S. Khorasani and K. Mehrany Vol. 20, No. 1 /January 2003 /J. Opt. Soc. Am. B 91

Differential transfer-matrix method for solutionof one-dimensional

linear nonhomogeneous optical structures

Sina Khorasani

School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0250, andDepartment of Electrical Engineering, Sharif University of Technology, P.O. Box 11365-9363, Tehran, Iran

Khashayar Mehrany

Department of Electrical Engineering, Sharif University of Technology, P.O. Box 11365-9363, Tehran, Iran

Received April 26, 2002; revised manuscript received July 26, 2002

We present an analytical method for solution of one-dimensional optical systems, based on the differentialtransfer matrices. This approach can be used for exact calculation of various functions including reflectionand transmission coefficients, band structures, and bound states. We show the consistency of the WKBmethod with our approach and discuss improvements for even symmetry and infinite periodic structures.Moreover, a general variational representation of bound states is introduced. As application examples, weconsider the reflection from a sinusoidal grating and the band structure of an infinite exponential grating. Anexcellent agreement between the results from our differential transfer-matrix method with other methods isobserved. The method can be equally applied to one-dimensional time-harmonic quantum-mechanical sys-tems. © 2003 Optical Society of America

OCIS codes: 000.3860, 260.0260, 260.2110, 130.0130.

1. INTRODUCTIONThe analytical solution of one-dimensional (1-D) wave-propagation problems in arbitrary nonhomogeneousquantum and optical systems has been studied in numer-ous reports. The existing known approaches include thetransfer-matrix function,1 the iterative eigenfunction ex-pansion method,2 and the domain integral equationmethod.3 There exist also approximate approaches, in-cluding semiclassical4 and the famous WKB5 methods.The transfer-matrix function has the capability to analyze2 1

2 -dimensional problems; however, the structure has tobe artificially placed in a virtual rectangular tube with in-finite potential barriers. The iterative eigenfunction ex-pansion method and the domain integral equationmethod both require elaborate numerical computationsand cannot be applied only by analytical methods. Thesemiclassical methods, though analytically simple, are tooapproximate and fail in correct derivation of lower-ordermodes. There is therefore still a need for development ofa mathematical approach capable of solving 1-D problemsanalytically with less effort.

The solution of the wave equation is studied for nonho-mogeneous layered media with piecewise constant pro-files of refractive index through the well known transfer-matrix method6–8 (TMM). In the standard TMM,6,8 theup- and down-traveling plane waves in layers arematched to those of adjacent layers, in compliance withthe continuity of the total electromagnetic field and its de-rivative, and the resulting equations are written in ma-trix form with each layer having its own reference of po-sition. The TMM has been recently improved7 to includethe effect of interface conductivity (which appears as dis-

0740-3224/2003/010091-06$15.00 ©

continuities in the derivative of electric field), and thesame reference for addressing the position has been em-ployed in all layers.

In this paper, a new analytical method is described forcalculation of the scalar wave function in 1-D optical sys-tems. Here, we take the benefit of having the same ref-erence of position in the improved TMM,7 while neglect-ing the presence of conducting interfaces through settingthe interface conductivities to zero. As discussed below,this has enabled us to evolve the transfer matrices intodifferential forms. Our formalism can be used for deri-vation of reflection and transmission coefficients of thesystem, as well as its bound states. The proposedmethod is mathematically exact within the validity rangeof the 1-D approximation of the realistic physical system.We demonstrate the compatibility of the WKB methodwith our approach and discuss improvements under evensymmetry and infinite periodic structures. We show theapplications to several examples and also present a varia-tional form for the bound states. As application ex-amples, we solve the reflection coefficient from a 50-period sinusoidal grating and compare the results withthe coupled-wave solution. Also, we compute the bandstructure of an infinite exponential grating and compareit with the exact solution based on Bessel functions. Inboth cases, an excellent agreement is observed, as dis-cussed below.

2. FORMULATIONThe propagation of a scalar TE polarized electromagneticwave in a 1-D nonhomogeneous time-harmonic system isgiven by

2003 Optical Society of America

92 J. Opt. Soc. Am. B/Vol. 20, No. 1 /January 2003 S. Khorasani and K. Mehrany

]2

]x2 A~x ! 1 k2~x !A~x ! 5 0, (1)

in which A(x) and k(x) represent the field amplitude andwave vector, respectively. We further accept that eitherk(x) or jk(x) are real and positive with j 5 A21; thiscondition corresponds to lossless media. For the mo-ment, we limit ourselves to TE polarized waves, since thetreatment of TM modes is slightly different.

We propose a solution to Eq. (1) of the form

A~x ! 5 A1~x !exp@2jk~x !x# 1 A2~x !exp@1jk~x !x#,(2)

with A1(x) and A2(x) as unknown functions, associatedwith the forward and backward waves, to be determined.

According to the transfer-matrix method,6–8 if k(x) ispiecewise constant equal to k1 and k2 for x , X and x. X, respectively, the functions A6(x) would be also con-stants equal to A1

6 or A26 . In this case, one has

A2 5 Q1→2A1 , (3)

in which Amt 5 @Am

1 Am2 # and

is the transfer matrix from layer 1 to layer 2. This trans-fer matrix resembles the wave-amplitude transmissionmatrix9 in microwave engineering, where the determi-nant equals the ratio between impedances in both sides ofthe junction. In contrast, by direct inspection of Eq. (4)one can verify that uQ1→2u 5 k1 /k2 .

As stated above, this approach follows the modifiedTMM7 in which the same reference is used for addressingthe position in all layers. This special treatment oftransfer matrices enables us to proceed with the analyti-cal calculations, mainly because of the solution form pre-scribed in Eq. (2), which requires a single reference at x5 0.

Now if k(x) is an analytic function of x, one can write

A~x 1 dx ! 5 Qx→x1dx • A~x !, (5)

where A(x) t 5 @A1(x) A2(x)#. Equivalently,

dA~x ! 5 ~Qx→x1dx 2 I! • A~x ! 5 U~x ! • A~x !dx, (6)

in which I is the identity matrix. Also differentiating Eq.(4) results in

U~x ! 51

2k~x !

dk~x !

dxF 21 1 j2k~x !x exp@1j2k~x !x#

exp@2j2k~x !x# 21 2 j2k~x !x G ,(7a)

which can be equally rewritten as

Q1→2 5 F k2 1 k1

2k2exp@1j~k2 2 k1!x#

k2 2 k1

2k2exp@1j~k2

k2 2 k1

2k2exp@2j~k2 1 k1!x#

k2 1 k1

2k2exp@2j~k2

U~x ! 5 jdk~x !

dxxF1 0

0 21G 21

2k~x !

dk~x !

dx

3 F 1 exp@1j2k~x !x#

exp@2j2k~x !x# 1 G , (7b)

in which the first term represents the phase accumulationalong the propagation path, and the second term repre-sents the interaction between up- and down-travelingwaves. Below, we observe that the second expression canbe neglected under the assumption of gradual variationsfor k(x).

However, Eq. (6) has a solution of the form

A~x2! 5 expF Ex1

x2

U~x !dxGA~x1! 5 exp~M! • A~x1!

5 Qx1→x2• A~x1!, (8)

in which exp(M) is defined as

exp~M! 5 I 1 (n51

` 1

n!Mn. (9)

It thus can be easily observed that exp(0) 5 I andexp(M)21 5 exp(2M). Meanwhile, the transfer matrixQx1→x2

has the properties

Qx→x 5 I, (10a)

Qx2→x3• Qx1→x2

5 Qx1→x3, (10b)

Qx2→x15 Qx1→x2

21 , (10c)

uQx1→x2u 5 k~x1!/k~x2!. (10d)

Equation (10a) is directly followed by Eqs. (7)–(9), and thevalidity of Eqs. (10b) and (10c) originates from the defini-tion of transfer matrices. Equation (10d) holds, since thedeterminant of the basic transfer matrix as given by Eq.(4) is uQ1→2u 5 k1 /k2 . It should be mentioned thatQx1→x2

Qx2→x35 Qx1→x3

is not necessarily true, because inorder for exp(A)exp(B) 5 exp(B)exp(A) 5 exp(A 1 B),the necessary and sufficient condition is that @A,B# 5 0.

3. ADDITIONAL CONSIDERATIONSA. Constant Wave VectorIn this case, k(x) 5 k corresponds to a homogeneous me-dium. Therefore the vector A(x) would be also indepen-dent of x. In other words, for any x1 and x2 , Qx1→x2

5 I. However, this condition is automatically satisfiedby Eqs. (7) and (8), by taking dk(x)/dx [ 0.

1!x#

1!x#G (4)

1 k

2 k

S. Khorasani and K. Mehrany Vol. 20, No. 1 /January 2003 /J. Opt. Soc. Am. B 93

B. Single Interface Across Two Homogeneous MediaIn this case, the wave vector k(x) is given by

k~x ! 5 H k1 x , X

k2 x . X. (11)

Therefore the parameter dk(x)/dx is nonzero only at theinterface x 5 X, being singular. According to the discus-sions in the previous subsection, one has

Qx1→x25 H I x1 , x2 , X

Q1→2 x1 , X , x2

I X , x1 , x2

, (12)

with Q1→2 5 exp(M) and

M 5 F Ek1

k2

~21 1 j2kX !dk

2kE

k1

k2

exp~1j2kX !dk

2k

Ek1

k2

exp~2j2kX !dk

2kE

k1

k2

~21 2 j2kX !dk

2kG .

(13)In derivation of Eq. (13) we notice that the integrals run-ning from x1 , X , x2 are nonzero where dk(x)/dx issingular, i.e., at the interface x 5 X. Subsequently, theintegrals can be evaluated by a change of variables from xto k.

Unfortunately, it is not possible to evaluate the off-diagonal elements of M analytically. However, in thecase that X 5 0, Eq. (13) can be simplified as

M 51

2lnS k1

k2D F 1 21

21 1 G . (14)

This matrix has the property

Mn 5 lnn21S k1

k2DM, (15)

so that

exp~M! 5 I 11

2 (n51

` 1

n!lnnS k1

k2D F 1 21

21 1 G5 I 1

1

2 FexpS lnk1

k2D 2 1G F 1 21

21 1 G

5 F k2 1 k1

2k2

k2 2 k1

2k2

k2 2 k1

2k2

k2 1 k1

2k2

G . (16)

The result is in agreement with Eq. (4) when X 5 0.

C. Reflection and Transmission CoefficientsIn this subsection, we derive the reflection and transmis-sion coefficients of a nonhomogeneous medium placed inthe region X1 , x , X2 and bounded by vacuum. Weconsider two cases: when a down-traveling plane wave isincident upon the medium from the lower boundary x5 X1 , and when an up-traveling wave is incident upon it

from upper boundary x 5 X2 . Furthermore, we identifythe lower and upper vacuums with the indices 1 and 2,respectively.

In these cases, one has A22 5 0 or A1

1 5 0, respectively.Therefore from the definition of the transfer matrices, it isfound that Rd 5 A1

2/A11 5 2q21 /q22 , Td 5 A2

1/A11

5 D/q22 for the down-traveling wave, and Ru 5 A21/A2

2

5 q12 /q22 , Tu 5 A12/A2

2 5 1/q22 for the up-travelingwave, where qij are elements of Q1→2 . Here D5 uQ1→2u, which from Eq. (10d) it turns out to be equal tounity.

The expressions for the up- and down-reflection andtransmission coefficients simplify considerably when theassociated interface coincides with the origin at x 5 0.For instance, we take X1 5 0, resulting in Rd 5 (k12 k2)/(k1 1 k2) and Td 5 2k1 /(k1 1 k2), being inagreement with well known expressions.

D. Periodic SystemsIn a periodic system with the period L, one has k(x)5 k(x 1 L), and the solutions according to Floquet’stheorem are given by Bloch waves10 as

A~x ! 5 Fk~x !exp~2jkx !, (17)

in which the envelope function Fk(x) 5 Fk(x 1 L) is alsoperiodic. Here, the reader is reminded that the analysisis not valid whenever the field includes an axial TM com-ponent, as prescribed in the development of formulation.Comparing the above equation with Eq. (2) and using theperiodicity of the envelope function Fk(x) and k(x) re-veals that

A~x 1 L !

5 Fexp$2j@k 2 k~x !#L% 0

0 exp$2j@k 1 k~x !#L%G

3 A~x ! [ P – A~x !. (18)

However, from Eq. (8), one has A(x 1 L) 5 Qx→x1L• A(x), and therefore the dispersion equation for Blochstates is obtained as

uI 2 Qx→x11 • P21u 5 0. (19)

Expanding and simplifying Eq. (19) gives

exp~2j2kL ! 2 exp~2jkL !bq11 exp@2jk~x !L#

1 q22 exp@1jk~x !L# c 1 q11q22 2 q12q21 5 0,

(20)

where qij are the elements of Qx→x1L . But uQx→x1Lu5 k(x 1 L)/k(x) 5 1, and Eq. (20) can be solved forexp(2jkL) and further simplified as

cos~kL ! 5q11

2exp@2jk~x !L# 1

q22

2exp@1jk~x !L#.

(21)

The right-hand side depends on x, whereas the left-handside does not. In order to eliminate this dependence, onecan easily integrate both sides from 0 to L, giving rise tothe general dispersion equation for Bloch states,

94 J. Opt. Soc. Am. B/Vol. 20, No. 1 /January 2003 S. Khorasani and K. Mehrany

cos~kL ! 5 E0

L q11 1 q22

2Lcos@k~x !L#dx

2 jE0

L q11 2 q22

2Lsin@k~x !L#dx. (22)

It is easy to verify that in the case of k(0 , x , a) 5 k1and k(a , x , L) 5 k2 , this equation simplifies to thewell-known Kronig–Penney dispersion equation for Blochstates.10

E. Bounded ModesSuppose that a system has bounded eigenmodes so thatA1(2`) 5 A2(1`) 5 0. In this case, solutions existonly for some special values of eigenvalues. From Eq.(8), A(1`) 5 Q2`→` • A(2`). Therefore the character-istic equation for obtaining eigenvalues would be8,11

q22 5 0, (23)

where q22 is the fourth element of Q2`→` . Please noticethat Eq. (23) is actually an equation in terms of the propa-gation constant in the waveguide.

Under even symmetry in bounded modes, one hask(x) 5 k(2x), and thus k8(x) 5 2k8(2x). In this case,it is possible to verify that

m11 5 2m22 5 2 jE0

`

k8~x !xdx, (24a)

m12 5 2m21 5 2 jE0

` sin@2k~x !x#

2k~x !k8~x !dx,

(24b)

in which mij are the elements of the M matrix, as definedin Eq. (8). In this case, the transfer matrix Q2`→`

5 exp(M) can be greatly simplified by direct expansion ofthe exponential as given in Eq. (9), since from Eq. (24) itfollows that M2 5 2DI, in which D 5 uMu. Here, onecan show that

Q2`→` 5 cosADI 1sinAD

ADM. (25)

From the above, the dispersion equation of boundedmodes, Eq. (23), would reduce to the simple expression

m11 5 AD cot AD, (26)

where m11 is defined in Eq. (24a).

F. Variational Extraction of Bounded ModesAccording to Eqs. (10b) and (23), it is found that Eq. (23)can be rewritten as either

q22 5 a21b12 1 a22b22 5 0 (27)

or

2a21

a223

b12

b225 1. (28)

Here, aij and bij are the elements of the Q2`→x andQx→` , respectively, in which x is an arbitrary referencepoint within the medium. Naturally, the solutions of Eq.(28) would be independent of x.

Comparing Eq. (28) to the results of Subsection 3.C re-veals that Eq. (28) consists of the production of two reflec-tion coefficients: the upward reflection coefficient from xto 1` and the downward reflection coefficient from x to2`. However, each of these reflection coefficients has amagnitude less than unity. Therefore the functional11

J 51

22

1

2ReH 2

a21b12

a22b22J (29)

should have minima coinciding with the bounded modes.Moreover, its minima must be independent of the choiceof the reference point x. It is also possible to define theclosely associated functional

J 5 ReHA2a21b12

a22 b22J , (30)

which has extrema coinciding with the bounded modes.We have assessed this variational method for a layered

lossless waveguide11 and shown its superior efficiency andaccuracy compared with the other existing mode-solvingmethods. The most interesting features of the functional(29) are that it is bounded and also has no local minima,but it coincides with the eigenmodes.11

It is also possible to rearrange Eq. (28) as

ln Rd 1 ln Ru 5 j2pn, (31)

where Rd 5 2a21 /a22 and Ru 5 b12 /b22 , and n is an in-teger. Since the reflection coefficients Rd and Ru involvephase-accumulation integrations from x to infinities, Eq.(31) can be regarded as an equivalent form of the Wilson–Bohr–Sommerfeld quantization law.5 In the case of slabwaveguides, Eq. (31) can be shown to be exactly the wellknown dispersion equation for the eigenmodes.11

G. Slowly Varying PotentialIf the refractive-index function varies slowly, then k(x)would also vary gradually. In this case, Eq. (8) can besimplified considerably. If the distance between thestarting and ending points x1 and x2 in Eq. (8) is muchlarger than k(x)21, being of the order of local wavelength,then the integrands of the off-diagonal elements of M os-cillate so rapidly that the magnitude of off-diagonal ele-ments becomes much smaller than the magnitude of diag-onal ones; here, we further assume that k(x) is realvalued over the integration from x1 to x2 , and its valuedoes not change significantly over this domain. Underthese circumstances, the magnitude of the second matrixon the right-hand side of Eq. (7b), expressing the interac-tion of forward and backward waves, is small comparedwith the diagonal matrix in the first one. It would bethen legitimate to make the approximations

u11 ' 1jdk~x !/dxx, (32a)

u12 ' u21 ' 0, (32b)

u22 ' 2jdk~x !/dxx. (32c)

Here uij are the approximate elements of U(x), and theelements of the transfer matrix can be approximated asq12 5 q21 5 0, q11 5 exp(m11), and q22 5 exp(m22),where mij 5 *x1

x2 uijdx.

S. Khorasani and K. Mehrany Vol. 20, No. 1 /January 2003 /J. Opt. Soc. Am. B 95

Now, consider a wave propagating in the 1x direction atx1 . The wave amplitude at x2 is given by

A~x2! 5 q11A1~x1!exp@2jk~x2!x2#

5 q11A~x1!exp$2j@k~x2!x2 2 k~x1!x1#%. (33)

But from the above discussions and Eq. (8), we have

q11 5 expF Ex1

x2

1jk8~x !xdxG5 expF1jE

x1

x2

@k~x !x#8dx 2 jEx1

x2

k~x !dxG . (34)

By plugging Eq. (33) into Eq. (32) and simplifying, it isfound that

A~x2! 5 A~x1!expF2jEx1

x2

k~x !dxG , (35)

which is exactly the WKB solution of Eq. (1).5

4. EXAMPLESIn this section we present application examples to assessthe validity of the proposed approach. The examples be-long to optical structures; however, quantum-mechanicalsystems can be also analyzed in the same way just bychange of dimensions and redefinition of parameters.

A. Reflection from Sinusoidal GratingConsider the structure shown in Fig. 1. Here we have asinusoidal grating with the spatial period L5 0.1375 mm, mean refractive index na 5 2, and modu-lation amplitude np/2 5 0.05. The grating is obtained byreproduction of the single cosine to 50 times. It is sup-posed that the structure is illuminated by a normal inci-dent TE light. In Fig. 2, the reflection coefficient of thestructure versus wavelength is plotted. We computedthe reflection coefficient by two different approaches.

The first is the coupled-mode solution,6,12 shown as asolid curve, and the next is the solution obtained by thedifferential transfer-matrix method, shown by a dashed

Fig. 1. Illustration of sinusoidal grating cell (L 5 0.1375mm, na 5 2, np 5 0.1).

curve. One can observe a reasonable agreement betweenthese two solutions, in justification of our proposedmethod.

The computation time for the coupled-mode solutionhas been ;1 s, whereas it has been ;5 s for the differen-tial transfer matrix. Please notice that the coupled-modesolution is indeed approximate and suitable only for smallmodulation indices, whereas the differential transfer-matrix approach is exact. Moreover, the coupled-modesolution for nonsinusoidal gratings usually is hard to de-rive, whereas our method can solve the system for anyother profile at the same efficiency. Please notice thatone can still compute the reflection coefficient by dividingthe entire grating into many small layers with constantrefraction indices and employ the multiplication of corre-sponding transfer matrices. This approach has beenused12; however, this method is hardly converging andtime consuming. We also tested this method and foundthat the efficiency reduces drastically to 1000 times oreven more.

B. Band Structure of Exponential GratingIn this example we consider an infinite grating obtainedby periodic reproduction of the exponential profile asshown in Fig. 3. In each period, the refractive index in-creases exponentially from n0 5 1 to ns 5 2. The exactsolution of the wave equation for this structure in termsof the Bessel functions is given in Ref. 6, p. 173. The ex-act frequency dispersion of the grating is shown in Fig. 4as a solid curve. Here, the first three forbidden fre-quency gaps can be observed. The band structure is dis-played through the reduced-zone scheme so that all gapsoccur either at k 5 0 or k 5 kmax .

The band structure of this grating has been also com-puted by the differential transfer-matrix method, which isshown in Fig. 4 as dashed lines. The result shows excel-lent agreement with the exact solution, again confirmingthe proposed approach.

Please notice that there exists an abrupt fall of the pro-file of the refractive index from ns to n0 at x 5 nL, as canbe seen in Fig. 3. This discontinuity can be easily in-cluded in the computation by multiplying a jump transfer

Fig. 2. Reflection coefficient of the grating obtained by repro-duction of the structure in Fig. 1, versus wavelength: coupled-mode solution (solid curve); differential transfer-matrix solution(dashed curve).

96 J. Opt. Soc. Am. B/Vol. 20, No. 1 /January 2003 S. Khorasani and K. Mehrany

matrix QL2→L1 from left to the differential transfer ma-trix Q01→L2. Here n(L2) 5 ns and n(L1) 5 n(01)5 n0 , and Eq. (4) can be used to find the jump transfermatrix QL2→L1.

5. CONCLUSIONSA new analytical tool has been presented for an exact so-lution of linear time-harmonic 1-D optical systems, based

Fig. 3. Illustration of exponential profile, generating the expo-nential grating (ns 5 2, n0 5 1).

Fig. 4. Band structure of the exponential grating obtained byreproduction of the unit cell in Fig. 3: exact solution from Besselfunctions (solid curves); differential transfer-matrix solution(dashed curves). Here, three first forbidden energy gaps (FEGs)can be seen.

on the differential transfer-matrix method. This ap-proach has been based on the generalization of a modifiedtransfer-matrix method. The derivation of reflection andtransmission coefficients and bound states have been dis-cussed. The compatibility of this approach with theWKB method and improvements under even symmetryand infinite periodic structures has been shown. A varia-tional form for the bound states has been also presented.Two application examples have been solved and comparedwith the solutions obtained from other approaches, and inboth cases an excellent agreement has been observed.Although the optical interpretation of the method is lim-ited to TE polarization, it is possible to extend the ap-proach to include the TM polarization and even to nonho-mogeneous anisotropic media with arbitrary profiles forpermittivity and permeability tensors.13

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