Abstract

We present an analytical method for solution of one-dimensional optical systems, based on the differential transfer matrices. This approach can be used for exact calculation of various functions including reflection and transmission coefficients, band structures, and bound states. We show the consistency of the WKB method 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, we consider the reflection from a sinusoidal grating and the band structure of an infinite exponential grating. An excellent agreement between the results from our differential transfer-matrix method with other methods is observed. The method can be equally applied to one-dimensional time-harmonic quantum-mechanical systems.

© 2003 Optical Society of America

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References

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  1. Z. Menachem and E. Jerby, “Matrix transfer function (MTF) for wave propagation in dielectric waveguides with arbitrary transverse profiles,” IEEE Trans. Microwave Theory Tech. 46, 975–982 (1998).
    [CrossRef]
  2. T. F. Jablonski, “Complex modes in open lossless dielectric waveguides,” J. Opt. Soc. Am. A 11, 1272–1282 (1994).
    [CrossRef]
  3. J. S. Bagby, D. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 906–915 (1985).
    [CrossRef]
  4. M. Brack and R. K. Bhaduri, Semiclassical Physics, Vol. 96 of Frontiers in Physics (Addison-Wesley, Reading, Mass., 1997).
  5. J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison-Wesley, Reading, Mass., 1994).
  6. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  7. S. Khorasani and B. Rashidian, “Modified transfer matrices for conducting interfaces,” J. Opt. A: Pure Appl. Opt. 4, 251–256 (2002).
    [CrossRef]
  8. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. 17, 929–941 (1999).
    [CrossRef]
  9. R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (McGraw-Hill, New York, 1992), p. 259.
  10. S. G. Davison and M. Steslicka, Basic Theory of Surface States (Clarendon, Oxford, 1992), p. 12.
  11. K. Mehrany, S. Khorasani, and B. Rashidian, “Variational approach for extraction of eigenmodes in layered waveguides,” J. Opt. Soc. Am. B 19, 1978–1981 (2002).
    [CrossRef]
  12. W. H. Southwell, “Spectral response calculations of rugate filters using coupled-wave theory,” J. Opt. Soc. Am. A 5, 1558–1564 (1998).
    [CrossRef]
  13. K. Mehrany and S. Khorasani, “Analytical solution of non-homogeneous anisotropic wave equation based on differential transfer matrices,” J. Opt. A: Pure Appl. Opt. 4, 624–635 (2002).
    [CrossRef]

2002 (3)

S. Khorasani and B. Rashidian, “Modified transfer matrices for conducting interfaces,” J. Opt. A: Pure Appl. Opt. 4, 251–256 (2002).
[CrossRef]

K. Mehrany, S. Khorasani, and B. Rashidian, “Variational approach for extraction of eigenmodes in layered waveguides,” J. Opt. Soc. Am. B 19, 1978–1981 (2002).
[CrossRef]

K. Mehrany and S. Khorasani, “Analytical solution of non-homogeneous anisotropic wave equation based on differential transfer matrices,” J. Opt. A: Pure Appl. Opt. 4, 624–635 (2002).
[CrossRef]

1999 (1)

1998 (2)

Z. Menachem and E. Jerby, “Matrix transfer function (MTF) for wave propagation in dielectric waveguides with arbitrary transverse profiles,” IEEE Trans. Microwave Theory Tech. 46, 975–982 (1998).
[CrossRef]

W. H. Southwell, “Spectral response calculations of rugate filters using coupled-wave theory,” J. Opt. Soc. Am. A 5, 1558–1564 (1998).
[CrossRef]

1994 (1)

1985 (1)

J. S. Bagby, D. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 906–915 (1985).
[CrossRef]

Anemogiannis, E.

Bagby, J. S.

J. S. Bagby, D. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 906–915 (1985).
[CrossRef]

Drachman, B. C.

J. S. Bagby, D. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 906–915 (1985).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Jablonski, T. F.

Jerby, E.

Z. Menachem and E. Jerby, “Matrix transfer function (MTF) for wave propagation in dielectric waveguides with arbitrary transverse profiles,” IEEE Trans. Microwave Theory Tech. 46, 975–982 (1998).
[CrossRef]

Khorasani, S.

S. Khorasani and B. Rashidian, “Modified transfer matrices for conducting interfaces,” J. Opt. A: Pure Appl. Opt. 4, 251–256 (2002).
[CrossRef]

K. Mehrany and S. Khorasani, “Analytical solution of non-homogeneous anisotropic wave equation based on differential transfer matrices,” J. Opt. A: Pure Appl. Opt. 4, 624–635 (2002).
[CrossRef]

K. Mehrany, S. Khorasani, and B. Rashidian, “Variational approach for extraction of eigenmodes in layered waveguides,” J. Opt. Soc. Am. B 19, 1978–1981 (2002).
[CrossRef]

Mehrany, K.

K. Mehrany, S. Khorasani, and B. Rashidian, “Variational approach for extraction of eigenmodes in layered waveguides,” J. Opt. Soc. Am. B 19, 1978–1981 (2002).
[CrossRef]

K. Mehrany and S. Khorasani, “Analytical solution of non-homogeneous anisotropic wave equation based on differential transfer matrices,” J. Opt. A: Pure Appl. Opt. 4, 624–635 (2002).
[CrossRef]

Menachem, Z.

Z. Menachem and E. Jerby, “Matrix transfer function (MTF) for wave propagation in dielectric waveguides with arbitrary transverse profiles,” IEEE Trans. Microwave Theory Tech. 46, 975–982 (1998).
[CrossRef]

Nyquist, D.

J. S. Bagby, D. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 906–915 (1985).
[CrossRef]

Rashidian, B.

S. Khorasani and B. Rashidian, “Modified transfer matrices for conducting interfaces,” J. Opt. A: Pure Appl. Opt. 4, 251–256 (2002).
[CrossRef]

K. Mehrany, S. Khorasani, and B. Rashidian, “Variational approach for extraction of eigenmodes in layered waveguides,” J. Opt. Soc. Am. B 19, 1978–1981 (2002).
[CrossRef]

Southwell, W. H.

IEEE Trans. Microwave Theory Tech. (2)

J. S. Bagby, D. Nyquist, and B. C. Drachman, “Integral formulation for analysis of integrated dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 906–915 (1985).
[CrossRef]

Z. Menachem and E. Jerby, “Matrix transfer function (MTF) for wave propagation in dielectric waveguides with arbitrary transverse profiles,” IEEE Trans. Microwave Theory Tech. 46, 975–982 (1998).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. A: Pure Appl. Opt. (2)

S. Khorasani and B. Rashidian, “Modified transfer matrices for conducting interfaces,” J. Opt. A: Pure Appl. Opt. 4, 251–256 (2002).
[CrossRef]

K. Mehrany and S. Khorasani, “Analytical solution of non-homogeneous anisotropic wave equation based on differential transfer matrices,” J. Opt. A: Pure Appl. Opt. 4, 624–635 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Other (5)

R. E. Collin, Foundations for Microwave Engineering, 2nd ed. (McGraw-Hill, New York, 1992), p. 259.

S. G. Davison and M. Steslicka, Basic Theory of Surface States (Clarendon, Oxford, 1992), p. 12.

M. Brack and R. K. Bhaduri, Semiclassical Physics, Vol. 96 of Frontiers in Physics (Addison-Wesley, Reading, Mass., 1997).

J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison-Wesley, Reading, Mass., 1994).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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Figures (4)

Fig. 1
Fig. 1

Illustration of sinusoidal grating cell (L=0.1375μm, na=2, np=0.1).

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Illustration of exponential profile, generating the exponential grating (ns=2, n0=1).

Fig. 4
Fig. 4

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

Equations (43)

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2x2 A(x)+k2(x)A(x)=0,
A(x)=A+(x)exp[-jk(x)x]+A-(x)exp[+jk(x)x],
A2=Q12A1,
Q12=k2+k12k2exp[+j(k2-k1)x]k2-k12k2exp[+j(k2+k1)x]k2-k12k2exp[-j(k2+k1)x]k2+k12k2exp[-j(k2-k1)x]
A(x+dx)=Qxx+dxA(x),
dA(x)=(Qxx+dx-I)A(x)=U(x)A(x)dx,
U(x)=12k(x)dk(x)dx-1+j2k(x)xexp[+j2k(x)x]exp[-j2k(x)x]-1-j2k(x)x,
U(x)=j dk(x)dx x100-1-12k(x)dk(x)dx×1exp[+j2k(x)x]exp[-j2k(x)x]1,
A(x2)=expx1x2 U(x)dxA(x1)=exp(M)A(x1)=Qx1x2A(x1),
exp(M)=I+n=11n!Mn.
Qxx=I,
Qx2x3Qx1x2=Qx1x3,
Qx2x1=Qx1x2-1,
|Qx1x2|=k(x1)/k(x2).
k(x)=k1x<Xk2x>X.
Qx1x2=Ix1<x2<XQ12x1<X<x2IX<x1<x2,
M=k1k2(-1+j2kX) dk2kk1k2exp(+j2kX) dk2kk1k2exp(-j2kX) dk2kk1k2(-1-j2kX) dk2k.
M=12lnk1k2 1-1-11.
Mn=lnn-1k1k2M,
exp(M)=I+12n=11n!lnnk1k2 1-1-11=I+12explnk1k2-1 1-1-11=k2+k12k2k2-k12k2k2-k12k2k2+k12k2.
A(x)=Φκ(x)exp(-jκx),
A(x+L)
=exp{-j[κ-k(x)]L}00exp{-j[κ+k(x)]L}×A(x)PA(x).
|I-Qxx+1P-1|=0.
exp(-j2κL)-exp(-jκL)q11exp[-jk(x)L]+q22exp[+jk(x)L]+q11q22-q12q21=0,
cos(κL)=q112exp[-jk(x)L]+q222exp[+jk(x)L].
cos(κL)=0Lq11+q222Lcos[k(x)L]dx-j0Lq11-q222Lsin[k(x)L]dx.
q22=0,
m11=-m22=2 j0k(x)xdx,
m12=-m21=2 j0sin[2k(x)x]2k(x) k(x)dx,
Q-=cosΔI+sinΔΔM.
m11=ΔcotΔ,
q22=a21b12+a22b22=0
-a21a22×b12b22=1.
J=12-12Re-a21b12a22b22
J=Re-a21b12a22 b22,
ln Rd+ln Ru=j2πn,
u11+jdk(x)/dxx,
u12u210,
u22-jdk(x)/dxx.
A(x2)=q11A+(x1)exp[-jk(x2)x2]=q11A(x1)exp{-j[k(x2)x2-k(x1)x1]}.
q11=expx1x2+jk(x)xdx=exp+jx1x2[k(x)x]dx-jx1x2k(x)dx.
A(x2)=A(x1)exp-jx1x2k(x)dx,

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