light modulator; BS – beam splitter; M – mirror; DP – Dove prism.
(b)
Fig. 3. Binary masks for generating the light string (a) and light capillary (b) beams.
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135
Table 1 Parameters of binary mask I Ring number n
J0 zero a0;n
Ring radius q00;n (mm)
Ring width Dq00;n (mm)
1 2 3 4 5
2.404 5.520 8.653 11.791 14.930
0.805 1.848 2.898 3.949 5.000
1.000 0.436 0.278 0.204 0.161
Table 2 Parameters of binary mask II Fig. 4. Different states of a random phase screen used for generating the light string (a) and light capillary (b) beams. The phase delay is presented by grey level.
results in violation of the conditions of propagation invariance. Lastly, as can be seen from Fig. 1, the finite sum approximation of the cross spectral density function WS originating from the finite number of rings used restricts the minimum transverse dimensions of the generated beams.
4. Experimental verification To verify the capacity of the proposed technique of generating light string and light capillary beams, we carried out some optical experiments. In these experiments as a primary source we used a linearly polarized laser beam (k ¼ 0:63 lm, He–Ne laser). To produce the needed secondary sources we used the binary masks with a finite number of transparent rings of variable width, which were specially manufactured by means of photolitographic technique. Taking into account a very fast decrease of the values 1=a20;n and 1=a21;n with n, to simplify our experiments, we limited the number of the transparent rings to N ¼ 5. The geometrical parameters of these masks are given by Tables 1 and 2. As a dynamic phase modulator we had available a commercial CRL’s SVGA2VX-model computer-controlled LCD panel, which works in transmissive mode and has a 33 mm diagonal active display area with 800 600 pixels. Since this device has a 90°-twisted nematic structure, to provide the phase-only modulation, we aligned the polarization direction of the incident light with
Ring number n
J1 zero a1;n
Ring radius q01;n (mm)
Ring width Dq01;n (mm)
1 2 3 4 5
3.831 7.015 10.173 13.323 16.470
1.163 2.129 3.088 4.044 5.000
1.000 0.546 0.377 0.287 0.232
that of the front director of the LCD and used in its output an analyzer with an orthogonal orientation [11]. For monitoring the results of the experiments, we used a digital camera connected to a computer. A brief description of the experiments follows. First of all, to verify the possibility of phase-only spatial modulation using the available LCD, we carried out an experiment sketched schematically in Fig. 5. The LCD together with the analyzer was placed in one of the arms of the Mach–Zehnder interferometric setup while, to conjugate the di-
Fig. 5. Experimental setup used for investigating the phase transmittance of LCD: L – laser; BE – beam expander; BS – beam splitter; M – mirror; A – analyzer; DC – digital camera; PC – personal computer. The symbols ! and correspond to linear polarization in the orthogonal directions.
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Fig. 6. Changes in the phase transmittance of LCD versus applied video signal. (a) Video pattern applied on the LCD. The half-tone strips correspond to different levels of a grey scale of video signal. (b) Interference fringes observed in experiment. (c) Dependence of phase shift on grey level of video signal.
rections of polarization of the interfered waves, a half-wave plate was inserted into the other arm. We formed a video pattern in the form of strips with different levels of a grey scale (Fig. 6(a)). Changes in the phase transmittance were monitored by examining shifts in the interference fringes (Fig. 6(b)). The measured phase behavior of the LCD is plotted in Fig. 6(c). As one can see from this figure, the phase delay produced by the LCD varies nearly linearly from 0 to 2p within the available grey scale of the applied video signal. At the same time the LCD remains practically transparent beginning from the grey level of 32. Both of these facts show the fitness of the used LCD for our purposes.
Fig. 7. Experimental setup used for generating the light string beam: FL – Fourier transforming lens; the rest of the notations are the same as in Fig. 5.
A.S. Ostrovsky et al. / Optics Communications 207 (2002) 131–138
137
Fig. 8. Results of experiment on generating the light string beam.
In the second experiment, sketched schematically in Fig. 7, we tried to generate a light string beam in accordance with the theory given in the previous section. In this experiment, together with the LCD we used the binary mask with the parameters given by Table 1. To control the LCD, we formed periodically changing with frequency 60 Hz random video patterns of the form shown in Fig. 3(a). As a Fourier transforming lens we used the high-quality single lens with 50 mm diameter and a focal length of 300 mm. We registered the transverse intensity distribution of the generated beam at different distances z behind the back focal plane of the Fourier transforming lens. To provide a solid statistical averaging the registered data, we used the exposure time of several seconds. The results of experiment are shown in Fig. 8. As one can see from this figure the generated beam within the limits given by Eq. (17) may be identified as the light string beam predicted by our theory of partially coherent propagation-invariant fields.
Fig. 9. Experimental setup used for generating the light capillary beam: the notations are the same as in Figs. 2, 5 and 7.
In our third experiment, sketched schematically in Fig. 9, we tried to generate the light capillary beam. This time we used the binary mask with the parameters given by Table 2 and controlled the LCD by applying the random video patterns of the form shown in Fig. 3(b). The results of experiment
Fig. 10. Results of experiment on generating the light capillary beam.
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A.S. Ostrovsky et al. / Optics Communications 207 (2002) 131–138
are shown in Fig. 10. It is necessary to note that these results are somewhat worse than the previous ones, which can be explain by the substantial difficulties of tuning the interferometric setup. Nevertheless, the beam generated in this experiment may be confidently identified as the light capillary beam predicted by our theory.
light capillary beam is explained by the presence of the vortex phase expðih0 Þ in the realization of the secondary source (see, e.g., [12]). Finally, it may be noted that the physical realization of the light string and light capillary beams, demonstrated in this paper, leads us to reflect upon possible practical applications of these remarkable beams. We reserve this task for a future publication.
5. Final remarks Acknowledgements The optical technique proposed in this paper for generating light string and light capillary beams at first sight is quite similar to the one used in [4] to generate the nondiffracting Bessel beam or to the one used in [8] to generate the propagation-invariant Bessel-correlated beam. But there are two essential differences in our technique. Firstly, to produce the needed secondary source, in place of the only annular aperture we use a set of annular apertures with specially chosen radii. Secondly, to illuminate the set of annular apertures, we use a partially coherent source with the special properties and not a completely coherent source or completely incoherent source as it is done in [4] and [8], respectively. In our technique, we completely destroy the spatial coherence of illumination in the radial direction, conserving it (for the case of light string beam) or partially destroying it (for the case of light capillary beam) in the azimuthal direction [see Eqs. (11) and (16), respectively]. These differences allow us to throw light on the physical nature of the light string and light capillary beams. Indeed, now it becames clear that these beams are produced by a superposition of plane waves whose wave vectors lie on a family of conic surfaces and which are completely uncorrelated for different cones and completely correlated or cosine-correlated in each cone. Besides, we may suppose that the hole in the
This work was supported by the Autonomous University of Puebla (Mexico) under the project no. II68G01 and by the Council for Science and Technology (CONACYT) of Mexico under project no. 36875-E.
References [1] A.S. Ostrovsky, G. Martınez-Niconoff, J.C. Ramırez-SanJuan, Opt. Commun. 195 (2001) 27. [2] G. Indebetouw, J. Opt. Soc. Am. A 6 (1989) 150. [3] J. Durnin, J. Opt. Soc. Am. A 4 (1987) 651. [4] J. Durnin, J.J. Miceli Jr., J.H. Eberly, Phys. Rev. Lett. 58 (1987) 1499. [5] R.M. Herman, T.A. Wiggins, J. Opt. Soc. Am. A 8 (1991) 932. [6] A. Vasara, J. Turunen, A.T. Friberg, J. Opt. Soc. Am. A 6 (1989) 1748. [7] F. Gori, G. Guattari, C. Padovani, Opt. Commun. 64 (1987) 311. [8] J. Turunen, A. Vasara, A.T. Friberg, J. Opt. Soc. Am. A 8 (1991) 282. [9] E.W. Marchand, E. Wolf, J. Opt. Soc. Am. 62 (1972) 379. [10] G. Arfken, Mathematical Methods for Physicists, third ed., Academic Press, New York, 1970, p. 576, see also pp. 587 and 797. [11] K. Lu, B.E.A. Saleh, Opt. Eng. 29 (1990) 240. [12] S. Ramee, R. Simon, J. Opt. Soc. Am. A 17 (2000) 84.
Optics Communications 207 (2002) 131–138 www.elsevier.com/locate/optcom
Generation of light string and light capillary beams Andrey S. Ostrovsky a,b,*, Gabriel Martınez-Niconoff b, Julio C. Ramırez-San-Juan b b
a Facultad de Ciencias Fısico Matem aticas, Universidad Aut onoma de Puebla, 72570 Puebla, Pue., Mexico Departamento de Optica, Instituto Nacional de Astrofısica, Optica y Electr onica, 72000 Puebla, Pue., Mexico
Received 20 February 2002; accepted 3 April 2002
Abstract Recently we reported a new modal theory of propagation-invariant optical fields [Opt. Commun. 195 (2001) 27–34]. Within the framework of this theory we predicted the existence of so-called light string and light capillary beams. Here, we propose an optical technique for generating these beams and demonstrate it through experimental results. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.25.F; 42.25.K Keywords: Propagation-invariant field; Coherent-mode structure; Bessel beam; Bessel-correlated beam; Light string beam; Light capillary beam
1. Introduction Recently, we studied the coherent-mode structure of a so-called propagation-invariant field characterized by the invariance of its transverse cross-spectral density function in the propagation direction [1]. We have found three fundamentally different coherent-mode representations of the cross-spectral density function which define three possible classes of the propagation-invariant fields termed by us as propagation-invariant fields of the first, the second and the third kind. The propagation-invariant fields of the first kind represent
*
Corresponding author. Fax: +52-222-247-29-40. E-mail address: [email protected] (A.S. Ostrovsky).
completely coherent so-called nondiffracting beams [2], including the fundamental Bessel beam [3]. Good approximations of such beams can be readily produced in the laboratory by means of a rather simple optical system with an annular aperture [4], an axicon [5] or computer-generated hologram [6]. The propagation-invariant fields of the second and the third kind are partially coherent fields with fundamentally different coherentmode structures. A typical representative of the propagation-invariant fields of the second kind is the so-called Bessel-correlated beam [7], which can be also readily produced in laboratory conditions [8]. Within the class of the propagation-invariant fields of the third kind, we have predicted in theory the existence of the peculiar beams termed by us as light string beam and light capillary beam, in view
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 5 3 1 - 6
132
A.S. Ostrovsky et al. / Optics Communications 207 (2002) 131–138
of their very specific extremely sharply localized energy distribution in the transverse section. However, up to now, the physical realization of such beams remained undone. In this paper, we propose two possible optical techniques for generating light string and light capillary beams and demonstrate these through experimental results.
where ðr; uÞ are the polar coordinates in the space– ðlÞ frequency domain. On substituting for WIII from Eq. (1) into Eq. (2) and realizing the integration, firstly in azimuthal and then in radial directions, with due regard for the relations [10] Z il 2p exp½iðx cos a þ laÞ da ¼ Jl ðxÞ; ð3Þ 2p 0
2. Light string and light capillary beams
Jl ðxÞ ¼ Jl ðxÞ ¼ ð1Þ Jl ðxÞ; Z 1 b Jl ðbxÞJl ðb0 xÞx dx ¼ dðb b0 Þ;
l
The light string and light capillary beams are characterized by the transverse cross-spectral density function given by [1] ðlÞ
WIII ðq1 ; h1 ; q2 ; h2 ; zÞ ¼ cos½lðh1 h2 Þ
1 X n¼1
q q Jl al;n 1 Jl al;n 2 ; R R
we find ð0Þ
AIII ðr1 ; u1 ; r2 ; u2 Þ 2 1 a X R a0;n 0;n ¼ d r1 d r2 a0;n 2pR 2pR n¼1
ð6Þ
and
where l ¼ 0 for the light string, l ¼ 1 for the light capillary, Jl denotes the Bessel function of the first kind and of order l, al;n is the nth zero of the function Jl , R is a real positive constant, and ðq; hÞ are the polar coordinates in the plane z ¼ const. The transverse intensity distributions of these ð0Þ ð1Þ beams, WIII ðq; h; q; hÞ and WIII ðq; h; q; hÞ, calculated with truncating the summation in Eq. (1) by the different values of the number N of zeros of the Bessel function are shown in Fig. 1. As one may conclude from this figure, when N tends to infinity, the field with an intensity distribution ð0Þ WIII ðq; h; q; hÞ represents the infinitely thin light beam and the field with an intensity distribution ð1Þ WIII ðq; h; q; hÞ represents the infinitely thin light tube, a fact that explains the origin of the names given to the corresponding fields. For further consideration it will be useful to know the angular correlation functions or the cross-angular spectra [9] of light string and light capillary beams defined as ðlÞ
AIII ðr1 ; u1 ; r2 ; u2 Þ Z Z 1 Z Z 2p ðlÞ q1 q2 WIII ðq1 ; h1 ; q2 ; h2 ; 0Þ ¼ 0
expfi2p½r2 q2 cosðu2 h2 Þ r1 q1 cosðu1 h1 Þg dq1 dq2 dh1 dh2 ;
ð5Þ
0
ð1Þ
0
ð4Þ
ð2Þ
ð1Þ AIII ðr1 ; u1 ; r2 ; u2 Þ
2 1 X R ¼ cosðu1 u2 Þ a1;n n¼1 a a 1;n 1;n d r1 d r2 : 2pR 2pR ð7Þ
As will be shown further, Eqs. (6) and (7) together with Eq. (2) can be taken as a principle of optical generating the light string and light capillary beams. 3. Optical synthesis of light string and light capillary beams As one may see from Eq. (2), the cross-angular ðlÞ spectrum AIII ðr1 ; u1 ; r2 ; u2 Þ represents a four-dimensional Fourier transform of the cross-spectral density function characterizing a propagation-invariant field of the third kind. On the other hand, it is well known that the cross-spectral densities of the wave fields in the front and the back focal planes of a thin spherical lens are also related by a four-dimensional Fourier transform. Both these facts can be used to realize an optical synthesis of the light string and light capillary beams by producing a secondary planar source with a cross-spectral ðlÞ density function WS ¼ AIII in the front focal plane of a Fourier-transforming lens. We will show now that such a secondary source can be produced by
A.S. Ostrovsky et al. / Optics Communications 207 (2002) 131–138
133
Fig. 1. Transverse intensity distributions of light string (a) and light capillary (b) beams calculated for different values of truncating parameter N.
means of an appropriate spatial-time modulation of the homogeneous plane monochromatic wave. To produce the secondary source with a crossð0Þ spectral density function WS ¼ AIII , one may use a spatial light modulator with a complex amplitude transmittance that can be approximated as follows: N X 1 T ðq ; h Þ ¼ dða0;n q0 q0 Þ expðiWn Þ; a 0;n n¼1 0
0
0
where ðq ; h Þ are the polar coordinates in the front focal plane of a Fourier-transforming lens, q0 is a constant and, for each n, Wn is a real random variable that is uniformly distributed in the interval ð0; 2pÞ and Wn and Wm are statistically independent when n 6¼ m. When illuminating this modulator by a homogeneous plane wave with amplitude V0 the cross-spectrum density function in the output of modulator is given by
WS ðq01 ; h01 ; q02 ; h02 Þ ¼ V02
N X 1 dða0;n q0 q01 Þ 2 a 0;n n¼1
dða0;n q0 q02 Þ:
ð11Þ
Eq. (11) represents the finite sum approximation of ð0Þ the cross-angular spectrum AIII (with due regard for corresponding substitution of arguments), that had to be shown. To produce the secondary source with a crossð1Þ spectral density function WS ¼ AIII , one may use a spatial light modulator with complex amplitude transmittance T 0 ðq0 ; h0 Þ ¼
ð9Þ
where the angle brackets denote the statistical average. Substituting for T ðq0 ; h0 Þ from Eq. (8) into Eq. (9) and taking into account that, with due regard for the accepted statistical properties of random variables Wn ,
ð10Þ
where dnm is the Kronecker symbol, we obtain
ð8Þ
0
WS ðq01 ; h01 ; q02 ; h02 Þ ¼ V02 hT ðq01 ; h01 ÞT ðq02 ; h02 Þi;
hexpðiWn Þ expðiWm Þi ¼ dnm ;
N X 1 dða1;n q0 q0 Þ exp½iUn ðh0 Þ; a 1;n n¼1
ð12Þ where ( 0
Un ðh Þ ¼
2p þ h0 Wn h 0 Wn
for 0 6 h0 6 Wn ; for Wn < h0 6 2p
ð13Þ
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A.S. Ostrovsky et al. / Optics Communications 207 (2002) 131–138
and Wn has the same meaning as in Eq. (8), followed by a Mach–Zehnder interferometer with a Dove prism in one of its arms as is shown in Fig. 2. The Dove prism realizes the mirror mapping of the incident wave field, providing thereby the phase conjugation of the distribution given by Eq. (12). The whole of the system in Fig. 2 acts as a new spatial light modulator with equivalent amplitude transmittance T ðq0 ; h0 Þ ¼
N 1X 1 dða1;n q0 q0 Þ 2 n¼1 a1;n
fexp½iUn ðh0 Þ þ exp½iUn ðh0 Þg ¼
N X 1 dða1;n q0 q0 Þ cosðh0 Wn Þ: a 1;n n¼1
ð14Þ 0
Substituting for T ðq0 ; h Þ from Eq. (14) into Eq. (9) and taking into account that hcosðh01 þ Wn Þ cosðh02 þ Wm Þi ¼ dnm cosðh01 h02 Þ;
ð15Þ
we come to the result WS ðq01 ; h01 ; q02 ; h02 Þ ¼ V02 cosðh01 h02 Þ
N X 1 2 a 1;n n¼1
dða1;n q0 q01 Þdða1;n q0 q02 Þ; ð16Þ which represents the finite sum approximation of ð1Þ the cross-angular spectrum AIII : Now, we will discuss briefly the possibility of physical realization of the space light modulators
with the complex amplitude transmittances given by Eqs. (8) and (12) (for more details see the next section). It is obvious that each of this modulators may be realized as a combination of the static purely amplitude modulator and the dynamic phase modulator with appropriate transmittance functions. The static amplitude modulator may be realized in a good approximation as a binary mask in the form of the transparent rings with the required radii and the widths which are proportional to the weight coefficients attached to the delta functions in Eqs. (8) and (12) (Fig. 3). Such a mask may be easily manufactured by means of a standard photolithographic technique. The dynamic phase modulator must represent a transparent plane screen that introduces the required azimuthal phase delays in the annular zones corresponding to the transparent rings of the binary mask that are changed randomly and independently in discrete moments of time (Fig. 4). As a such random phase screen, the liquid crystal display (LCD) controlled by a computer may be successfully used [11]. Concluding this section, we note that the proposed optical techniques allow us to realize only an approximate synthesis of the propagation-invariant beams described by Eq. (1). There are three main reasons for this. Firstly, as it is well known (see, e.g., [8]), if the source is bounded within a circle of diameter DS and the Fourier transforming lens with the focal length f has a circle input pupil of diameter DF ; the maximum distance at which the propagation-invariant behavior of the generated field can be expected is limited by the value zmax ¼ fDF =DS :
ð17Þ
Secondly, approximating the weighted delta functions by transparent rings of finite width, also
(a) Fig. 2. Secondary source for generating the light capillary beam: SLM – spatial light modulator; BS – beam splitter; M – mirror; DP – Dove prism.
(b)
Fig. 3. Binary masks for generating the light string (a) and light capillary (b) beams.
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Table 1 Parameters of binary mask I Ring number n
J0 zero a0;n
Ring radius q00;n (mm)
Ring width Dq00;n (mm)
1 2 3 4 5
2.404 5.520 8.653 11.791 14.930
0.805 1.848 2.898 3.949 5.000
1.000 0.436 0.278 0.204 0.161
Table 2 Parameters of binary mask II Fig. 4. Different states of a random phase screen used for generating the light string (a) and light capillary (b) beams. The phase delay is presented by grey level.
results in violation of the conditions of propagation invariance. Lastly, as can be seen from Fig. 1, the finite sum approximation of the cross spectral density function WS originating from the finite number of rings used restricts the minimum transverse dimensions of the generated beams.
4. Experimental verification To verify the capacity of the proposed technique of generating light string and light capillary beams, we carried out some optical experiments. In these experiments as a primary source we used a linearly polarized laser beam (k ¼ 0:63 lm, He–Ne laser). To produce the needed secondary sources we used the binary masks with a finite number of transparent rings of variable width, which were specially manufactured by means of photolitographic technique. Taking into account a very fast decrease of the values 1=a20;n and 1=a21;n with n, to simplify our experiments, we limited the number of the transparent rings to N ¼ 5. The geometrical parameters of these masks are given by Tables 1 and 2. As a dynamic phase modulator we had available a commercial CRL’s SVGA2VX-model computer-controlled LCD panel, which works in transmissive mode and has a 33 mm diagonal active display area with 800 600 pixels. Since this device has a 90°-twisted nematic structure, to provide the phase-only modulation, we aligned the polarization direction of the incident light with
Ring number n
J1 zero a1;n
Ring radius q01;n (mm)
Ring width Dq01;n (mm)
1 2 3 4 5
3.831 7.015 10.173 13.323 16.470
1.163 2.129 3.088 4.044 5.000
1.000 0.546 0.377 0.287 0.232
that of the front director of the LCD and used in its output an analyzer with an orthogonal orientation [11]. For monitoring the results of the experiments, we used a digital camera connected to a computer. A brief description of the experiments follows. First of all, to verify the possibility of phase-only spatial modulation using the available LCD, we carried out an experiment sketched schematically in Fig. 5. The LCD together with the analyzer was placed in one of the arms of the Mach–Zehnder interferometric setup while, to conjugate the di-
Fig. 5. Experimental setup used for investigating the phase transmittance of LCD: L – laser; BE – beam expander; BS – beam splitter; M – mirror; A – analyzer; DC – digital camera; PC – personal computer. The symbols ! and correspond to linear polarization in the orthogonal directions.
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Fig. 6. Changes in the phase transmittance of LCD versus applied video signal. (a) Video pattern applied on the LCD. The half-tone strips correspond to different levels of a grey scale of video signal. (b) Interference fringes observed in experiment. (c) Dependence of phase shift on grey level of video signal.
rections of polarization of the interfered waves, a half-wave plate was inserted into the other arm. We formed a video pattern in the form of strips with different levels of a grey scale (Fig. 6(a)). Changes in the phase transmittance were monitored by examining shifts in the interference fringes (Fig. 6(b)). The measured phase behavior of the LCD is plotted in Fig. 6(c). As one can see from this figure, the phase delay produced by the LCD varies nearly linearly from 0 to 2p within the available grey scale of the applied video signal. At the same time the LCD remains practically transparent beginning from the grey level of 32. Both of these facts show the fitness of the used LCD for our purposes.
Fig. 7. Experimental setup used for generating the light string beam: FL – Fourier transforming lens; the rest of the notations are the same as in Fig. 5.
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Fig. 8. Results of experiment on generating the light string beam.
In the second experiment, sketched schematically in Fig. 7, we tried to generate a light string beam in accordance with the theory given in the previous section. In this experiment, together with the LCD we used the binary mask with the parameters given by Table 1. To control the LCD, we formed periodically changing with frequency 60 Hz random video patterns of the form shown in Fig. 3(a). As a Fourier transforming lens we used the high-quality single lens with 50 mm diameter and a focal length of 300 mm. We registered the transverse intensity distribution of the generated beam at different distances z behind the back focal plane of the Fourier transforming lens. To provide a solid statistical averaging the registered data, we used the exposure time of several seconds. The results of experiment are shown in Fig. 8. As one can see from this figure the generated beam within the limits given by Eq. (17) may be identified as the light string beam predicted by our theory of partially coherent propagation-invariant fields.
Fig. 9. Experimental setup used for generating the light capillary beam: the notations are the same as in Figs. 2, 5 and 7.
In our third experiment, sketched schematically in Fig. 9, we tried to generate the light capillary beam. This time we used the binary mask with the parameters given by Table 2 and controlled the LCD by applying the random video patterns of the form shown in Fig. 3(b). The results of experiment
Fig. 10. Results of experiment on generating the light capillary beam.
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are shown in Fig. 10. It is necessary to note that these results are somewhat worse than the previous ones, which can be explain by the substantial difficulties of tuning the interferometric setup. Nevertheless, the beam generated in this experiment may be confidently identified as the light capillary beam predicted by our theory.
light capillary beam is explained by the presence of the vortex phase expðih0 Þ in the realization of the secondary source (see, e.g., [12]). Finally, it may be noted that the physical realization of the light string and light capillary beams, demonstrated in this paper, leads us to reflect upon possible practical applications of these remarkable beams. We reserve this task for a future publication.
5. Final remarks Acknowledgements The optical technique proposed in this paper for generating light string and light capillary beams at first sight is quite similar to the one used in [4] to generate the nondiffracting Bessel beam or to the one used in [8] to generate the propagation-invariant Bessel-correlated beam. But there are two essential differences in our technique. Firstly, to produce the needed secondary source, in place of the only annular aperture we use a set of annular apertures with specially chosen radii. Secondly, to illuminate the set of annular apertures, we use a partially coherent source with the special properties and not a completely coherent source or completely incoherent source as it is done in [4] and [8], respectively. In our technique, we completely destroy the spatial coherence of illumination in the radial direction, conserving it (for the case of light string beam) or partially destroying it (for the case of light capillary beam) in the azimuthal direction [see Eqs. (11) and (16), respectively]. These differences allow us to throw light on the physical nature of the light string and light capillary beams. Indeed, now it becames clear that these beams are produced by a superposition of plane waves whose wave vectors lie on a family of conic surfaces and which are completely uncorrelated for different cones and completely correlated or cosine-correlated in each cone. Besides, we may suppose that the hole in the
This work was supported by the Autonomous University of Puebla (Mexico) under the project no. II68G01 and by the Council for Science and Technology (CONACYT) of Mexico under project no. 36875-E.
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