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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 23 — Nov. 13, 2006
  • pp: 11385–11391
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Matrix method for the study of wave propagation in one-dimensional general media

L. Carretero, M. Perez-Molina, P. Acebal, S. Blaya, and A. Fimia  »View Author Affiliations


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

Let us consider a plane monochromatic wave propagating in the z-direction normal-incident upon a one dimensional medium of arbitrary dielectric constant 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 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]

] but considering the linear substitution given by:

ξ=Lz
(1)

This linear transformation is the main difference respect to the method developed in reference [3

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

], and it can be helpful to determine the transfer matrix numerically for profiles 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.

The modified Helmholtz equation is then given by:

2Ex(ξ)ξ2+(2πλ)2ε(ξ)Ex(ξ)=0
(2)

Now we consider the solutions E c1(ξ) and E c2(ξ) of Eq. (2) satisfying that:

Ec1(0)=1,Ec1(0)=0,Ec2(0)=0andEc2(0)=1
(3)

The solution to Eq. (2) will be a linear combiantion of Eqs. (3):

Ec(ξ)=c1Ec1(ξ)+c2Ec2(ξ)
(4)

The transfer matrix M for one layer [0,L] is then given by:

M=(Ec1(L)Ec2(L)Ec1ξ|ξ=LEc2ξ|ξ=L)
(5)

MT=i=1NMNi+1
(6)

In particular, if wave N periodic media with a unit cell transfer matrix Mi and thickness Li , then the total transfer matrix will be MT =MiN .

Following the same procedure of reference [3

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

] and denoting by MT(ij) the matrix coefficients of MT , we can determine the amplitude reflectance ρ and transmittance τ for each wavelength λ as follows:

ρ=2πiε1(λMT(11)2πiε3MT(12))+λ(λMT(21)2πiε3MT(22))2πiε1(λMT(11)2πiε3MT(12))λ(λMT(21)2πiε3MT(22))
(7)
τ=4λπiε12πiε1(λMT(11)2πiε3MT(12))λ(λMT(21)2πiε3MT(22))
(8)

Since MT is unimodular, we can evaluate it in terms of Chebyshev polynomials of the second kind [1

1. M. Born and E. Wolf, Principles of Optics (Macmillan, London, 1964).

, 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

For an elementary layer of dielectric constant e and refractive index n=√ε, the general solution of the (modified) Helmholtz Eq. (2) is given by:

Ec(ξ)=c1cos(2πξnλ)+c2sin(2πξnλ)
(9)

where 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:

My(λ,n,L)=(cos(2πLnλ)λ2πnsin(2πLnλ)2πnλsin(2πLnλ)cos(2πLnλ))
(10)

As it can be observed in Eq. (10) the tranfer matrix has real coefficients, that are the same as those deduced in reference [2

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

ε(ξ)=ε1exp(Bξ)
(11)

where B is a constant given by: B=ln(ε3ε1)L . Introducing the parameters a1=4πε1λB and a3=4πε3λB , the solution to the (modified) Helmholtz Eq. (2) is given by:

Ec(ξ)=c1J0(a1exp(Bξ))+c2Y0(a1exp(Bξ))
(12)

where 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:

Mga(λ,L,ε1,ε1)=(πa12(J1(a1).Y0(a3)J0(a3).Y1(a1))λa34ε3(J0(a3).Y0(a1)J0(a1).Y0(a3))π2ε3a1λ(J1(a3).Y1(a1)J1(a1).Y1(a3))πa32(J1(a3).Y0(a1)J0(a1).Y1(a3)))
(13)

where 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)

For an elementary Fiber Bragg Grating (FBG) the dielectric constant is given by:

ε(ξ)=ε0+εmcos(2πξΛ)
(14)

where ε 0 is the average relative permittivity of the fiber and εm the maximum value of dielectric constant modulation. Λ=λ0(2ε0) is the spatial frequency of the grating and λ 0 the design wavelength of the grating. To simplify the notation we are going to introduce the parameters:

L=πLΛ,ζ1=λ02λ2andζ2=2εmΛ2λ2
(15)

The exact solution of the (modified) Helmholtz Eq. (2) is given by a linear combination of Mathieu functions [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]

]:

Ec(ξ)=c1mc(ζ1,ζ2,πξΛ)+c2ms(ζ1,ζ2,πξΛ)
(16)

where 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:

Mg(λ,ε0,εm,Λ,L)=1p(βc0.αsLαcL.βs0Λπ(αc0.αs0αc0.αsL)πΛ(βcL.βs0+βc0.βsL)βcL.αs0αc0.βsL)
(17)

where p is a constant given by: β c0.α s0-α c0.β s0 and:

αqw=mq(ζ1,ζ2,w)andβqw=[Λπmq(ζ1,ζ2,πξΛ)ξ]ξ=w
(18)

(where the indices q and w could take the values: q=(s,c), w=(0,L′)).

The matrix My of Eq. (10) is a particular case of Eq. (17):

My(λ,n,L)=limεm0Mg(λ,ε0,εm,Λ,L)

This transfer matrix Mg 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

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

In the following numerical study, we choose SiO2 (L) and TiO2 (H) as two elementary layers with dielectric constants nL =1.45 and nH =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 My (λ,nH ,dH ) and L 1=My (λ,nL ,dL ). For the exponential profile we have chosen L 2=Mga (λ,dL ,nL2 ,nH2 ). The optical thickness of the layer H and L are the same and equal to λ 0/4, so nL dL =nH dH =λ 0/4. If the central wavelength λ 0 is chosen as 500 nm, then dL =86.2 nm and dH =54.35 nm. Now let the recurrence relations of periodic sequences (P) be:

Pn(i)=(H.Li)n,i=1,2
(19)

quasi-periodic of Fibonaci (F) (see reference [5

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]

]):

Fn(i)=Fn1(i).Fn2(i)withF0=H,F1=Li,i=1,2
(20)

and Thue-Morse (TM) (see reference [6

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

]):

TMn(i)=TMn1(i).TMn1(i)
TMn(i)=TMn1(i).TMn1(i)
TM0=HandTM0=Li,i=1,2
(21)

Figure 1 shows the transmission spectra of the sequences P 16(i), F 8(i) and TM 5(i) (i=1,2) obtained by using the transfer matrix method previously described.

Fig. 1. Transmission curve vs. λ(nm) for (A) P 16(1) gray lines, P 16(2) black lines,(B) TM 5(1) gray lines, TM 5(2) black lines, (C) F 8(1) gray lines, F 8(2) black color

As it can be observed, the transmission curve for sequence 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

For this example we are going to consider a Fabry-Perot fiber cavity [8

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

],[10

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]

]. Figure 2 shows the transmittance spectra of a Fabry-Perot fiber cavity formed by two equal FBG uniform of length 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]

]).The gratings are centered at 1531 nm and for the fiber index we have chosen 1.444, so the parameter Λ=1886 lines/mm. For this case the transfer matrix of the Fabry-Perot cavity MFP is given by:

MFP(λ,2.085,εm,d1,d2)=Mg(λ,2.085,εm,Λ,d1).My(λ,1.444,d2).Mg(λ,2.085,εm,Λ,d1)
(22)
Fig. 2. Transmittance vs. λ(µm) of a Fabry-Perot fiber cavity formed by two equal uniform 4-cm FBG separated by 5 cm with R=99 % (dotted lines, εm =0.0001), R=95 % (black lines εm =0.000075) and R=66 % (grey lines, εm =0.00004)

As can be seen in Fig. 2 the distance between the transmittance peaks varies with the reflectivity of the gratings, so a lower diffraction efficiency implies lower mode spacing and vice versa. It can also be seen that the mode-spacing increases as the diffraction efficiency increases. These results are the same as those obtained recently in reference [10

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]

] using a different theoretical method based on coupled wave theory.

More complex devices can be simulated, for example, Fig. 3 shows the transmittance spectra of fiber cavity formed by the juxtaposition of two Fabry-Perot fiber cavities identical to those simulated previously separated by 5cm. For this case the transfer matrix of the Fabry-Perot cavity MFP2 is given by:

MFP2(λ,2.085,εm,d1,d2)=MFP(λ,2.085,εm,Λ,d1,d2).My(λ,1.444,d2).MFP(λ,2.085,εm,Λ,d1,d2)
(23)

As it can be seen the response of this device is more complex that the case analyzed in Fig. 2. In this case instead of the main peaks for one Fabry-Perot fiber cavity, now three different narrower peaks can be seen (Fig. 3).

Fig. 3. Transmittance vs. λ(µm ) of a double Fabry-Perot fiber cavity formed by two equal uniform 4-cm FBG separated by 5 cm with R=99 % (εm =0.0001)

4. Conclusions

A matrix method is presented for the study of wave propagation that can be applied to any type of one-dimensional media. The transfer matrix is obtained from the exact canonical solutions of the Helmholtz equation at normal incidence. The method has been applied to very different photonic systems such as 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. We believe that the method is a powerful tool to analyze complex fiber Bragg systems formed by any juxtaposition of continuous or discontinuous dielectric media.

Acknowledgments

This work has received financial support from the Comision Interministerial de Ciencia y Tecnologia (CICYT) of Spain (Project No. MAT2004-04643-C03-03) and the Generalitat Valenciana (Project No. ACOMP06/022).

References and links

1.

M. Born and E. Wolf, Principles of Optics (Macmillan, London, 1964).

2.

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

3.

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

4.

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization in Optics: Quasiperiodic Media,” Phys. Rev. Lett. 58, 2436 (1987). [CrossRef] [PubMed]

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]

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]

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]

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]

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

  1. M. Born and E. Wolf, Principles of Optics (Macmillan, London, 1964).
  2. J. Lekner and M. Dorf, "Matrix methods for the calculation of reflection amplitudes," J. Opt. Soc. Am. A 4, 2092 (1987). [CrossRef]
  3. J. Lekner, "Light in periodically stratified media," J. Opt. Soc. Am. A 11, 2892 (1994). [CrossRef]
  4. M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in Optics: Quasiperiodic Media," Phys. Rev. Lett. 58, 2436 (1987). [CrossRef] [PubMed]
  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]
  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]
  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]
  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]

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