Abstract

In an exact treatment of the Maxwell equations, we derive form and structure factors for reflection from periodic layers, and we show that these factors are significantly different from their analogs in kinematic x-ray diffraction. Quite generally, we show that reflection and impedance can be written precisely as the sum of an additive form factor and the product of a structure factor and a second form factor. This additive form factor does not have an analog in kinematic x-ray diffraction. It is demonstrated that the form factors are found by analytic continuation to an arbitrary wavelength of expressions for the impedance both at long wavelengths and at quarter wavelengths. A correction to the Bragg law relating fringe spacing to the total structure thickness is derived. We go beyond previous numerical work by deriving simple analytic exact expressions for reflection and impedance of periodic layers for all frequencies within the reflection passband, and for an arbitrary number of periods in the structure, an arbitrary index profile within each period, arbitrary layer thicknesses (not just quarter-wave layers), and for arbitrary sizes of the refractive index differences.

© 2007 Optical Society of America

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References

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  1. M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
    [CrossRef]
  2. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).
  3. P. Yeh, A. Yariv, and C.-S. Hong, "Electromagenetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977).
    [CrossRef]
  4. D. L. MacFarlane and E. M. Dowling, "Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures," J. Opt. Soc. Am. A 11, 236-245 (1994).
    [CrossRef]
  5. D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE J. Quantum Electron. 28, 514-524 (1992).
    [CrossRef]
  6. F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
    [CrossRef]
  7. N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum. Electron. 33, 295-302 (1997).
    [CrossRef]
  8. S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum. Electron. 27, 2086-2090 (1991).
    [CrossRef]
  9. M. A. Parent, S. S. Murtaza, and J. C. Campbell, "Analytic solution for the peak reflectivity of an asymmetric mirror," Appl. Opt. 36, 4265-4268 (1997).
    [CrossRef] [PubMed]
  10. S. S. Murtaza, M. A. Parent, J. C. Bean, and J. C. Campbell, "Theory of reflectivity of an asymmetric mirror," Appl. Opt. 35, 2054-2059 (1996).
    [CrossRef] [PubMed]
  11. D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
    [CrossRef]
  12. R. Maulini, A. Mohan, M. Giovannini, J. Faist, and E. Gini, "External cavity quantum-cascade laser tunable from 8.2 to 10.4 μm using a gain element with a heterogeneous cascade," Appl. Phys. Lett. 88, 201113 (2006).
  13. T. Maier, H. Schneider, H. C. Liu, M. Walther, and P. Koidl, "Quantum-well infrared photodetector with voltage-switchable quadratic and linear response," Appl. Phys. Lett. 88, 51117 (2006).
  14. J. C. Caylor, K. Coonley, J. Stuart, T. Colpitts, and R. Venkatasubramanian, "Enhanced thermoelectric performance in PbTe-based superlattice structures from reduction of lattice thermal conductivity," Appl. Phys. Lett. 87, 23105 (2005).
  15. R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O'Quinn, "Thin-film thermoelectric devices with high room-temperature figures of merit," Nature 413, 597-602 (2001).
    [CrossRef] [PubMed]
  16. A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, "Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance," Phys. Rev. Lett. 96, 215504 (2006).
  17. B. E. Warrren, X-Ray Diffraction (Addison-Wesley, 1990).

2001 (1)

R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O'Quinn, "Thin-film thermoelectric devices with high room-temperature figures of merit," Nature 413, 597-602 (2001).
[CrossRef] [PubMed]

1999 (1)

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

1997 (2)

N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum. Electron. 33, 295-302 (1997).
[CrossRef]

M. A. Parent, S. S. Murtaza, and J. C. Campbell, "Analytic solution for the peak reflectivity of an asymmetric mirror," Appl. Opt. 36, 4265-4268 (1997).
[CrossRef] [PubMed]

1996 (1)

1994 (1)

1993 (1)

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

1992 (1)

D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE J. Quantum Electron. 28, 514-524 (1992).
[CrossRef]

1991 (1)

S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum. Electron. 27, 2086-2090 (1991).
[CrossRef]

1983 (1)

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

1977 (1)

Arai, S.

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

Babic, D. I.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE J. Quantum Electron. 28, 514-524 (1992).
[CrossRef]

Bean, J. C.

Benson, T. M.

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

Bowers, J. E.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

Campbell, J. C.

Caylor, J. C.

J. C. Caylor, K. Coonley, J. Stuart, T. Colpitts, and R. Venkatasubramanian, "Enhanced thermoelectric performance in PbTe-based superlattice structures from reduction of lattice thermal conductivity," Appl. Phys. Lett. 87, 23105 (2005).

Chen, L. R.

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

Chung, Y.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

Coldren, L. A.

S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum. Electron. 27, 2086-2090 (1991).
[CrossRef]

Colpitts, T.

R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O'Quinn, "Thin-film thermoelectric devices with high room-temperature figures of merit," Nature 413, 597-602 (2001).
[CrossRef] [PubMed]

J. C. Caylor, K. Coonley, J. Stuart, T. Colpitts, and R. Venkatasubramanian, "Enhanced thermoelectric performance in PbTe-based superlattice structures from reduction of lattice thermal conductivity," Appl. Phys. Lett. 87, 23105 (2005).

Coonley, K.

J. C. Caylor, K. Coonley, J. Stuart, T. Colpitts, and R. Venkatasubramanian, "Enhanced thermoelectric performance in PbTe-based superlattice structures from reduction of lattice thermal conductivity," Appl. Phys. Lett. 87, 23105 (2005).

Corzine, S. W.

D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE J. Quantum Electron. 28, 514-524 (1992).
[CrossRef]

S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum. Electron. 27, 2086-2090 (1991).
[CrossRef]

Dagli, N.

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

Dowling, E. M.

Faist, J.

R. Maulini, A. Mohan, M. Giovannini, J. Faist, and E. Gini, "External cavity quantum-cascade laser tunable from 8.2 to 10.4 μm using a gain element with a heterogeneous cascade," Appl. Phys. Lett. 88, 201113 (2006).

Gini, E.

R. Maulini, A. Mohan, M. Giovannini, J. Faist, and E. Gini, "External cavity quantum-cascade laser tunable from 8.2 to 10.4 μm using a gain element with a heterogeneous cascade," Appl. Phys. Lett. 88, 201113 (2006).

Giovannini, M.

R. Maulini, A. Mohan, M. Giovannini, J. Faist, and E. Gini, "External cavity quantum-cascade laser tunable from 8.2 to 10.4 μm using a gain element with a heterogeneous cascade," Appl. Phys. Lett. 88, 201113 (2006).

Glavin, B. A.

A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, "Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance," Phys. Rev. Lett. 96, 215504 (2006).

Henini, M.

A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, "Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance," Phys. Rev. Lett. 96, 215504 (2006).

Hong, C.-S.

Kaertner, F.

N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum. Electron. 33, 295-302 (1997).
[CrossRef]

Keller, U.

N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum. Electron. 33, 295-302 (1997).
[CrossRef]

Kent, A. J.

A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, "Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance," Phys. Rev. Lett. 96, 215504 (2006).

Kini, R. N.

A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, "Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance," Phys. Rev. Lett. 96, 215504 (2006).

Kochelap, V. A.

A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, "Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance," Phys. Rev. Lett. 96, 215504 (2006).

Koidl, P.

T. Maier, H. Schneider, H. C. Liu, M. Walther, and P. Koidl, "Quantum-well infrared photodetector with voltage-switchable quadratic and linear response," Appl. Phys. Lett. 88, 51117 (2006).

Koyama, F.

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

Linnik, T. L.

A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, "Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance," Phys. Rev. Lett. 96, 215504 (2006).

Liu, H. C.

T. Maier, H. Schneider, H. C. Liu, M. Walther, and P. Koidl, "Quantum-well infrared photodetector with voltage-switchable quadratic and linear response," Appl. Phys. Lett. 88, 51117 (2006).

MacFarlane, D. L.

Maier, T.

T. Maier, H. Schneider, H. C. Liu, M. Walther, and P. Koidl, "Quantum-well infrared photodetector with voltage-switchable quadratic and linear response," Appl. Phys. Lett. 88, 51117 (2006).

Matin, M. A.

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

Matuschek, N.

N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum. Electron. 33, 295-302 (1997).
[CrossRef]

Maulini, R.

R. Maulini, A. Mohan, M. Giovannini, J. Faist, and E. Gini, "External cavity quantum-cascade laser tunable from 8.2 to 10.4 μm using a gain element with a heterogeneous cascade," Appl. Phys. Lett. 88, 201113 (2006).

Mohan, A.

R. Maulini, A. Mohan, M. Giovannini, J. Faist, and E. Gini, "External cavity quantum-cascade laser tunable from 8.2 to 10.4 μm using a gain element with a heterogeneous cascade," Appl. Phys. Lett. 88, 201113 (2006).

Murtaza, S. S.

O'Quinn, B.

R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O'Quinn, "Thin-film thermoelectric devices with high room-temperature figures of merit," Nature 413, 597-602 (2001).
[CrossRef] [PubMed]

Parent, M. A.

Schneider, H.

T. Maier, H. Schneider, H. C. Liu, M. Walther, and P. Koidl, "Quantum-well infrared photodetector with voltage-switchable quadratic and linear response," Appl. Phys. Lett. 88, 51117 (2006).

Siivola, E.

R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O'Quinn, "Thin-film thermoelectric devices with high room-temperature figures of merit," Nature 413, 597-602 (2001).
[CrossRef] [PubMed]

Smith, P. W. E.

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

Stanton, N. M.

A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, "Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance," Phys. Rev. Lett. 96, 215504 (2006).

Stuart, J.

J. C. Caylor, K. Coonley, J. Stuart, T. Colpitts, and R. Venkatasubramanian, "Enhanced thermoelectric performance in PbTe-based superlattice structures from reduction of lattice thermal conductivity," Appl. Phys. Lett. 87, 23105 (2005).

Suematsu, Y.

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

Tawee, T.

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

Thelen, A.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

Venkatasubramanian, R.

R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O'Quinn, "Thin-film thermoelectric devices with high room-temperature figures of merit," Nature 413, 597-602 (2001).
[CrossRef] [PubMed]

J. C. Caylor, K. Coonley, J. Stuart, T. Colpitts, and R. Venkatasubramanian, "Enhanced thermoelectric performance in PbTe-based superlattice structures from reduction of lattice thermal conductivity," Appl. Phys. Lett. 87, 23105 (2005).

Walther, M.

T. Maier, H. Schneider, H. C. Liu, M. Walther, and P. Koidl, "Quantum-well infrared photodetector with voltage-switchable quadratic and linear response," Appl. Phys. Lett. 88, 51117 (2006).

Warrren, B. E.

B. E. Warrren, X-Ray Diffraction (Addison-Wesley, 1990).

Yan, R. H.

S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum. Electron. 27, 2086-2090 (1991).
[CrossRef]

Yariv, A.

Yeh, P.

Appl. Opt. (2)

IEEE J. Quantum Electron. (3)

D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE J. Quantum Electron. 28, 514-524 (1992).
[CrossRef]

F. Koyama, Y. Suematsu, S. Arai, and T. Tawee, "1.5-1.6 μm GaInAsP/InP dynamic-single-mode (DSM) lasers with distributed Bragg reflector," IEEE J. Quantum Electron. 19, 1042-1051 (1983).
[CrossRef]

D. I. Babic, Y. Chung, N. Dagli, and J. E. Bowers, "Modal reflection of quarter-wave mirrors in vertical-cavity lasers," IEEE J. Quantum Electron. 29, 1950-1962 (1993).
[CrossRef]

IEEE J. Quantum. Electron. (2)

N. Matuschek, F. Kaertner, and U. Keller, "Exact coupled-mode theories for multilayer interference coating with arbitrary strong index modulation," IEEE J. Quantum. Electron. 33, 295-302 (1997).
[CrossRef]

S. W. Corzine, R. H. Yan, and L. A. Coldren, "A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks," IEEE J. Quantum. Electron. 27, 2086-2090 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Microwave Opt. Technol. Lett. (1)

M. A. Matin, T. M. Benson, L. R. Chen, and P. W. E. Smith, "Analysis of distributed Bragg reflectors using thin-film optics," Microwave Opt. Technol. Lett. 21, 11-15 (1999).
[CrossRef]

Nature (1)

R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O'Quinn, "Thin-film thermoelectric devices with high room-temperature figures of merit," Nature 413, 597-602 (2001).
[CrossRef] [PubMed]

Other (6)

A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, and T. L. Linnik, "Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance," Phys. Rev. Lett. 96, 215504 (2006).

B. E. Warrren, X-Ray Diffraction (Addison-Wesley, 1990).

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

R. Maulini, A. Mohan, M. Giovannini, J. Faist, and E. Gini, "External cavity quantum-cascade laser tunable from 8.2 to 10.4 μm using a gain element with a heterogeneous cascade," Appl. Phys. Lett. 88, 201113 (2006).

T. Maier, H. Schneider, H. C. Liu, M. Walther, and P. Koidl, "Quantum-well infrared photodetector with voltage-switchable quadratic and linear response," Appl. Phys. Lett. 88, 51117 (2006).

J. C. Caylor, K. Coonley, J. Stuart, T. Colpitts, and R. Venkatasubramanian, "Enhanced thermoelectric performance in PbTe-based superlattice structures from reduction of lattice thermal conductivity," Appl. Phys. Lett. 87, 23105 (2005).

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Figures (6)

Fig. 1
Fig. 1

Magnitude of the reflection coefficient Γ BIG ( N ) as a function of incident wave vector ( K A Z or K G Z ) for an AlAs∕GaAs periodic layered structure. Here we assume that K A Z L A = K G Z L G , N = 40 , and the impedance Z x and the index n x are related through Z x = η 0 / n x , where η 0 is the free-space impedance, n A = 2.91 (for AlAs), n G = 3.25 (for GaAs), and n INC = 1.0 . Both the upper and the lower bounds on the magnitude of Γ BIG ( N ) are determined by the form factors that are derived in this paper. Equation (53) shows that the upper bound on | Γ BIG ( N ) | is | Γ A V | + | Γ R | , and the lower bound on | Γ BIG ( N ) | is Γ A V | | Γ R .

Fig. 2
Fig. 2

An example of a periodic layered structure. Layers having characteristic impedance Z A and thickness L A are alternated with layers having characteristic impedance Z G and thickness L G on top of a substrate of characteristic impedance Z S . The impedance of N periods of the layered structure (with the layers of impedance Z A on top) is denoted as Z BIG ( N ) . The impedance of N ( 1 / 2 ) periods of the layered structure (with the layers of impedance Z G on top) is denoted as Z SMALL ( N ) . The layer from which the radiation is incident has characteristic impedance Z INC . The reflection coefficient of the entire structure, as seen from the layer of impedance Z INC , is denoted Γ BIG ( N ) .

Fig. 3
Fig. 3

The black-filled circles denote calculations of the reflection coefficient Γ BIG ( N ) in the complex plane as a function of the number of periods N of an AlAs∕GaAs periodic layered structure. Here we assume that N is the parametric variable that varies between 1 and 13 (as labeled), K A Z L A = K G Z L G is fixed at 82°, the impedance Z x and the index n x are related through Z x = η 0 / n x , where η 0 is the free-space impedance, n A = 2.91 (for AlAs), n G = 3.25 (for GaAs), and n INC = 1.0 . Equation (53) shows that the Γ BIG ( N ) are displaced from their average value Γ A V by the complex number Γ R S Γ ( N ) , which is denoted in the figure as a vector (pointing to the lower right) in the complex Γ plane. In the text we show that this complex number, Γ R S Γ ( N ) , has magnitude | Γ R | . Thus the ΓBIG(N) are located on a circle of radius | Γ R | and center at Γ A V . The phase of Γ BIG ( N ) is a function of N through its dependence on the structure factor S Γ ( N ) .

Fig. 4
Fig. 4

The black filled circles denote calculations of the impedance Z BIG ( N ) in the complex plane as a function of the number of periods N of an AlAs∕GaAs periodic layered structure. Here we assume that N is the parametric variable that varies between 1 and 13 (as labeled), K A Z L A = K G Z L G is fixed at 82°, and the impedance Z x and the index n x are related through Z x = η 0 / n x , where η 0 is the free-space impedance, n A = 2.91 (for AlAs), n G = 3.25 (for GaAs). Equation (25) shows that the Z BIG ( N ) are displaced from their average value Z A V by the complex number Z R S Z ( N ) , which is denoted in the figure as a vector (pointing to the lower right) in the complex Z plane. In the text we show that this complex number, Z R S Z ( N ) , has magnitude | Z R | . Thus the Z BIG ( N ) are located on a circle of radius | Z R | and center at Z A V . The phase of Z BIG ( N ) is a function of N through its dependence on the structure factor S Z ( N ) .

Fig. 5
Fig. 5

In the limit where the wavelength approaches the center Bragg wavelength of the mirror, the impedance of the entire structure Z BIG ( N ) approaches the value of Z BIG , MC near K A Z L A = K G Z L G 90 ° . At these wavelengths near the center Bragg wavelength, each period of the layered structure is an integral number of quarter wavelengths. This is indicated schematically here. The evaluation of Z BIG , MC in Eq. (40) at arbitrary wavelengths is denoted in the text as the analytic continuation to arbitrary wavelengths of the expression for the impedance ( Z BIG , MC ) at the center Bragg wavelength of the reflection stop band. This analytic continuation is an important component of the form factors Z A V , Z R , Γ A V , and Γ R .

Fig. 6
Fig. 6

In the limit where the wavelength approaches infinity, Z A V + Z R approaches the value of ( Z CHAR 2 / Z S ) near K A Z L A K G Z L G 0 ° . (In the limit of long wavelengths, Z BIG, MC approaches zero.) This makes sense because Z CHAR is the characteristic impedance of the entire layered structure in the limit of long wavelengths. Thus ( Z CHAR 2 / Z S ) is the impedance when the entire structure is a quarter wavelength, and the radiation is incident from the material having characteristic impedance Z I N C . This is indicated schematically here. Similarly, ( Z CHAR 2 / Z INC ) is the impedance when the entire structure is a quarter wavelength, and the radiation is incident from the material having characteristic impedance Z S . The evaluation of ( Z CHAR 2 / Z S ) using Eq. (41) at arbitrary wavelengths is denoted in the text as the analytic continuation to arbitrary wavelengths of the expression for the impedance at long wavelengths when the entire structure is a quarter wavelength. This analytic continuation is an important component of the form factors Z A V , Z R , Γ A V , and Γ R .

Equations (79)

Equations on this page are rendered with MathJax. Learn more.

M = [ M 11 M 12 M 21 M 22 ] ,
[ E t , BIG ( N ) η 0 H t , BIG ( N ) ] = [ M 11 M 12 M 21 M 22 ] [ E t , BIG ( N 1 ) η 0 H t , BIG ( N 1 ) ] ,
[ E t , BIG ( N ) η 0 H t , BIG ( N ) ] = [ M 11 M 12 M 21 M 22 ] N [ E t , BIG ( 0 ) η 0 H t , BIG ( 0 ) ] .
M N = [ M 11 M 12 M 21 M 22 ] N ,
M N = S N 1 ( τ ) M S N 2 ( τ ) I ,
τ = Trace   M = M 11 + M 22 ,
S N 1 ( τ ) = sin N α sin α ,
α = arccos ( τ 2 ) ,
e i α   and   e i α = eigenvalues   of   M .
Z BIG ( 1 ) Z BIG ( 0 ) = M 11 + M 12 [ η 0 Z BIG ( 0 ) ] M 21 [ Z BIG ( 0 ) η 0 ] + M 22 N Z , 1 P D Z , 1 P ,
N Z , 1 P = M 11 + M 12 [ η 0 Z BIG ( 0 ) ] ,
D Z , 1 P = M 21 [ Z BIG ( 0 ) η 0 ] + M 22 .
Z BIG ( N ) = E t , BIG ( N ) H t , BIG ( N ) .
Z BIG ( N ) Z BIG ( 0 ) = ( M N ) 11 + ( M N ) 12 [ η 0 Z BIG ( 0 ) ] ( M N ) 21 [ Z BIG ( 0 ) η 0 ] + ( M N ) 22 ,
Z BIG ( N ) Z BIG ( 0 ) = N Z , 1 P S N 1 ( τ ) S N 2 ( τ ) D Z , 1 P S N 1 ( τ ) S N 2 ( τ ) .
M G = [ cos K G Z L G ( Z G / η 0 ) ( i sin K G Z L G ) ( η 0 / Z G ) ( i sin K G Z L G ) cos K G Z L G ] ,
M A = [ cos K A Z L A ( Z A / η 0 ) ( i sin K A Z L A ) ( η 0 / Z A ) ( i sin K A Z L A ) cos K A Z L A ] ,
Z A , TE = μ A ω K A Z   and   Z A , TM = K A Z ϵ A ω ,
M = M A M G .
Z BIG ( 1 ) Z S = N Z , 1 P D Z , 1 P .
Z BIG ( N ) Z S = N Z , 1 P sin N α sin ( N 1 ) α D Z , 1 P sin N α sin ( N 1 ) α ,
Z BIG ( N ) Z S = a 1 exp ( i 2 N α ) a 2 b 1 exp ( i 2 N α ) b 2 = a 1 v a 2 b 1 v b 2 ,
a 1 = N Z , 1 P exp ( i α ) ,
a 2 = N Z , 1 P exp ( i α ) ,
b 1 = D Z , 1 P exp ( i α ) ,
b 2 = D Z , 1 P exp ( i α ) ,
v = exp ( i 2 N α ) .
Z BIG ( N ) Z R S Z ( N ) + Z A V ,
S Z ( N ) = { b ¯ 2 v b ¯ 1 b 2 b 1 v } = { b ¯ 2 e i 2 N α b ¯ 1 b 2 b 1 e i 2 N α } ,
Z R Z S = a 1 b 2 + a 2 b 1 | b 2 | 2 | b 1 | 2 ,
Z A V Z S = a 2 b ¯ 2 a 1 b ¯ 1 | b 2 | 2 | b 1 | 2 ,
S Z ( N ) exp [ i S Z ( N ) ] ,
S Z ( N ) = S Z ( N + π α ) .
Z R Z S = N Z , 1 P D Z , 1 P D Z , 1 P D ¯ Z , 1 P ,
Z A V Z S = N Z , 1 P D ¯ Z , 1 P D Z , 1 P D ¯ Z , 1 P .
Z R = Z A V Z S .
N Z , 1 P = [ cos K G Z L G ] [ cos K A Z L A ] + Z A Z S [ cos K G Z L G ] ×  [ i sin K A Z L A ] + Z G Z S [ i sin K G Z L G ] [ cos K A Z L A ] + Z A Z G [ i sin K G Z L G ] [ i sin K A Z L A ] ,
D Z , 1 P = [ cos K G Z L G ] [ cos K A Z L A ] + Z S Z A [ cos K G Z L G ] × [ i sin K A Z L A ] + Z S Z G [ i sin K G Z L G ] [ cos K A Z L A ] + Z G Z A [ i sin K G Z L G ] [ i sin K A Z L A ] ,
cos α = [ cos K G Z L G ] [ cos K A Z L A ] + 1 2 [ Z A Z G + Z G Z A ] × [ i sin K G Z L G ] [ i sin K A Z L A ] ,
S Z ( N ) = [ D ¯ Z , 1 P sin N α sin ( N 1 ) α D Z , 1 P sin N α sin ( N 1 ) α ] ,
Z A V = [ Z S Z S + Z ¯ S ] [ Z BIG , MC + Z CHAR 2 Z S + Z ¯ S ] ,
Z R = [ Z S Z S + Z ¯ S ] [ Z BIG , MC + Z CHAR 2 Z S Z S ] ,
Z BIG , MC = [ Z A 2 Z G 2 ] [ Z G i tan K G Z L G + Z A i tan K A Z L A ] ,
Z CHAR 2 = [ Z A Z G ] 2 [ 1 Z G i tan K G Z L G + 1 Z A i tan K A Z L A ] [ Z G i tan K G Z L G + Z A i tan K A Z L A ] ,
Z A V = 1 2 [ Z BIG , MC + Z CHAR 2 Z S + Z S ] ,
Z R = 1 2 [ Z BIG , MC + Z CHAR 2 Z S Z S ] .
Z A V Z R = Z S ,
Z A V + Z R = [ Z BIG , MC + Z CHAR 2 Z S ] .
K A Z L A = K G Z L G ,
Z BIG , MC = [ Z A Z G ] i tan K A Z L A ,
Z CHAR 2 = [ Z A Z G ] .
Γ BIG ( N ) = Z BIG ( N ) Z INC Z BIG ( N ) + Z INC .
z i , s Z INC Z S ,
c 1 ( a 1 z i , s b 1 ) ,
c 2 ( a 2 z i , s b 2 ) ,
d 1 ( a 1 + z i , s b 1 ) ,
d 2 ( a 2 + z i , s b 2 ) ,
Γ BIG ( N ) = c 1 v c 2 d 1 v d 2 ,
Γ BIG ( N ) Γ R S Γ ( N ) + Γ A V ,
S Γ ( N ) = { d ¯ 2 v d ¯ 1 d 2 d 1 v } = { d ¯ 2 e i 2 N α d ¯ 1 d 2 d 1 e i 2 N α } ,
Γ R = c 1 d 2 + c 2 d 1 | d 2 | 2 | d 1 | 2 ,
Γ A V = c 2 d ¯ 2 c 1 d ¯ 1 | d 2 | 2 | d 1 | 2 .
S Γ ( N ) exp [ i S Γ ( N ) ] ,
S Γ ( N ) = S Γ ( N + π α ) .
Γ R = 2 Z INC ( Z ¯ S / Z S ) ( Z A V Z S ) Z A V Z ¯ S + Z ¯ A V Z S | Z S | 2 + Z A V Z ¯ INC + ( Z ¯ A V + Z ¯ INC ) Z INC ,
Γ A V = Z A V Z ¯ S + Z ¯ A V Z S | Z S | 2 + Z A V Z ¯ INC ( Z ¯ A V + Z ¯ INC ) Z INC Z A V Z ¯ S + Z ¯ A V Z S | Z S | 2 + Z A V Z ¯ INC + ( Z ¯ A V + Z ¯ INC ) Z INC .
1 Γ A V Γ R = Z S Z ¯ S [ Z ¯ A V + Z ¯ INC Z A V Z S ] .
S Γ ( N ) = [ ( N ¯ Z , 1 P + ( Z ¯ INC / Z ¯ S ) D ¯ Z , 1 P ) sin N α ( 1 + Z ¯ INC / Z ¯ S ) sin ( N 1 ) α ( N Z , 1 P + ( Z INC / Z S ) D Z , 1 P ) sin N α ( 1 + Z INC / Z S ) sin ( N 1 ) α ] ,
Γ A V = [ Z CHAR 2 | Z INC | 2 ] [ Z S + Z ¯ S ] + Z BIG , MC [ Z S Z ¯ INC + Z ¯ S Z INC ] + [ Z CHAR 2 + | Z S | 2 ] [ Z ¯ INC Z INC ] [ Z CHAR 2 + | Z INC | 2 ] [ Z S + Z ¯ S ] + Z BIG , MC [ Z S Z ¯ INC Z ¯ S Z INC ] + [ Z CHAR 2 + | Z S | 2 ] [ Z ¯ INC + Z INC ] ,
Γ R = 2 Z INC Z ¯ S [ Z BIG , MC + ( Z CHAR 2 / Z S ) Z S ] [ Z CHAR 2 + | Z INC | 2 ] [ Z S + Z ¯ S ] + Z BIG , MC [ Z S Z ¯ INC Z ¯ S Z INC ] + [ Z CHAR 2 + | Z S | 2 ] [ Z ¯ INC + Z INC ] ,
Γ A V = [ Z INC Z INC + Z S ] [ Z BIG , MC + ( Z CHAR 2 / Z INC ) Z INC ] [ ( Z CHAR 2 / Z S ) + Z INC ] ,
Γ R = [ Z INC Z INC + Z S ] [ Z BIG , MC + ( Z CHAR 2 / Z S ) Z S ] [ ( Z CHAR 2 / Z S ) + Z INC ] .
Γ A V + Γ R = [ ( Z CHAR 2 / Z S ) Z INC ( Z CHAR 2 / Z S ) + Z INC ] , ( limit   of   long   wavelengths ) ,
Γ A V Γ R = [ Z S Z INC Z S + Z INC ] , ( limit   of   long   wavelengths ) ,
( K A Z L A + K G Z L G ) = m π N ,   ( Bragg   law ) ,
α = C α ( K A Z L A + K G Z L G ) ,
C α = [ 1 + ( { Z A Z G + Z G Z A } 2 ) f ( 1 f ) ] 1 / 2 ,
f = K A Z L A ( K A Z L A + K G Z L G ) .
α = C α ( K A Z L A + K G Z L G ) = m π / N , ( corrected   Bragg   law ) ,

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