- Email: [email protected]
Photonic crystal circuits
Physica B 296 (2001) 201}209
Photonic crystal circuits A.R. McGurn* Department of Physics, Western Michigan University, Kalamazoo, MI 49008-5151, USA
Abstract A theory is presented for the #ow of electromagnetic energy in photonic crystal circuits. Photonic crystal circuits are waveguide networks in photonic crystals. The theory is based on a set of di!erence equations which describe the propagation of electromagnetic waves in particular types of photonic crystal waveguides. Solutions are given for the transmission and re#ection of electromagnetic waves from barriers in an otherwise in"nitely long straight photonic crystal waveguide, from a junction of an in"nite and a semi-in"nite photonic crystal waveguide, and from an optical switching circuit. The theory developed is applicable to extended circuits with complex topologies. It is applicable to circuits in two-dimensional and three-dimensional photonic crystals. 2001 Elsevier Science B.V. All rights reserved. Keywords: Photonic crystal; Impurity modes; Waveguides; Optical circuits
1. Introduction Recently there has been considerable interest in photonic crystal waveguides [1}14]. Photonic crystal waveguides are formed in photonic crystals by the addition of a line of site impurities to the photonic crystal. The line of site impurities forms a waveguide channel which binds electromagnetic modes to it and along which, the electromagnetic waveguide modes propagate. The waveguide channel can only transport electromagnetic energy of waveguide modes parallel to the channel, and photonic crystal waveguides are of interest for applications requiring the e$cient transportation and channeling of the #ow of electromagnetic energy through space [1,2]. Photonic crystal waveguides are designed to support electromagnetic waveguide
* Tel.: 616-387-4950; fax.: 616-387-4939. E-mail address: [email protected] (A.R. McGurn).
modes which propagate in the waveguide channel at frequencies in the stop gaps of the photonic crystal. Waveguide modes at stop gap frequencies of the photonic crystal are stable against radiative loss from the waveguide channel even in the presence of bends or junctions with other waveguide channels [1,2,4}6]. A recent review of waveguide structures in photonic crystals has been given by Joannopoulos et al. [1,2]. More recently, the existence in nonlinear photonic crystals of static and propagating intrinsic localized modes have also been investigated [13,14]. One interest in photonic crystal waveguides has been in forming them into networks of branching waveguide structures [8]. These types of networks of photonic crystal waveguides are referred to as photonic crystal circuits. They transport electromagnetic energy in space in a manner analogous to the transportation of electrical current through electrical circuits. An interesting recent work on networks of photonic crystal waveguides which
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 8 0 1 - 2
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channel the #ow of light through photonic crystals has appeared in Phys. Rev. Lett. [8]. Most theoretical work on photonic crystal waveguides and their networks been based on supercell computer simulation methods. The focus of these studies has been on networks with waveguide channels formed by removing rows of dielectric rods from photonic crystals. Two major limitations of the supercell approach are: First, the size of the network which can be treated is limited by the computer memory; and second, it becomes time consuming (CPU time) to treat many di!erent types of network geometries for a wide variety of di!erent combinations of materials forming the waveguide channels. In this paper, a theory which is not limited by either of these conditions will be presented. We study the theory of branching structures of photonic waveguides using a method based on studies of the Green's functions of the electromagnetic modes of the photonic crystal. Unlike supercell simulation techniques, our method does not restrict the size of the circuit network to which it can be applied and can be applied to general networks in photonic crystals of any dimension. The Green's function-based method can also be easily applied to waveguide channels formed from di!erent dielectric materials or to junctions formed from a number of di!erent interconnecting waveguide channels all composed of di!erent types of dielectric materials. In this approach, Green's functions techniques are applied [3,9}14] to particular types of photonic crystal waveguides for which the equations describing the propagation of light in the waveguide channels reduce to a set of di!erence equations. These di!erence equations are solved using standard methods to obtain closed-form analytic expressions for the propagation characteristics of photonic crystal circuits. The order of this paper will be as follows: In Section 2, we discuss the network geometry and forms of electromagnetic solutions in photonic crystal circuits. The derivation of the di!erence equations describing the waveguide modes is presented. In Section 3, a discussion is given of waveguides which contain dielectric barriers, waveguide junctions and switches. In Section 4, conclusions are given.
2. Di4erence equations for photonic crystal waveguides As in Ref. [13], the E-polarized electromagnetic modes of a two-dimensional photonic crystal formed as a square lattice array of in"nitely long, parallel, identical dielectric rods are considered [11,13,15,16]. The rods are of circular cross section and are characterized by a dielectric constant [11,13]. They are embedded in vacuum. The periodic dielectric constant of the system as a function of position, r "xiK #yjK , in the x}y plane is , r !na iK !ma jK )R A A for n and m integers, (1) (r )" 1, otherwise.
Here, a is the lattice constant of the square lattice, A and R(a /2 is the radius of the dielectric rods. The A E-polarized electromagnetic modes of the photonic crystal which propagate in the x}y plane are solutions of the matrix eigenvalue equation [11,13,15] (k #G )e(k G ) " ( (G !G )e(k G ). c G
(2)
Here the eigenvalue /c gives the frequency of the electromagnetic mode, G is a reciprocal lattice vector of the square lattice, (r )"G ( (G )e G r , and e(k G ) are related to the electric "eld, E(r ), of the mode of frequency by E(r )" e(k G )e k >G r . (3) % A waveguide impurity is formed in the system de"ned in Eq. (1) by adding impurity material to a row of rods along one of the directions of the square lattice [11,13]. A waveguide with a straight in"nitely long channel is formed when impurity material is added to the sites (nra , nsa ) where A A r and s are "xed integers and n"!R,2,!2, !1, 0, 1, 2,2R ranges over the integers. The slope of the waveguide channel in the x}y plane is then s/r and the separation between nearest neighbor impurity sites is ((r#s)a . A "nite segment A is formed for the case in which impurity material is added to the sites (nra , nsa ) where r and s are A A
A.R. McGurn / Physica B 296 (2001) 201}209
"xed integers and n"0, 1, 2,2, m where m is an integer. A branching system of waveguides (photonic circuits) can be formed by piecing together various waveguide segments having a variety of di!erent slopes in the x}y plane. The total dielectric constant of a waveguide segment, (r ), is given by (r )"(r )#(r ) 2 2 where (r ) is the change in the dielectric constant of the photonic crystal upon the addition of impurity dielectric material. For a waveguide segment composed of an array of identical single-site impurities of square cross-sectional area in the x}y plane, (r ) is [11,13] , x!nra , y!nsa )t A A for n integers, (4) (r )" 0, otherwise.
Here r and s are "xed integers, the length of the waveguide segment depends on the range of the consecutive integers n, and 2t is the length of a side of one of the single-site impurities. For the impurities considered in this paper 2t;R(a /2. A The electric "eld of the frequency modes associated with waveguide segments and their branchings can be written in the form E(r t)"E(r , ) exp(!it). (5) Using standard techniques [3,9}13], the electric "eld amplitude, E(r, ), of waveguide segments and their branchings is expressed as the integral equation
E(r , )" dr G(r , r )(r ) ;
E(r , ). c
(6)
Here (r ) is the change in the photonic crystal dielectric constant due to all of the waveguide segments and their branchings in the photonic crystal, and G(r , r ) is the Green's function of the Hel mholtz operator for the photonic crystal in Eq. (1), i.e., #(r )(/c). If t is small enough so that the electromagnetic "eld at each square cross section rod of impurity material is constant over that volume of the impu-
203
rity material, Eq. (6) for the "elds in the rods becomes a di!erence equation. For example, consider (r ) for a waveguide segment formed from identi cal single-site impurities. Let the electric "eld in the rod of impurity material of the waveguide labeled by (nr, ns) in Eq. (4) be denoted by E where LP LQ E "E(n(riK #sjK )a , ), then from Eq. (6) LP LQ A E " B (m(riK #sjK )a )E LP LQ LP LQ_KP KQ A KP KQ K
(7)
for this segment. Here
B " LP LQ_KP KQ c
dr KP KQ
;G(n(riK #sjK )a , r ), A
(8)
m in the sum in Eq. (7) runs over the integers denoting the waveguide channel sites, and is the frequency of the impurity mode. The impurity mode frequency is chosen to be in one of the stop gaps of the photonic crystal described in Eq. (1) so that the waveguide modes are bound to the waveguide channel. From Eqs. (7) and (8) then E "A B E , LP LQ LP LQ_KP KQ KP KQ KP KQ
(9)
where A". Eq. (9) determines the "elds in the impurity rods forming the photonic crystal waveguide. As will be seen in the next sections, Eqs. (7) and (9) are easily and naturally generalized to obtain from Eq. (6) a series of interconnecting di!erence equations describing more general photonic crystal circuits. The mathematics of our treatment is simpli"ed by restricting the recursion relation in Eq. (9) to consider only the same site and the nearest-neighbor site couplings [11,13,14]. The couplings, as seen in Eq. (8), are related to the Green's function G(r , r ) which decays with increasing r !r for in a stop gap of the photonic crystal. For directions of high Miller indices this decay is expected to be large. With this provision Eq. (9) becomes E " [ (0, 0)E # (r, s)(E LP LQ LP LQ L>P L>Q #E
L\P L\Q
)].
(10)
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Here (0, 0)"B /(4t), (r, s)"B /(4t) _ _P Q where r and s are de"ned in Eq. (4), and "4t. The electromagnetic mode solutions of Eq. (10) which are bound to the waveguide are obtained by "rst choosing to be a frequency in the stop band of the photonic crystal and computing (0, 0) and
(r, s). Eq. (10) can then be solved for E and LP LQ (i.e., 4t) characterizing these modes.
3. Dielectric barriers, junctions, and switches A dielectric barrier is created in an in"nitely long straight waveguide by changing the dielectric impurity material in 2n#1 consecutive sites in the channel of the dielectric waveguide to a new type of impurity material. The di!erence equations describing this system are from Eq. (6): E " [ (0, 0)E # (r, s)(E JP JQ JP JQ J>P J>Q #E )], J\P J\Q where l"$(n#2),$(n#3),2, E " [ (0, 0)E # (r, s)(E GP GQ GP GQ G>P G>Q #E )], G\P G\Q with i"0,$1,2,$(n!1),
(11)
E "be\ OJ#ce OJ, JP JQ where l"0,$1,$2,2,$n,
"[ (0, 0)#2 (r, s) cos k]\ (18) which expresses in terms of k and the couplings in the di!erence equations, and substituting Eq. (16) in Eq. (12) yields a similar relation given by "[ (0, 0)#2 (r, s) cos q]\ (19) for in terms of q and the couplings in the di!er ence equations. In the following, it is assumed that kOq so that from Eqs. (18) and (19) it follows that O . Using the boundary conditions, Eqs. (13) and (14), to match the solutions at both edges of the barrier, we obtain
(12)
(15)
(16)
E "de\ IJ#e e IJ, (17) JP JQ where l"!(n#1),!(n#2),2 and substituting into Eqs. (11)}(14). Substituting Eqs. (15) and (17) into Eq. (11) yields
f
E " [ (0, 0)E !LP !LQ !LP !LQ # (r, s)E ] !L\P !L\Q # (r, s)E , (13) !L>P !L>Q E " [ (0, 0)E !L>P !L>Q !L>P !L>Q # (r, s)E ] !L>P !L>Q # (r, s)E . (14) !LP !LQ Here, the slope of the waveguide channel in the x}y plane is s/r, and and in Eqs. (11)}(14) charac terize, respectively, the impurity material forming the waveguide channel and the impurity material forming the barrier in the waveguide channel. A solution of Eqs. (11)}(14) can be obtained by assuming a solution of the form E "fe\ IJ#ae IJ, JP JQ
where l"(n#1), (n#2),2,
b " a bH where
b d , bH e
b "!e IL>(e\ OL[1!e\ I>O] !e OL[1!e\ I\O])/(4 sin k sin q)
(20)
(21)
and b "!(e\ OL[1!e\ I>O][1!e I\O] !c.c.)/(4 sin k sin q).
(22)
Eq. (20) represents a transform (transfer matrix equation) between the states in Eqs. (15) and (17) on opposite sides of the barrier. This transform conserves the total energy #ux along the length of the waveguide. As an example, to treat a scattering problem involving a wave incident on the barrier from the right we can evaluate Eq. (20) for d"1 and e "0. This gives the incident wave amplitude, f"b , and the re#ected wave amplitude, a"bH, on the right of the barrier for there to be a unit transmitted wave on the left of the barrier. The barrier transmission coe$cient, de"ned as the ratio of the transmitted to incident power #ux is then
A.R. McGurn / Physica B 296 (2001) 201}209
Fig. 1. Results for the barrier problem. Plots for the transmission coe$cient vs. frequency in units of a /2c are shown for A (q , k )"(/2, ) (solid) and (/2, /100) (dashed). Each curve A A presents results for "xed and .
given by
Fig. 2. Schematic drawings of waveguide channels. Only the sites of the waveguide channels are represented and the channels lie along the [1 0] and [0 1] directions of the square lattice. (a) Seen above at the left. Represents the waveguide junction. (b) Seen above at the right. Represents the U-loop or waveguide switch.
mode frequency over the stop gap of the photonic crystal. Speci"cally, if we denote the values of q and k at the mid-stop band frequency, a /2c"0.440, A by q and k , we present plots for the pairs A A (q , k )"(/2,) and (/2,/100). These correspond A A
2 sin q sin k ¹" , (1!cos k cos q)#sin k sin q!(cos q!cos k) cos((4n#2)q) and the re#ection coe$cient is R"1!¹. In Fig. 1 results are plotted for the barrier transmission coe$cient, Eq. (23), vs. the frequency, , for a square lattice photonic crystal. The frequency, , is taken to be in one of the stop gaps of the photonic crystal. The photonic crystal used to generate this plot is an array of cylindrical dielectric rods of dielectric constant "9 and radius R"0.37796a A where a is the lattice constant of the square lattice. A The rods are of in"nite length and are surrounded by vacuum. This particular photonic crystal has a stop band which includes the frequency region 0.425(a /2c(0.455 used to generate the plot. A The waveguide channel is taken to be formed from impurity materials with t"0.01a and to lie in the A [1,0] direction. The impurities forming the waveguide channel have a nearest neighbor separation of a lattice constant so that in Eqs. (18) and (19)
(r, s)" (1, 0). For the results presented in Fig. 1 we have taken n"10. The plots in Fig. 1 are obtained by "xing the values of and and varying the waveguide
205
(23)
to plots of "xed and for which / "1.20 and 0.86, respectively. A branched waveguide or waveguide junction is formed by attaching an end of a semi-in"nite waveguide to a site of an in"nite waveguide. (See Fig. 2a.) For our discussions of this problem, it is assumed that the channels of the two waveguides are of the same type of impurity material. The di!erence equations describing the waveguide junction in Fig. 2a are, from Eq. (6), E " [ (0, 0)E # (r, 0)(E JP JP J\P #E )], J>P where l"2, 3, 4,2,
(24)
E " [ (0, 0)E # (r, 0)(E JP JP J\P #E )], J>P where l"$2,$3,$4,2,
(25)
E " [ (0, 0)E # (r, 0)(E #E ) P \P # (r,0)E ], P
(26)
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A.R. McGurn / Physica B 296 (2001) 201}209
E " [( (0, 0)E # (r, 0)E ) !P !P !P # (r,0)E ], and
(27)
E " [( (0, 0)E # (r, 0)E )# (r,0)E ]. P P P (28) In these equations, we have used the notation of Eq. (10). A solution of Eqs. (24)}(28) can be obtained by assuming a solution of the general form: E "ae\ IJ#be IJ JP for l"1, 2, 3,2,
(29)
E "ce\ IJ#de IJ JP for l"!1,!2,!3,2,
(30)
E "e e\ IJ#fe IJ JP for l"1, 2, 3,2, and
(31)
E "h. (32) Substituting Eqs. (29), (30), and (31), respectively, in Eqs. (24) and (25), gives "[ (0, 0)#2 (r, 0) cos k]\
(33)
which expresses in terms of k and the coe$cients of the di!erence equations. The boundary condition equations in Eq. (26)}(28), then gives the matrix equation
a
e\ I e I e I e\ I e\ I e I !2 cos k 1
1
0
0
0
0
!1
0
0
1
1
0
0
!1
0
0
0
0
1
1
!1
b c
d "0,
e f h
(34)
which relates the coe$cients in Eqs. (29)}(32) to one another. The solution of the scattering problem for a wave of unit amplitude incident on the junction from (0,!R) is obtained by solving Eq. (34) for the case in which a"0, e "0, and d"1. De"ning the
Fig. 3. Results for the junction problem. Plots for the transmission coe$cient vs. frequency in units of a /2c are shown for A k ": (a) /2, (b) , (c) /100, (d) 3/4. Each curve presents results A for "xed .
re#ection and transmission coe$cients for this case as the ratios of the re#ected and transmitted #ux, respectively, to the incident #ux, gives the re#ection coe$cient 1 R" cos k#9 sin k
(35)
and the transmission coe$cients in each of the two branches leaving the vertex 4 sin k ¹" . cos k#9 sin k
(36)
In Fig. 3, plots of ¹ in Eq. (36) vs. for the junction are presented. The plots are made for the same photonic crystal and range of as considered in Fig. 1. The waveguide channels have t"0.01a A and a nearest-neighbor impurity separation of a lattice constant, a . Curves are generated from A Eq. (36) in conjunction with the dispersion relation "[ (0, 0)#2 (1, 0) cos k]\, and are presented as a function of mode frequency for a variety of "xed values of . The studied are chosen so that the value of k for the mid-band frequency a /2c"0.440, which we denote as k , assumes A A a set of representative values k ", 3/4, /2, and A /100 corresponding, respectively, to "0.342, 0.323, 0.284, and 0.243.
A.R. McGurn / Physica B 296 (2001) 201}209
Our theory of waveguide junctions can be easily extended to solve for the transportation of electromagnetic waves in more complex waveguide networks, i.e., photonic crystal circuits. As an example, we now treat a waveguide which contains a closed loop (see Fig. 2b). This circuit behaves like an optical switch, and a similar arrangement of optical "ber waveguides is used in optical switching applications in "ber optics. Consider an in"nitely long straight waveguide in a square lattice photonic crystal. The impurity sites of the waveguide channel are labeled by (0, mr) where m ranges over the integers. The waveguide is formed of only one type of impurity material. Now attach to the waveguide channel at its (0, 0) and (0, (n#1)r) sites for n'1 a U-shaped channel of impurities. The impurity material forming the channel of the U can be di!erent from the impurity material forming the channel of the in"nitely long waveguide. Let us use the above di!erence equation formulation to compute the transmission and re#ection characteristics of a waveguide mode incident on the U from (0,!R). The set of di!erence equations which describe this branching geometry consists of ten separate equations. The equations associated with the in"nitely long straight waveguide channel are given by the channel equation E " [ (0, 0)E # (r, 0) JP JP ;(E #E )], (37) J>P J\P where lO0 or n#1 otherwise ranges over the integers, and two equations connecting the channel to the U-shaped channel E " [ (0, 0)E # (r, 0)(E #E )] P \P # (r,0)E P and
(38)
E " [ (0, 0)E # (r, 0)[E L>P L>P L>P #E )]# (r, 0)E . (39) LP P L>P There are seven equations which describe the U-shaped channel. These are E " [ (0, 0)E # (r, 0)(E JP JP J>P #E )], J\P
(40)
207
where l"2, 3,2, l !1, E " [ (0, 0)E # (r, 0)(E J P JP J P J>P J P JP #E )], J P J\P where l"1, 2, 3,2, n,
(41)
E " [ (0, 0)E # (r, 0) JP L>P JP L>P ;(E #E )], (42) J>P L>P J\P L>P where l"2, 3,2, l !1, E " [ (0, 0)E # (r, 0)(E #E )], J P J P J P P J \P (43) " [ (0, 0)E # (r, 0)(E E J P L>P J P LP J P L>P .#E )], (44) J \P L>P E " [ (0, 0)E # (r, 0)E ]# (r,0)E , P P P (45) and E " [ (0, 0)E # (r, 0)E ] P L>P P L>P P L>P # (r, 0)E . (46) L>P A solution of this system of di!erence equations can be assumed to be of the form E "ae\ IJ#be IJ, JP where l"1, 2, 3,2, n,
(47)
E "ce\ IJ#de IJ, JP where l"!1,!2,!3,2,
(48)
E "e e\ OJ#fe OJ, JP where l"1, 2,2, l , E "e e\ OJ >J#fe OJ >J, J P JP where l"1, 2, 3,2, n,
(49)
(50)
E "e e\ OJ >L>J#fe OJ >L>J, J >\JP L>P where l"1, 2,2, l , E "h, E "h, L>P and
(52)
E "re\ IJ#se IJ JP
(54)
(51)
(53)
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A.R. McGurn / Physica B 296 (2001) 201}209
where l"n#2, n#3, n#4,2 . Substituting Eqs. (47)}(54) in Eqs. (37)}(46), we "nd a system of 10 algebraic equations which can be used to relate a, b, c, d, e , f, h, h, r, s to one another. An interesting case of these equations is to use them to relate the coe$cients c, d to r, s. This then gives the transmission and re#ection of the waveguide modes from the U-loop. Assuming that the waveguide channel of the in"nitely long channel and the U-shaped channel resonate at the same frequency (i.e., "[ (0, 0)#2 (r, 0) cos k]\ and "[ (0, 0)#2 (r, 0) cos q]\), after some algebra gives
c
"
d
a aH
a
aH
r s
,
(55)
where e\ IL> (a!a), a " P Q d
(56)
e IL> a " (a!a ), P Q d
(57)
sin k sin q a" # , P sin k(n#1) sin q(2l #n#1)
(58)
sin kn sin q(2l #n) ! , a "e I! Q sin k(n#1) sin q(2l #n#1)
(59)
Fig. 4. Results for the U-loop problem. Plots for the transmission coe$cient vs. frequency in units of a /2c are shown for A (q , k )"(/2, /2). The curve corresponds to results for "xed A A values of and .
Fig. 1. The waveguide channels of the circuit are taken to have t"0.01a with a nearest-neighbor A impurity separation equal to the lattice constant of the square lattice. Let q and k be wavenumbers in the U-loop and A A the waveguide channel at the mid-stop band frequency, a /2c"0.440. We have chosen (q , k )" A A A (/2, /2) for the plot in Fig. 4. This corresponds to a plot for "xed and for which / "1.00.
and
sin k sin q d "!2i sin k # . sin k(n#1) sin q(2l #n#1) (60) For there to be a unit transmitted #ux in the region above the U-loop, we need r"0 and s"1 in Eq. (55). The relative intensities of the incident and re#ected #ux in the region below the U-loop are then a and a , respectively. In Fig. 4, results are presented for the U-loop system with n"l "10. The transmission coe$c ient, ¹"1/a , de"ned as the ratio of the trans mitted to incident #ux, vs. the frequency, , in the stop gap are plotted. The parameters used in this plot are those of the "9, R"0.37796a square A lattice photonic crystal used to obtain the results in
4. Conclusions The propagation of electromagnetic waves in various types of waveguide circuits in photonic crystals have been studied. It has been demonstrated that under certain conditions, the mathematics describing this propagation can be reduced to a set of di!erence equations which can be solved exactly. As seen in the text, the phase coherent e!ects in photonic crystal circuits can cause the transmission and re#ection coe$cients of photonic circuits to be complicated functions of the mode frequencies. These e!ects may be important in the design of optical switches and other types of optical circuit devices similar to those used in "ber optics. An example of this is the optical switch discussed in Section 3.
A.R. McGurn / Physica B 296 (2001) 201}209
A feature of the di!erence equation theory is that very complicated and extended branching waveguide circuits can be solved exactly. The physics of these photonic crystal circuits is found to be described by closed form expressions involving elementary functions of mathematical physics. An additional advantage of the theory is that, for a given circuit network geometry, solutions for the properties of the circuit can be easily obtained for a wide variety of di!erent types of waveguide channel materials. References [1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals, Princeton University Press, Princeton, 1995. [2] J.D. Joannopoulos, P.R. Villeneuve, S. Fan, Photonic crystals: putting a new twist on light, Nature (London) 386 (1997) 143. [3] A.A. Maradudin, A.R. McGurn, in: C.M. Soukoulis (Ed.), Photonic Band Gaps and Localization, Plenum, New York, 1992. [4] R.D. Meade, A. Devenyi, J.D. Joannopoulos, O.L. Alerhand, D.A. Smith, K. Kash, Novel applications of photonic band gap materials: low-loss bends and high Q cavities, J. Appl. Phys. 75 (1994) 4753. [5] S. Fan, J.A. Winn, A. Devenyi, J.C. Chen, R.D. Meade, J.D. Joannopoulos, Guided and deafec modes in periodic dielectric waveguides, J. Opt. Soc. Am. B 12 (1995) 1267.
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[6] A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, J.D. Joannopoulos, High transmission through sharpe bends in photonic crystal waveguides, Phys. Rev. Lett. 77 (1996) 3787. [7] A. Mekis, S. Fan, J.D. Joannopoulos, Bound states in photonic crystal waveguides and waveguide bends, Phys. Rev. B 58 (1998) 4809. [8] S. Fan, P.R. Villeneuve, J.D. Joannopoulos, H.A. Haus, Channel drop tunneling through localized states, Phys. Rev. Lett. 80 (1998) 960. [9] H.G. Algul, M. Khazhinsky, A.R. McGurn, J. Kapenga, Impurity modes from impurity clusters in photonic band structures, J. Phys. 7 (1995) 447. [10] A.R. McGurn, M.G. Khazhinsky, in: C.M. Soukoulis (Ed.), Photonic Band Gap Materials, Kluwer Academic Press, Vol. 487, 1995. [11] A.R. McGurn, Green's-function theory for row and periodic defect arrays in photonic band structures, Phys. Rev. B 53 (1996) 7059. [12] K.G. Khazhinsky, A.R. McGurn, Greens's function method for waveguide and single impurity modes in 2D photonic crystals: H-polarization, Phys. Lett. A 237 (1998) 175. [13] A.R. McGurn, Intrinsic localized modes in nonlinear photonic crystal waveguides, Phys. Lett. A 251 (1999) 322. [14] A.R. McGurn, Intrinsic localized modes in nonlinear photonic crystal waveguides: dispersive modes, Phys. Lett. A 260 (1999) 314. [15] M. Phihal, A. Shambrook, A.A. Maradudin, P. Sheng, Two-dimensional photonic band structures, Opt. Commun. 80 (1991) 199. [16] C.M. Soukoulis (Ed.), Photonic Band Gap Materials, Kluwer, Dordrecht, 1996.
Photonic crystal circuits A.R. McGurn* Department of Physics, Western Michigan University, Kalamazoo, MI 49008-5151, USA
Abstract A theory is presented for the #ow of electromagnetic energy in photonic crystal circuits. Photonic crystal circuits are waveguide networks in photonic crystals. The theory is based on a set of di!erence equations which describe the propagation of electromagnetic waves in particular types of photonic crystal waveguides. Solutions are given for the transmission and re#ection of electromagnetic waves from barriers in an otherwise in"nitely long straight photonic crystal waveguide, from a junction of an in"nite and a semi-in"nite photonic crystal waveguide, and from an optical switching circuit. The theory developed is applicable to extended circuits with complex topologies. It is applicable to circuits in two-dimensional and three-dimensional photonic crystals. 2001 Elsevier Science B.V. All rights reserved. Keywords: Photonic crystal; Impurity modes; Waveguides; Optical circuits
1. Introduction Recently there has been considerable interest in photonic crystal waveguides [1}14]. Photonic crystal waveguides are formed in photonic crystals by the addition of a line of site impurities to the photonic crystal. The line of site impurities forms a waveguide channel which binds electromagnetic modes to it and along which, the electromagnetic waveguide modes propagate. The waveguide channel can only transport electromagnetic energy of waveguide modes parallel to the channel, and photonic crystal waveguides are of interest for applications requiring the e$cient transportation and channeling of the #ow of electromagnetic energy through space [1,2]. Photonic crystal waveguides are designed to support electromagnetic waveguide
* Tel.: 616-387-4950; fax.: 616-387-4939. E-mail address: [email protected] (A.R. McGurn).
modes which propagate in the waveguide channel at frequencies in the stop gaps of the photonic crystal. Waveguide modes at stop gap frequencies of the photonic crystal are stable against radiative loss from the waveguide channel even in the presence of bends or junctions with other waveguide channels [1,2,4}6]. A recent review of waveguide structures in photonic crystals has been given by Joannopoulos et al. [1,2]. More recently, the existence in nonlinear photonic crystals of static and propagating intrinsic localized modes have also been investigated [13,14]. One interest in photonic crystal waveguides has been in forming them into networks of branching waveguide structures [8]. These types of networks of photonic crystal waveguides are referred to as photonic crystal circuits. They transport electromagnetic energy in space in a manner analogous to the transportation of electrical current through electrical circuits. An interesting recent work on networks of photonic crystal waveguides which
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 8 0 1 - 2
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channel the #ow of light through photonic crystals has appeared in Phys. Rev. Lett. [8]. Most theoretical work on photonic crystal waveguides and their networks been based on supercell computer simulation methods. The focus of these studies has been on networks with waveguide channels formed by removing rows of dielectric rods from photonic crystals. Two major limitations of the supercell approach are: First, the size of the network which can be treated is limited by the computer memory; and second, it becomes time consuming (CPU time) to treat many di!erent types of network geometries for a wide variety of di!erent combinations of materials forming the waveguide channels. In this paper, a theory which is not limited by either of these conditions will be presented. We study the theory of branching structures of photonic waveguides using a method based on studies of the Green's functions of the electromagnetic modes of the photonic crystal. Unlike supercell simulation techniques, our method does not restrict the size of the circuit network to which it can be applied and can be applied to general networks in photonic crystals of any dimension. The Green's function-based method can also be easily applied to waveguide channels formed from di!erent dielectric materials or to junctions formed from a number of di!erent interconnecting waveguide channels all composed of di!erent types of dielectric materials. In this approach, Green's functions techniques are applied [3,9}14] to particular types of photonic crystal waveguides for which the equations describing the propagation of light in the waveguide channels reduce to a set of di!erence equations. These di!erence equations are solved using standard methods to obtain closed-form analytic expressions for the propagation characteristics of photonic crystal circuits. The order of this paper will be as follows: In Section 2, we discuss the network geometry and forms of electromagnetic solutions in photonic crystal circuits. The derivation of the di!erence equations describing the waveguide modes is presented. In Section 3, a discussion is given of waveguides which contain dielectric barriers, waveguide junctions and switches. In Section 4, conclusions are given.
2. Di4erence equations for photonic crystal waveguides As in Ref. [13], the E-polarized electromagnetic modes of a two-dimensional photonic crystal formed as a square lattice array of in"nitely long, parallel, identical dielectric rods are considered [11,13,15,16]. The rods are of circular cross section and are characterized by a dielectric constant [11,13]. They are embedded in vacuum. The periodic dielectric constant of the system as a function of position, r "xiK #yjK , in the x}y plane is , r !na iK !ma jK )R A A for n and m integers, (1) (r )" 1, otherwise.
Here, a is the lattice constant of the square lattice, A and R(a /2 is the radius of the dielectric rods. The A E-polarized electromagnetic modes of the photonic crystal which propagate in the x}y plane are solutions of the matrix eigenvalue equation [11,13,15] (k #G )e(k G ) " ( (G !G )e(k G ). c G
(2)
Here the eigenvalue /c gives the frequency of the electromagnetic mode, G is a reciprocal lattice vector of the square lattice, (r )"G ( (G )e G r , and e(k G ) are related to the electric "eld, E(r ), of the mode of frequency by E(r )" e(k G )e k >G r . (3) % A waveguide impurity is formed in the system de"ned in Eq. (1) by adding impurity material to a row of rods along one of the directions of the square lattice [11,13]. A waveguide with a straight in"nitely long channel is formed when impurity material is added to the sites (nra , nsa ) where A A r and s are "xed integers and n"!R,2,!2, !1, 0, 1, 2,2R ranges over the integers. The slope of the waveguide channel in the x}y plane is then s/r and the separation between nearest neighbor impurity sites is ((r#s)a . A "nite segment A is formed for the case in which impurity material is added to the sites (nra , nsa ) where r and s are A A
A.R. McGurn / Physica B 296 (2001) 201}209
"xed integers and n"0, 1, 2,2, m where m is an integer. A branching system of waveguides (photonic circuits) can be formed by piecing together various waveguide segments having a variety of di!erent slopes in the x}y plane. The total dielectric constant of a waveguide segment, (r ), is given by (r )"(r )#(r ) 2 2 where (r ) is the change in the dielectric constant of the photonic crystal upon the addition of impurity dielectric material. For a waveguide segment composed of an array of identical single-site impurities of square cross-sectional area in the x}y plane, (r ) is [11,13] , x!nra , y!nsa )t A A for n integers, (4) (r )" 0, otherwise.
Here r and s are "xed integers, the length of the waveguide segment depends on the range of the consecutive integers n, and 2t is the length of a side of one of the single-site impurities. For the impurities considered in this paper 2t;R(a /2. A The electric "eld of the frequency modes associated with waveguide segments and their branchings can be written in the form E(r t)"E(r , ) exp(!it). (5) Using standard techniques [3,9}13], the electric "eld amplitude, E(r, ), of waveguide segments and their branchings is expressed as the integral equation
E(r , )" dr G(r , r )(r ) ;
E(r , ). c
(6)
Here (r ) is the change in the photonic crystal dielectric constant due to all of the waveguide segments and their branchings in the photonic crystal, and G(r , r ) is the Green's function of the Hel mholtz operator for the photonic crystal in Eq. (1), i.e., #(r )(/c). If t is small enough so that the electromagnetic "eld at each square cross section rod of impurity material is constant over that volume of the impu-
203
rity material, Eq. (6) for the "elds in the rods becomes a di!erence equation. For example, consider (r ) for a waveguide segment formed from identi cal single-site impurities. Let the electric "eld in the rod of impurity material of the waveguide labeled by (nr, ns) in Eq. (4) be denoted by E where LP LQ E "E(n(riK #sjK )a , ), then from Eq. (6) LP LQ A E " B (m(riK #sjK )a )E LP LQ LP LQ_KP KQ A KP KQ K
(7)
for this segment. Here
B " LP LQ_KP KQ c
dr KP KQ
;G(n(riK #sjK )a , r ), A
(8)
m in the sum in Eq. (7) runs over the integers denoting the waveguide channel sites, and is the frequency of the impurity mode. The impurity mode frequency is chosen to be in one of the stop gaps of the photonic crystal described in Eq. (1) so that the waveguide modes are bound to the waveguide channel. From Eqs. (7) and (8) then E "A B E , LP LQ LP LQ_KP KQ KP KQ KP KQ
(9)
where A". Eq. (9) determines the "elds in the impurity rods forming the photonic crystal waveguide. As will be seen in the next sections, Eqs. (7) and (9) are easily and naturally generalized to obtain from Eq. (6) a series of interconnecting di!erence equations describing more general photonic crystal circuits. The mathematics of our treatment is simpli"ed by restricting the recursion relation in Eq. (9) to consider only the same site and the nearest-neighbor site couplings [11,13,14]. The couplings, as seen in Eq. (8), are related to the Green's function G(r , r ) which decays with increasing r !r for in a stop gap of the photonic crystal. For directions of high Miller indices this decay is expected to be large. With this provision Eq. (9) becomes E " [ (0, 0)E # (r, s)(E LP LQ LP LQ L>P L>Q #E
L\P L\Q
)].
(10)
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Here (0, 0)"B /(4t), (r, s)"B /(4t) _ _P Q where r and s are de"ned in Eq. (4), and "4t. The electromagnetic mode solutions of Eq. (10) which are bound to the waveguide are obtained by "rst choosing to be a frequency in the stop band of the photonic crystal and computing (0, 0) and
(r, s). Eq. (10) can then be solved for E and LP LQ (i.e., 4t) characterizing these modes.
3. Dielectric barriers, junctions, and switches A dielectric barrier is created in an in"nitely long straight waveguide by changing the dielectric impurity material in 2n#1 consecutive sites in the channel of the dielectric waveguide to a new type of impurity material. The di!erence equations describing this system are from Eq. (6): E " [ (0, 0)E # (r, s)(E JP JQ JP JQ J>P J>Q #E )], J\P J\Q where l"$(n#2),$(n#3),2, E " [ (0, 0)E # (r, s)(E GP GQ GP GQ G>P G>Q #E )], G\P G\Q with i"0,$1,2,$(n!1),
(11)
E "be\ OJ#ce OJ, JP JQ where l"0,$1,$2,2,$n,
"[ (0, 0)#2 (r, s) cos k]\ (18) which expresses in terms of k and the couplings in the di!erence equations, and substituting Eq. (16) in Eq. (12) yields a similar relation given by "[ (0, 0)#2 (r, s) cos q]\ (19) for in terms of q and the couplings in the di!er ence equations. In the following, it is assumed that kOq so that from Eqs. (18) and (19) it follows that O . Using the boundary conditions, Eqs. (13) and (14), to match the solutions at both edges of the barrier, we obtain
(12)
(15)
(16)
E "de\ IJ#e e IJ, (17) JP JQ where l"!(n#1),!(n#2),2 and substituting into Eqs. (11)}(14). Substituting Eqs. (15) and (17) into Eq. (11) yields
f
E " [ (0, 0)E !LP !LQ !LP !LQ # (r, s)E ] !L\P !L\Q # (r, s)E , (13) !L>P !L>Q E " [ (0, 0)E !L>P !L>Q !L>P !L>Q # (r, s)E ] !L>P !L>Q # (r, s)E . (14) !LP !LQ Here, the slope of the waveguide channel in the x}y plane is s/r, and and in Eqs. (11)}(14) charac terize, respectively, the impurity material forming the waveguide channel and the impurity material forming the barrier in the waveguide channel. A solution of Eqs. (11)}(14) can be obtained by assuming a solution of the form E "fe\ IJ#ae IJ, JP JQ
where l"(n#1), (n#2),2,
b " a bH where
b d , bH e
b "!e IL>(e\ OL[1!e\ I>O] !e OL[1!e\ I\O])/(4 sin k sin q)
(20)
(21)
and b "!(e\ OL[1!e\ I>O][1!e I\O] !c.c.)/(4 sin k sin q).
(22)
Eq. (20) represents a transform (transfer matrix equation) between the states in Eqs. (15) and (17) on opposite sides of the barrier. This transform conserves the total energy #ux along the length of the waveguide. As an example, to treat a scattering problem involving a wave incident on the barrier from the right we can evaluate Eq. (20) for d"1 and e "0. This gives the incident wave amplitude, f"b , and the re#ected wave amplitude, a"bH, on the right of the barrier for there to be a unit transmitted wave on the left of the barrier. The barrier transmission coe$cient, de"ned as the ratio of the transmitted to incident power #ux is then
A.R. McGurn / Physica B 296 (2001) 201}209
Fig. 1. Results for the barrier problem. Plots for the transmission coe$cient vs. frequency in units of a /2c are shown for A (q , k )"(/2, ) (solid) and (/2, /100) (dashed). Each curve A A presents results for "xed and .
given by
Fig. 2. Schematic drawings of waveguide channels. Only the sites of the waveguide channels are represented and the channels lie along the [1 0] and [0 1] directions of the square lattice. (a) Seen above at the left. Represents the waveguide junction. (b) Seen above at the right. Represents the U-loop or waveguide switch.
mode frequency over the stop gap of the photonic crystal. Speci"cally, if we denote the values of q and k at the mid-stop band frequency, a /2c"0.440, A by q and k , we present plots for the pairs A A (q , k )"(/2,) and (/2,/100). These correspond A A
2 sin q sin k ¹" , (1!cos k cos q)#sin k sin q!(cos q!cos k) cos((4n#2)q) and the re#ection coe$cient is R"1!¹. In Fig. 1 results are plotted for the barrier transmission coe$cient, Eq. (23), vs. the frequency, , for a square lattice photonic crystal. The frequency, , is taken to be in one of the stop gaps of the photonic crystal. The photonic crystal used to generate this plot is an array of cylindrical dielectric rods of dielectric constant "9 and radius R"0.37796a A where a is the lattice constant of the square lattice. A The rods are of in"nite length and are surrounded by vacuum. This particular photonic crystal has a stop band which includes the frequency region 0.425(a /2c(0.455 used to generate the plot. A The waveguide channel is taken to be formed from impurity materials with t"0.01a and to lie in the A [1,0] direction. The impurities forming the waveguide channel have a nearest neighbor separation of a lattice constant so that in Eqs. (18) and (19)
(r, s)" (1, 0). For the results presented in Fig. 1 we have taken n"10. The plots in Fig. 1 are obtained by "xing the values of and and varying the waveguide
205
(23)
to plots of "xed and for which / "1.20 and 0.86, respectively. A branched waveguide or waveguide junction is formed by attaching an end of a semi-in"nite waveguide to a site of an in"nite waveguide. (See Fig. 2a.) For our discussions of this problem, it is assumed that the channels of the two waveguides are of the same type of impurity material. The di!erence equations describing the waveguide junction in Fig. 2a are, from Eq. (6), E " [ (0, 0)E # (r, 0)(E JP JP J\P #E )], J>P where l"2, 3, 4,2,
(24)
E " [ (0, 0)E # (r, 0)(E JP JP J\P #E )], J>P where l"$2,$3,$4,2,
(25)
E " [ (0, 0)E # (r, 0)(E #E ) P \P # (r,0)E ], P
(26)
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E " [( (0, 0)E # (r, 0)E ) !P !P !P # (r,0)E ], and
(27)
E " [( (0, 0)E # (r, 0)E )# (r,0)E ]. P P P (28) In these equations, we have used the notation of Eq. (10). A solution of Eqs. (24)}(28) can be obtained by assuming a solution of the general form: E "ae\ IJ#be IJ JP for l"1, 2, 3,2,
(29)
E "ce\ IJ#de IJ JP for l"!1,!2,!3,2,
(30)
E "e e\ IJ#fe IJ JP for l"1, 2, 3,2, and
(31)
E "h. (32) Substituting Eqs. (29), (30), and (31), respectively, in Eqs. (24) and (25), gives "[ (0, 0)#2 (r, 0) cos k]\
(33)
which expresses in terms of k and the coe$cients of the di!erence equations. The boundary condition equations in Eq. (26)}(28), then gives the matrix equation
a
e\ I e I e I e\ I e\ I e I !2 cos k 1
1
0
0
0
0
!1
0
0
1
1
0
0
!1
0
0
0
0
1
1
!1
b c
d "0,
e f h
(34)
which relates the coe$cients in Eqs. (29)}(32) to one another. The solution of the scattering problem for a wave of unit amplitude incident on the junction from (0,!R) is obtained by solving Eq. (34) for the case in which a"0, e "0, and d"1. De"ning the
Fig. 3. Results for the junction problem. Plots for the transmission coe$cient vs. frequency in units of a /2c are shown for A k ": (a) /2, (b) , (c) /100, (d) 3/4. Each curve presents results A for "xed .
re#ection and transmission coe$cients for this case as the ratios of the re#ected and transmitted #ux, respectively, to the incident #ux, gives the re#ection coe$cient 1 R" cos k#9 sin k
(35)
and the transmission coe$cients in each of the two branches leaving the vertex 4 sin k ¹" . cos k#9 sin k
(36)
In Fig. 3, plots of ¹ in Eq. (36) vs. for the junction are presented. The plots are made for the same photonic crystal and range of as considered in Fig. 1. The waveguide channels have t"0.01a A and a nearest-neighbor impurity separation of a lattice constant, a . Curves are generated from A Eq. (36) in conjunction with the dispersion relation "[ (0, 0)#2 (1, 0) cos k]\, and are presented as a function of mode frequency for a variety of "xed values of . The studied are chosen so that the value of k for the mid-band frequency a /2c"0.440, which we denote as k , assumes A A a set of representative values k ", 3/4, /2, and A /100 corresponding, respectively, to "0.342, 0.323, 0.284, and 0.243.
A.R. McGurn / Physica B 296 (2001) 201}209
Our theory of waveguide junctions can be easily extended to solve for the transportation of electromagnetic waves in more complex waveguide networks, i.e., photonic crystal circuits. As an example, we now treat a waveguide which contains a closed loop (see Fig. 2b). This circuit behaves like an optical switch, and a similar arrangement of optical "ber waveguides is used in optical switching applications in "ber optics. Consider an in"nitely long straight waveguide in a square lattice photonic crystal. The impurity sites of the waveguide channel are labeled by (0, mr) where m ranges over the integers. The waveguide is formed of only one type of impurity material. Now attach to the waveguide channel at its (0, 0) and (0, (n#1)r) sites for n'1 a U-shaped channel of impurities. The impurity material forming the channel of the U can be di!erent from the impurity material forming the channel of the in"nitely long waveguide. Let us use the above di!erence equation formulation to compute the transmission and re#ection characteristics of a waveguide mode incident on the U from (0,!R). The set of di!erence equations which describe this branching geometry consists of ten separate equations. The equations associated with the in"nitely long straight waveguide channel are given by the channel equation E " [ (0, 0)E # (r, 0) JP JP ;(E #E )], (37) J>P J\P where lO0 or n#1 otherwise ranges over the integers, and two equations connecting the channel to the U-shaped channel E " [ (0, 0)E # (r, 0)(E #E )] P \P # (r,0)E P and
(38)
E " [ (0, 0)E # (r, 0)[E L>P L>P L>P #E )]# (r, 0)E . (39) LP P L>P There are seven equations which describe the U-shaped channel. These are E " [ (0, 0)E # (r, 0)(E JP JP J>P #E )], J\P
(40)
207
where l"2, 3,2, l !1, E " [ (0, 0)E # (r, 0)(E J P JP J P J>P J P JP #E )], J P J\P where l"1, 2, 3,2, n,
(41)
E " [ (0, 0)E # (r, 0) JP L>P JP L>P ;(E #E )], (42) J>P L>P J\P L>P where l"2, 3,2, l !1, E " [ (0, 0)E # (r, 0)(E #E )], J P J P J P P J \P (43) " [ (0, 0)E # (r, 0)(E E J P L>P J P LP J P L>P .#E )], (44) J \P L>P E " [ (0, 0)E # (r, 0)E ]# (r,0)E , P P P (45) and E " [ (0, 0)E # (r, 0)E ] P L>P P L>P P L>P # (r, 0)E . (46) L>P A solution of this system of di!erence equations can be assumed to be of the form E "ae\ IJ#be IJ, JP where l"1, 2, 3,2, n,
(47)
E "ce\ IJ#de IJ, JP where l"!1,!2,!3,2,
(48)
E "e e\ OJ#fe OJ, JP where l"1, 2,2, l , E "e e\ OJ >J#fe OJ >J, J P JP where l"1, 2, 3,2, n,
(49)
(50)
E "e e\ OJ >L>J#fe OJ >L>J, J >\JP L>P where l"1, 2,2, l , E "h, E "h, L>P and
(52)
E "re\ IJ#se IJ JP
(54)
(51)
(53)
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A.R. McGurn / Physica B 296 (2001) 201}209
where l"n#2, n#3, n#4,2 . Substituting Eqs. (47)}(54) in Eqs. (37)}(46), we "nd a system of 10 algebraic equations which can be used to relate a, b, c, d, e , f, h, h, r, s to one another. An interesting case of these equations is to use them to relate the coe$cients c, d to r, s. This then gives the transmission and re#ection of the waveguide modes from the U-loop. Assuming that the waveguide channel of the in"nitely long channel and the U-shaped channel resonate at the same frequency (i.e., "[ (0, 0)#2 (r, 0) cos k]\ and "[ (0, 0)#2 (r, 0) cos q]\), after some algebra gives
c
"
d
a aH
a
aH
r s
,
(55)
where e\ IL> (a!a), a " P Q d
(56)
e IL> a " (a!a ), P Q d
(57)
sin k sin q a" # , P sin k(n#1) sin q(2l #n#1)
(58)
sin kn sin q(2l #n) ! , a "e I! Q sin k(n#1) sin q(2l #n#1)
(59)
Fig. 4. Results for the U-loop problem. Plots for the transmission coe$cient vs. frequency in units of a /2c are shown for A (q , k )"(/2, /2). The curve corresponds to results for "xed A A values of and .
Fig. 1. The waveguide channels of the circuit are taken to have t"0.01a with a nearest-neighbor A impurity separation equal to the lattice constant of the square lattice. Let q and k be wavenumbers in the U-loop and A A the waveguide channel at the mid-stop band frequency, a /2c"0.440. We have chosen (q , k )" A A A (/2, /2) for the plot in Fig. 4. This corresponds to a plot for "xed and for which / "1.00.
and
sin k sin q d "!2i sin k # . sin k(n#1) sin q(2l #n#1) (60) For there to be a unit transmitted #ux in the region above the U-loop, we need r"0 and s"1 in Eq. (55). The relative intensities of the incident and re#ected #ux in the region below the U-loop are then a and a , respectively. In Fig. 4, results are presented for the U-loop system with n"l "10. The transmission coe$c ient, ¹"1/a , de"ned as the ratio of the trans mitted to incident #ux, vs. the frequency, , in the stop gap are plotted. The parameters used in this plot are those of the "9, R"0.37796a square A lattice photonic crystal used to obtain the results in
4. Conclusions The propagation of electromagnetic waves in various types of waveguide circuits in photonic crystals have been studied. It has been demonstrated that under certain conditions, the mathematics describing this propagation can be reduced to a set of di!erence equations which can be solved exactly. As seen in the text, the phase coherent e!ects in photonic crystal circuits can cause the transmission and re#ection coe$cients of photonic circuits to be complicated functions of the mode frequencies. These e!ects may be important in the design of optical switches and other types of optical circuit devices similar to those used in "ber optics. An example of this is the optical switch discussed in Section 3.
A.R. McGurn / Physica B 296 (2001) 201}209
A feature of the di!erence equation theory is that very complicated and extended branching waveguide circuits can be solved exactly. The physics of these photonic crystal circuits is found to be described by closed form expressions involving elementary functions of mathematical physics. An additional advantage of the theory is that, for a given circuit network geometry, solutions for the properties of the circuit can be easily obtained for a wide variety of di!erent types of waveguide channel materials. References [1] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals, Princeton University Press, Princeton, 1995. [2] J.D. Joannopoulos, P.R. Villeneuve, S. Fan, Photonic crystals: putting a new twist on light, Nature (London) 386 (1997) 143. [3] A.A. Maradudin, A.R. McGurn, in: C.M. Soukoulis (Ed.), Photonic Band Gaps and Localization, Plenum, New York, 1992. [4] R.D. Meade, A. Devenyi, J.D. Joannopoulos, O.L. Alerhand, D.A. Smith, K. Kash, Novel applications of photonic band gap materials: low-loss bends and high Q cavities, J. Appl. Phys. 75 (1994) 4753. [5] S. Fan, J.A. Winn, A. Devenyi, J.C. Chen, R.D. Meade, J.D. Joannopoulos, Guided and deafec modes in periodic dielectric waveguides, J. Opt. Soc. Am. B 12 (1995) 1267.
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