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Munguía-Arvayo et al.
Bloch modes at the surface of a photonic crystal interacting with a waveguide R. Munguía-Arvayo,1 R. García-Llamas,2,* and J. Gaspar-Armenta2 2
1 Posgrado en Física, Universidad de Sonora, Apdo. Postal 5-88, Hermosillo, Sonora, Mexico Centro de Investigación en Física, Universidad de Sonora, Apdo. Postal 5-88, Hermosillo, Sonora, Mexico *Corresponding author: [email protected]
Received February 18, 2014; revised May 7, 2014; accepted May 19, 2014; posted May 21, 2014 (Doc. ID 206482); published June 24, 2014 We theoretically studied the influence of an infinite set of waveguides on the evanescent field of Bloch waves at the surface of a one-dimensional photonic crystal (PC) excited by a TE Gaussian beam undergoing total internal reflection. The set of waveguides is regarded as a periodic inhomogeneous medium with period a. Numerical results are presented for the case in which a is greater than λ, which is the wavelength used to excite the surface mode. When the waveguide tip is very close to the surface of the PC, a fraction of the surface wave is reflected by the tip, producing an interference pattern that can be observed in the near field. In this case, the system simulates scanning tunneling optical microscopy in 2D geometry, and an image of the field distribution of the surface mode is obtained by quantifying the flux energy throughout the waveguide. © 2014 Optical Society of America OCIS codes: (180.0180) Microscopy; (230.7390) Waveguides, planar; (240.6690) Surface waves; (260.6970) Total internal reflection; (310.2790) Guided waves; (230.5298) Photonic crystals. http://dx.doi.org/10.1364/JOSAA.31.001588
1. INTRODUCTION A Bloch surface mode (SM) is an electromagnetic wave that travels on the interface of a truncated photonic crystal (PC), and its amplitude decays exponentially as a function of the distance perpendicular to the surface. The existence of surface modes at the surface of a finite-size two-dimensional (2D) PC was predicted by Meade et al. [1]. The experimental observation of surface modes in a 2D PC was measured by Robertson et al. [2]. For the case of one-dimensional (1D) PCs [3], it has been theoretically shown that surface waves can exist at specific frequencies inside the forbidden band gap. This characteristic frequency depends on the thickness of the top layer of the PC [4,5], as experimentally demonstrated by Robertson and May [4]. The influence of the roughness on the surface mode in a 1D PC was recently studied, and the scattered light was found to be enhanced in resonant conditions [6]. In contrast, scanning near-field optical microscopy (SNOM) [7–9] is employed for imaging objects with subwavelength dimensions or a field distribution of surface modes. SNOM uses a sharpened optical fiber that is scanned over a surface region containing the nano-objects under study. The resolution depends on the diameter of the aperture of the sharpened optical fiber, and it exceeds the optical Rayleigh limit. The SNOM response is the convolution of the diffracted field produced by the subwavelength structure and that produced by the fiber tip that acts as a diffractive element. Several reports have considered this effect; for example, the authors of [10] used a probe based on the fiber Fabry–Perot interferometer. In this study, an exact solution of the diffraction of light from a finite and truncated 1D PC near a set of equally spaced planar waveguides is obtained. The primary motivation is to distinguish between the electromagnetic responses produced by a Bloch SM at the surface of the 1D PC and those produced 1084-7529/14/071588-07$15.00/0
by the waveguide’s tip, which acts as a diffractive element. The solution for the electromagnetic diffracted field in the inhomogeneous medium (IM) is a multimodal expansion as proposed by Burckhardt [11,12]. The scattering problem was addressed by following the approach used in [13,14]. Using this methodology, a matrix equation of infinite dimension was obtained for the amplitudes of the diffracted field. The focus of this study is the case where the distance between the tip of the waveguide and the interface of the 1D PC is greater than the wavelength (λ) used to excite the SM, and also the width of the waveguide is smaller than λ.
2. THE SYSTEM The geometry of the aforementioned system is shown in Fig. 1(b). A semi-infinite homogeneous, isotropic, nonabsorbing dielectric and linear medium with dielectric constant ε1 fills the region given by z < 0, followed by a finite and truncated 1D PC for 0 < z < d11 and a vacuum gap of thickness d12 . The region given by z > d12 is filled with a semi-infinite periodic IM, where a is the distance between the equally spaced waveguides and d is the width of one of them. The parameters εc and εe are the dielectric constants of the core (waveguide) and the envelope, respectively, vacuum in this case. A Gaussian beam with width w is incident obliquely, from the homogeneous side, on the dielectric/1D PC interface. When a is greater than both the width of the waveguide and waist of the beam, the set of waveguides behaves as an isolated waveguide up to a distance d12 of separation from the 1D PC. A. TE Modes Supported for a Set of Waveguides It is assumed that the IM, i.e., the set of waveguides, is periodic with period a in the y direction and homogeneous in x and z directions. Then, according to the Floquet–Bloch theorem, © 2014 Optical Society of America
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Substituting Eqs. (3) and (1) into Eq. (2) and using the orthogonal properties of the Fourier functions, the following equation is obtained: ∞ X n0 −∞
13 0 0 q20 δn;n0 − ε−1 13 n−n0 qy gy n qy gy n An0 ;k
13 ξ13 k ξk
∞ X n0 −∞
13 ε−1 13 n−n0 An0 ;k ;
(4)
where n 0, 1, 2…. and q20 μ0 ε0 ω2 . If Eq. (4) is multiplied by the inverse matrix of ε−1 13 n−n0 , then ∞ X n0 −∞
q20 ηn;n0 −
∞ X
13 0 0 ηn;l ε−1 13 l−n0 qy gy n qy gy n An0 ;k
l−∞
13 13 ξ13 k ξk An0 ;k ;
(5)
where ηn;n0 are the elements of the inverse matrix of ε−1 13 n−n0 .
By solving Eq. (5), a set of discrete eigenvalues ξ13 and its k k associated eigenvector E⃗ ⃗r are found. Here, A13 is the 13t
n;k
nth component of the kth eigenvector.
Fig. 1. (a) Set of equally spaced waveguides with dielectric constant εc and width d surrounded by nonabsorbing medium with dielectric constant εe < εc . The distance of separation between waveguides is a. (b) The PC is composed of 10 films, and it has a period p d2 d3 , with d2 d4 d6 d8 d10 and d3 d5 d7 d9 . The thickness of the last film, d11 , breaks the symmetry of the PC in order to obtain the surface mode. The waveguides are separated by a distance d12 from the truncated 1D PC for a vacuum gap.
the x component of the electric field can be written as a Burckhardt–Fourier expansion,
E 13tx y; z
∞ X
13 A13 n;k expiqy gy ny expiξk z;
B. Light Diffraction from a Photonic Crystal Near a Set of Waveguides The problem of the diffraction is addressed using the boundary conditions at each interface, which are shown in Fig. 1(b). A Rayleigh–Fourier expansion for the electric field in each film and a Burckhardt–Fourier expansion for the corresponding electric field in the IM are proposed. Finally, recursive formalism is found, which allows calculating the amplitudes of the expansion for the field in a medium in relationship to those of the contiguous one. The electric fields in each of the homogenous media are written as Rayleigh–Fourier series:
E lx y; z
∞ X
eltx βn expfiβn y γ l βn z − zl−1 g
n−∞ ∞ X
elrx βn expfiβn y − γ l βn z − zl−1 g; (6)
n−∞
n−∞
(1) where gy 2π∕a and qy is the Bloch wave number in the first Brillouin zone. Using the Maxwell equations, the wave equation for the TE modes in an IM are given as 1 ∇ × ∇ × E⃗ 13t ⃗r ω2 μ0 ε0 E⃗ 13t ⃗r: ε13 y
(2)
If the medium is periodic, ε13 y ε13 y a), the inverse of the dielectric constant can be written as a Fourier series: ∞ X 1 ε−1 expigy ny: ε13 y n−∞ 13 n
(3)
where l 1 represents the homogeneous semi-infinite medium, l 2 represents the first film, l 3 represents the second film, and this pattern continues, with l 11 representing the tenth film; l 12 represents the vacuum gap, and z1 0, z2 z1 d2 ; …; z11 z10 d11 , correspond to the localization of the interfaces. The parallel component of the diffracted wave vector is βn
p μ1 ε1 ω sin θi 2π∕an;
(7)
p and γ l βn μ1 ε1 ω2 − β2n are the perpendicular components. The Fourier coefficients of the Gaussian beam, which are displaced a horizontal distance −yd from the origin of the coordinate system, are given by
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e1tx βn
Munguía-Arvayo et al.
exp−β2n − β20 w2 ∕2 expiβn yd for jβn j ≤ β0 : 0 for jβn j > β0 (8)
e3tx βn e3rx βn e2tx βn expiγ 2 βn z2 e2rx βn exp−iγ 2 βn z2 ;
(16)
The electric field components E 1x y; z and E 2x y; z must satisfy the boundary conditions at the interface z1 0, that is, E 1x y; z E 2x y; z and μ2 ∂E1x y; z1 ∕∂z μ1 ∂E 2x y; z1 ∕∂z. According to the orthogonal properties of the Rayleigh functions, this yields the following pair of equations:
γ 3 βn γ β e β − 3 n e3rx βn μ3 3tx n μ3 γ 2 βn e β expiγ 2 βn z2 μ2 2tx n γ β − 2 n e2rx βn exp−iγ 2 βn z2 : μ2
e2tx βn e2rx βn e1tx βn e1rx βn ;
Using Eqs. (16) and (17), the following relations are obtained:
(9)
(17)
e3tx βn
C 2 βn e2tx βn expiγ 2 βn z2 − D2 βn e2rx βn exp−iγ 2 βn z2 ; F 3 βn
(18)
e3rx βn
D2 βn e2tx βn expiγ 2 βn z2 − C 2 βn e2rx βn exp−iγ 2 βn z2 ; F 3 βn
(19)
γ 2 βn γ β e β − 2 n e2rx βn μ2 2tx n μ2 γ 1 βn γ β e β − 1 n e1rx βn : μ1 1tx n μ1
(10)
where C 2 βn , D2 βn , and F 3 βn have the same form as in Eqs. (13), (14), and (15) respectively, and only the subindexes are incremented by one. Substituting Eqs. (11) and (12) into Eqs. (18) and (19), the following expressions are obtained:
Using Eqs. (9) and (10), the following relations are obtained: C β e β − D1 βn e1rx βn e2tx βn 1 n 1tx n ; F 2 βn D β e β − C 1 βn e1rx βn e2rx βn − 1 n 1tx n ; F 2 βn
e3tx βn
U 02 βn e1tx βn − V 02 βn e1rx βn ; F 3r βn
(20)
e3rx βn
U 2 βn e1tx βn − V 2 βn e1rx βn ; F 3r βn
(21)
(11)
(12) where
where V 2 βn D1 βn D2 βn expiγ 2 βn d2
γ β γ β C 1 βn 1 n 2 n ; μ1 μ2
(13)
γ 1 βn γ 2 βn − ; μ1 μ2
(14)
2γ 2 βn : μ2
(15)
D1 βn
F 2 βn
The formalism continues in the next interface. The electric field’s components and their derivatives must satisfy the boundary conditions at the interface z2 z1 d2 . Then, according to the corresponding boundary conditions, two equations are obtained:
C 1 βn C 2 βn exp−iγ 2 βn d2 ;
(22)
U 2 βn C 1 βn D2 βn expiγ 2 βn d2 D1 βn C 2 βn exp−iγ 2 βn d2 ;
(23)
U 02 βn C 1 βn C 2 βn expiγ 2 βn d2 D1 βn D2 βn exp−iγ 2 βn d2 ;
(24)
V 02 βn D1 βn C 2 βn expiγ 2 βn d2 C 1 βn D2 βn exp−iγ 2 βn d2 ;
(25)
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F 3r βn F 2 βn F 3 βn :
(26)
∞ 13 X ξ k
k1
This procedure continues until the last interface between homogeneous media. Then, using the boundary conditions at the interface z11 z10 d11 and the corresponding electric fields, the Fourier coefficients of the electric field in the vacuum gap are determined: e12tx βn
U 011 βn e1tx βn − V 011 βn e1rx βn ; F 12r βn
μ13
13tx A13 n;k ek
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γ 12 βn e β expiγ 12 βn z12 μ12 12tx n −
γ 12 βn e β exp−iγ 12 βn z12 : μ12 12rx n (36)
By substituting Eqs. (27) and (28) into Eqs. (35) and (36), two equations are obtained:
(27) F 12 βn F 11r βn
∞ X
13tx A13 n;k ek
k1
e12rx βn
U 11 βn e1tx βn − V 11 βn e1rx βn ; F 12r βn
H 12 βn e1tx βn − H 12 r βn e1rx β n ; t
(28)
F 11r βn
where
∞ X 2ξ13 k
μ13
k1
V 11 βn V 010 βn D11 βn expiγ 11 βn d11 V 10 βn C 11 βn exp−iγ 11 βn d11 ;
(29)
U 011 βn
G12 βn e1tx βn t
− G12 r βn e1rx β n ;
(38)
βn U 011 βn expiγ 12 βn d12 G12 t U 11 βn exp−iγ 12 βn d12 ;
(30)
(39)
0 G12 r βn V 11 βn expiγ 12 βn d12
U 010 βn C 11 βn expiγ 11 βn d11 U 10 βn D11 βn exp−iγ 11 βn d11 ;
13tx A13 n;k ek
where
U 11 βn U 010 βn D11 βn expiγ 11 βn d11 U 10 βn C 11 βn exp−iγ 11 βn d11 ;
(37)
V 11 βn exp−iγ 12 βn d12 ;
(31)
(40)
0 H 12 r βn V 11 β n expiγ 12 βn d12
− V 11 βn exp−iγ 12 βn d12 ;
V 011 βn V 010 βn C 11 βn expiγ 11 βn d11 V 10 βn D11 βn exp−iγ 11 βn d11 ;
(41)
(32) H 12 βn U 011 βn expiγ 12 βn d12 t
F 12r βn F 11r βn F 12 βn :
− U 11 βn exp−iγ 12 βn d12 :
(33)
It is helpful to note that Eqs. (29–33) have a recursive form, which enables the determination of the amplitudes given by Eqs. (27) and (28) for any number of films. The field in the IM can be written as the modal (Burckhardt–Rayleigh) expansion
(42)
Multiplying the terms of Eq. (37) by G12 r βn and those of Eq. (38) by H 12 r β n and subtracting the results, a matrix equation called the “matrix scattering equation” is obtained: ∞ X
13tx T 12 U 12 n ; n;k ek
(43)
k1
E13tx ⃗r
∞ X ∞ X
13tx A13 expfiβn y ξ13 n;k ek k z − z12 g;
where
k1 n−∞
2γ 12 βn 2ξ13 12 12 k A13 G β − H β T 12 r r n n n;k n;k μ12 μ13
(34) where the subindex k iterates through all of the solutions obtained using Eq. (5). Using the boundary condition at the interface between the vacuum gap and IM z12 z11 d12 , the equations below are obtained: ∞ X
13tx A13 e12tx βn expiγ 12 βn z12 n;k ek
k1
e12rx βn exp−iγ 12 βn z12 ;
(35)
(44)
and U 12 n
12 12 G12 βn − H 12 r β n H t r β n Gr βn e1tx βn : (45) F 11r βn
The procedure to obtain the different coefficients for the electric field in each homogeneous medium is as follows. First, Eq. (43) is inverted to determine the coefficients
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P 13wz
μ1 13 X 13 13tx 13 13tx ξu An;u eu An0 ;u eu μ13 n;n0 × f sincπn − n0 f ∕P i ;
where the power of the incident P 2 n γ 12 βn je1tx β n j and f d∕a.
(46) beam
is
Pi
3. NUMERICAL RESULTS In order to obtain numerical results, some parameters are fixed. The waist of the s-polarized Gaussian beam is fixed as w 64λ0 40.5 μm, with λ0 633 nm, and the beam is incident at angle θi 60.13° with respect to the normal to the interface between the homogeneous medium and the PC. This angle is adequate to resonantly excite the surface mode as shown in [6]. The dielectric constant of the nonabsorbing homogeneous, semi-infinite medium is ε1 1.502 . The waveguide has width d λ0 ∕10 63.0 nm and dielectric constant εc 1.50, and it is surrounded by the core, whose dielectric constant is εc 1.0. Its thickness is a − d, where the equally spaced distance between waveguides is a 256λ0 162.0 μm. For this selection of parameters, the waveguides behave as a single one; because a ≫ d, the crosstalk among the set of waveguides is negligible. The infinite dimension of the matrix-scattering equation was truncated to 2N 2048 in order to obtain the amplitudes of the diffracted fields. The value of this parameter is sufficient to reach the energy conservation within 0.1% of variation. The 1D PC has a period of 0.25 μm and comprises 10 films. The materials are considered nondispersive, and their refractive index and thickness are given in Table 1. Figure 2(a) shows the reflection as a function of the wavelength for a fixed angle of incidence θi 60.13°. The curve with red solid circles (blue solid squares) corresponds to the reflection from the 1D PC very near to (very far from) the planar waveguide, whose propagation axis is perpendicular to the surface of the truncated 1D PC. The minimum in the reflection corresponds to the excitation of the SM at the interface between the 1D PC and the vacuum, which was efficiently excited for a wavelength of λ 644.8 nm inside the forbidden band. The distance of separation between the 1D PC and the waveguide is d12 ∼ λ0 ∕63 10.0 nm. The closeness of the waveguide modifies the optical response of the 1D PC, and the waveguide diminishes the strength coupling between the surface mode and the photon. The finite width of the Gaussian beam introduces additional wave vectors in its spatial spectrum, and a weak coupling is observed as two almost symmetric dips. This effect tends to disappear as the width of the Gaussian beam is increased. The minimum of the resonance is shifted to smaller energies by approximately 0.00012 eV. Two secondary minima can be observed. The small width, approximately 0.0023 eV,
Table 1. 1D PC is Composed of Alternate Materials That Have High and Low Refractive Indexesa Medium
Film
Refraction Index (n)
Absorption Index (k)
Thickness d (μm)
2 3 4 5 6 7 8 9 10 11
1 2 3 4 5 6 7 8 9 10
1.37 2.14 1.37 2.14 1.37 2.14 1.37 2.14 1.37 2.14
0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003
0.120 0.130 0.120 0.130 0.120 0.130 0.120 0.130 0.120 0.051
a The pair of materials correspond to MgF2 (the low index 1.37) and TiO2 (the high index 2.14). The thickness of the last film breaks the symmetry of the 1D PC and induces the existence of the surface mode.
is related to the propagation distance of the SM and is associated to the radiative losses of the SM. The energy balance establishes that 80% of the energy is transferred to the surface modes and the remainder is lost
1.0 0.9
(h/2π)ω =1.91325 eV
(h/2π)ω =1.93300 eV
0.8 0.7 0.6
R (TE-pol)
e13tx . Next, these values are substituted into Eq. (37) to obk tain the Fourier coefficients of the reflected beam. Then, the values of e1rx are substituted into Eqs. (27) and (28) to determine the coefficients e12tx βn and e12rx βn . Finally, the recursive form is used to obtain the remainder of the amplitudes. The normalized power in the waveguide, assuming that only one mode (the uth mode) is supported for it, is
0.5 0.4 0.3 0.2 0.1
(h/2π)ω =1.92325 eV
(a)
0.0 1.90
1.91
1.92
1.93
1.94
1.95
E (eV) 1E-4
Pw Pt
1E-5
Power
1592
1E-6
1E-7
(b) 1E-8 1.89
1.90
1.91
1.92
1.93
1.94
1.95
E (eV) Fig. 2. (a) Reflection as a function of the energy (wavelength) for a fixed angle of incidence θi 60.13°. The curve with red solid circles (blue solid squares) corresponds to the reflection of the 1D PC near to (far from) a waveguide. The air gap is λ0 ∕63, where λ is the wavelength of 633 nm. The curve was calculated with energy steps of 0.00025 eV. (b) Log total power (red solid squares) in the IM and the power in the waveguide (blue solid circles) as functions of the incident energy at a 60.13° fixed angle of incidence.
Munguía-Arvayo et al.
as scattering energy because the waveguide acts as an efficient diffractive element. Figure 2(b) shows the total power (red solid squares) in the IM and the power in the waveguide (blue solid circles) as functions of the incident energy. The resonant coupling of the SM appears as a maximum in both plots. The amplitude of the total power is greater than the power in the waveguide because the former includes the scattering light. The measurement of the energy in the waveguide is the basic principle supporting the SNOM technique, and this result shows that the energy is easily measured at resonant conditions; however, the ease of measurement depends on the distance from the waveguide. In the case of Fig. 3, several effects are included with the waveguide, but they cannot be detected using the SNOM technique; they can only be visualized theoretically. Figure 3 shows the modulus of the electric field projected into the y–z plane. It was calculated at resonant conditions, that is, h∕2πω 1.92325 eV and λ 644.8 nm, and angle of incidence θι 60.13°. The 1D PC is approximately 1.2 μm thick. The excitation of the surface mode in the last TiO2 film can be clearly seen. As expected, its field decays exponentially approaching both sides of the interface, in the 1D PC and in the vacuum, and it oscillates in the 1D PC. Two effects are observed: (a) the propagation of the surface mode, which extends far from the waveguide, and (b) the oscillations of the surface modes along the last film. For the region z < 0, it is possible to observe the interference between the incident beam and the reflected one, and the diffraction produced for the very narrow waveguide exhibits zones of high and low intensity near y 0. In Fig. 4, the near-field power, that is, the power inside of the waveguide, is plotted as a function of both distances, perpendicular (z) and parallel (y) to the surface, at resonant conditions, i.e., E 1.92325 eV and θi 60.13°. The end waveguide is positioned at different distances from the real surface. The discrete points are measured with resolution of Δz 10 nm and Δy 5 μm. It is possible to observe the field distribution of the surface mode. Two main characteristics evident in the figure are the decay distance of approximately 50 nm and the propagation distance of approximately 25 μm.
Fig. 3. Modulus of the electric field projected into the y–z plane. The horizontal black lines correspond to the interfaces between each film that form the 1D PC. The vertical lines above the top of the 1D PC help to visualize one of the waveguides in the set; the remainder are outside of the limits of this graph. The image was calculated using steps of Δy 80.6 nm and Δz 20.1 nm.
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Fig. 4. Near-field power (power flowing inside of the waveguide) plotted as a function of the waveguide end-tip position for both distances, perpendicular (z) and parallel (y) to the surface, at resonant conditions, i.e., E 1.92325 eV and θi 60.13°.
Our intent was not necessarily to use a sharp or rounded tip or an optical fiber covered with metal; instead, in this work, the waveguide end-tip was “rectangular” and a pure dielectric. A bare dielectric waveguide 63 nm in width was used. It was impossible to observe the oscillations of the surface mode, owing to the width of the waveguide.
4. CONCLUSIONS The influence of a waveguide on the evanescent field of an SM was studied. When a small waveguide is placed near (
J. Opt. Soc. Am. A / Vol. 31, No. 7 / July 2014
Munguía-Arvayo et al.
Bloch modes at the surface of a photonic crystal interacting with a waveguide R. Munguía-Arvayo,1 R. García-Llamas,2,* and J. Gaspar-Armenta2 2
1 Posgrado en Física, Universidad de Sonora, Apdo. Postal 5-88, Hermosillo, Sonora, Mexico Centro de Investigación en Física, Universidad de Sonora, Apdo. Postal 5-88, Hermosillo, Sonora, Mexico *Corresponding author: [email protected]
Received February 18, 2014; revised May 7, 2014; accepted May 19, 2014; posted May 21, 2014 (Doc. ID 206482); published June 24, 2014 We theoretically studied the influence of an infinite set of waveguides on the evanescent field of Bloch waves at the surface of a one-dimensional photonic crystal (PC) excited by a TE Gaussian beam undergoing total internal reflection. The set of waveguides is regarded as a periodic inhomogeneous medium with period a. Numerical results are presented for the case in which a is greater than λ, which is the wavelength used to excite the surface mode. When the waveguide tip is very close to the surface of the PC, a fraction of the surface wave is reflected by the tip, producing an interference pattern that can be observed in the near field. In this case, the system simulates scanning tunneling optical microscopy in 2D geometry, and an image of the field distribution of the surface mode is obtained by quantifying the flux energy throughout the waveguide. © 2014 Optical Society of America OCIS codes: (180.0180) Microscopy; (230.7390) Waveguides, planar; (240.6690) Surface waves; (260.6970) Total internal reflection; (310.2790) Guided waves; (230.5298) Photonic crystals. http://dx.doi.org/10.1364/JOSAA.31.001588
1. INTRODUCTION A Bloch surface mode (SM) is an electromagnetic wave that travels on the interface of a truncated photonic crystal (PC), and its amplitude decays exponentially as a function of the distance perpendicular to the surface. The existence of surface modes at the surface of a finite-size two-dimensional (2D) PC was predicted by Meade et al. [1]. The experimental observation of surface modes in a 2D PC was measured by Robertson et al. [2]. For the case of one-dimensional (1D) PCs [3], it has been theoretically shown that surface waves can exist at specific frequencies inside the forbidden band gap. This characteristic frequency depends on the thickness of the top layer of the PC [4,5], as experimentally demonstrated by Robertson and May [4]. The influence of the roughness on the surface mode in a 1D PC was recently studied, and the scattered light was found to be enhanced in resonant conditions [6]. In contrast, scanning near-field optical microscopy (SNOM) [7–9] is employed for imaging objects with subwavelength dimensions or a field distribution of surface modes. SNOM uses a sharpened optical fiber that is scanned over a surface region containing the nano-objects under study. The resolution depends on the diameter of the aperture of the sharpened optical fiber, and it exceeds the optical Rayleigh limit. The SNOM response is the convolution of the diffracted field produced by the subwavelength structure and that produced by the fiber tip that acts as a diffractive element. Several reports have considered this effect; for example, the authors of [10] used a probe based on the fiber Fabry–Perot interferometer. In this study, an exact solution of the diffraction of light from a finite and truncated 1D PC near a set of equally spaced planar waveguides is obtained. The primary motivation is to distinguish between the electromagnetic responses produced by a Bloch SM at the surface of the 1D PC and those produced 1084-7529/14/071588-07$15.00/0
by the waveguide’s tip, which acts as a diffractive element. The solution for the electromagnetic diffracted field in the inhomogeneous medium (IM) is a multimodal expansion as proposed by Burckhardt [11,12]. The scattering problem was addressed by following the approach used in [13,14]. Using this methodology, a matrix equation of infinite dimension was obtained for the amplitudes of the diffracted field. The focus of this study is the case where the distance between the tip of the waveguide and the interface of the 1D PC is greater than the wavelength (λ) used to excite the SM, and also the width of the waveguide is smaller than λ.
2. THE SYSTEM The geometry of the aforementioned system is shown in Fig. 1(b). A semi-infinite homogeneous, isotropic, nonabsorbing dielectric and linear medium with dielectric constant ε1 fills the region given by z < 0, followed by a finite and truncated 1D PC for 0 < z < d11 and a vacuum gap of thickness d12 . The region given by z > d12 is filled with a semi-infinite periodic IM, where a is the distance between the equally spaced waveguides and d is the width of one of them. The parameters εc and εe are the dielectric constants of the core (waveguide) and the envelope, respectively, vacuum in this case. A Gaussian beam with width w is incident obliquely, from the homogeneous side, on the dielectric/1D PC interface. When a is greater than both the width of the waveguide and waist of the beam, the set of waveguides behaves as an isolated waveguide up to a distance d12 of separation from the 1D PC. A. TE Modes Supported for a Set of Waveguides It is assumed that the IM, i.e., the set of waveguides, is periodic with period a in the y direction and homogeneous in x and z directions. Then, according to the Floquet–Bloch theorem, © 2014 Optical Society of America
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Substituting Eqs. (3) and (1) into Eq. (2) and using the orthogonal properties of the Fourier functions, the following equation is obtained: ∞ X n0 −∞
13 0 0 q20 δn;n0 − ε−1 13 n−n0 qy gy n qy gy n An0 ;k
13 ξ13 k ξk
∞ X n0 −∞
13 ε−1 13 n−n0 An0 ;k ;
(4)
where n 0, 1, 2…. and q20 μ0 ε0 ω2 . If Eq. (4) is multiplied by the inverse matrix of ε−1 13 n−n0 , then ∞ X n0 −∞
q20 ηn;n0 −
∞ X
13 0 0 ηn;l ε−1 13 l−n0 qy gy n qy gy n An0 ;k
l−∞
13 13 ξ13 k ξk An0 ;k ;
(5)
where ηn;n0 are the elements of the inverse matrix of ε−1 13 n−n0 .
By solving Eq. (5), a set of discrete eigenvalues ξ13 and its k k associated eigenvector E⃗ ⃗r are found. Here, A13 is the 13t
n;k
nth component of the kth eigenvector.
Fig. 1. (a) Set of equally spaced waveguides with dielectric constant εc and width d surrounded by nonabsorbing medium with dielectric constant εe < εc . The distance of separation between waveguides is a. (b) The PC is composed of 10 films, and it has a period p d2 d3 , with d2 d4 d6 d8 d10 and d3 d5 d7 d9 . The thickness of the last film, d11 , breaks the symmetry of the PC in order to obtain the surface mode. The waveguides are separated by a distance d12 from the truncated 1D PC for a vacuum gap.
the x component of the electric field can be written as a Burckhardt–Fourier expansion,
E 13tx y; z
∞ X
13 A13 n;k expiqy gy ny expiξk z;
B. Light Diffraction from a Photonic Crystal Near a Set of Waveguides The problem of the diffraction is addressed using the boundary conditions at each interface, which are shown in Fig. 1(b). A Rayleigh–Fourier expansion for the electric field in each film and a Burckhardt–Fourier expansion for the corresponding electric field in the IM are proposed. Finally, recursive formalism is found, which allows calculating the amplitudes of the expansion for the field in a medium in relationship to those of the contiguous one. The electric fields in each of the homogenous media are written as Rayleigh–Fourier series:
E lx y; z
∞ X
eltx βn expfiβn y γ l βn z − zl−1 g
n−∞ ∞ X
elrx βn expfiβn y − γ l βn z − zl−1 g; (6)
n−∞
n−∞
(1) where gy 2π∕a and qy is the Bloch wave number in the first Brillouin zone. Using the Maxwell equations, the wave equation for the TE modes in an IM are given as 1 ∇ × ∇ × E⃗ 13t ⃗r ω2 μ0 ε0 E⃗ 13t ⃗r: ε13 y
(2)
If the medium is periodic, ε13 y ε13 y a), the inverse of the dielectric constant can be written as a Fourier series: ∞ X 1 ε−1 expigy ny: ε13 y n−∞ 13 n
(3)
where l 1 represents the homogeneous semi-infinite medium, l 2 represents the first film, l 3 represents the second film, and this pattern continues, with l 11 representing the tenth film; l 12 represents the vacuum gap, and z1 0, z2 z1 d2 ; …; z11 z10 d11 , correspond to the localization of the interfaces. The parallel component of the diffracted wave vector is βn
p μ1 ε1 ω sin θi 2π∕an;
(7)
p and γ l βn μ1 ε1 ω2 − β2n are the perpendicular components. The Fourier coefficients of the Gaussian beam, which are displaced a horizontal distance −yd from the origin of the coordinate system, are given by
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e1tx βn
Munguía-Arvayo et al.
exp−β2n − β20 w2 ∕2 expiβn yd for jβn j ≤ β0 : 0 for jβn j > β0 (8)
e3tx βn e3rx βn e2tx βn expiγ 2 βn z2 e2rx βn exp−iγ 2 βn z2 ;
(16)
The electric field components E 1x y; z and E 2x y; z must satisfy the boundary conditions at the interface z1 0, that is, E 1x y; z E 2x y; z and μ2 ∂E1x y; z1 ∕∂z μ1 ∂E 2x y; z1 ∕∂z. According to the orthogonal properties of the Rayleigh functions, this yields the following pair of equations:
γ 3 βn γ β e β − 3 n e3rx βn μ3 3tx n μ3 γ 2 βn e β expiγ 2 βn z2 μ2 2tx n γ β − 2 n e2rx βn exp−iγ 2 βn z2 : μ2
e2tx βn e2rx βn e1tx βn e1rx βn ;
Using Eqs. (16) and (17), the following relations are obtained:
(9)
(17)
e3tx βn
C 2 βn e2tx βn expiγ 2 βn z2 − D2 βn e2rx βn exp−iγ 2 βn z2 ; F 3 βn
(18)
e3rx βn
D2 βn e2tx βn expiγ 2 βn z2 − C 2 βn e2rx βn exp−iγ 2 βn z2 ; F 3 βn
(19)
γ 2 βn γ β e β − 2 n e2rx βn μ2 2tx n μ2 γ 1 βn γ β e β − 1 n e1rx βn : μ1 1tx n μ1
(10)
where C 2 βn , D2 βn , and F 3 βn have the same form as in Eqs. (13), (14), and (15) respectively, and only the subindexes are incremented by one. Substituting Eqs. (11) and (12) into Eqs. (18) and (19), the following expressions are obtained:
Using Eqs. (9) and (10), the following relations are obtained: C β e β − D1 βn e1rx βn e2tx βn 1 n 1tx n ; F 2 βn D β e β − C 1 βn e1rx βn e2rx βn − 1 n 1tx n ; F 2 βn
e3tx βn
U 02 βn e1tx βn − V 02 βn e1rx βn ; F 3r βn
(20)
e3rx βn
U 2 βn e1tx βn − V 2 βn e1rx βn ; F 3r βn
(21)
(11)
(12) where
where V 2 βn D1 βn D2 βn expiγ 2 βn d2
γ β γ β C 1 βn 1 n 2 n ; μ1 μ2
(13)
γ 1 βn γ 2 βn − ; μ1 μ2
(14)
2γ 2 βn : μ2
(15)
D1 βn
F 2 βn
The formalism continues in the next interface. The electric field’s components and their derivatives must satisfy the boundary conditions at the interface z2 z1 d2 . Then, according to the corresponding boundary conditions, two equations are obtained:
C 1 βn C 2 βn exp−iγ 2 βn d2 ;
(22)
U 2 βn C 1 βn D2 βn expiγ 2 βn d2 D1 βn C 2 βn exp−iγ 2 βn d2 ;
(23)
U 02 βn C 1 βn C 2 βn expiγ 2 βn d2 D1 βn D2 βn exp−iγ 2 βn d2 ;
(24)
V 02 βn D1 βn C 2 βn expiγ 2 βn d2 C 1 βn D2 βn exp−iγ 2 βn d2 ;
(25)
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Vol. 31, No. 7 / July 2014 / J. Opt. Soc. Am. A
F 3r βn F 2 βn F 3 βn :
(26)
∞ 13 X ξ k
k1
This procedure continues until the last interface between homogeneous media. Then, using the boundary conditions at the interface z11 z10 d11 and the corresponding electric fields, the Fourier coefficients of the electric field in the vacuum gap are determined: e12tx βn
U 011 βn e1tx βn − V 011 βn e1rx βn ; F 12r βn
μ13
13tx A13 n;k ek
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γ 12 βn e β expiγ 12 βn z12 μ12 12tx n −
γ 12 βn e β exp−iγ 12 βn z12 : μ12 12rx n (36)
By substituting Eqs. (27) and (28) into Eqs. (35) and (36), two equations are obtained:
(27) F 12 βn F 11r βn
∞ X
13tx A13 n;k ek
k1
e12rx βn
U 11 βn e1tx βn − V 11 βn e1rx βn ; F 12r βn
H 12 βn e1tx βn − H 12 r βn e1rx β n ; t
(28)
F 11r βn
where
∞ X 2ξ13 k
μ13
k1
V 11 βn V 010 βn D11 βn expiγ 11 βn d11 V 10 βn C 11 βn exp−iγ 11 βn d11 ;
(29)
U 011 βn
G12 βn e1tx βn t
− G12 r βn e1rx β n ;
(38)
βn U 011 βn expiγ 12 βn d12 G12 t U 11 βn exp−iγ 12 βn d12 ;
(30)
(39)
0 G12 r βn V 11 βn expiγ 12 βn d12
U 010 βn C 11 βn expiγ 11 βn d11 U 10 βn D11 βn exp−iγ 11 βn d11 ;
13tx A13 n;k ek
where
U 11 βn U 010 βn D11 βn expiγ 11 βn d11 U 10 βn C 11 βn exp−iγ 11 βn d11 ;
(37)
V 11 βn exp−iγ 12 βn d12 ;
(31)
(40)
0 H 12 r βn V 11 β n expiγ 12 βn d12
− V 11 βn exp−iγ 12 βn d12 ;
V 011 βn V 010 βn C 11 βn expiγ 11 βn d11 V 10 βn D11 βn exp−iγ 11 βn d11 ;
(41)
(32) H 12 βn U 011 βn expiγ 12 βn d12 t
F 12r βn F 11r βn F 12 βn :
− U 11 βn exp−iγ 12 βn d12 :
(33)
It is helpful to note that Eqs. (29–33) have a recursive form, which enables the determination of the amplitudes given by Eqs. (27) and (28) for any number of films. The field in the IM can be written as the modal (Burckhardt–Rayleigh) expansion
(42)
Multiplying the terms of Eq. (37) by G12 r βn and those of Eq. (38) by H 12 r β n and subtracting the results, a matrix equation called the “matrix scattering equation” is obtained: ∞ X
13tx T 12 U 12 n ; n;k ek
(43)
k1
E13tx ⃗r
∞ X ∞ X
13tx A13 expfiβn y ξ13 n;k ek k z − z12 g;
where
k1 n−∞
2γ 12 βn 2ξ13 12 12 k A13 G β − H β T 12 r r n n n;k n;k μ12 μ13
(34) where the subindex k iterates through all of the solutions obtained using Eq. (5). Using the boundary condition at the interface between the vacuum gap and IM z12 z11 d12 , the equations below are obtained: ∞ X
13tx A13 e12tx βn expiγ 12 βn z12 n;k ek
k1
e12rx βn exp−iγ 12 βn z12 ;
(35)
(44)
and U 12 n
12 12 G12 βn − H 12 r β n H t r β n Gr βn e1tx βn : (45) F 11r βn
The procedure to obtain the different coefficients for the electric field in each homogeneous medium is as follows. First, Eq. (43) is inverted to determine the coefficients
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Munguía-Arvayo et al.
P 13wz
μ1 13 X 13 13tx 13 13tx ξu An;u eu An0 ;u eu μ13 n;n0 × f sincπn − n0 f ∕P i ;
where the power of the incident P 2 n γ 12 βn je1tx β n j and f d∕a.
(46) beam
is
Pi
3. NUMERICAL RESULTS In order to obtain numerical results, some parameters are fixed. The waist of the s-polarized Gaussian beam is fixed as w 64λ0 40.5 μm, with λ0 633 nm, and the beam is incident at angle θi 60.13° with respect to the normal to the interface between the homogeneous medium and the PC. This angle is adequate to resonantly excite the surface mode as shown in [6]. The dielectric constant of the nonabsorbing homogeneous, semi-infinite medium is ε1 1.502 . The waveguide has width d λ0 ∕10 63.0 nm and dielectric constant εc 1.50, and it is surrounded by the core, whose dielectric constant is εc 1.0. Its thickness is a − d, where the equally spaced distance between waveguides is a 256λ0 162.0 μm. For this selection of parameters, the waveguides behave as a single one; because a ≫ d, the crosstalk among the set of waveguides is negligible. The infinite dimension of the matrix-scattering equation was truncated to 2N 2048 in order to obtain the amplitudes of the diffracted fields. The value of this parameter is sufficient to reach the energy conservation within 0.1% of variation. The 1D PC has a period of 0.25 μm and comprises 10 films. The materials are considered nondispersive, and their refractive index and thickness are given in Table 1. Figure 2(a) shows the reflection as a function of the wavelength for a fixed angle of incidence θi 60.13°. The curve with red solid circles (blue solid squares) corresponds to the reflection from the 1D PC very near to (very far from) the planar waveguide, whose propagation axis is perpendicular to the surface of the truncated 1D PC. The minimum in the reflection corresponds to the excitation of the SM at the interface between the 1D PC and the vacuum, which was efficiently excited for a wavelength of λ 644.8 nm inside the forbidden band. The distance of separation between the 1D PC and the waveguide is d12 ∼ λ0 ∕63 10.0 nm. The closeness of the waveguide modifies the optical response of the 1D PC, and the waveguide diminishes the strength coupling between the surface mode and the photon. The finite width of the Gaussian beam introduces additional wave vectors in its spatial spectrum, and a weak coupling is observed as two almost symmetric dips. This effect tends to disappear as the width of the Gaussian beam is increased. The minimum of the resonance is shifted to smaller energies by approximately 0.00012 eV. Two secondary minima can be observed. The small width, approximately 0.0023 eV,
Table 1. 1D PC is Composed of Alternate Materials That Have High and Low Refractive Indexesa Medium
Film
Refraction Index (n)
Absorption Index (k)
Thickness d (μm)
2 3 4 5 6 7 8 9 10 11
1 2 3 4 5 6 7 8 9 10
1.37 2.14 1.37 2.14 1.37 2.14 1.37 2.14 1.37 2.14
0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003
0.120 0.130 0.120 0.130 0.120 0.130 0.120 0.130 0.120 0.051
a The pair of materials correspond to MgF2 (the low index 1.37) and TiO2 (the high index 2.14). The thickness of the last film breaks the symmetry of the 1D PC and induces the existence of the surface mode.
is related to the propagation distance of the SM and is associated to the radiative losses of the SM. The energy balance establishes that 80% of the energy is transferred to the surface modes and the remainder is lost
1.0 0.9
(h/2π)ω =1.91325 eV
(h/2π)ω =1.93300 eV
0.8 0.7 0.6
R (TE-pol)
e13tx . Next, these values are substituted into Eq. (37) to obk tain the Fourier coefficients of the reflected beam. Then, the values of e1rx are substituted into Eqs. (27) and (28) to determine the coefficients e12tx βn and e12rx βn . Finally, the recursive form is used to obtain the remainder of the amplitudes. The normalized power in the waveguide, assuming that only one mode (the uth mode) is supported for it, is
0.5 0.4 0.3 0.2 0.1
(h/2π)ω =1.92325 eV
(a)
0.0 1.90
1.91
1.92
1.93
1.94
1.95
E (eV) 1E-4
Pw Pt
1E-5
Power
1592
1E-6
1E-7
(b) 1E-8 1.89
1.90
1.91
1.92
1.93
1.94
1.95
E (eV) Fig. 2. (a) Reflection as a function of the energy (wavelength) for a fixed angle of incidence θi 60.13°. The curve with red solid circles (blue solid squares) corresponds to the reflection of the 1D PC near to (far from) a waveguide. The air gap is λ0 ∕63, where λ is the wavelength of 633 nm. The curve was calculated with energy steps of 0.00025 eV. (b) Log total power (red solid squares) in the IM and the power in the waveguide (blue solid circles) as functions of the incident energy at a 60.13° fixed angle of incidence.
Munguía-Arvayo et al.
as scattering energy because the waveguide acts as an efficient diffractive element. Figure 2(b) shows the total power (red solid squares) in the IM and the power in the waveguide (blue solid circles) as functions of the incident energy. The resonant coupling of the SM appears as a maximum in both plots. The amplitude of the total power is greater than the power in the waveguide because the former includes the scattering light. The measurement of the energy in the waveguide is the basic principle supporting the SNOM technique, and this result shows that the energy is easily measured at resonant conditions; however, the ease of measurement depends on the distance from the waveguide. In the case of Fig. 3, several effects are included with the waveguide, but they cannot be detected using the SNOM technique; they can only be visualized theoretically. Figure 3 shows the modulus of the electric field projected into the y–z plane. It was calculated at resonant conditions, that is, h∕2πω 1.92325 eV and λ 644.8 nm, and angle of incidence θι 60.13°. The 1D PC is approximately 1.2 μm thick. The excitation of the surface mode in the last TiO2 film can be clearly seen. As expected, its field decays exponentially approaching both sides of the interface, in the 1D PC and in the vacuum, and it oscillates in the 1D PC. Two effects are observed: (a) the propagation of the surface mode, which extends far from the waveguide, and (b) the oscillations of the surface modes along the last film. For the region z < 0, it is possible to observe the interference between the incident beam and the reflected one, and the diffraction produced for the very narrow waveguide exhibits zones of high and low intensity near y 0. In Fig. 4, the near-field power, that is, the power inside of the waveguide, is plotted as a function of both distances, perpendicular (z) and parallel (y) to the surface, at resonant conditions, i.e., E 1.92325 eV and θi 60.13°. The end waveguide is positioned at different distances from the real surface. The discrete points are measured with resolution of Δz 10 nm and Δy 5 μm. It is possible to observe the field distribution of the surface mode. Two main characteristics evident in the figure are the decay distance of approximately 50 nm and the propagation distance of approximately 25 μm.
Fig. 3. Modulus of the electric field projected into the y–z plane. The horizontal black lines correspond to the interfaces between each film that form the 1D PC. The vertical lines above the top of the 1D PC help to visualize one of the waveguides in the set; the remainder are outside of the limits of this graph. The image was calculated using steps of Δy 80.6 nm and Δz 20.1 nm.
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Fig. 4. Near-field power (power flowing inside of the waveguide) plotted as a function of the waveguide end-tip position for both distances, perpendicular (z) and parallel (y) to the surface, at resonant conditions, i.e., E 1.92325 eV and θi 60.13°.
Our intent was not necessarily to use a sharp or rounded tip or an optical fiber covered with metal; instead, in this work, the waveguide end-tip was “rectangular” and a pure dielectric. A bare dielectric waveguide 63 nm in width was used. It was impossible to observe the oscillations of the surface mode, owing to the width of the waveguide.
4. CONCLUSIONS The influence of a waveguide on the evanescent field of an SM was studied. When a small waveguide is placed near (
