## Matrix method for the study of wave propagation in one-dimensional general media

Optics Express, Vol. 14, Issue 23, pp. 11385-11391 (2006)

http://dx.doi.org/10.1364/OE.14.011385

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### Abstract

A matrix method which relates the field and its derivative is presented for the study of wave propagation in any type of one-dimensional media. The transfer matrix is obtained from the canonical solutions of Helmholtz equations at normal incidence. The method is applied to different optical systems like a Fabry-Perot cavity formed by uniform fiber Bragg gratings, periodic dielectric structures and different quasi-periodic structures based on Fibonacci and Thue-Morse sequences of layers with constant and variable refractive index.

© 2006 Optical Society of America

## 1. Introduction

## 2. Theory

*ε*(

*z*) confined between the planes

*z*=0 and

*z*=

*L*and surrounded by two homogeneous media of dielectric constants

*ε*

_{1}and

*ε*

_{3}. In order to calculate the electric field in the material as well as the transmission and reflection efficiencies (assuming an harmonic electric field polarization in the

*x̂*direction and an isotropic linear material), we shall apply the procedure developed in reference [3

3. J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A **11**, 2892 (1994). [CrossRef]

3. J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A **11**, 2892 (1994). [CrossRef]

*ε*(

*z*) (which do not admit analytical solutions), by solving a boundary condition problem with boundary conditions for the electric field and its derivative at

*z*=0 only.

*M*for one layer [0,

*L*] is then given by:

*N*periodic media with a unit cell transfer matrix

*M*

_{i}and thickness

*L*

_{i}, then the total transfer matrix will be

*M*

_{T}=

3. J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A **11**, 2892 (1994). [CrossRef]

*M*

_{T}, we can determine the amplitude reflectance

*ρ*and transmittance

*τ*for each wavelength

*λ*as follows:

*M*

_{T}is unimodular, we can evaluate it in terms of Chebyshev polynomials of the second kind [1, 6

6 . E. Macia, “Exploiting quasiperiodic order in design of optical devices,” Phys. Rev. B **63**, 205,421 (2001). [CrossRef]

### 2.1. Matrix of an elementary layer

*ε*, the general solution of the (modified) Helmholtz Eq. (2) is given by:

*c*

_{1}and

*c*

_{2}are the constants of integration. Imposing the boundary conditions given in Eq. (3) for the solution 9 we obtain

*E*

_{c1}and

*E*

_{c2}, and according to Eq. (5), the transfer matrix is given by:

2. J. Lekner and M. Dorf, “Matrix methods for the calculation of reflection amplitudes,” J. Opt. Soc. Am. A **4**, 2092 (1987). [CrossRef]

### 2.2. Matrix of a layer with an exponential profile

2. J. Lekner and M. Dorf, “Matrix methods for the calculation of reflection amplitudes,” J. Opt. Soc. Am. A **4**, 2092 (1987). [CrossRef]

*B*is a constant given by:

*c*

_{1}and

*c*

_{2}are the constants of integration and

*J*

_{0},

*Y*

_{1}are the Bessel functions of order 1 of the first and second kind respectively. Imposing the boundary conditions given in Eq. (3) for the solution 12 we obtain

*E*

_{c1}and

*E*

_{c2}. So according to Eq. (5), the transfer matrix in this case is given by:

*J*

_{p}and

*Y*

_{p}are the Bessel functions of p-order of the first and second kind respectively p=(0,1). Therefore the transfer matrix has been obtained without sectioning into thin layers.

### 2.3. Matrix of a Fiber Bragg Grating (FBG)

*ε*

_{0}is the average relative permittivity of the fiber and

*ε*

_{m}the maximum value of dielectric constant modulation.

*λ*

_{0}the design wavelength of the grating. To simplify the notation we are going to introduce the parameters:

9. L. Carretero, M. Perez-Molina, S. Blaya, R. Madrigal, P. Acebal, and A. Fimia, “Application of the Fixed Point Theorem for the solution of the 1D wave equation: comparison with exact Mathieu solutions,” Opt. Express **13**, 9078 (2005). [CrossRef] [PubMed]

*c*

_{1}and

*c*

_{2}are the constants of integration and mc, ms are the even and odd Mathieu functions respectively. Imposing the boundary conditions given in Eq. (3) for the solution 16 we obtain

*E*

_{c1}and

*E*

_{c2}. So according to Eq. (3), the transfer matrix of the (FBG) is given by:

*β*

_{c0}.

*α*

_{s0}-

*α*

_{c0}.

*β*

_{s0}and:

*q*=(

*s,c*),

*w*=(0,

*L*′)).

_{g}and the previously described method can be used as an exact alternative to those obtained from approximated coupled wave equations in references [7

7. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via fundamental matrix approach,” Applied Optics **26**, 3474–3478 (1987). [CrossRef] [PubMed]

8 . T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol . **15**(8), 1277 (1997). [CrossRef]

## 3. Results and discussion

### 3.1. Periodic and quasi-periodic dielectric structures

_{2}(L) and TiO

_{2}(H) as two elementary layers with dielectric constants

*n*

_{L}=1.45 and

*n*

_{H}=2.30 at 700 nm, respectively. The constants

*ε*

_{1}and

*ε*

_{3}are assumed equal to 1. We are going to study the optical transmission properties for normal incidence through the juxtaposition of periodic and quasi-periodic multi-layers of constant refractive index and constant refractive index with an exponential profile. The transfer matrices of the single layers

**H**will be

**M**

_{y}(

*λ*,

*n*

_{H},

*d*

_{H}) and

*L*

_{1}=

**M**

_{y}(

*λ*,

*n*

_{L},

*d*

_{L}). For the exponential profile we have chosen

*L*

_{2}=

**M**

_{ga}(

*λ*,

*d*

_{L},

*λ*

_{0}/4, so

*n*

_{L}

*d*

_{L}=

*n*

_{H}

*d*

_{H}=

*λ*

_{0}/4. If the central wavelength

*λ*

_{0}is chosen as 500 nm, then

*d*

_{L}=86.2 nm and

*d*

_{H}=54.35 nm. Now let the recurrence relations of periodic sequences (P) be:

5. W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. **72**, 633 (1994). [CrossRef] [PubMed]

6 . E. Macia, “Exploiting quasiperiodic order in design of optical devices,” Phys. Rev. B **63**, 205,421 (2001). [CrossRef]

*P*

_{16}(

*i*),

*F*

_{8}(

*i*) and

*TM*

_{5}(

*i*) (

*i*=1,2) obtained by using the transfer matrix method previously described.

*P*

_{16}(1) shows a band gap centered at 500 nm. The transmission spectra of

*F*

_{8}(1) and

*TM*

_{5}(1) are more complex, but it is important to note that in the band gap zone of

*P*

_{16}(1) maximal values of transmitted intensity appear. Two gaps are also seen for

*F*

_{8}(1) centered at 400 and 650 nm, and

*TM*

_{5}(1) shows four gaps at 350, 450, 600 nm and infrared region. When the second fundamental layer has an exponential profile, the results are different. For example, in Fig. 1(a) it can be seen that the gap is displaced toward the red region, and is narrower for the sequence

*P*

_{16}(2). In Figs. 1(b) and 1(c), it can be seen that the gap regions are quite different for

*TM*

_{5}(1) and

*TM*

_{5}(2), and the same occurs for

*F*

_{8}(1) and

*F*

_{8}(2).

### 3.2. Fabry-Perot fiber cavity

8 . T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol . **15**(8), 1277 (1997). [CrossRef]

10. Y. O. Barmenkov, D. Zalvidea, S. Torres-Peiro, J. Cruz, and M. Andres, “Effective length of short Fabry-Perot cavity formed by uniform fiber Bragg gratings,” Opt. Express **14**, 6394 (2006). [CrossRef] [PubMed]

*d*

_{1}=4-cm spaced by a length

*d*

_{2}=5 cm with different values of Reflectance (R) of the grating (obtained by using Eq. (19) of Ref. [9

9. L. Carretero, M. Perez-Molina, S. Blaya, R. Madrigal, P. Acebal, and A. Fimia, “Application of the Fixed Point Theorem for the solution of the 1D wave equation: comparison with exact Mathieu solutions,” Opt. Express **13**, 9078 (2005). [CrossRef] [PubMed]

**M**

_{FP}is given by:

10. Y. O. Barmenkov, D. Zalvidea, S. Torres-Peiro, J. Cruz, and M. Andres, “Effective length of short Fabry-Perot cavity formed by uniform fiber Bragg gratings,” Opt. Express **14**, 6394 (2006). [CrossRef] [PubMed]

_{MFP2}is given by:

## 4. Conclusions

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | J. Lekner and M. Dorf, “Matrix methods for the calculation of reflection amplitudes,” J. Opt. Soc. Am. A |

3. | J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A |

4. | M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in Optics: Quasiperiodic Media,” Phys. Rev. Lett. |

5. | W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. |

6 . | E. Macia, “Exploiting quasiperiodic order in design of optical devices,” Phys. Rev. B |

7. | M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via fundamental matrix approach,” Applied Optics |

8 . | T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol . |

9. | L. Carretero, M. Perez-Molina, S. Blaya, R. Madrigal, P. Acebal, and A. Fimia, “Application of the Fixed Point Theorem for the solution of the 1D wave equation: comparison with exact Mathieu solutions,” Opt. Express |

10. | Y. O. Barmenkov, D. Zalvidea, S. Torres-Peiro, J. Cruz, and M. Andres, “Effective length of short Fabry-Perot cavity formed by uniform fiber Bragg gratings,” Opt. Express |

**OCIS Codes**

(050.2230) Diffraction and gratings : Fabry-Perot

(230.1480) Optical devices : Bragg reflectors

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 8, 2006

Revised Manuscript: September 29, 2006

Manuscript Accepted: October 8, 2006

Published: November 13, 2006

**Citation**

L. Carretero, M. Perez-Molina, P. Acebal, S. Blaya, and A. Fimia, "Matrix method for the study of wave propagation in one-dimensional general media," Opt. Express **14**, 11385-11391 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11385

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### References

- M. Born and E. Wolf, Principles of Optics (Macmillan, London, 1964).
- J. Lekner and M. Dorf, "Matrix methods for the calculation of reflection amplitudes," J. Opt. Soc. Am. A 4, 2092 (1987). [CrossRef]
- J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892 (1994). [CrossRef]
- M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic Media," Phys. Rev. Lett. 58, 2436 (1987). [CrossRef] [PubMed]
- W. Gellermann, M. Kohmoto, B. Sutherland, and P. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633 (1994). [CrossRef] [PubMed]
- E. Macia, "Exploiting quasiperiodic order in design of optical devices," Phys. Rev. B 63, 205,421 (2001). [CrossRef]
- M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed feedback slab waveguides via fundamental matrix approach," Applied Optics 26, 3474 - 3478 (1987). [CrossRef] [PubMed]
- T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15(8), 1277 (1997). [CrossRef]
- L. Carretero, M. Perez-Molina, S. Blaya, R. Madrigal, P. Acebal, and A. Fimia, "Application of the fixed point theorem for the solution of the 1D wave equation: comparison with exact Mathieu solutions," Opt. Express 13, 9078 (2005). [CrossRef] [PubMed]
- Y. O. Barmenkov, D. Zalvidea, S. Torres-Peiro, J. Cruz, and M. Andres, "Effective length of short Fabry-Perot cavity formed by uniform fiber Bragg gratings," Opt. Express 14, 6394 (2006). [CrossRef] [PubMed]

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