Atomic spontaneous decay near a finite-length dielectric cylinder

Optics Communications 355 (2015) 27–32

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Atomic spontaneous decay near a finite-length dielectric cylinder Ho Trung Dung a,b,n, Tran Minh Hien c a b c

Theoretical Physics Research Group, Ton Duc Thang University, 19 Nguyen Huu Tho St., District 7, Ho Chi Minh city, Vietnam Faculty of Applied Sciences, Ton Duc Thang University, 19 Nguyen Huu Tho St., District 7, Ho Chi Minh city, Vietnam Institute of Physics, Academy of Science and Technology, 1 Mac Dinh Chi St., District 1, Ho Chi Minh city, Vietnam

art ic l e i nf o

a b s t r a c t

Article history: Received 21 August 2013 Received in revised form 25 May 2015 Accepted 5 June 2015 Available online 15 June 2015

Using a truncated Born expansion of the Green tensor, we consider the spontaneous decay rate of an excited two-level atom placed in the vicinity of a finite-length dielectric cylinder. It is shown that the spontaneous decay rate experiences most pronounced modifications in the edge regions. By comparing the results obtained for finite-length cylinders with those obtained for an infinitely extended one, we establish the parameters ranges where the latter is a valid approximation. We discuss how some typical singularities in the Born series, which arise when the integral variables coincide, can be removed. & 2015 Elsevier B.V. All rights reserved.

Keywords: Spontaneous decay Finite-length cylinder Atom-photon interactions Born expansion

1. Introduction It is well known that the spontaneous decay process can be controlled by tailoring the environment surrounding the emitters, a fact that holds the key to powerful applications in optoelectronic devices. Characteristics of the decay process can be formulated in terms of the Green tensor [1,2], which, unfortunately, is known only for highly symmetric geometries such as infinitely extended planar and cylindrical structures, and spherical structures [3]. In reality, planar and cylindrical structures have finite dimensions in all three spatial directions. When the sizes of these structures decrease, or when the host emitters are located too close to the edges, the effects of finite sizes have to be treated with care. Methods heavily relying on numerics such as the finite difference time domain algorithm can handle arbitrary geometries, but often require extensive computational resources and have their own limitations such as errors introduced by the finite sizes of the computational grids [4]. A simple and straightforward approach to realistically sized structures is clearly desirable. Even when such an approach is not strictly applicable, it still can provide a rough estimation of the effects of the boundaries. The approach we have in mind is based on expanding the Green tensor around some known background Green tensor as a Born series. Under appropriate conditions, the series can be truncated, preserving only one n Corresponding author at: Theoretical Physics Research Group, Ton Duc Thang University, 19 Nguyen Huu Tho St., District 7, Ho Chi Minh city, Vietnam. Fax: þ84 8 37 755 055. E-mail address: [email protected] (H.T. Dung).

http://dx.doi.org/10.1016/j.optcom.2015.06.019 0030-4018/& 2015 Elsevier B.V. All rights reserved.

or a few leading-order terms. This method has been employed for a consideration of the Casimir–Polder interaction [5,6], local-field corrections [7], the decay rate of an atom placed near a rectangular plate [8], Casimir forces [9,10], and the spontaneous decay rate near lithographed surfaces [6]. Cylindrically symmetric structures appear in practical applications such as optical fibers, carbon nanotubes, and two-dimensional photonic crystals. They are a key component in recent considerations of atom chips [11], spontaneous decay rates and level shifts [12–19], zero-point energy [20], Casimir–Polder interactions [21–23], and intermolecular resonance energy transfer [24]. In all these works, the cylinders are assumed to be infinitely long. The spontaneous emission of a two-level system coupled to a single-end, one-dimensional photonic waveguide has been investigated in Ref. [25] and the efficiency of the single-mode photon coupling near a nanofiber tip has been studied in Ref. [26], using the finite difference time domain method. Here we employ the Born expansion approach to address the problem of atomic spontaneous decay in the presence of a finitelength dielectric cylinder. The Green tensor of the cylinder is expanded around the vacuum Green tensor and is truncated based on the assumption of optical diluteness of the medium. Boundary conditions appear in the formulas as integral limits. There is a kind of common singularity in the Born series, which arises because some denominators vanish when the integral variables coincide with each other or with the atomic coordinates. We present attempts to deal with these singularities in the first-order term, when the atom is embedded in the medium, and in the second-order term, when the atom is located outside the medium.

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H.T. Dung, T.M. Hien / Optics Communications 355 (2015) 27–32

The atomic decay rate near a finite-length cylinder and that occurring near an infinitely long one are compared, and the limitations of the latter as an approximate model are discussed. The effects of the boundaries in the direction of the cylinder axis on the atomic decay rate are investigated.

∞

G (r, r ′, ω) = G¯ (r, r ′, ω) +

∑ G k (r, r′, ω),

(6)

k=1

⎛ k G k (r, r ′, ω) = ⎜⎜∏ ⎝ j=1

⎞

∫ d3s j ⎟⎟ ⎠

× G¯ (r, s1, ω) G˜ (s1, s2 , ω)⋯G˜ (s k , r ′, ω), 2. Born expansion of the decay rate The system under consideration consists of an initially excited two-level atom (atomic dipole moment d A , transition frequency ω A ) located at position rA in the proximity of a dielectric cylinder of radius R and height H. We choose a coordinate system such that its origin is located at the center of the cylinder, as sketched in Fig. 1. In the electric-dipole and rotating-wave approximations, the atomic decay rate can be expressed in terms of the imaginary part of the Green tensor as [1,2]

Γ=

2k A2 d A Im G (rA, rA, ω A ) d A =ε 0

(1)

(k A = ω A /c ). In the rigorous quantization scheme that leads to Eq. (1), the dielectric medium surrounding the host atom is described macroscopically, via a frequency- and space-dependent complex permittivity ε (r, ω) which obeys the Kramers–Kronig relations [2]. With the Born expansion in mind, we decompose the permittivity as

ε (r, ω) = ε¯ (r, ω) + χε (r, ω),

(2)

where χε is a perturbation to ε¯ . This is relevant in description of, say, weakly dielectric bodies for which

ε¯ (r, ω) = 1,

|χε (r, ω)|⪡1.

(3)

(7)

ω2 G˜ (r, r ′, ω) = χε (r, ω) G¯ (r, r ′, ω). c2

(8)

Substituting Eqs. (6) and (7) in Eq. (1), one obtains the Born series for the normalized spontaneous decay rate

Γ =1+ Γ0

⎛Γ⎞ ⎟ , ⎝ Γ0 ⎠k k=1 ∞

∑⎜

(9)

⎛Γ⎞ 6π ^ ^ d A Im G k (rA, rA, ω A ) d A , ⎜ ⎟ = ⎝ Γ0 ⎠k k A where Γ0 = k A3 d A2 /(3π=ε0 ) ^ d A = d A /dA .

(10)

is the free-space decay rate and

2.1. First-order term Using Eqs. (7) and (8) in Eq. (10), we can write the term linearly dependent on χε as

⎛Γ⎞ ^ ⎜ ⎟ = 6πk A d A Im ⎝ Γ0 ⎠1

∫ d3sχε (s, ω A ) G¯ (rA, s, ω A ) G¯ (s, rA, ω A ) d^ A .

(11)

The unperturbed Green tensor is then the vacuum Green tensor

kA δ (u) ^ ⊗u ^ ) eiq , G¯ (r, r ′, ω A ) = − I+ (aI − bu 4π 3k A2

(4)

where

a ≡ a (q) =

i 1 1 + − , q q2 q3

b ≡ b (q) =

1 3i 3 + − , q q2 q3

(5)

^ = u/u , q = k A u. The Green tensor can be exand u = r − r′, u panded around the vacuum one with the susceptibility χε (r, ω) being the small parameter

z R

H

rA y

x Fig. 1. A dipole emitter in the vicinity of a finite-length cylinder.

If the atom is located outside the macroscopic body, the deltafunction in the free-space Green tensor, Eq. (4), does not contribute, and one obtains the following expression for the first-order term in the Born series of the spontaneous decay rate

⎧ ⎛Γ⎞ ⎪ 3k A3 ⎨ Im ⎪ ⎜ ⎟ = 8π ⎝ Γ0 ⎠1 ⎩

∫ d3sχε (s, ω A )

⎫ ^ ^ 2 2iq⎪ ⎬, ×[a2 + (b2 − 2ab)(d A u ) ]e ⎪ ⎭

(12)

where a and b are given as in Eqs. (5), with u = s − rA . For an atom positioned outside the macroscopic body, the integrand in Eq. (12) is well-behaved. Eq. (12) holds for an arbitrarily shaped body and just like Eq. (1), allows for both material dispersion and absorption. Specific shape and size of the bodies are taken care of via the integral limits. It has been employed for an investigation of the atomic spontaneous decay rate in the presence of a rectangular plate [8]. For the geometry of the material body under consideration, it is convenient to use circular cylindrical coordinates rA = (ρA , φA , z A ) and s = (ρ , φ , z ). It is assumed from now on that the cylinder is uniform, then the integral in Eq. (12) becomes H /2

∫ d3sχε (s, ω A )⋯ → χε (ω A ) ∫−H/2

dz

∫0

R

ρ dρ

∫0

2π

d φ⋯ .

(13)

The expression under the integral in Eq. (12) can be written explicitly in the cylindrical coordinates via

u = [(z A − z )2 + ρ A2 + ρ2 − 2ρ A ρ cos (φ A − φ)]1/2 ,

(14)

H.T. Dung, T.M. Hien / Optics Communications 355 (2015) 27–32

and

zA − z ^ ^ = dA u u

(15)

^ for a dipole moment oriented along the z-axis d A = z^ A ,

ρ A − ρ cos (φ A − φ) ^ ^ = dA u u

(16)

^ for a dipole moment oriented in the radial direction d A = ρ^A , and

ρ sin (φ A − φ) ^ ^ = dA u u

(17)

^ ^A. for a dipole moment oriented in the tangential direction d A = φ When the atom is embedded in the macroscopic body, denominators in Eq. (11) vanish when the integral variables coincide with the coordinates of the atom s = rA (u¼ 0). One way to avoid this complication is to choose the bulk Green tensor as the reference point around which the Green tensor of the cylinder is expanded. Assuming for simplicity that the medium is nonabsorbing, then the first term in Eq. (9) becomes ε¯ instead of one. The linear term is of the same form as in Eq. (12), with q = ε¯ k A u. Since now inside the cylinder and χε (s, ω A ) = 0 χε (s, ω A ) = − χε (ω A ) outside of it, the integral region lies outside the cylinder and can be decomposed into three parts, in cylindrical coordinates,

⎛

H /2

∫ d3sχε (s, ω A )⋯ → − χε (ω A ) ⎜⎝ ∫−H/2 ∞

+

∫H/2

dz

∫0

∞

ρ dρ

∫0

2π

dφ +

dz

− H /2

∫−∞

∫R dz

∞

ρ dρ

∫0

∞

∫0

ρ dρ

2π

∫0

dφ 2π

⎞ d φ⎟ ⋯ . ⎠ (18)

A small positive imaginary part can be added in the permittivity to ensure that the integrals converge. This is consistent with the boundary condition that the Green tensor has to satisfy at infinity. Note that here we ignore local-field corrections. The local-field corrections for atoms embedded in a finite-size body can be taken into account using, e.g., the prescription given in Ref. [7]. 2.2. Second-order term From Eq. (10), the second-order term can be written as

⎛Γ⎞ ^ ⎜ ⎟ = 6πk A3 d A Im ⎝ Γ0 ⎠2

∫ d3s1 ∫ d3s2 χε (s1, ω A ) χε (s2, ω A )

^ × G¯ (rA, s1, ω A ) G¯ (s1, s2 , ω A ) G¯ (s2 , rA, ω A ) d A .

(19)

We restrict ourselves to the situation where the atom is positioned outside the macroscopic body, for which case the function under integral contains singularities at s1 = s2. Even though these singularities are superficial, they make it difficult to compute (Γ /Γ0 )2 numerically. Next, an example is given on how the singularities can be tackled. Using Eq. (4) in Eq. (19), one can decompose the second-order term into two parts

⎛Γ⎞ ⎛Γ⎞ ⎛Γ⎞ ⎜ ⎟ =⎜ ⎟ +⎜ ⎟ , ⎝ Γ0 ⎠2 ⎝ Γ0 ⎠21 ⎝ Γ0 ⎠22

⎞ ⎛Γ⎞ ⎛ 1 ⎜ ⎟ ⎜⎜χε (s , ω A ) → − χε2 (s , ω A ) ⎟⎟, 3 ⎝ Γ0 ⎠1⎝ ⎠

(Γ /Γ0 )1 being given as in Eq. (12), and

∫ d3s1 ∫ d3s2 χε (s1, ω A ) χε (s2, ω A )

^ ^ 2 × ei (q1+ q2 + q12 ) {a1a2 [a12 − b12 (d A u 12 ) ] ^ ^ ^ ^ ^ ^ ^ u ^ − a1b2 (d A u2 )[a12 (d A u2 ) − b12 (d A u12 )(u 2 12 )] ^ ^ ^ ^ ^ ^ ^ ^ − a2 b1 (d A u 1)[a12 (d A u1) − b12 (d A u12 )(u1u12 )] ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ + b1b2 (d A u 1)(d A u2 )[a12 (u1u2 ) − b12 (u1u12 )(u2 u12 )]} . (22) Here u2 = s2 − rA , q1,2 = k A u1,2 , u1 = rA − s1, u12 = s1 − s2, q12 = k A u12, a1,2 = a (q1,2 ), a12 = a (q12 ), and similarly for b. In cylindrical coordinates, denoting s1 = (ρ1, φ1, z1), s2 = (ρ2 , φ2, z2 ) one can write 2 − 2ρ1,2 ρ A cos (φ A − φ1,2 )]1/2 , u1,2 = [(u1,2 )2z + ρ A2 + ρ1,2

(23)

u12 = [(u12 )2z + ρ12 + ρ22 − 2ρ1 ρ2 cos (φ1 − φ2 )]1/2 ,

(24)

^ u ^ u 1 2 = [(u1) z (u2 ) z + (ρA − ρ1 )(ρ2 − ρA )]/u1u2 ,

(25)

^ u ^ u 1 12 = [(u1) z (u12 ) z + (ρA − ρ1 )(ρ1 − ρ2 )]/u1u12 ,

(26)

^ u ^ u 2 12 = [(u2 ) z (u12 ) z + (ρ2 − ρA )(ρ1 − ρ2 )]/u2 u12 .

(27)

^ ^ ^ ^ The inner products d A u 1,2 and d A u12 can be expressed in terms of the cylindrical coordinates in the same manner as in Eqs. (15)–(17) for the three independent directions of the atomic dipole moment. In the spirit of Ref. [5], where the Casimir–Polder interaction between a ground-state atom and a dielectric ring has been considered, we assume that the atom is sufficiently far from the dielectric cylinder R⪡ρa . Then ρA − ρ1,2 ≃ ρA . Further, s1 is replaced by its average over the cross section of the cylinder, which implies ρ1 = 0, and the integral over the cross section can be evaluated H /2

∫ d3s1⋯ → πR2 ∫−H /2 dz1⋯ . Eqs. (23)–(27) now become u1,2 ≃ [(u1,2 )2z + ρ A2 ]1/2 ,

(28)

u12 ≃ [(u12 )2z + ρ22 ]1/2 ,

(29)

2 ^ u ^ u 1 2 ≃ [(u1) z (u2 ) z − ρ A ]/u1u2 ,

(30)

^ u ^ u 1 12 ≃ [(u1) z (u12 ) z − ρ A ρ2 cos (φ A − φ2 ))]/u1u12 ,

(31)

^ u ^ u 2 12 ≃ [(u2 ) z (u12 ) z + ρ A ρ2 cos (φ A − φ2 ))]/u2 u12 .

(32)

We can assume without loss of generality that the atom is located on the x-axis, φA = 0. After taking the φ2-integral, we derive, for an atomic dipole moment oriented in the z-direction,

⎛Γ⎞ 3k A6 R2 Im χε2 (ω A ) ⎜ ⎟ = 16 ⎝ Γ0 ⎠22

H /2

∫−H/2

dz1

H /2

∫−H/2

dz 2

∫0

R

ρ2 dρ2

(20)

⎧ ⎡ (u2 ) 2 (u1) 2 ⎪ × ei (q1+ q2 + q12 ) ⎨ ⎢a1a2 − a1b2 2 z − a2 b1 2 z ⎪⎢ u2 u1 ⎩⎣ 2 ⎞⎤ (u1) z (u2 ) z ⎛⎜ ρ A ⎟⎥ +b1b2 ⎜ 2 + (u1) z (u2 ) z ⎟ ⎥ 2 (u1u2 ) ⎝ ⎠⎦

(21)

⎛ ⎪ (u12 ) 2z ⎞ ρ2⎫ ⎟ − b1b2 (u1) z (u2 ) z A ⎬ . × ⎜⎜a12 − b12 2 ⎟ 2 ⎪ (u1u2 ) q12 ⎭ u12 ⎠ ⎝

where

⎛Γ⎞ ⎜ ⎟ = ⎝ Γ0 ⎠21

⎛Γ⎞ 3k A6 Im ⎜ ⎟ = ⎝ Γ0 ⎠22 32π 2

29

(33)

30

H.T. Dung, T.M. Hien / Optics Communications 355 (2015) 27–32

Changing the integral variable ρ2 /λ A → q12, the integral over q12 can be evaluated and we end up with

dz1

H /2

∫−H/2

χεR = 0.01 dz 2

Γ/Γ 0

H /2

∫−H/2

⎧⎡ ⎪ (u2 ) 2 (u1) 2 × ei (q1+ q2 ) ⎨ ⎢a1a2 − a1b2 2 z − a2 b1 2 z u2 u1 ⎪ ⎢⎣ ⎩ 2 ⎞⎤ (u1) z (u2 ) z ⎛⎜ ρ A ⎥ u u +b1b2 + ( ) ( ) 1 z 2 z⎟ ⎜ ⎟⎥ (u1u2 )2 ⎝ 2 ⎠ ⎦⎥ ⎞⎛ ⎛ 1 (q12 ) l ⎞ ⎟⎟ eiq12 × ⎜⎜ − i⎟⎟ ⎜⎜1 − q q12 ⎠ ⎠⎝ ⎝ 12 +iρ A2 b1b2

(q12 )l = k A |(u12 ) z | ,

(u1) z (u2 ) z (u1u2 )2

1 χεR= 0.1 χεR = 0.5

0.98 0.1 1.04

q12 =(q12 ) u

eiq12 q12=(q12 ) l

⎪ ⎬, ⎪ ⎭

(q12 )u = k A (u12 )2z + R2 .

1.1

2.1

χεR = 0.5

q12=(q12 ) u ⎫

χεR = 0.1

1.02 (34)

(35)

It can be seen that the second-order term in the form of Eqs. (35) and (35) no longer contain any singularity and can be readily subject to numerical computation.

3. Numerical results In this section, we give some illustrative examples of the atomic spontaneous decay rate outside a finite-length dielectric cylinder. Despite the simple look of Eq. (12), numerical computations of the integrals are necessary in order to gain insight into the behavior of the decay rate. In Fig. 2 the spontaneous decay rate in accordance with the truncated Born expansion Eq. (12) is plotted as a function of the atom-cylinder axis distance for three different values of the permittivity. For comparison, the infinitely long-cylinder counterpart is plotted using the exact Green tensor in accordance with Eq. (A.5) (dashed lines). For the sizes of the cylinder H/λ A = 10 and R/λ A = 0.1 [H⪢R , Fig. 2(a)], and the atomic z coordinate z A = 0, the atom practically sees the cylinder as being infinitely long. Indeed, for χεR = 0.01 and χεR = 0.1, solid curves and dashed curves are visually indistinguishable. The linear-order Born expansion becomes less reliable as χεR increases. As can be seen from the figure, for the larger value of χεR = 0.5, noticeable discrepancies occur. In Fig. 2(b) the radius of the cylinder is increased to R/λ A = 2. For very small χεR = 0.01, the agreement between the infinitely long cylinder model and the finite-length cylinder described by the linear Born expansion is still good. However when χεR increases, the disagreement between the two sets in earlier and is in general more significant than that in the case of smaller cylinder radii [compare curves for χεR = 0.1 and 0.5 in Fig. 2(b) with their counterparts in Fig. 2(a)]. This can be attributed to the fact that as long as the cylinder height is fixed, an increasing value of the radius will gradually invalidate the assumption of infinite length. As the atom–surface distance increases, the decay rate exhibits oscillating behavior, interchanging between being enhanced and being inhibited. This results from the constructive and destructive interference between waves reflected by the surface. Far from the surface, the normalized spontaneous decay rate tends to the freespace value Γ /Γ0 = 1, as it should. Though in Fig. 2, we restrict ourselves to χεR ≤ 0.5, the firstorder Born expansion performs surprisingly well for larger values of χεR . At χεR ∼ 3, when a truncated Born series is not expected to make a good approximation, it still produces curves closely

Γ/Γ 0

⎛Γ⎞ 3k A4 R2 Im χε2 (ω A ) ⎜ ⎟ = 16 ⎝ Γ0 ⎠22

1.02

1 χεR = 0.01

0.98 2

3 rA/λA

4

Fig. 2. The normalized spontaneous decay rate of an excited atom positioned in the (z A = 0 )-plane as a function of the atom–z-axis distance for different values of χε = χεR + i10−8. All distances are scaled with respect to the atomic transition wavelength λ A = 2πc /ω A . The atomic dipole moment is oriented in the z direction. The length of the cylinder is H/λ A = 10 and its radius R/λ A is equal to (a) 0.1 and (b) 2. Results for a finite-length cylinder are represented by solid curves while those for an infinitely long cylinder of same radius are represented by dashed curves.

resembling the infinitely extended-cylinder ones. When the second-order term, Eq. (34), is included, for the ranges of parameters as in Fig. 2, we get curves which oscillate in phase with the first-order-term counterparts while having amplitudes about twice as large. The approximation (34) thus somehow overestimates the second-order corrections. We speculate that the replacement ρ1 − ρ2 → − ρ2 we made in going from Eqs. (24) to (29) is the reason behind this overestimation. To explore further the limitations of the assumption of an infinite cylinder length, instead of varying the radius of the cylinder as in Fig. 2, in Fig 3 we directly vary the height of the cylinder while keeping its radius constant. The atom is positioned in the middle plane of z A = 0 and χε = 0.1 + i10−8. As can be seen from Fig 2 (very long cylinders H/λ A ⪢1), the two models produce well agreed results. For H/λ A as small as 2, the decay rate remains essentially the same as those for H/λ A = 10 shown in Fig. 2(a). Noticeable deference between the two models occurs when the length of the cylinder is reduced to values comparable with or smaller than the atomic transition wavelength (H/λ A = 0.5 – dashed curve in Fig. 3). For H/λ A ⪡1, the infinitely long cylinder approximation fails completely in the region close to the surface (compare the solid curve with the dotted and dash-dotted ones in Fig. 3). In particular, the infinitely long cylinder model predicts an enhancement of the atomic decay rate instead of an inhibition. From the numerical results, it can be inferred that the Born expansion truncated at the linear term works well for |χε |⪡1 and >

remains more or less valid even for |χε | ∼ 1. Staying within the limit |χε |⪡1, we move on now to a consideration of the effects of the edge of the cylinder top and bottom, which is beyond the reach of the infinitely extended cylinder model. In Fig. 4, where the cylinder top and bottom has z-coordinate

H.T. Dung, T.M. Hien / Optics Communications 355 (2015) 27–32

Γ/Γ 0

1.12

1.06

1 -5

0

5

0 rA/λA

5

1.0002 Γ/Γ 0

z/λ A = 5 and z/λ A = − 5, respectively, the effects of the edge are illustrated by varying the position of the atom along a line parallel to the cylinder axis. It can be seen that the atomic decay rate exhibits oscillations on both sides of the edge, the magnitudes of the oscillations being most significant close to the edge. On the material side, the oscillations are about some mean value which can be larger or smaller than the free-space value of unity, indicating enhancement or inhibition of the decay rate, respectively. Particularly steep changes occur across the edge at z/λ A = 5. On the vacuum side, the normalized decay rate oscillates about one with a damping tail. In Fig. 5, the atom is moved along a straight line parallel to the top surface of the cylinder. The radius of the cylinder is chosen such that R/λ A = 0.1⪡1. For atomic transition wavelengths being in the optical range, this value of R/λ A means that the finite-length cylinder is a piece of a nanofiber. In the lateral directions, apparently the distance between neighboring interference fringes is comparable with the radius of the cylinder, so that within the region rA < R just a single bright [Fig. 5(a)] or dark [Fig. 5(b)] fringe is observed. The magnitude of the peak/dip reduces with an increasing atom-surface distance. When rA increases beyond R, the decay rate exhibits oscillations which are damped to the value of unity. In Fig. 5(a) where the atom is located close to the surface, the influence of the surface dominates that of the edge in the region rA < R and the magnitude of the decay-rate modifications occurring in this region is much more significant than that occurring outside. In Fig. 5(b) where the atom is located far from the surface, the magnitude of the decay-rate modifications occurring

31

1 0.9998 -5

1

in the region rA < R is comparable with or even smaller than that occurring outside.

Γ/Γ0

1.02

Fig. 5. The spontaneous decay rate over the top of a cylinder as a function of rA /λ A for χε = 0.1 + i10−8 , H/λ A = 10 , and R/λ A = 0.1. The atom is positioned in a plane that contains the cylinder axis and different curves are for z A /λ A = (a) 5.01 (solid line), 5.05 (dashed line), 5.1 (dotted line), and (b) 5.5 (solid line), 6.0 (dashed line), 7.0 (dotted line). The atomic dipole moment is oriented in the z direction.

0.98

4. Summary

0.998

In summary, using a truncated Born expansion of the Green tensor, we have considered the decay rate of an excited two-level atom in the presence of a dielectric cylinder of finite length, whose Green tensor is not known in closed form. Based on numerical computations, it has been shown that a cylinder can be treated as extending to infinity along the axis only when its length is much larger than the atomic transition wavelength, and when the atom is located sufficiently far from the edges. A boundary in the axial direction can give rise to significant modifications of the decay rate. Despite the fact that the convergence of the Born series requires optical diluteness of the medium, numerical results produced by the first-order Born series indicate that many predictions on the boundary effects hold qualitatively for media having higher optical densities as well. When the emitting atom is embedded in the medium, one can switch the center of the Born series from the vacuum Green tensor to the bulk Green tensor to avoid the singularity. We have also given a (crude) estimate of the second-order term in the Born series.

0.996

Appendix A. Green tensor of an infinitely extended cylinder

0.96 0.1

0.3 rA/λA

0.5

Fig. 3. The normalized spontaneous decay rate of an excited atom as a function of the atom–z-axis distance for different values of the cylinder height H/λ A = 0.5 (dashed line), 0.05 (dotted line), and 0.01 (dash-dotted line). The case of an infinitely long cylinder is shown by the solid line. The other parameters are z A = 0 , χε = 0.1 + i10−8 , R/λ A = 0.1. The atomic dipole moment is oriented in the z direction. In the last two cases, the cylinder is actually a disk.

Γ/ Γ0

1.002 1

3

5 zA/ λA

7

Fig. 4. The spontaneous decay rate near the top of the cylinder as a function of z A /λ A for χε = 0.1 + i10−8, H /λ A = 10, R/λ A = 0.1, and for various atom–axis distances rA /λ A = 0.26 (solid line), 0.3 (dashed line), 0.6 (dotted line), and 1.1 (dashdotted line). The atomic dipole moment is oriented in the z direction.

The Green tensor of an infinitely extended cylinder is known in closed form (see, e.g. Ref. [27]). Assuming that the atom is located outside the cylinder, the relevant Green tensor can be separated into two parts

G (r, r ′) = G¯ (r, r ′) + Gsc (r, r ′),

(A.1)

H.T. Dung, T.M. Hien / Optics Communications 355 (2015) 27–32

32

where G¯ (r, r′) is the vacuum Green tensor and Gsc (r, r′) is the scattering Green tensor

i Gsc (r, r ′) = 8π

∞

∫−∞

∞

dh

∑

J f = Jn (η f R),

(2 − δ 0n )

× [*1′ H M (e1n)η1(h) M′e(n1η)1 ( − h) + *1′ V N (e1n)η1 (h) N′e(n1η)1 ( − h) o

o

H1 = Hn(1) (η1R),

o

+ *2′ H N (o1n)η1 (h) M′e(n1η)1 ( − h) + *2′ V M (o1n)η1(h) N′e(n1η)1 ( − h)]. e

In

Eq.

o

(A.1),

δ0n

e

denotes

o

the

Kronecker

(A.2) delta,

r = (ρ , φ , z ) , r′ = (ρ′, φ′, z′),

⎡ nH (1) (η ρ) ⎤ ∂Hn(1) (η f ρ) cos n f sin ^⎥ M (e1n)η f (h) = ⎢∓ (nφ) ρ^ − (nφ) φ o ⎢ ⎥ cos sin ρ ∂ρ ⎣ ⎦ eihz ,

(A.3)

(1) ⎡ 1 ⎢ ∂Hn (η f ρ) cos N (e1n)η f (h) = ih (nφ) ρ^ o k f ⎢⎣ sin ∂ρ

∓

⎤ ihn (1) sin ^ + η 2 Hn(1) (η ρ) cos (nφ) z^⎥ eihz , Hn (η f ρ) (nφ) φ f f ⎥ cos sin ρ ⎦

where Hn(1) (z ) is ω k f = ε f c , and f

Hankel function of the first kind, η f =

∞

∞

dh

∑

− h2 ,

(2 − δ 0n ) *1′ V

n= 0

η12 k13

[Hn(1) (η1ρ A )]2 ,

(A.5)

where only the second term in Eq. (A.2) contributes. The coefficient *1′ V reads as

*1′ V = −

∂(η f R)

, (A.9)

A , D

(A.6)

⎡ ihn ⎤2 (k12 − k22 ) ⎥ A = H1J1 J22 ε2 ⎢ ⎣ k1k2 ⎦ ⎡ ⎤⎡ ⎤ η1 η2 η1 η2 + (Rη1η2 )2 ⎢J2′ J1 ε2 − J1′ J2 ⎥ ⎢J2′ H1 − H1′ J2 ε2 ⎥, ⎢⎣ k1 k2 ⎥⎦ ⎢⎣ k1 k2 ⎥⎦

(A.7)

⎡ ihn ⎤2 (k12 − k22 ) ⎥ D = H12 J22 ε2 ⎢ ⎣ k1k2 ⎦ ⎡ ⎤⎡ ⎤ η1 η2 η1 η2 + (Rη1η2 )2 ⎢J2′ H1 ε2 − H1′ J2 ⎥ ⎢J2′ H1 − H1′ J2 ε2 ⎥, ⎢⎣ k1 k2 ⎥⎦ ⎢⎣ k1 k2 ⎥⎦

H1′ =

∂Hn(1) (η1R) ∂(η1R)

,

(A.10)

Jn(z) being Bessel function of the first kind. It is well known that dielectric cylinders may sustain sharp resonances which are associated with the poles of the Green tensor in the complex plane. These poles lie very close to the real axis, resulting in that the h-integral presents a challenging numerical task. Following [28], we circumvent the poles by deforming the integration path and perform the integral along a half ellipse in the lower half of the complex plane. A similar trick where an integration contour consisting of a radial line and a circular arc has been employed in Ref. [24].

(A.4) k 2f

= 2 (1) for inside (outside) the cylinder. The prime

∫0

∂Jn (η f R)

References

superscript of M and N indicates that r′ is to be used in place of r. For an atomic dipole moment oriented in the z direction, using Eqs. (1) and (A1)–(A4), one finds for the decay rate

Γ 3 = 1 + Re Γ0 2

J ′f =

η12

n= 0

o

where

(A.8)

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