Abstract

The photonic bandgap of periodic multilayers with diffuse interfaces is calculated by considering an empirical model for the interdiffusion profile. The model for the diffuse profile is based on the error function Erf and a characteristic parameter σ. The model is valid for multilayer structures with an arbitrary layer thickness. It is shown that the width of the bandgap varies with the value of σ and the Brillouin zone boundary. Numerical examples are presented. It is suggested that measuring the ratio of the width of the bandgaps at different orders, that is, determining experimentally the ratio of the widths of the different Bragg peaks on a reflectivity curve, can allow estimation of the thickness of the interdiffusion layer.

© 2013 Optical Society of America

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References

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
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  3. S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009).
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    [CrossRef]
  5. V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “Wave spectrum of multilayers with finite thicknesses of interfaces,” Phys. Rev. B 62, 2181–2184 (2000).
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    [CrossRef]
  8. B. Pardo, T. Megademini, and J.-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
    [CrossRef]
  9. G. W. Hill, “On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon,” Acta Math. 8, 1–36 (1886).
    [CrossRef]
  10. J.-M. André and B. Pardo, “Onde électromagnétique sinusoidale dans un milieu périodiquement stratifié,” Opt. Commun. 66, 249–254 (1988).
    [CrossRef]
  11. O. Francescangeli and A. Morini, “On the limits of validity of the two-wave approximation in the dynamical theory of electromagnetic scattering by periodic dielectric media,” J. Phys. I 6, 705–723 (1996).
    [CrossRef]
  12. C. Fermon, F. Ott, and A. Menelle, “Neutron reflectometry,” in X-ray and Neutron Reflectivity, J. Daillant and A. Gibaud, eds., Lecture Notes in Physics No. 770 (Springer, 2009), pp. 183–234.
  13. P. Shewmon, Diffusion in Solids, 2nd ed. (Shewmon, 2010).
  14. H. Maury, P. Jonnard, J.-M. André, J. Gautier, F. Bridou, F. Delmotte, and M.-F. Ravet, “Interface characteristics of Mo/Si and B4C/Mo/Si multilayers using non-destructive X-ray techniques,” Surf. Sci. 601, 2315–2322 (2007).
    [CrossRef]
  15. H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
    [CrossRef]

2009 (1)

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

2007 (1)

H. Maury, P. Jonnard, J.-M. André, J. Gautier, F. Bridou, F. Delmotte, and M.-F. Ravet, “Interface characteristics of Mo/Si and B4C/Mo/Si multilayers using non-destructive X-ray techniques,” Surf. Sci. 601, 2315–2322 (2007).
[CrossRef]

2000 (1)

V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “Wave spectrum of multilayers with finite thicknesses of interfaces,” Phys. Rev. B 62, 2181–2184 (2000).

1999 (1)

V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “The spectrum and damping of waves in partially randomized multilayers,” J. Phys. Condens. Matter 11, 2773–2790 (1999).
[CrossRef]

1996 (1)

O. Francescangeli and A. Morini, “On the limits of validity of the two-wave approximation in the dynamical theory of electromagnetic scattering by periodic dielectric media,” J. Phys. I 6, 705–723 (1996).
[CrossRef]

1993 (1)

1988 (2)

B. Pardo, T. Megademini, and J.-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
[CrossRef]

J.-M. André and B. Pardo, “Onde électromagnétique sinusoidale dans un milieu périodiquement stratifié,” Opt. Commun. 66, 249–254 (1988).
[CrossRef]

1886 (1)

G. W. Hill, “On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon,” Acta Math. 8, 1–36 (1886).
[CrossRef]

André, J.-M.

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

H. Maury, P. Jonnard, J.-M. André, J. Gautier, F. Bridou, F. Delmotte, and M.-F. Ravet, “Interface characteristics of Mo/Si and B4C/Mo/Si multilayers using non-destructive X-ray techniques,” Surf. Sci. 601, 2315–2322 (2007).
[CrossRef]

B. Pardo, T. Megademini, and J.-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
[CrossRef]

J.-M. André and B. Pardo, “Onde électromagnétique sinusoidale dans un milieu périodiquement stratifié,” Opt. Commun. 66, 249–254 (1988).
[CrossRef]

Attwood, D.

D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge University, 2000).

Bovard, B. G.

Bridou, F.

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

H. Maury, P. Jonnard, J.-M. André, J. Gautier, F. Bridou, F. Delmotte, and M.-F. Ravet, “Interface characteristics of Mo/Si and B4C/Mo/Si multilayers using non-destructive X-ray techniques,” Surf. Sci. 601, 2315–2322 (2007).
[CrossRef]

Chuang, S. L.

S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009).

Delmotte, F.

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

H. Maury, P. Jonnard, J.-M. André, J. Gautier, F. Bridou, F. Delmotte, and M.-F. Ravet, “Interface characteristics of Mo/Si and B4C/Mo/Si multilayers using non-destructive X-ray techniques,” Surf. Sci. 601, 2315–2322 (2007).
[CrossRef]

Fermon, C.

C. Fermon, F. Ott, and A. Menelle, “Neutron reflectometry,” in X-ray and Neutron Reflectivity, J. Daillant and A. Gibaud, eds., Lecture Notes in Physics No. 770 (Springer, 2009), pp. 183–234.

Francescangeli, O.

O. Francescangeli and A. Morini, “On the limits of validity of the two-wave approximation in the dynamical theory of electromagnetic scattering by periodic dielectric media,” J. Phys. I 6, 705–723 (1996).
[CrossRef]

Gautier, J.

H. Maury, P. Jonnard, J.-M. André, J. Gautier, F. Bridou, F. Delmotte, and M.-F. Ravet, “Interface characteristics of Mo/Si and B4C/Mo/Si multilayers using non-destructive X-ray techniques,” Surf. Sci. 601, 2315–2322 (2007).
[CrossRef]

Giglia, A.

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

Hill, G. W.

G. W. Hill, “On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon,” Acta Math. 8, 1–36 (1886).
[CrossRef]

Ignatchenko, V. A.

V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “Wave spectrum of multilayers with finite thicknesses of interfaces,” Phys. Rev. B 62, 2181–2184 (2000).

V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “The spectrum and damping of waves in partially randomized multilayers,” J. Phys. Condens. Matter 11, 2773–2790 (1999).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Jonnard, P.

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

H. Maury, P. Jonnard, J.-M. André, J. Gautier, F. Bridou, F. Delmotte, and M.-F. Ravet, “Interface characteristics of Mo/Si and B4C/Mo/Si multilayers using non-destructive X-ray techniques,” Surf. Sci. 601, 2315–2322 (2007).
[CrossRef]

Le Guen, K.

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

Mahne, N.

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

Mankov, Y. I.

V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “Wave spectrum of multilayers with finite thicknesses of interfaces,” Phys. Rev. B 62, 2181–2184 (2000).

V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “The spectrum and damping of waves in partially randomized multilayers,” J. Phys. Condens. Matter 11, 2773–2790 (1999).
[CrossRef]

Maradudin, A. A.

V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “Wave spectrum of multilayers with finite thicknesses of interfaces,” Phys. Rev. B 62, 2181–2184 (2000).

V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “The spectrum and damping of waves in partially randomized multilayers,” J. Phys. Condens. Matter 11, 2773–2790 (1999).
[CrossRef]

Maury, H.

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

H. Maury, P. Jonnard, J.-M. André, J. Gautier, F. Bridou, F. Delmotte, and M.-F. Ravet, “Interface characteristics of Mo/Si and B4C/Mo/Si multilayers using non-destructive X-ray techniques,” Surf. Sci. 601, 2315–2322 (2007).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Megademini, T.

B. Pardo, T. Megademini, and J.-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
[CrossRef]

Menelle, A.

C. Fermon, F. Ott, and A. Menelle, “Neutron reflectometry,” in X-ray and Neutron Reflectivity, J. Daillant and A. Gibaud, eds., Lecture Notes in Physics No. 770 (Springer, 2009), pp. 183–234.

Morini, A.

O. Francescangeli and A. Morini, “On the limits of validity of the two-wave approximation in the dynamical theory of electromagnetic scattering by periodic dielectric media,” J. Phys. I 6, 705–723 (1996).
[CrossRef]

Nannarone, S.

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

Ott, F.

C. Fermon, F. Ott, and A. Menelle, “Neutron reflectometry,” in X-ray and Neutron Reflectivity, J. Daillant and A. Gibaud, eds., Lecture Notes in Physics No. 770 (Springer, 2009), pp. 183–234.

Pardo, B.

B. Pardo, T. Megademini, and J.-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
[CrossRef]

J.-M. André and B. Pardo, “Onde électromagnétique sinusoidale dans un milieu périodiquement stratifié,” Opt. Commun. 66, 249–254 (1988).
[CrossRef]

Ravet, M.-F.

H. Maury, P. Jonnard, J.-M. André, J. Gautier, F. Bridou, F. Delmotte, and M.-F. Ravet, “Interface characteristics of Mo/Si and B4C/Mo/Si multilayers using non-destructive X-ray techniques,” Surf. Sci. 601, 2315–2322 (2007).
[CrossRef]

Shewmon, P.

P. Shewmon, Diffusion in Solids, 2nd ed. (Shewmon, 2010).

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

Acta Math. (1)

G. W. Hill, “On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon,” Acta Math. 8, 1–36 (1886).
[CrossRef]

Appl. Opt. (1)

J. Phys. Condens. Matter (1)

V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “The spectrum and damping of waves in partially randomized multilayers,” J. Phys. Condens. Matter 11, 2773–2790 (1999).
[CrossRef]

J. Phys. I (1)

O. Francescangeli and A. Morini, “On the limits of validity of the two-wave approximation in the dynamical theory of electromagnetic scattering by periodic dielectric media,” J. Phys. I 6, 705–723 (1996).
[CrossRef]

Opt. Commun. (1)

J.-M. André and B. Pardo, “Onde électromagnétique sinusoidale dans un milieu périodiquement stratifié,” Opt. Commun. 66, 249–254 (1988).
[CrossRef]

Phys. Rev. B (1)

V. A. Ignatchenko, Y. I. Mankov, and A. A. Maradudin, “Wave spectrum of multilayers with finite thicknesses of interfaces,” Phys. Rev. B 62, 2181–2184 (2000).

Rev. Phys. Appl. (1)

B. Pardo, T. Megademini, and J.-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
[CrossRef]

Surf. Sci. (2)

H. Maury, P. Jonnard, J.-M. André, J. Gautier, F. Bridou, F. Delmotte, and M.-F. Ravet, “Interface characteristics of Mo/Si and B4C/Mo/Si multilayers using non-destructive X-ray techniques,” Surf. Sci. 601, 2315–2322 (2007).
[CrossRef]

H. Maury, J.-M. André, K. Le Guen, N. Mahne, A. Giglia, S. Nannarone, F. Bridou, F. Delmotte, and P. Jonnard, “Analysis of periodic Mo/Si multilayers: influence of the Mo thickness,” Surf. Sci. 603, 407–411 (2009).
[CrossRef]

Other (6)

C. Fermon, F. Ott, and A. Menelle, “Neutron reflectometry,” in X-ray and Neutron Reflectivity, J. Daillant and A. Gibaud, eds., Lecture Notes in Physics No. 770 (Springer, 2009), pp. 183–234.

P. Shewmon, Diffusion in Solids, 2nd ed. (Shewmon, 2010).

D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge University, 2000).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009).

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Figures (3)

Fig. 1.
Fig. 1.

Scheme of the bilayer system and geometry used. The period of the multilayer D is the sum of the thicknesses of the a and b layers. The γ-ratio is the ratio of the thickness of the a layer to the period.

Fig. 2.
Fig. 2.

In-depth profile of the dielectric function ε(z,σ) for the following parameters: D=1, γ=0.4, δ=0.04 (solid line), 0.1 (dashed line), and 0.3 (dotted line), εa=1.0, and Δε=0.2, computed from the Fourier series by means of Eqs. (15) and (16).

Fig. 3.
Fig. 3.

Ratios of the widths of the bandgap at different Brillouin zone boundaries calculated with D=1 and γ=0.3 for the TE case. The stars give the values obtained from a computation using the transfer-matrix method.

Equations (29)

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k2+k2=k2=ε(z)k02,
d2F(z)dz2+h2(z)F(z)=0,
U(z)=exp(iKz)u(z).
U(z)=exp(iKz)n=unexp(inGz).
ε(z)=p=εpexp(ipGz),
n=n=exp(inGz)[ε0k02((KnG)2+k2)]un=k02(p=+ϵpexp(ipGz))(n=+unexp(inGz)).
[ε0k02((KnG)2+k2)]un=k02(t=+utϵnt).
[ε0k02((KnG)2+k2)]un=k04t=+p=+upϵtpϵnt[ε0k2((KtG)2+k2)].
[ε0k02(K2+k2)]=k04t=+ϵtϵt[ε0k2((KtG)2+k2)].
[ε0k02(K2+k2)][ε0k02((KtG)2+k2)]=k04ϵtϵt.
ε˜(z)=ε(z)+12ε(z)k02d2ε(z)dz234ε(z)2k02(dε(z)dz)2.
ρ˜(z,σ)=12[1+Erf(z2σ)].
ε(z,δ)=εa+ΔεΩ(z,δ)
Ω(z,δ)=limN(n=N+N(ρ(+nD+γD2+z,δ)+ρ(nD+γD2z,δ))(2N+1)).
ε(z)=εa+Δεp=Ωpexp(ipGz),
Ωp(δ)=i4pπe(2p2π2δ2)[eipπγ(Erf((1+γ)/δ4ipπδ222)+Erf((1γ)/δ+4ipπδ222))+eipπγ(Erf((1+γ)/δ+4ipπδ222)Erf((1+γ)/δ+4ipπδ222))].
K=pG2.
[ε0ωc22(p2G24+k2)]2=(Δε)2ω4c4(Ωp(δ))2,
Ωp(δ)=Ωp(δ).
Δωp=Δεε0ωΩp(δ).
ε0ν±β=±αν±,
ν±=(ω±c)2,
α=ΔεΩp(δ),
β=(p2G42+k2).
ε0(ν+ν)=α(ν++ν)
ω+ω=αε0(ω+2+ω2ω++ω).
ω±=ω0±12Δω,
(ω+ω)=Δω0=αε02ω02+12Δω22ω0,
Δω0(Δεε0)ω0Ωp(δ).

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