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Increase of optical resolution by evanescent waves
Physics Letters A 328 (2004) 306–312 www.elsevier.com/locate/pla
Increase of optical resolution by evanescent waves Y. Ben-Aryeh Department of Physics, Technion—Israel Institute of Technology, Haifa 32000, Israel Received 6 January 2004; received in revised form 16 May 2004; accepted 16 May 2004
Communicated by P.R. Holland
Abstract High resolution effects are obtained in simple diffraction patterns from very narrow apertures producing evanescent waves. The quantization of such waves is described by the equations of motion for momentum operators in which the effective propagation wavelength is reduced by many photons cooperative effects, increasing the resolution far beyond the Rayleigh criterion. 2004 Elsevier B.V. All rights reserved. PACS: 42.50.Dv; 42.50.Lc; 42.25.Fx Keywords: High resolution; Quantization of evanescent waves; Momentum operators
1. Introduction The equation of motion for the momentum operator has been used in various articles [1–12] for the description of quantum propagation phenomena, in addition to the use of the more conventional Hamiltonian operator. It is well known from quantum field theory, that while the generator for time evolution is the Hamiltonian Hˆ satisfying the equation of motion
i h¯
ˆ t) ∂ O(z, ˆ t), Hˆ , = O(z, ∂t
(1)
E-mail address: [email protected] (Y. Ben-Aryeh). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.05.069
ˆ t), the corresponding generator for for operator O(z, ˆ satisspace propagation is the momentum operator G fying the equation of motion −i h¯
ˆ t) ∂ O(z, ˆ t), G ˆ . = O(z, ∂z
(2)
These equations give a complete description of the time and one-dimensional space evolution for any opˆ t). The relations between the momentum erator O(z, and the Hamiltonian operators and the quantum mechanical energy–momentum tensor has been discussed in previous works [2–6]. In comparison for the Hamiltonian which is always positive the momentum operator gets positive values for the codirectional propagation while relative to these positive values the momentum of the contradirectional propagation is negative
Y. Ben-Aryeh / Physics Letters A 328 (2004) 306–312
[5–12]. It has been shown that one can replace the equal time commutation relations (CR) by the equal space CR [1–12]. The use of the momentum operator has been shown to be very useful for treating photon statistics effects related to propagation in many quantum optical systems [6–12]. In the present Letter the momentum operator method [1–12] is used to analyze quantum mechanically the superresolution propagation effects obtained by evanescent waves. Although most of the investigations of superresolution obtained in the near field deal with microscopic imaging [13,14], to clarify the basic quantum mechanical (QM) principles by which superresolution is obtained it will suffice to analyze propagation phenomena from very narrow apertures in a screen plane. The fundamental quantum properties of high resolution are found also in simple diffraction patterns from very narrow apertures [15]. It has been shown [16,17] that improvement of the resolution beyond the Rayleigh criterion (claiming that the minimal spatial resolution xmin in the resolving axis is equal to λ/2, where λ is the photon wavelength [18]) can be obtained by the use of evanescent waves. As is well known the diffraction waves are composed of propagating modes (k ω/c) and evanescent modes (k > ω/c). The propagating modes correspond to radiative components which can be captured in the far field and give information on the low spatial frequencies. The nonradiative components are related to high spatial frequencies and yield information about the smallest sub-Rayleigh details. Usually the information included in the evanescent waves is lost. However, by applying small detectors of subwavelength dimensions one can convert part of this field to propagating waves which include information on the smallest details of the object. The existence of the Rayleigh standard quantum limit is related to a classical treatment of optical beams which is described by a stream of uncorrelated photons [16,17]. The origin of the superresolution obtained by evanescent waves can be visualized as photons clustering into N -photon states with de Broglie wavelengths which are proportional to the momentum of N -photon states [15–17]. Following this idea it has been claimed in the previous article [15] that the superresolution obtained by evanescent waves is similar to the superresolution effects obtained by nonclassical entangled electromagnetic (em) states which are referred by the common name of quantum
307
lithography [19–25]. However, it has not been shown explicitly how the evanescent waves can be quantized and it is the aim of the present article to present a simple model for quantization of evanescent waves produced by the diffraction from very narrow apertures in a plane screen. We give in Section 2 a classical analysis of the superresolution obtained by evanescent waves and following such analysis we describe in Section 3 a theoretical model for the quantization of evanescent waves, by the use of the momentum operator method [1–12]. In Section 4 we discuss our results.
2. Classical analysis of evanescent waves We consider a monochromatic scalar wavefield V (ˆr , t) = U (r )eiωt ,
(3)
where the space dependent part U (r ) satisfies the Helmholtz equation 2 ∇ + k02 U (r ) = 0.
(4)
Here, for simplicity, we assume index of refraction equal to 1 so that k0 = ω/c. The sources of the field are located outside the region in which Eq. (4) is valid. The wavefield in the half space z 0 which is diffracted from the screen with the narrow apertures may be represented in the form [26–28] 1 U (x, y, z) = 2π
∞ ∞ A(u, v) −∞ −∞
× exp i(ux + vy + wz) du dv,
(5)
where 1/2 w = k02 − u2 − v 2 when u2 + v 2 k02 , (6a) 1/2 when u2 + v 2 > k02 . (6b) w = i u2 + v 2 − k02 Each plane-wave mode (u, v) in the angular spectrum representation of the field given by Eq. (5) is uniquely specified by one spatial-frequency Fourier component (u, v) of the boundary value of the field U0 (x, y) = U (x, y, 0) in the plane z = 0.
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In order to treat evanescent waves we replace Eq. (5) by Fourier series expansion: ∞ U (x, y, z) = Am(1) exp −im(1)k0x m(1)=0 ∞
+
Am(2) exp im(2)k0x
U (x, y, 0) = T (x, y)C,
m(2)=0
×
∞
the plane z = 0 and is illuminated by a monochromatic plane wave propagating from the region z < 0 along the positive z direction, the space dependent part of the plane wave is represented by U (x, y, z) = Ce−ikz , where the amplitude C is constant. We suppose that the field which emerges from the screen at z = 0 is to a good approximation given by
An(1) exp −in(1)k0 y
n(1)=0 ∞
+
An(2) exp in(2)k0y
n(2)=0
× exp(−γm(i),n(j )z).
(7)
Here the decay in the z direction of the evanescent waves is fixed by the real constants γm(i),n(j ) given by the relation: 2 2 m(i)k0 + n(j )k0 − (γm(i),n(j ))2 = (k0 )2 for i, j = 1, 2; m(i)2 + n(j )2 > 1.
(8)
The special form of Eqs. (7), (8) is found later to be convenient for QM description of the superresolution phenomena obtained by the use of evanescent waves. One should notice that the solutions of Eq. (8) include both decreasing wave as represented in Eq. (7) and also increasing exponential wave which can be neglected only for unbounded region of the evanescent wave. More generally, the ratio between the magnitudes of these two waves is fixed by the boundary conditions, as will be discussed later. Eq. (7) includes Fourier series expansion of 4 momentum modes. The terms in the first curled brackets represent linear combinations of momentum states propagating in +x and −x directions, while the terms in the second curled brackets represent momentum states propagating in the +y and −y directions. Am(i) and An(j ) are the amplitudes for these terms. The decay constant for propagation in the z direction, for each term in the multiplications of Eq. (7), is given by Eq. (8). The high resolution effects are obtained by λ/m(i) and λ/n(j ) de Broglie wavelengths in the x and y directions, respectively. In order to explain and demonstrate the use of Eqs. (7), (8) we bring here a specific example taken from Ref. [28]. Assuming that the screen is placed in
(9)
where T (x, y) is the “transmission function” of the object (for normal incidence at the given temporal frequency of the light). Eq. (9) represents the boundary values of the field in the plane z = 0. This can be represented in terms of all possible two-dimensional spatial periodic components, labeled by two-dimensional spatial frequencies (u, v) [−∞ < u < ∞, −∞ < v < ∞]. The Fourier amplitudes A(u, v) are given by the Fourier inverse of U (x, y, 0) = CT (x, y): 1 A(u, v) = (2π)
∞ ∞ CT (x, y) −∞ −∞
× exp −i(ux + vy) dx dy.
(10)
The basic principle by which superresolution is obtained in the near field is related to the simple equation k02 = kx2 + ky2 + kz2 .
(11)
In the use of evanescent waves it is possible to increase one or two components of k0 leading to third component to be imaginary. Following the example of Ref. [28], by assuming a certain periodic condition, one spatial mode is obtained with a wavevector kx = 5k0 , ky = 0. For this case one gets [28]: k02 = kx2 + kz2 ∼ = (5k0)2 − (4.9k0)2 .
(12)
The above simple analysis corresponds to one Fourier series component of Eq. (7). According to Eq. (12) the propagation in the z direction is composed of linear combination of exponentially decreasing wave proportional to exp(−4.9k0z) and exponentially in increasing wave proportional to exp(4.9k0z). The linear combination of these two terms and the ratio between their magnitudes is fixed by the boundary conditions. For unbounded region of the evanescent waves the exponentially increasing wave should vanish. However, for a bounded length in the z direction, before the evanescent waves detectors, one should include linear com-
Y. Ben-Aryeh / Physics Letters A 328 (2004) 306–312
binations of these two waves as discussed in the next section. One should notice that for a spatial mode given by kx = −5k0 , ky = 0 one gets the same equation (12) (so that the superposition of spatial modes with opposite directions can lead to standing waves [15]). In principle, there are many high spatial modes in the evanescent waves and the Fourier series expansion of Eq. (7) can describe the phenomena obtained by the evanescent waves propagating from narrow apertures in the screen plane. The evanescent waves do not transport energy away from the surface but can transport energy parallel to the surface [27]. However, by applying detectors of subwavelength dimension propagating waves are obtained from the evanescent waves. This effect is related to “tunneling phenomena” which will be described by the quantized model developed in the next section, for a specific system. The same principles should apply also to quantization of more complicated systems.
3. Quantization of the evanescent waves Following the classical equation (7), and taking into account the boundary conditions in the plane z = 0, the initial momentum state is given by ∞ † m(1) † m(2) ˆ ˆ am(1) bx + am(2) b−x m(1)=0
×
∞
n(1)=0
∞ † n(1) † n(2) |0 an(1) bˆ y + an(2) bˆ−y n(2)=0
(13) † here bˆx† and bˆ−x are the creation operators for the momentum h¯ k0 in the +x and −x directions, while bˆy† † and bˆ−y are the corresponding creation operators for the +y and −y directions. The state |0 represents the vacuum state of the four-momentum modes. The propagation of the initial state (1) in the x and y directions is developed according to the momentum operators:
† ˆ x = h¯ k0 bˆx† bˆx − bˆ−x bˆ−x , G † ˆ y = h¯ k0 bˆy† bˆy − bˆ−y bˆ−y . G
(14)
309
Following Eq. (2) we get the equations of motion ∂ bˆx† ˆ † ˆ = bx , Gx = −h¯ k0 bˆx† ∂x −→ bˆx† (x) = bˆx† (0) exp(−ik0 x),
−i h¯
−i h¯
† ∂ bˆ−x
(15)
† † ˆ x = h¯ k0 bˆ −x = bˆ−x ,G
∂x † † −→ bˆ−x (x) = bˆ−x (0) exp(ik0x).
Using Eqs. (15), (16) we get: † m(1) bˆx (x) |0
1/2 exp −im(1)k0x m(1) , = m(1)! m(2) † |0 bˆ−x (x)
1/2 = m(2)! exp im(2)k0x m(2) .
(16)
(17)
(18)
Using in similar way the equations of motion for bˆy† † and bˆ−y we get: † n(1) bˆy (y) |0
1/2 exp −in(1)k0 y n(1) , = n(1)! † n(2) bˆ−y (y) |0 1/2
= n(2)! exp in(2)k0 y n(2) .
(19)
(20)
In deriving Eqs. (15)–(20) we have used the CR for the momentum operators, i.e., the operators belonging to different modes commute while the operators belonging to the same mode obey the equal space CR which are of the same form as the equal time CR [1–12]. We get according to Eqs. (17)–(20) cooperative photon states |m(i) and |n(j ) which have correspondingly λ/m(i) and λ/n(j ) wavelengths, representing sub-Rayleigh fine structures. In Eq. (13) we have quantized only the propagation modes in the x and y directions. The propagation in the z direction is coupled to the propagation in the x and y direction. One should notice that the pho† ˆ † ˆ b−x ; bˆy† bˆy ; bˆ−y b−y ton number operators bˆx† bˆx ; bˆ−x are constants of motion fixed by the Fourier series boundary conditions of the plane screen. For each cooperative photon numbers m(i), n(j ), representing one multiplication term of Eq. (13), we get the constant γm(i),n(j ) of Eq. (8). The phenomena related to Eq. (8) can be described as a special kind of “barrier penetration” effects [29]. Larger values of m(i)
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and n(j ), i.e., larger numbers of cooperative photons by which smaller effective wavelengths are obtained, lead to larger decay constants γm(i),n(j ) . By bringing the detectors very near to the object, i.e., in the region in which the evanescent wave has not completely decayed, the evanescent wave is transformed to (nondecaying) propagating wave. Expectation values for the decaying photon number momentum in the z direction, which is coupled to the propagation term m(i), n(i) of the x, y direction, is given by
† bˆz− (z)bˆz− (z)
† (0)bˆz− (0) exp(−2γm(i),n(j )z). = bˆ z− (21) In a similar calculation for the corresponding amplified photon number we get
† bˆz+ (z)bˆz+ (z) †
= bˆ z+ (22) (0)bˆz+ (0) exp(2γm(i),n(j )z). Eqs. (21), (22) have been given in correspondence to the dispersion relation (11) and to the momentum quantization in the screen plane. The decay of the evanescent momentum in the z direction is obtained by the use of Helmholtz–Kirchoff diffraction theory [28,30]. According to this theory the propagation in the half space region z > 0 is affected by the boundary conditions at the screen plane. The evanescent wave is totally reflected, after a certain depth of penetration beyond the screen (assuming that new boundary conditions are not inserted in this region). Such decay of momentum represents essentially accumulative reflection of the evanescent wave in this penetration region. By putting the detectors very near to the screen plane (in the region of penetration depth) the evanescent wave is affected by new boundary conditions transforming a small part of the evanescent wave to a propagating wave which preserves, however, the fine structure image in the x and y directions. Following this idea, we give here a semiclassical description for the “barrier” penetration effects obtained by the evanescent waves detectors. The general solution for the decay of each term of Eq. (13) is given by Um(i),n(j ) (z) = Bm(i),n(j ) exp(−γm(i),n(j ) z) + Cm(i),n(j ) exp(γm(i),n(j ) z).
(23)
Bm(i),n(j ) and Cm(i),n(j ) are certain amplitudes, respectively, for the decreasing and increasing wave propagating in the z direction, which are coupled to the m(i), n(j ) propagation term in the x, y directions. We require U and dU/dz to be continuous at the detector which is located at a very short distance a from the screen plane. Then we get the boundary conditions at the absorbing detector Um(i),n(j ) (a) = Bm(i),n(j ) exp(−γm(i),n(j ) a) + Cm(i),n(j ) exp(γm(i),n(j ) a) = Am(i),n(j ) exp(−ik0 a), dUm(i),n(j ) (z) (a) dz = Bm(i),n(j ) exp(−γm(i),n(j ) a) − Cm(i),n(j ) exp(γm(i),n(j )a)
ik0 exp(−ik0 a). = Am(i),n(j ) γm(i),n(j )
(24)
(25)
Under the approximation k0 γm(i),n(j ) we get: Bm(i),n(j ) exp(−γm(i),n(j )a) ∼ = Cm(i),n(j ) exp(γm(i),n(j ) a).
(26)
Using the above approximations we get: Am(i),n(j ) = 2Bm(i),n(j ) exp(ik0a) exp(−γm(i),n(j )a).
(27)
The propagating wave in the z direction obtained from the evanescent wave is given by Um(i),n(j ) (z) = Am(i),n(j ) exp(−ik0z),
(28)
where Am(i),n(j ) of Eqs. (27), (28) is proportional to the amplitude multiplication am(i) an(j ) of the m(i), n(j ) term of Eq. (13) and also proportional to a decay term. The propagation modes in the x and y directions are not changed by the boundary conditions at the detector and in this way enable the high resolution effects obtained by the detectors. We can summarize the above analysis as follows. The em field measured by the evanescent wave detectors is proportional to the light intensity: 2 I ∝ U (x, y, z) . (29) Due to the periodic boundary conditions at the screen plane the propagation of the em waves in the x, y plane is produced by cooperative photon momentum states
Y. Ben-Aryeh / Physics Letters A 328 (2004) 306–312
m(i), n(j ) described by Eq. (13) leading to effective wavelengths λ/m(i) and λ/n(j ) in the x and y directions, respectively. The description of cooperative photon states has been based on the use of momentum operators, in Eqs. (13)–(22). The small de Broglie wavelengths are accompanied with evanescent waves decay in the z direction, which are transformed into propagating waves by the evanescent waves detectors. This effect can be considered as “tunneling” from the momentum barrier given by Eq. (8).
4. Discussion and conclusions In the present Letter we have used a classical and corresponding QM model for analyzing superresolution effects obtained by evanescent waves propagating from very narrow apertures in a plane screen. Although we have treated a special system the principles discussed here should apply also to more complicated systems. QM analysis of evanescent waves, for the whole space of the optical system, has been given in previous works [31–33], but the previous analysis turned out to be very complicated without simple relations to superresolution effects. The present analysis for diffraction from very narrow apertures has been simplified by quantizing the scalar field only in the half space region (z > 0) taking into account boundary conditions. The following results are obtained from the present QM model: (a) Classically the one photon resolution is bounded by Rayleigh criterion [18]. We have shown that high resolution effects obtained by evanescent waves are produced by various cooperative photon states |m(i), |n(j ), by which the fine structure of the object is mapped into Fourier series expansion of effective wavelengths λ/m(i), λ/n(j ) (see Eqs. (7) and (13), and for comparison, Refs. [16,17]). The high resolution effects resemble, therefore, those obtained in quantum lithography [19–25]. However, the quantization of the evanescent waves, which have special propagation characteristics, is done in momentum space and therefore momentum operators [1–12] are used in the present analysis, rather than the Hamiltonian. (b) An important result which follows from the QM analysis is that the cooperative photon numbers for propagation in the x and y directions are constants of motion and are not changed by the propagation
311
processes in the z directions (including decay and amplifying processes), and also are not by the transformation of the evanescent wave into a propagating wave at the detector. This fact explains the experimental results [13,14] that the fine structure of the objects are preserved during the measurement process. (c) The measurement of the evanescent wave by detectors of subwavelength dimension located very near the object is related to a “tunneling effect”. In the evanescent waves the exponential decreasing wave, and the increasing one, propagating in the z direction are coupled to the propagation in the x and y directions. The “barrier” which leads to the decay of the evanescent wave is given by the semiclassical Eq. (8). The tunneling effect is fixed by the boundary conditions at the detector, producing a wave propagating in the z direction which is uncoupled to the propagation in the x, y directions. We have demonstrated such effect by assuming specific boundary conditions. One can check experimentally the ideas presented in the present Letter by introducing certain patterns of spatially periodic very narrow apertures (with spatial periods much smaller than wavelength) in a screen plane and observe by detectors of subwavelength dimensions located very near the screen (in distances much smaller than wavelength) the image of the screen. It should be interesting to examine experimentally, by photon statistics methods [34], if the evanescent waves, which have N -photon effects for getting a superresolution, have also other nonclassical properties. This can be realized, for example, by using an optical fiber near the object as a tip-detector which will transmit the propagating wave extracted from the evanescent wave, to photon statistics detectors.
Acknowledgements The author would like to thank Prof. S.G. Lipson for interesting discussions.
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[4] N. Imoto, J.R. Jeffers, R. Loudon, in: P. Tombesi, D.F. Walls (Eds.), Quantum Measurements in Optics, Plenum Press, New York, 1992, p. 295. [5] Y. Ben-Aryeh, A. Lux, V. Perinova, Phys. Lett. A 165 (1992) 19. [6] M. Toren, Y. Ben-Aryeh, Quantum Opt. 6 (1994) 425. [7] J. Perina, J. Mod. Opt. 42 (1995) 1517. [8] J. Perina, J. Perina Jr., Quantum Semiclass. Opt. 7 (1995) 849; J. Perina, J. Perina Jr., Quantum Semiclass. Opt. 7 (1995) 863. [9] J. Perina, J. Perina Jr., J. Mod. Opt. 43 (1996) 1951. [10] A. Luis, J. Perina, Quantum Semiclass. Opt. 8 (1996) 39. [11] G. Ariunbold, J. Perina, Opt. Commun. 176 (2000) 149. [12] G. Ariunbold, J. Perina, J. Mod. Opt. 48 (2001) 1005. [13] D. Courjon, Near Field Microscopy and Near Field Optics, Imperial College Press, London, 2003; D.W. Pohl, D. Courjon (Eds.), Near Field Optics, Kluwer, London, 1992. [14] M.A. Paesler, P.J. Moyer, Near Field Optics, Wiley, New York, 1996. [15] Y. Ben-Aryeh, Quantum Semiclass. Opt. 5 (2003) S553. [16] J.M. Vigoureux, F. Depasse, C. Girard, Appl. Opt. 31 (1992) 3036. [17] J.M. Vigoureux, D. Courjon, Appl. Opt. 31 (1992) 3170. [18] Lord Rayleigh, Philos. Mag. 8 (1879) 261. [19] E.J.S. Fonseca, C.H. Monken, S. Padua, Phys. Rev. Lett. 82 (1999) 2868.
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Increase of optical resolution by evanescent waves Y. Ben-Aryeh Department of Physics, Technion—Israel Institute of Technology, Haifa 32000, Israel Received 6 January 2004; received in revised form 16 May 2004; accepted 16 May 2004
Communicated by P.R. Holland
Abstract High resolution effects are obtained in simple diffraction patterns from very narrow apertures producing evanescent waves. The quantization of such waves is described by the equations of motion for momentum operators in which the effective propagation wavelength is reduced by many photons cooperative effects, increasing the resolution far beyond the Rayleigh criterion. 2004 Elsevier B.V. All rights reserved. PACS: 42.50.Dv; 42.50.Lc; 42.25.Fx Keywords: High resolution; Quantization of evanescent waves; Momentum operators
1. Introduction The equation of motion for the momentum operator has been used in various articles [1–12] for the description of quantum propagation phenomena, in addition to the use of the more conventional Hamiltonian operator. It is well known from quantum field theory, that while the generator for time evolution is the Hamiltonian Hˆ satisfying the equation of motion
i h¯
ˆ t) ∂ O(z, ˆ t), Hˆ , = O(z, ∂t
(1)
E-mail address: [email protected] (Y. Ben-Aryeh). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.05.069
ˆ t), the corresponding generator for for operator O(z, ˆ satisspace propagation is the momentum operator G fying the equation of motion −i h¯
ˆ t) ∂ O(z, ˆ t), G ˆ . = O(z, ∂z
(2)
These equations give a complete description of the time and one-dimensional space evolution for any opˆ t). The relations between the momentum erator O(z, and the Hamiltonian operators and the quantum mechanical energy–momentum tensor has been discussed in previous works [2–6]. In comparison for the Hamiltonian which is always positive the momentum operator gets positive values for the codirectional propagation while relative to these positive values the momentum of the contradirectional propagation is negative
Y. Ben-Aryeh / Physics Letters A 328 (2004) 306–312
[5–12]. It has been shown that one can replace the equal time commutation relations (CR) by the equal space CR [1–12]. The use of the momentum operator has been shown to be very useful for treating photon statistics effects related to propagation in many quantum optical systems [6–12]. In the present Letter the momentum operator method [1–12] is used to analyze quantum mechanically the superresolution propagation effects obtained by evanescent waves. Although most of the investigations of superresolution obtained in the near field deal with microscopic imaging [13,14], to clarify the basic quantum mechanical (QM) principles by which superresolution is obtained it will suffice to analyze propagation phenomena from very narrow apertures in a screen plane. The fundamental quantum properties of high resolution are found also in simple diffraction patterns from very narrow apertures [15]. It has been shown [16,17] that improvement of the resolution beyond the Rayleigh criterion (claiming that the minimal spatial resolution xmin in the resolving axis is equal to λ/2, where λ is the photon wavelength [18]) can be obtained by the use of evanescent waves. As is well known the diffraction waves are composed of propagating modes (k ω/c) and evanescent modes (k > ω/c). The propagating modes correspond to radiative components which can be captured in the far field and give information on the low spatial frequencies. The nonradiative components are related to high spatial frequencies and yield information about the smallest sub-Rayleigh details. Usually the information included in the evanescent waves is lost. However, by applying small detectors of subwavelength dimensions one can convert part of this field to propagating waves which include information on the smallest details of the object. The existence of the Rayleigh standard quantum limit is related to a classical treatment of optical beams which is described by a stream of uncorrelated photons [16,17]. The origin of the superresolution obtained by evanescent waves can be visualized as photons clustering into N -photon states with de Broglie wavelengths which are proportional to the momentum of N -photon states [15–17]. Following this idea it has been claimed in the previous article [15] that the superresolution obtained by evanescent waves is similar to the superresolution effects obtained by nonclassical entangled electromagnetic (em) states which are referred by the common name of quantum
307
lithography [19–25]. However, it has not been shown explicitly how the evanescent waves can be quantized and it is the aim of the present article to present a simple model for quantization of evanescent waves produced by the diffraction from very narrow apertures in a plane screen. We give in Section 2 a classical analysis of the superresolution obtained by evanescent waves and following such analysis we describe in Section 3 a theoretical model for the quantization of evanescent waves, by the use of the momentum operator method [1–12]. In Section 4 we discuss our results.
2. Classical analysis of evanescent waves We consider a monochromatic scalar wavefield V (ˆr , t) = U (r )eiωt ,
(3)
where the space dependent part U (r ) satisfies the Helmholtz equation 2 ∇ + k02 U (r ) = 0.
(4)
Here, for simplicity, we assume index of refraction equal to 1 so that k0 = ω/c. The sources of the field are located outside the region in which Eq. (4) is valid. The wavefield in the half space z 0 which is diffracted from the screen with the narrow apertures may be represented in the form [26–28] 1 U (x, y, z) = 2π
∞ ∞ A(u, v) −∞ −∞
× exp i(ux + vy + wz) du dv,
(5)
where 1/2 w = k02 − u2 − v 2 when u2 + v 2 k02 , (6a) 1/2 when u2 + v 2 > k02 . (6b) w = i u2 + v 2 − k02 Each plane-wave mode (u, v) in the angular spectrum representation of the field given by Eq. (5) is uniquely specified by one spatial-frequency Fourier component (u, v) of the boundary value of the field U0 (x, y) = U (x, y, 0) in the plane z = 0.
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In order to treat evanescent waves we replace Eq. (5) by Fourier series expansion: ∞ U (x, y, z) = Am(1) exp −im(1)k0x m(1)=0 ∞
+
Am(2) exp im(2)k0x
U (x, y, 0) = T (x, y)C,
m(2)=0
×
∞
the plane z = 0 and is illuminated by a monochromatic plane wave propagating from the region z < 0 along the positive z direction, the space dependent part of the plane wave is represented by U (x, y, z) = Ce−ikz , where the amplitude C is constant. We suppose that the field which emerges from the screen at z = 0 is to a good approximation given by
An(1) exp −in(1)k0 y
n(1)=0 ∞
+
An(2) exp in(2)k0y
n(2)=0
× exp(−γm(i),n(j )z).
(7)
Here the decay in the z direction of the evanescent waves is fixed by the real constants γm(i),n(j ) given by the relation: 2 2 m(i)k0 + n(j )k0 − (γm(i),n(j ))2 = (k0 )2 for i, j = 1, 2; m(i)2 + n(j )2 > 1.
(8)
The special form of Eqs. (7), (8) is found later to be convenient for QM description of the superresolution phenomena obtained by the use of evanescent waves. One should notice that the solutions of Eq. (8) include both decreasing wave as represented in Eq. (7) and also increasing exponential wave which can be neglected only for unbounded region of the evanescent wave. More generally, the ratio between the magnitudes of these two waves is fixed by the boundary conditions, as will be discussed later. Eq. (7) includes Fourier series expansion of 4 momentum modes. The terms in the first curled brackets represent linear combinations of momentum states propagating in +x and −x directions, while the terms in the second curled brackets represent momentum states propagating in the +y and −y directions. Am(i) and An(j ) are the amplitudes for these terms. The decay constant for propagation in the z direction, for each term in the multiplications of Eq. (7), is given by Eq. (8). The high resolution effects are obtained by λ/m(i) and λ/n(j ) de Broglie wavelengths in the x and y directions, respectively. In order to explain and demonstrate the use of Eqs. (7), (8) we bring here a specific example taken from Ref. [28]. Assuming that the screen is placed in
(9)
where T (x, y) is the “transmission function” of the object (for normal incidence at the given temporal frequency of the light). Eq. (9) represents the boundary values of the field in the plane z = 0. This can be represented in terms of all possible two-dimensional spatial periodic components, labeled by two-dimensional spatial frequencies (u, v) [−∞ < u < ∞, −∞ < v < ∞]. The Fourier amplitudes A(u, v) are given by the Fourier inverse of U (x, y, 0) = CT (x, y): 1 A(u, v) = (2π)
∞ ∞ CT (x, y) −∞ −∞
× exp −i(ux + vy) dx dy.
(10)
The basic principle by which superresolution is obtained in the near field is related to the simple equation k02 = kx2 + ky2 + kz2 .
(11)
In the use of evanescent waves it is possible to increase one or two components of k0 leading to third component to be imaginary. Following the example of Ref. [28], by assuming a certain periodic condition, one spatial mode is obtained with a wavevector kx = 5k0 , ky = 0. For this case one gets [28]: k02 = kx2 + kz2 ∼ = (5k0)2 − (4.9k0)2 .
(12)
The above simple analysis corresponds to one Fourier series component of Eq. (7). According to Eq. (12) the propagation in the z direction is composed of linear combination of exponentially decreasing wave proportional to exp(−4.9k0z) and exponentially in increasing wave proportional to exp(4.9k0z). The linear combination of these two terms and the ratio between their magnitudes is fixed by the boundary conditions. For unbounded region of the evanescent waves the exponentially increasing wave should vanish. However, for a bounded length in the z direction, before the evanescent waves detectors, one should include linear com-
Y. Ben-Aryeh / Physics Letters A 328 (2004) 306–312
binations of these two waves as discussed in the next section. One should notice that for a spatial mode given by kx = −5k0 , ky = 0 one gets the same equation (12) (so that the superposition of spatial modes with opposite directions can lead to standing waves [15]). In principle, there are many high spatial modes in the evanescent waves and the Fourier series expansion of Eq. (7) can describe the phenomena obtained by the evanescent waves propagating from narrow apertures in the screen plane. The evanescent waves do not transport energy away from the surface but can transport energy parallel to the surface [27]. However, by applying detectors of subwavelength dimension propagating waves are obtained from the evanescent waves. This effect is related to “tunneling phenomena” which will be described by the quantized model developed in the next section, for a specific system. The same principles should apply also to quantization of more complicated systems.
3. Quantization of the evanescent waves Following the classical equation (7), and taking into account the boundary conditions in the plane z = 0, the initial momentum state is given by ∞ † m(1) † m(2) ˆ ˆ am(1) bx + am(2) b−x m(1)=0
×
∞
n(1)=0
∞ † n(1) † n(2) |0 an(1) bˆ y + an(2) bˆ−y n(2)=0
(13) † here bˆx† and bˆ−x are the creation operators for the momentum h¯ k0 in the +x and −x directions, while bˆy† † and bˆ−y are the corresponding creation operators for the +y and −y directions. The state |0 represents the vacuum state of the four-momentum modes. The propagation of the initial state (1) in the x and y directions is developed according to the momentum operators:
† ˆ x = h¯ k0 bˆx† bˆx − bˆ−x bˆ−x , G † ˆ y = h¯ k0 bˆy† bˆy − bˆ−y bˆ−y . G
(14)
309
Following Eq. (2) we get the equations of motion ∂ bˆx† ˆ † ˆ = bx , Gx = −h¯ k0 bˆx† ∂x −→ bˆx† (x) = bˆx† (0) exp(−ik0 x),
−i h¯
−i h¯
† ∂ bˆ−x
(15)
† † ˆ x = h¯ k0 bˆ −x = bˆ−x ,G
∂x † † −→ bˆ−x (x) = bˆ−x (0) exp(ik0x).
Using Eqs. (15), (16) we get: † m(1) bˆx (x) |0
1/2 exp −im(1)k0x m(1) , = m(1)! m(2) † |0 bˆ−x (x)
1/2 = m(2)! exp im(2)k0x m(2) .
(16)
(17)
(18)
Using in similar way the equations of motion for bˆy† † and bˆ−y we get: † n(1) bˆy (y) |0
1/2 exp −in(1)k0 y n(1) , = n(1)! † n(2) bˆ−y (y) |0 1/2
= n(2)! exp in(2)k0 y n(2) .
(19)
(20)
In deriving Eqs. (15)–(20) we have used the CR for the momentum operators, i.e., the operators belonging to different modes commute while the operators belonging to the same mode obey the equal space CR which are of the same form as the equal time CR [1–12]. We get according to Eqs. (17)–(20) cooperative photon states |m(i) and |n(j ) which have correspondingly λ/m(i) and λ/n(j ) wavelengths, representing sub-Rayleigh fine structures. In Eq. (13) we have quantized only the propagation modes in the x and y directions. The propagation in the z direction is coupled to the propagation in the x and y direction. One should notice that the pho† ˆ † ˆ b−x ; bˆy† bˆy ; bˆ−y b−y ton number operators bˆx† bˆx ; bˆ−x are constants of motion fixed by the Fourier series boundary conditions of the plane screen. For each cooperative photon numbers m(i), n(j ), representing one multiplication term of Eq. (13), we get the constant γm(i),n(j ) of Eq. (8). The phenomena related to Eq. (8) can be described as a special kind of “barrier penetration” effects [29]. Larger values of m(i)
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and n(j ), i.e., larger numbers of cooperative photons by which smaller effective wavelengths are obtained, lead to larger decay constants γm(i),n(j ) . By bringing the detectors very near to the object, i.e., in the region in which the evanescent wave has not completely decayed, the evanescent wave is transformed to (nondecaying) propagating wave. Expectation values for the decaying photon number momentum in the z direction, which is coupled to the propagation term m(i), n(i) of the x, y direction, is given by
† bˆz− (z)bˆz− (z)
† (0)bˆz− (0) exp(−2γm(i),n(j )z). = bˆ z− (21) In a similar calculation for the corresponding amplified photon number we get
† bˆz+ (z)bˆz+ (z) †
= bˆ z+ (22) (0)bˆz+ (0) exp(2γm(i),n(j )z). Eqs. (21), (22) have been given in correspondence to the dispersion relation (11) and to the momentum quantization in the screen plane. The decay of the evanescent momentum in the z direction is obtained by the use of Helmholtz–Kirchoff diffraction theory [28,30]. According to this theory the propagation in the half space region z > 0 is affected by the boundary conditions at the screen plane. The evanescent wave is totally reflected, after a certain depth of penetration beyond the screen (assuming that new boundary conditions are not inserted in this region). Such decay of momentum represents essentially accumulative reflection of the evanescent wave in this penetration region. By putting the detectors very near to the screen plane (in the region of penetration depth) the evanescent wave is affected by new boundary conditions transforming a small part of the evanescent wave to a propagating wave which preserves, however, the fine structure image in the x and y directions. Following this idea, we give here a semiclassical description for the “barrier” penetration effects obtained by the evanescent waves detectors. The general solution for the decay of each term of Eq. (13) is given by Um(i),n(j ) (z) = Bm(i),n(j ) exp(−γm(i),n(j ) z) + Cm(i),n(j ) exp(γm(i),n(j ) z).
(23)
Bm(i),n(j ) and Cm(i),n(j ) are certain amplitudes, respectively, for the decreasing and increasing wave propagating in the z direction, which are coupled to the m(i), n(j ) propagation term in the x, y directions. We require U and dU/dz to be continuous at the detector which is located at a very short distance a from the screen plane. Then we get the boundary conditions at the absorbing detector Um(i),n(j ) (a) = Bm(i),n(j ) exp(−γm(i),n(j ) a) + Cm(i),n(j ) exp(γm(i),n(j ) a) = Am(i),n(j ) exp(−ik0 a), dUm(i),n(j ) (z) (a) dz = Bm(i),n(j ) exp(−γm(i),n(j ) a) − Cm(i),n(j ) exp(γm(i),n(j )a)
ik0 exp(−ik0 a). = Am(i),n(j ) γm(i),n(j )
(24)
(25)
Under the approximation k0 γm(i),n(j ) we get: Bm(i),n(j ) exp(−γm(i),n(j )a) ∼ = Cm(i),n(j ) exp(γm(i),n(j ) a).
(26)
Using the above approximations we get: Am(i),n(j ) = 2Bm(i),n(j ) exp(ik0a) exp(−γm(i),n(j )a).
(27)
The propagating wave in the z direction obtained from the evanescent wave is given by Um(i),n(j ) (z) = Am(i),n(j ) exp(−ik0z),
(28)
where Am(i),n(j ) of Eqs. (27), (28) is proportional to the amplitude multiplication am(i) an(j ) of the m(i), n(j ) term of Eq. (13) and also proportional to a decay term. The propagation modes in the x and y directions are not changed by the boundary conditions at the detector and in this way enable the high resolution effects obtained by the detectors. We can summarize the above analysis as follows. The em field measured by the evanescent wave detectors is proportional to the light intensity: 2 I ∝ U (x, y, z) . (29) Due to the periodic boundary conditions at the screen plane the propagation of the em waves in the x, y plane is produced by cooperative photon momentum states
Y. Ben-Aryeh / Physics Letters A 328 (2004) 306–312
m(i), n(j ) described by Eq. (13) leading to effective wavelengths λ/m(i) and λ/n(j ) in the x and y directions, respectively. The description of cooperative photon states has been based on the use of momentum operators, in Eqs. (13)–(22). The small de Broglie wavelengths are accompanied with evanescent waves decay in the z direction, which are transformed into propagating waves by the evanescent waves detectors. This effect can be considered as “tunneling” from the momentum barrier given by Eq. (8).
4. Discussion and conclusions In the present Letter we have used a classical and corresponding QM model for analyzing superresolution effects obtained by evanescent waves propagating from very narrow apertures in a plane screen. Although we have treated a special system the principles discussed here should apply also to more complicated systems. QM analysis of evanescent waves, for the whole space of the optical system, has been given in previous works [31–33], but the previous analysis turned out to be very complicated without simple relations to superresolution effects. The present analysis for diffraction from very narrow apertures has been simplified by quantizing the scalar field only in the half space region (z > 0) taking into account boundary conditions. The following results are obtained from the present QM model: (a) Classically the one photon resolution is bounded by Rayleigh criterion [18]. We have shown that high resolution effects obtained by evanescent waves are produced by various cooperative photon states |m(i), |n(j ), by which the fine structure of the object is mapped into Fourier series expansion of effective wavelengths λ/m(i), λ/n(j ) (see Eqs. (7) and (13), and for comparison, Refs. [16,17]). The high resolution effects resemble, therefore, those obtained in quantum lithography [19–25]. However, the quantization of the evanescent waves, which have special propagation characteristics, is done in momentum space and therefore momentum operators [1–12] are used in the present analysis, rather than the Hamiltonian. (b) An important result which follows from the QM analysis is that the cooperative photon numbers for propagation in the x and y directions are constants of motion and are not changed by the propagation
311
processes in the z directions (including decay and amplifying processes), and also are not by the transformation of the evanescent wave into a propagating wave at the detector. This fact explains the experimental results [13,14] that the fine structure of the objects are preserved during the measurement process. (c) The measurement of the evanescent wave by detectors of subwavelength dimension located very near the object is related to a “tunneling effect”. In the evanescent waves the exponential decreasing wave, and the increasing one, propagating in the z direction are coupled to the propagation in the x and y directions. The “barrier” which leads to the decay of the evanescent wave is given by the semiclassical Eq. (8). The tunneling effect is fixed by the boundary conditions at the detector, producing a wave propagating in the z direction which is uncoupled to the propagation in the x, y directions. We have demonstrated such effect by assuming specific boundary conditions. One can check experimentally the ideas presented in the present Letter by introducing certain patterns of spatially periodic very narrow apertures (with spatial periods much smaller than wavelength) in a screen plane and observe by detectors of subwavelength dimensions located very near the screen (in distances much smaller than wavelength) the image of the screen. It should be interesting to examine experimentally, by photon statistics methods [34], if the evanescent waves, which have N -photon effects for getting a superresolution, have also other nonclassical properties. This can be realized, for example, by using an optical fiber near the object as a tip-detector which will transmit the propagating wave extracted from the evanescent wave, to photon statistics detectors.
Acknowledgements The author would like to thank Prof. S.G. Lipson for interesting discussions.
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