- Email: [email protected]
Propagating mode in the photonic gap of 1D resonant Bragg reflector
Journal of Luminescence 100 (2002) 283–289
Propagating mode in the photonic gap of 1D resonant Bragg reflector Kikuo Cho*, Takahiko Hirai, Tomoe Ikawa Department of Materials Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, 560-8531 Japan Received 26 July 2002
Abstract Concerning ‘‘1D resonant photonic crystals’’, i.e., 1D periodic arrays of resonant layers with a spacing satisfying the Bragg condition for resonant light, recent topics are summarized together with additional new findings. The central points are about (1) the recent theoretical finding of a new branch in the middle of photonic gap, (2) an approachdependent controversy about its existence, (3) its consequence as sharp dips in reflectance spectrum and the corresponding internal field patters, indicating its propagating mode character, (4) the comparison between macroscopic and microscopic treatment of resonance effect, and (5) corrected transfer matrix method and its analytical equivalence with nonlocal response theory. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Resonant photonic crystal; Gap mode; Standing wave; Nonlocal response theory
1. Introduction The study of photonic crystals is now very popular because of its potential utility for device applications [1]. They are periodic structures consisting of two or more different dielectrics, and the theoretical treatment of them is essentially the macroscopic electromagnetism for finite frequency. Apart from the validity condition of using macroscopic dielectric constants for these structures, it is physically a well-defined problem within the macroscopic electromagnetic (EM) theory. The main problem exists in obtaining mathematical *Corresponding author. Tel.: +81-6-6850-6400; fax: +81-66850-6400. E-mail address: [email protected] (K. Cho).
solutions of the Maxwell equations with complicated boundary conditions for EM field. One could consider other types of photonic crystals, i.e., those containing materials with resonances, for example, periodic structures consisting of quantum wells (QWs), wires or dots. In such a system, there is, in addition to mathematical problem, a physical one as to the description of the resonant effect whether one may use a macroscopic approach or should use a microscopic one. It is possible that the macro- and microscopic approaches may give different results. In this article, we consider a particular type of resonant photonic crystals, namely, a 1D resonant Bragg reflector, or a 1D photonic crystal consisting of N resonant layers (N QWs for example), with the lattice interval d satisfying the Bragg
0022-2313/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 0 2 ) 0 0 4 2 2 - 2
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
284
condition for the resonant wavelength l0 : In each layer we assume a single resonant level. This system has a peculiar excited state, called superradiant (SR) mode, with optimal coupling with the light for small values of N; and the system behaves as a photonic crystal for larger N; where SR mode no more exists. Examples of such systems are multiple QWs or a fictitious lattice of atomic . planes [2–5]. The crystals of Mossbauer nuclei are examples of 3D version in gamma ray region [6]. The outline of the nonlocal formalism, to be used as the main tool, is as follows [7]. The induced current density is written as X # # Jðr; oÞ ¼ ð/0jIðrÞjnSX n0 þ /njIðrÞj0SX0n Þ; ð1Þ n
# where IðrÞ is the current density operator, and jnS; j0S are the matter excited state and the ground state, respectively. In applying to the present model, the number of the excited states is restricted to just one, and for resonant process, the terms proportional to fX0n g can be neglected. The vector potential A is obtained from the Maxwell equations with this current density as the source term as Z 1 Aðr; oÞ ¼ A0 ðr; oÞ þ dr0 Gðr; r0 ; oÞ Jðr0 ; oÞ; ð2Þ c where A0 is an incident field, and G is the (tensor) radiation Green’s function in vacuum. The equations to determine fX g are given, in the rotating wave approximation, as X ð0Þ ½ðEm0 _oÞdmn þ An0;0m Xm0 ¼ Fn0 ; ð3Þ m
where Em0 ¼ Em E0 is the excitation energy of matter, and F ð0Þ is defined as Z ð0Þ # ¼ dr/njIðrÞj0S A0 ðr; oÞ: ð4Þ Fn0 The radiative correction, A; i.e., the interaction energy between the components of induced current density via transverse EM field, is defined as Z Z 1 # Am0;0n ðoÞ ¼ 2 dr dr0 /mjIðrÞj0S c # 0 ÞjnS: ð5Þ Gðr; r0 ; oÞ /0jIðr
The real and imaginary parts of An0;0n ðoÞ give the radiative shift and width of the excitation energy En0 of matter.
2. Photonic band dispersion The dispersion relation of light oðkÞ in a resonant Bragg reflector has been studied by Deych and Lisyansky [3] in terms of the dispersion equations derived from the Kronig–Penny model type of consideration, or transfer matrix method [2]. The dispersion equation used in Ref. [3] has the form cos kd ¼ cos qd G0
sin qd ; o0 o ig
ð6Þ
where q ¼ o=c; o0 is the resonance frequency, g the nonradiative decay width, and G0 the radiative width of a single resonant layer. If the Bragg condition d ¼ l0 =2 is not exactly satisfied and if g ¼ 0; this equation has three real roots for each k in the neighborhood of the lowest photonic gap. They are the usual photonic bands coupled with the resonant level. For the exact Bragg condition d ¼ l0 =2; however, the solution for the resonant level is lost. Since the vanishing of the resonant level solution for a particular value of d is unphysical, we tried to examine the dispersion curve from a different point of view. Namely, applying the nonlocal theory to a periodic lattice, we have derived a new dispersion equation and obtained the dispersion curves of the 1D resonant Bragg reflector [4,8]. For the ideal case of ‘‘the exact Bragg condition d ¼ l0 =2 and g ¼ 0’’, the dispersion equation has the form N jSð0Þj2 X _cðq0 qÞ ¼ 2 þ jSð2jq0 Þj2 k q2 j¼1 1 1
þ ; ðk þ 2jq0 Þ2 q2 ðk 2jq0 Þ2 q2 ð7Þ where q0 is the wave number of the resonant light in vacuum, which is equal to the half of the 1D Brillouin zone, and SðgÞ is the k þ g Fourier component of the induced current density
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
accompanying the resonant excitation of this system. In this equation the background part of polarization is neglected. (The effect of background dielectric constant could be easily included via the 1D radiation Green’s function to calculate the radiative correction.) This equation is derived from the condition for the nontrivial solution, in the absence of the external force, of the fundamental equation of the nonlocal response theory Eq. (3), applied to the 1D resonant Bragg reflector, as 0 ¼ Ek _o þ Ak0;0k ðoÞ:
ð8Þ
Here, Ak0;0k ðoÞ is the radiative correction calculated for the resonant level with the wave number k: This is a diagonal element of the matrix equation det jSj ¼ 0; where S is the coefficient matrix of Eq. (3). The assumption of a single excited level in each layer leads to the block diagonalization of the matrix S into the product of 1 1 blocks for the system with N ¼ N: The above Eq. (7) can be solved graphically. Drawing the values of the right- and left-hand
285
sides as functions of q for each value of k; we obtain the solutions from the crossing points of the straight line passing through q ¼ q0 (l.h.s.) and the monotonously increasing function between the poles (r.h.s.). Obviously, we get three real roots near the lowest photonic gap, which is true also for the exact Bragg condition d ¼ l0 =2 and g ¼ 0 (Fig. 1). Namely, the solution of the resonant level always exists irrespective of the exactness of the Bragg condition, and this is a more reasonable conclusion. Later it turned out that the vanishing of the resonant level solution from Eq. (6) is due to the neglect of the radiative shift term in the denominator of Eq. (6). By including the radiative shift term as a constant in the denominator of Eq. (6), we can numerically reproduce the three band dispersion curves. Though this term is included in the original expression of Ivchenko [9], it was for some reason neglected, probably because it is very small. Actually, the bandwidth of the gap mode is the magnitude of the radiative shift of a single resonant layer. However, this does not necessarily mean the equivalence of the
hω
E0 + ∆ 0
E0
E00 0.0
0.2
0.4
k
0
0
0.2
0.4
k
0.6 π
0.8
0.6 π
0.8
1.0
d
1
d
Fig. 1. Photonic bands of the 1D resonant Bragg reflector near the lowest gap calculated for the fictitious lattice of hydrogen atoms [4]. The flat band in the gap is enlarged on the right-hand side. D0 is the radiative shift of a single layer, and E0 the 1s-2p transition energy corrected by the dipole–dipole interaction energy of the single lattice plane.
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
286
transfer matrix method and the nonlocal theory, as will be discussed in Section 4.
hω = hω* 1600
N = 101
The dispersion relation of the gap mode with real oðkÞ for the ideal condition mentioned above allows a light propagating channel in the middle of the total reflection range of the photonic gap. Though the bandwidth is quite narrow, one would expect some indications of its propagating character in observable quantities. Thus, we inspected the calculated reflectance spectrum in great detail, and then, we found out very sharp dips in the total reflection range for systems with finite number of resonant layers (Fig. 2). For an N layer system, there are N 1 dips. The positions of the dips well correspond to the frequency range of the gap mode. Furthermore, the internal fields at these dip frequencies have standing wave patterns (Fig. 3), which corresponds well to the propagating character of the gap mode [4]. These results are obtained from the analysis in terms of the nonlocal framework, which requires the diagonalization of N N matrix to obtain the self-consistent field and induced polarization for an N-layered system. From the results of finite N systems, we guess that the N 1 dips will merge into a single
* Reflectivity
1.0
0.99
N = 101
4.0
5.0
6.0
7.0
hω E 0 µeV
8.0
9.0
Fig. 2. Sharp dips in the total reflection range for finite lattice ðN ¼ 101Þ; corresponding to the flat band region of Fig. 1.
E (z) E inc
2
3. The consequence of the gap mode
7.2µeV
800
0
20
40
60
80
100
site Fig. 3. Internal field pattern for the dip frequency with * mark of Fig. 2 as a function of position in the N ¼ 101 lattice. The intensity depends very sensitively on the frequency within the dip.
broader dip in the limit of N-N; and the width of the merged dip will coincide with the band width of the gap mode. However, the calculation of the spectrum for N ¼ N system is hard to carry out because of the N N diagonalization. As an alternative, we have made use of the transfer matrix method for the macroscopic response of the corresponding model, because it allows the calculation of the case N ¼ N by the use of Cayley– Hamilton method [10]. The model is an infinite array of the layers with resonant dielectric constant. The reflectivity spectrum is calculated via the transfer matrix method, and the photonic bands are calculated from the Fourier transform of the Maxwell equations with 1D periodicity. The dispersion curves for N ¼ N contains a very flat gap mode in the photonic gap region. For the reflectivity of finite N layers, we have obtained a very similar result as before about the sharp dips in the range of total reflection. There are N 1 dips for an N layer system, and their positions correspond well to the gap mode frequencies. The case of N ¼ N was calculated with the help of Cayley–Hamilton method, which shows the expected result about the form of the dip for the gap mode. Namely, all the sharp dips for finite N systems merge into a single dip with the band width of the gap mode.
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
287
4. Equivalence of the transfer matrix method and nonlocal scheme
1.00 0.99
Reflectivity
0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 -1.0
-0.8
-0.6
-0.4
-0.2
0
hω – E 0 (µeV) Fig. 4. Spurious dip structure just below E0 due to the use of macroscopic response theory in the resonant region of very thin layers. The value of nonradiative damping g ¼ 1:0 10 9 eV is used.
There is a series of spurious structure just below the resonant frequency o0 as shown in Fig. 4. This is due to the multiple internal reflection of light in each of the resonant layer treated as a macroscopic dielectric slab. Since we have used a very small value of g ð¼ 1:0 10 12 eVÞ; the refractive index increases almost indefinitely as o approaches o0 from below. This means that, in this limited frequency region, there are an infinite number of Fabry–Pe! rot interferences with more and more short wavelength components as the resonance is approached. This is totally different from the result of the nonlocal scheme, where the spatial structure of internal field near a resonance is characterized by the structure of the resonant polarization. Thus, the feature in Fig. 4 clearly indicates the invalid use of the macroscopic approach in a small space. The use of the macroscopic framework has brought about both positive and negative results. The feature about the merging of the narrow dips into a single one as N-N belongs to the former, and the spurious structure below the resonance to the latter. A similar calculation should be possible in terms of the transfer matrix method applied to the microscopic model. After giving an argument about the equivalence of such a transfer method and the nonlocal theory in the next section, we will show the result of such a numerical calculation.
Though the nonlocal response theory has more general applicability, the transfer matrix method applied to the microscopic model is also useful in the present 1D problem. Numerical results seem to indicate the equivalence of the two schemes. In this section, we consider whether or not they are analytically equivalent. For this purpose, we give the following set of considerations: (A) We first determine the elements of the transfer matrix via the solution of the nonlocal response for a single resonant layer. (B) From the fundamental equations of the nonlocal theory, we determine the weight Xn of the emitted field from the nth resonant layer, which provides the field amplitude at an arbitrary position. (C) Then, we relate the field amplitudes at neighboring interlayer positions by means of the transfer matrix. (D) We compare the set of equations obtained from (C), which are written in terms of fXn g; with the equations for fXn g of the nonlocal theory, and see if they are equivalent or not. For simplicity, we assume the normal incidence of external light and neglect the background polarization. These assumptions do not affect the equivalence of the two schemes. According to the nonlocal theory, the induced current density by the excitation of the resonant level of a single layer ð0-nÞ is, given as 1 JðzÞ ¼ Xn0 I0n ðzÞ; c
ð9Þ
where we consider only one resonant level ðEn0 Þ in the layer. For this current density, the vector potential is given as Z 1 AðzÞ ¼ A0 ðzÞ þ 2 Xn0 dz0 Gðz; z0 ; oÞI0n ðz0 Þ: ð10Þ c The coefficient Xn0 satisfies ð0Þ ðEn0 _o þ An0;0n ÞXn0 ¼ Fn0 ;
ð11Þ
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
288
where ð0Þ Fn0 ¼
Z
dz In0 ðzÞA0 ðzÞ:
ð12Þ
The (1D) radiation Green’s function Gðz; z0 ; oÞ is defined as 2pi expðiqjz z0 jÞ; ð13Þ Gðz; z0 ; oÞ ¼ q so that the plane waves going to the þz and z directions in (10) represent the transmitted and reflected light, respectively. For an incident plane wave A0 ðzÞ ¼ A% 0 expðiqzÞ we have reflection amplitude r¼
2pi I*n0n ðqÞI*0n ð qÞ ; qc2 En0 _o þ An0;0n
ð14Þ
This is satisfied for any real induced current density I0n ðzÞ with arbitrary phase factor. Thus, we may generally expect the analytical equivalence of the transfer matrix method and the nonlocal framework with respect to the present 1D problem, if we start with the corrected reflection amplitude mentioned above. The details of this comparison will be published elsewhere. Now that the equivalence is shown between the transfer matrix method and the nonlocal theory for the present 1D problem, we can calculate the reflectivity spectrum in the limit of N ¼ N via the transfer matrix method with the help of the Cayley–Hamilton method. (For the parameter values of the present model, the correction for 1.00
where Z
0.99
dz expð iqzÞI0n ðzÞ
ð15Þ
and the transmission amplitude 2pi jI*0n ðqÞj2 t¼1þ 2 : qc En0 _o þ An0;0n
Reflectivity
I*0n ðqÞ ¼
ð16Þ
This result does not allow us to write t ¼ 1 þ r as Ivchenko did [9]. This is because the numerator of * The result of r cannot be written as i ImðAÞ: Ivchenko is correct in the limit of q-0; or for thin enough layer. In spite of this complication, it is amusing to see that jrj2 þ jtj2 ¼ 1 is satisfied, which can be shown by making use of the Fourier transform of the Green’s function. Thus step (A) requires the above-mentioned change in the reflection amplitude. The transfer matrix is given in terms of r and t as ! 1 ðt2 r2 Þg2 rg2 T¼ ; ð17Þ tg r 1 where the phase factor g is defined as g ¼ expðiqdÞ for the period of the 1D lattice d: Considerations (B)–(D) for an N layer system shows that the equations for the field amplitudes between the neighboring interlayer positions can be rewritten into the equations of fXn g; which agree with Eqs. (3), if the following equality holds: ð18Þ jI*0n ðqÞj2 ¼ jI*0n ð qÞj2 :
N = 5000
0.98 0.97 0.96 0.95 0.94 0.93 -20
-10
(a)
0
10
20
hω– E0 (µeV)
1.0000
N =∞
0.9999
-20 (b)
-10
0
10
20
hω – E0 (µeV)
Fig. 5. Reflectivity due to the gap mode for (a) finite ðN ¼ 5000Þ and (b) infinite lattices, calculated in terms of the transfer matrix method for microscopic model. The value of nonradiative damping g ¼ 1:0 10 12 eV is used.
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
r is not important.) Such a calculation shows the merging of the sharp dips for finite N systems into a single dip for N ¼ N; and the series of the sharp Fabry–Pe! rot interference just below o0 ; mentioned in the previous section as a spurious effect, does not exist in this microscopic calculation, as expected. Fig. 5 shows the comparison between the cases of N ¼ 5000 and N: To summarize, we have shown the existence of a gap mode in a 1D resonant Bragg reflector. This leads to a dip structure in the total reflection range, and the corresponding internal field has a standing wave pattern. The equivalence of the transfer matrix method and the nonlocal theory for this 1D problem is shown, and the validity limit of a macroscopic model for this problem is explicitly demonstrated.
Acknowledgements This paper is dedicated to late Prof. S. Shionoya, who gave a unique contribution to the field of optical physics and chemistry by editing a comprehensive Handbook of ‘‘Optical Properties of Solids’’ (in Japanese), in addition to his active research works. This work was supported in part by the Grant-in-Aid for Specially Promoted
289
Research (10CE2004) of the Ministry of Education, Culture, Sports, Science and Technology of Japan.
References [1] K. Sakoda, Optical Properties of Photonic Crystals, Springer Series of Optical Sciences, Springer, Berlin, 2001; C.M. Soukoulis (Ed.), Photonic Crystals and Light Localization in the 21st Century, Nato Science Series C: Mathematical and Physical Science, Vol. 563, Kluwer Academic Publishers, Dordrecht, 2001. [2] E.L. Ivchenko, A.I. Nesvizhskii, S. Jorda, Fiz. Tverd. Tela (St. Petersburg) 36 (1994) 2118 [Phys. Solid State 36 (1994) 1156]. [3] L.I. Deych, A.A. Lisyansky, Phys. Rev. B 62 (2000) 4242. [4] T. Ikawa, K. Cho, Phys. Rev. B 66 (2002) 085338; T. Ikawa, K. Cho, J. Phys. Soc. Japan 71 (2002) 1381. [5] I.H. Deutsch, R.J.C. Spreeuw, S.L. Rolston, W.D. Phillips, Phys. Rev. A 52 (1995) 1394. . [6] U. van Burck, . R.L. Mossbauer, E. Gerdau, R. Ruffer, . R. Hollatz, G.V. Smirnov, J.P. Hannon, Phys. Rev. Lett. 59 (1987) 355. [7] K. Cho, Prog. Theor. Phys. Suppl. 106 (1991) 225; K. Cho, J. Phys. Soc. Japan 66 (1997) 2496. [8] K. Cho, T. Ikawa, Phys. Stat. Sol. A 190 (2002) 401. [9] E.L. Ivchenko, Fiz. Tverd Tela (St. Petersburg) 33 (1991) 2388 [Sov. Phys. Solid State 33 (1991) 1344]. [10] F. Abel!es, Ann. Phys. 5 (1950) 706.
Propagating mode in the photonic gap of 1D resonant Bragg reflector Kikuo Cho*, Takahiko Hirai, Tomoe Ikawa Department of Materials Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, 560-8531 Japan Received 26 July 2002
Abstract Concerning ‘‘1D resonant photonic crystals’’, i.e., 1D periodic arrays of resonant layers with a spacing satisfying the Bragg condition for resonant light, recent topics are summarized together with additional new findings. The central points are about (1) the recent theoretical finding of a new branch in the middle of photonic gap, (2) an approachdependent controversy about its existence, (3) its consequence as sharp dips in reflectance spectrum and the corresponding internal field patters, indicating its propagating mode character, (4) the comparison between macroscopic and microscopic treatment of resonance effect, and (5) corrected transfer matrix method and its analytical equivalence with nonlocal response theory. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Resonant photonic crystal; Gap mode; Standing wave; Nonlocal response theory
1. Introduction The study of photonic crystals is now very popular because of its potential utility for device applications [1]. They are periodic structures consisting of two or more different dielectrics, and the theoretical treatment of them is essentially the macroscopic electromagnetism for finite frequency. Apart from the validity condition of using macroscopic dielectric constants for these structures, it is physically a well-defined problem within the macroscopic electromagnetic (EM) theory. The main problem exists in obtaining mathematical *Corresponding author. Tel.: +81-6-6850-6400; fax: +81-66850-6400. E-mail address: [email protected] (K. Cho).
solutions of the Maxwell equations with complicated boundary conditions for EM field. One could consider other types of photonic crystals, i.e., those containing materials with resonances, for example, periodic structures consisting of quantum wells (QWs), wires or dots. In such a system, there is, in addition to mathematical problem, a physical one as to the description of the resonant effect whether one may use a macroscopic approach or should use a microscopic one. It is possible that the macro- and microscopic approaches may give different results. In this article, we consider a particular type of resonant photonic crystals, namely, a 1D resonant Bragg reflector, or a 1D photonic crystal consisting of N resonant layers (N QWs for example), with the lattice interval d satisfying the Bragg
0022-2313/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 0 2 ) 0 0 4 2 2 - 2
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
284
condition for the resonant wavelength l0 : In each layer we assume a single resonant level. This system has a peculiar excited state, called superradiant (SR) mode, with optimal coupling with the light for small values of N; and the system behaves as a photonic crystal for larger N; where SR mode no more exists. Examples of such systems are multiple QWs or a fictitious lattice of atomic . planes [2–5]. The crystals of Mossbauer nuclei are examples of 3D version in gamma ray region [6]. The outline of the nonlocal formalism, to be used as the main tool, is as follows [7]. The induced current density is written as X # # Jðr; oÞ ¼ ð/0jIðrÞjnSX n0 þ /njIðrÞj0SX0n Þ; ð1Þ n
# where IðrÞ is the current density operator, and jnS; j0S are the matter excited state and the ground state, respectively. In applying to the present model, the number of the excited states is restricted to just one, and for resonant process, the terms proportional to fX0n g can be neglected. The vector potential A is obtained from the Maxwell equations with this current density as the source term as Z 1 Aðr; oÞ ¼ A0 ðr; oÞ þ dr0 Gðr; r0 ; oÞ Jðr0 ; oÞ; ð2Þ c where A0 is an incident field, and G is the (tensor) radiation Green’s function in vacuum. The equations to determine fX g are given, in the rotating wave approximation, as X ð0Þ ½ðEm0 _oÞdmn þ An0;0m Xm0 ¼ Fn0 ; ð3Þ m
where Em0 ¼ Em E0 is the excitation energy of matter, and F ð0Þ is defined as Z ð0Þ # ¼ dr/njIðrÞj0S A0 ðr; oÞ: ð4Þ Fn0 The radiative correction, A; i.e., the interaction energy between the components of induced current density via transverse EM field, is defined as Z Z 1 # Am0;0n ðoÞ ¼ 2 dr dr0 /mjIðrÞj0S c # 0 ÞjnS: ð5Þ Gðr; r0 ; oÞ /0jIðr
The real and imaginary parts of An0;0n ðoÞ give the radiative shift and width of the excitation energy En0 of matter.
2. Photonic band dispersion The dispersion relation of light oðkÞ in a resonant Bragg reflector has been studied by Deych and Lisyansky [3] in terms of the dispersion equations derived from the Kronig–Penny model type of consideration, or transfer matrix method [2]. The dispersion equation used in Ref. [3] has the form cos kd ¼ cos qd G0
sin qd ; o0 o ig
ð6Þ
where q ¼ o=c; o0 is the resonance frequency, g the nonradiative decay width, and G0 the radiative width of a single resonant layer. If the Bragg condition d ¼ l0 =2 is not exactly satisfied and if g ¼ 0; this equation has three real roots for each k in the neighborhood of the lowest photonic gap. They are the usual photonic bands coupled with the resonant level. For the exact Bragg condition d ¼ l0 =2; however, the solution for the resonant level is lost. Since the vanishing of the resonant level solution for a particular value of d is unphysical, we tried to examine the dispersion curve from a different point of view. Namely, applying the nonlocal theory to a periodic lattice, we have derived a new dispersion equation and obtained the dispersion curves of the 1D resonant Bragg reflector [4,8]. For the ideal case of ‘‘the exact Bragg condition d ¼ l0 =2 and g ¼ 0’’, the dispersion equation has the form N jSð0Þj2 X _cðq0 qÞ ¼ 2 þ jSð2jq0 Þj2 k q2 j¼1 1 1
þ ; ðk þ 2jq0 Þ2 q2 ðk 2jq0 Þ2 q2 ð7Þ where q0 is the wave number of the resonant light in vacuum, which is equal to the half of the 1D Brillouin zone, and SðgÞ is the k þ g Fourier component of the induced current density
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
accompanying the resonant excitation of this system. In this equation the background part of polarization is neglected. (The effect of background dielectric constant could be easily included via the 1D radiation Green’s function to calculate the radiative correction.) This equation is derived from the condition for the nontrivial solution, in the absence of the external force, of the fundamental equation of the nonlocal response theory Eq. (3), applied to the 1D resonant Bragg reflector, as 0 ¼ Ek _o þ Ak0;0k ðoÞ:
ð8Þ
Here, Ak0;0k ðoÞ is the radiative correction calculated for the resonant level with the wave number k: This is a diagonal element of the matrix equation det jSj ¼ 0; where S is the coefficient matrix of Eq. (3). The assumption of a single excited level in each layer leads to the block diagonalization of the matrix S into the product of 1 1 blocks for the system with N ¼ N: The above Eq. (7) can be solved graphically. Drawing the values of the right- and left-hand
285
sides as functions of q for each value of k; we obtain the solutions from the crossing points of the straight line passing through q ¼ q0 (l.h.s.) and the monotonously increasing function between the poles (r.h.s.). Obviously, we get three real roots near the lowest photonic gap, which is true also for the exact Bragg condition d ¼ l0 =2 and g ¼ 0 (Fig. 1). Namely, the solution of the resonant level always exists irrespective of the exactness of the Bragg condition, and this is a more reasonable conclusion. Later it turned out that the vanishing of the resonant level solution from Eq. (6) is due to the neglect of the radiative shift term in the denominator of Eq. (6). By including the radiative shift term as a constant in the denominator of Eq. (6), we can numerically reproduce the three band dispersion curves. Though this term is included in the original expression of Ivchenko [9], it was for some reason neglected, probably because it is very small. Actually, the bandwidth of the gap mode is the magnitude of the radiative shift of a single resonant layer. However, this does not necessarily mean the equivalence of the
hω
E0 + ∆ 0
E0
E00 0.0
0.2
0.4
k
0
0
0.2
0.4
k
0.6 π
0.8
0.6 π
0.8
1.0
d
1
d
Fig. 1. Photonic bands of the 1D resonant Bragg reflector near the lowest gap calculated for the fictitious lattice of hydrogen atoms [4]. The flat band in the gap is enlarged on the right-hand side. D0 is the radiative shift of a single layer, and E0 the 1s-2p transition energy corrected by the dipole–dipole interaction energy of the single lattice plane.
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
286
transfer matrix method and the nonlocal theory, as will be discussed in Section 4.
hω = hω* 1600
N = 101
The dispersion relation of the gap mode with real oðkÞ for the ideal condition mentioned above allows a light propagating channel in the middle of the total reflection range of the photonic gap. Though the bandwidth is quite narrow, one would expect some indications of its propagating character in observable quantities. Thus, we inspected the calculated reflectance spectrum in great detail, and then, we found out very sharp dips in the total reflection range for systems with finite number of resonant layers (Fig. 2). For an N layer system, there are N 1 dips. The positions of the dips well correspond to the frequency range of the gap mode. Furthermore, the internal fields at these dip frequencies have standing wave patterns (Fig. 3), which corresponds well to the propagating character of the gap mode [4]. These results are obtained from the analysis in terms of the nonlocal framework, which requires the diagonalization of N N matrix to obtain the self-consistent field and induced polarization for an N-layered system. From the results of finite N systems, we guess that the N 1 dips will merge into a single
* Reflectivity
1.0
0.99
N = 101
4.0
5.0
6.0
7.0
hω E 0 µeV
8.0
9.0
Fig. 2. Sharp dips in the total reflection range for finite lattice ðN ¼ 101Þ; corresponding to the flat band region of Fig. 1.
E (z) E inc
2
3. The consequence of the gap mode
7.2µeV
800
0
20
40
60
80
100
site Fig. 3. Internal field pattern for the dip frequency with * mark of Fig. 2 as a function of position in the N ¼ 101 lattice. The intensity depends very sensitively on the frequency within the dip.
broader dip in the limit of N-N; and the width of the merged dip will coincide with the band width of the gap mode. However, the calculation of the spectrum for N ¼ N system is hard to carry out because of the N N diagonalization. As an alternative, we have made use of the transfer matrix method for the macroscopic response of the corresponding model, because it allows the calculation of the case N ¼ N by the use of Cayley– Hamilton method [10]. The model is an infinite array of the layers with resonant dielectric constant. The reflectivity spectrum is calculated via the transfer matrix method, and the photonic bands are calculated from the Fourier transform of the Maxwell equations with 1D periodicity. The dispersion curves for N ¼ N contains a very flat gap mode in the photonic gap region. For the reflectivity of finite N layers, we have obtained a very similar result as before about the sharp dips in the range of total reflection. There are N 1 dips for an N layer system, and their positions correspond well to the gap mode frequencies. The case of N ¼ N was calculated with the help of Cayley–Hamilton method, which shows the expected result about the form of the dip for the gap mode. Namely, all the sharp dips for finite N systems merge into a single dip with the band width of the gap mode.
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
287
4. Equivalence of the transfer matrix method and nonlocal scheme
1.00 0.99
Reflectivity
0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 -1.0
-0.8
-0.6
-0.4
-0.2
0
hω – E 0 (µeV) Fig. 4. Spurious dip structure just below E0 due to the use of macroscopic response theory in the resonant region of very thin layers. The value of nonradiative damping g ¼ 1:0 10 9 eV is used.
There is a series of spurious structure just below the resonant frequency o0 as shown in Fig. 4. This is due to the multiple internal reflection of light in each of the resonant layer treated as a macroscopic dielectric slab. Since we have used a very small value of g ð¼ 1:0 10 12 eVÞ; the refractive index increases almost indefinitely as o approaches o0 from below. This means that, in this limited frequency region, there are an infinite number of Fabry–Pe! rot interferences with more and more short wavelength components as the resonance is approached. This is totally different from the result of the nonlocal scheme, where the spatial structure of internal field near a resonance is characterized by the structure of the resonant polarization. Thus, the feature in Fig. 4 clearly indicates the invalid use of the macroscopic approach in a small space. The use of the macroscopic framework has brought about both positive and negative results. The feature about the merging of the narrow dips into a single one as N-N belongs to the former, and the spurious structure below the resonance to the latter. A similar calculation should be possible in terms of the transfer matrix method applied to the microscopic model. After giving an argument about the equivalence of such a transfer method and the nonlocal theory in the next section, we will show the result of such a numerical calculation.
Though the nonlocal response theory has more general applicability, the transfer matrix method applied to the microscopic model is also useful in the present 1D problem. Numerical results seem to indicate the equivalence of the two schemes. In this section, we consider whether or not they are analytically equivalent. For this purpose, we give the following set of considerations: (A) We first determine the elements of the transfer matrix via the solution of the nonlocal response for a single resonant layer. (B) From the fundamental equations of the nonlocal theory, we determine the weight Xn of the emitted field from the nth resonant layer, which provides the field amplitude at an arbitrary position. (C) Then, we relate the field amplitudes at neighboring interlayer positions by means of the transfer matrix. (D) We compare the set of equations obtained from (C), which are written in terms of fXn g; with the equations for fXn g of the nonlocal theory, and see if they are equivalent or not. For simplicity, we assume the normal incidence of external light and neglect the background polarization. These assumptions do not affect the equivalence of the two schemes. According to the nonlocal theory, the induced current density by the excitation of the resonant level of a single layer ð0-nÞ is, given as 1 JðzÞ ¼ Xn0 I0n ðzÞ; c
ð9Þ
where we consider only one resonant level ðEn0 Þ in the layer. For this current density, the vector potential is given as Z 1 AðzÞ ¼ A0 ðzÞ þ 2 Xn0 dz0 Gðz; z0 ; oÞI0n ðz0 Þ: ð10Þ c The coefficient Xn0 satisfies ð0Þ ðEn0 _o þ An0;0n ÞXn0 ¼ Fn0 ;
ð11Þ
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
288
where ð0Þ Fn0 ¼
Z
dz In0 ðzÞA0 ðzÞ:
ð12Þ
The (1D) radiation Green’s function Gðz; z0 ; oÞ is defined as 2pi expðiqjz z0 jÞ; ð13Þ Gðz; z0 ; oÞ ¼ q so that the plane waves going to the þz and z directions in (10) represent the transmitted and reflected light, respectively. For an incident plane wave A0 ðzÞ ¼ A% 0 expðiqzÞ we have reflection amplitude r¼
2pi I*n0n ðqÞI*0n ð qÞ ; qc2 En0 _o þ An0;0n
ð14Þ
This is satisfied for any real induced current density I0n ðzÞ with arbitrary phase factor. Thus, we may generally expect the analytical equivalence of the transfer matrix method and the nonlocal framework with respect to the present 1D problem, if we start with the corrected reflection amplitude mentioned above. The details of this comparison will be published elsewhere. Now that the equivalence is shown between the transfer matrix method and the nonlocal theory for the present 1D problem, we can calculate the reflectivity spectrum in the limit of N ¼ N via the transfer matrix method with the help of the Cayley–Hamilton method. (For the parameter values of the present model, the correction for 1.00
where Z
0.99
dz expð iqzÞI0n ðzÞ
ð15Þ
and the transmission amplitude 2pi jI*0n ðqÞj2 t¼1þ 2 : qc En0 _o þ An0;0n
Reflectivity
I*0n ðqÞ ¼
ð16Þ
This result does not allow us to write t ¼ 1 þ r as Ivchenko did [9]. This is because the numerator of * The result of r cannot be written as i ImðAÞ: Ivchenko is correct in the limit of q-0; or for thin enough layer. In spite of this complication, it is amusing to see that jrj2 þ jtj2 ¼ 1 is satisfied, which can be shown by making use of the Fourier transform of the Green’s function. Thus step (A) requires the above-mentioned change in the reflection amplitude. The transfer matrix is given in terms of r and t as ! 1 ðt2 r2 Þg2 rg2 T¼ ; ð17Þ tg r 1 where the phase factor g is defined as g ¼ expðiqdÞ for the period of the 1D lattice d: Considerations (B)–(D) for an N layer system shows that the equations for the field amplitudes between the neighboring interlayer positions can be rewritten into the equations of fXn g; which agree with Eqs. (3), if the following equality holds: ð18Þ jI*0n ðqÞj2 ¼ jI*0n ð qÞj2 :
N = 5000
0.98 0.97 0.96 0.95 0.94 0.93 -20
-10
(a)
0
10
20
hω– E0 (µeV)
1.0000
N =∞
0.9999
-20 (b)
-10
0
10
20
hω – E0 (µeV)
Fig. 5. Reflectivity due to the gap mode for (a) finite ðN ¼ 5000Þ and (b) infinite lattices, calculated in terms of the transfer matrix method for microscopic model. The value of nonradiative damping g ¼ 1:0 10 12 eV is used.
K. Cho et al. / Journal of Luminescence 100 (2002) 283–289
r is not important.) Such a calculation shows the merging of the sharp dips for finite N systems into a single dip for N ¼ N; and the series of the sharp Fabry–Pe! rot interference just below o0 ; mentioned in the previous section as a spurious effect, does not exist in this microscopic calculation, as expected. Fig. 5 shows the comparison between the cases of N ¼ 5000 and N: To summarize, we have shown the existence of a gap mode in a 1D resonant Bragg reflector. This leads to a dip structure in the total reflection range, and the corresponding internal field has a standing wave pattern. The equivalence of the transfer matrix method and the nonlocal theory for this 1D problem is shown, and the validity limit of a macroscopic model for this problem is explicitly demonstrated.
Acknowledgements This paper is dedicated to late Prof. S. Shionoya, who gave a unique contribution to the field of optical physics and chemistry by editing a comprehensive Handbook of ‘‘Optical Properties of Solids’’ (in Japanese), in addition to his active research works. This work was supported in part by the Grant-in-Aid for Specially Promoted
289
Research (10CE2004) of the Ministry of Education, Culture, Sports, Science and Technology of Japan.
References [1] K. Sakoda, Optical Properties of Photonic Crystals, Springer Series of Optical Sciences, Springer, Berlin, 2001; C.M. Soukoulis (Ed.), Photonic Crystals and Light Localization in the 21st Century, Nato Science Series C: Mathematical and Physical Science, Vol. 563, Kluwer Academic Publishers, Dordrecht, 2001. [2] E.L. Ivchenko, A.I. Nesvizhskii, S. Jorda, Fiz. Tverd. Tela (St. Petersburg) 36 (1994) 2118 [Phys. Solid State 36 (1994) 1156]. [3] L.I. Deych, A.A. Lisyansky, Phys. Rev. B 62 (2000) 4242. [4] T. Ikawa, K. Cho, Phys. Rev. B 66 (2002) 085338; T. Ikawa, K. Cho, J. Phys. Soc. Japan 71 (2002) 1381. [5] I.H. Deutsch, R.J.C. Spreeuw, S.L. Rolston, W.D. Phillips, Phys. Rev. A 52 (1995) 1394. . [6] U. van Burck, . R.L. Mossbauer, E. Gerdau, R. Ruffer, . R. Hollatz, G.V. Smirnov, J.P. Hannon, Phys. Rev. Lett. 59 (1987) 355. [7] K. Cho, Prog. Theor. Phys. Suppl. 106 (1991) 225; K. Cho, J. Phys. Soc. Japan 66 (1997) 2496. [8] K. Cho, T. Ikawa, Phys. Stat. Sol. A 190 (2002) 401. [9] E.L. Ivchenko, Fiz. Tverd Tela (St. Petersburg) 33 (1991) 2388 [Sov. Phys. Solid State 33 (1991) 1344]. [10] F. Abel!es, Ann. Phys. 5 (1950) 706.