Abstract

It may be shown that, even when a Fabry–Pérot interferometer is used with plane waves propagating at normal incidence, the variations of the intensity reflected by it with respect to the phase difference (induced by the distance between the two mirrors) are generally not symmetrical around its extrema. We study this problem and express the necessary and general conditions for obtaining a symmetrical optical response in the reflection mode. We analyze the simple case of a Fabry–Pérot interferometer the first mirror of which is constituted by a thin layer of metal.

© 2000 Optical Society of America

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References

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  1. C. F. Bruce, W. K. Clothier, “Optical properties of thin chromium films,” J. Opt. Soc. Am. 64, 823–829 (1974).
    [CrossRef]
  2. J. J. Monzon, L. L. Sanchez-Soto, E. Bernadeu, “Influence of coating thickness on the performance of a Fabry–Pérot interferometer,” Appl. Opt. 30, 4126–4132 (1991).
    [CrossRef]
  3. J. J. Monzon, L. L. Sanchez-Soto, “On the concept of absorption for a Fabry–Pérot interferometer,” Am. J. Phys. 64, 156–163 (1996).
    [CrossRef]
  4. L. M. Brekovskikh, Waves in Layered Media (Academic, New York, 1960).
  5. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1997).
  6. J. Lekner, Theory of Reflection (Martinus Nijhoff, Dordrecht, The Netherlands, 1987).
  7. F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1963), Vol. II, pp. 249–288. “Recherches sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. Paris 5; part I, 596–640; part II, 706–782 (1950).
  8. J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structure,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
    [CrossRef]
  9. J. M. Vigoureux, D. Van Labeke, “A geometrical phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
    [CrossRef]
  10. Ph. Grossel, J. M. Vigoureux, F. Baı̈da, “Nonlocal approach to scattering in a one-dimensional problem,” Phys. Rev. A 50, 3627–3637 (1994).
    [CrossRef] [PubMed]
  11. J. M. Vigoureux, “The reflection of light by planar stratified media: the groupoı̈d of amplitude and phase ‘Thomas precession,’” J. Phys. A Math. Gen. 26, 385–393 (1993).
    [CrossRef]
  12. J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
    [CrossRef]

1998 (1)

J. M. Vigoureux, D. Van Labeke, “A geometrical phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

1996 (1)

J. J. Monzon, L. L. Sanchez-Soto, “On the concept of absorption for a Fabry–Pérot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

1994 (1)

Ph. Grossel, J. M. Vigoureux, F. Baı̈da, “Nonlocal approach to scattering in a one-dimensional problem,” Phys. Rev. A 50, 3627–3637 (1994).
[CrossRef] [PubMed]

1993 (1)

J. M. Vigoureux, “The reflection of light by planar stratified media: the groupoı̈d of amplitude and phase ‘Thomas precession,’” J. Phys. A Math. Gen. 26, 385–393 (1993).
[CrossRef]

1992 (1)

1991 (2)

1974 (1)

Abelès, F.

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1963), Vol. II, pp. 249–288. “Recherches sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. Paris 5; part I, 596–640; part II, 706–782 (1950).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1997).

Bai¨da, F.

Ph. Grossel, J. M. Vigoureux, F. Baı̈da, “Nonlocal approach to scattering in a one-dimensional problem,” Phys. Rev. A 50, 3627–3637 (1994).
[CrossRef] [PubMed]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1997).

Bernadeu, E.

Brekovskikh, L. M.

L. M. Brekovskikh, Waves in Layered Media (Academic, New York, 1960).

Bruce, C. F.

Clothier, W. K.

Grossel, Ph.

Ph. Grossel, J. M. Vigoureux, F. Baı̈da, “Nonlocal approach to scattering in a one-dimensional problem,” Phys. Rev. A 50, 3627–3637 (1994).
[CrossRef] [PubMed]

Lekner, J.

J. Lekner, Theory of Reflection (Martinus Nijhoff, Dordrecht, The Netherlands, 1987).

Monzon, J. J.

J. J. Monzon, L. L. Sanchez-Soto, “On the concept of absorption for a Fabry–Pérot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

J. J. Monzon, L. L. Sanchez-Soto, E. Bernadeu, “Influence of coating thickness on the performance of a Fabry–Pérot interferometer,” Appl. Opt. 30, 4126–4132 (1991).
[CrossRef]

Sanchez-Soto, L. L.

J. J. Monzon, L. L. Sanchez-Soto, “On the concept of absorption for a Fabry–Pérot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

J. J. Monzon, L. L. Sanchez-Soto, E. Bernadeu, “Influence of coating thickness on the performance of a Fabry–Pérot interferometer,” Appl. Opt. 30, 4126–4132 (1991).
[CrossRef]

Van Labeke, D.

J. M. Vigoureux, D. Van Labeke, “A geometrical phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

Vigoureux, J. M.

J. M. Vigoureux, D. Van Labeke, “A geometrical phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

Ph. Grossel, J. M. Vigoureux, F. Baı̈da, “Nonlocal approach to scattering in a one-dimensional problem,” Phys. Rev. A 50, 3627–3637 (1994).
[CrossRef] [PubMed]

J. M. Vigoureux, “The reflection of light by planar stratified media: the groupoı̈d of amplitude and phase ‘Thomas precession,’” J. Phys. A Math. Gen. 26, 385–393 (1993).
[CrossRef]

J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structure,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
[CrossRef]

J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structures,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
[CrossRef]

Am. J. Phys. (1)

J. J. Monzon, L. L. Sanchez-Soto, “On the concept of absorption for a Fabry–Pérot interferometer,” Am. J. Phys. 64, 156–163 (1996).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

J. M. Vigoureux, D. Van Labeke, “A geometrical phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. A Math. Gen. (1)

J. M. Vigoureux, “The reflection of light by planar stratified media: the groupoı̈d of amplitude and phase ‘Thomas precession,’” J. Phys. A Math. Gen. 26, 385–393 (1993).
[CrossRef]

Phys. Rev. A (1)

Ph. Grossel, J. M. Vigoureux, F. Baı̈da, “Nonlocal approach to scattering in a one-dimensional problem,” Phys. Rev. A 50, 3627–3637 (1994).
[CrossRef] [PubMed]

Other (4)

L. M. Brekovskikh, Waves in Layered Media (Academic, New York, 1960).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1997).

J. Lekner, Theory of Reflection (Martinus Nijhoff, Dordrecht, The Netherlands, 1987).

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1963), Vol. II, pp. 249–288. “Recherches sur la propagation des ondes électromagnétiques dans les milieux stratifiés,” Ann. Phys. Paris 5; part I, 596–640; part II, 706–782 (1950).

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Figures (5)

Fig. 1
Fig. 1

Study of the reflectivity of the Fabry–Pérot |R(β)|2 with respect to the distance between the two mirrors. We denote d as the thickness of the first metallic mirror. (a) nI=8.4,λ=1.4 µm,d=3 nm. Dashed curve, nR=0.05; dashed–dotted curve, nR=0.1; dotted curve, nR=0.3; solid curve, nR=0.4 (in this range of values of nR, the curves remain asymmetrical). (b) nR=0.4,λ=1.4 µm,d=3 nm. Dashed curve, nI=2; dashed–dotted curve, nI=4; dotted curve, nI=6; solid curve, nI=8.4 (the smaller the value of nI, the more symmetrical the reflectivity). (c) n=0.4+i8.4,λ=1.4 µm. Dashed curve, d=1 nm; dashed–dotted curve, d=2.5 nm; dotted curve, d=4 nm; solid curve, d=6 nm (the smaller the thickness, the more symmetrical the reflectivity).

Fig. 2
Fig. 2

Definition of our notation in the case of a Fabry–Pérot interferometer. The instrument consists essentially of two stacks made of glass or quartz plates, the inner surfaces of which are coated with partially transparent films of high reflectivity.

Fig. 3
Fig. 3

Conditions for obtaining a symmetrical reflectivity. The reflectivity is symmetrical only when the imaginary part of μ is zero, that is, when the curves cut the x-axis. (a) λ=1.4 µm,d=3 nm. Dashed curve, nI=2, dashed–dotted curve, nI=4; dotted curve, nI=6; solid curve, nI=8.4 (the intersection between the curves and the axis decreases with nI; it tends toward +1 when nI tends toward 0). (b) λ=1.4 µm,d=3 nm. Dashed curve, nR=0.05; dashed–dotted curve, nR=0.1; dotted curve, nR=0.3; solid curve, nR=0.4. When nR<1, the curves never cut the axis. (c) λ=1.4 µm,d=3nm. Dashed curve, nR=0.5; dashed–dotted curve, nR=0.9; dotted curve, nR=1; solid curve, nR=1.5. When nR>1, the curves cut the axis.

Fig. 4
Fig. 4

Representation of the index values corresponding to a symmetrical reflectance curve when λ=1.4 µm,d=6 nm. Although they are experimentally impossible, we use here very high values of nI and of nR to show the influence of the phase of light inside the first layer.

Fig. 5
Fig. 5

Variations of the asymmetry parameter and of the contrast in the (nR, nI) space when λ=1.4 µm,d=3 nm. (a) Representation of the asymmetry parameter Iasym. Some contour lines are represented: Iasym=0, totally symmetrical response; Iasym=1, totally asymmetrical response. (b) Representation of the contrast C. Some contour lines are represented: C=0, no contrast; C=1, maximum contrast.

Equations (45)

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R1,N+1T1,N+1=R1,2T1,2R2,3T2,3RN-1,NTN-1,NRN,N+1TN,N+1,
Rx,x+1Tx,x+1Ry,y+1Ty,y+1=Rx,x+1+Ry,y+11+R¯x,x+1Ry,y+1Tx,x+1Ty,y+11+R¯x,x+1Ry,y+1,
R1,N+1T1,N+1=R1,M+1T1,M+1RM+1,N+1Ө1,M+1TM+1,N+1.
R=|R1,N+1|2=R1,M+1+RM+1,N+1Ө1,M+11+R¯1,M+1RM+1,N+1Ө1,M+12.
R(β)=R1+R2Ө exp(-2iβ)1+R¯1R2Ө exp(-2iβ)2=|R1|2+|R2Ө|2+2|R1R2Ө|cos(2β+φ1-φ2)1+|R¯1R2Ө|2+2|R¯1R2Ө|cos(2β-φ¯1-φ2).
R(x)=A+B cos(x+x1)C+D cos(x+x2),
BD sin(x2-x1)=BC sin(x0+x1)-AD sin(x0+x2).
R(x)=R(2x0-x).
BD sin(x2-x1)=0;
B=0,and/orD=0,and/orx2=x1+qπ,
|R1R2Ө|=0,and/or|R¯1R2Ө|=0,and/orφ2-φ1=φ2+φ¯1+qπ.
R1*=μR¯1,
T(β)=|T1,N+1|2=|T1,M+1|2|TM+1,N+1|2|1+R¯1R2Ө exp(-2iβ)|2.
T(x)=EC+D cos(x+x2),
ED sin(x0+x2)=0.
T(x)=T(2x0-x),
-2ED sin(x-x0)sin 2(x0+x2)=0.
r1+r2ρ=0,
ρ+r1r2=0,
r2+r1ρ=0.
C=|Rmax|2-|Rmin|2|Rmax|2+|Rmin|2
[M]=[R1,2][R2,3][RN,N+1][β1+β2++βN],
[Rj,j+1]=1tj,j+11R¯j,j+1Rj,j+11.
[D1,N+1]=D11D12D21D22=1j=1Ntj,j+1S¯2m1,N+1S¯2m+11,N+1S2m+11,N+1S2m1,N+1,
S2m1,N+1=m0S2m1,N+1,
S2m+11,N+1=m0S2m+11,N+1,
S¯2m1,N+1=m0S¯2m1,N+1,
S¯2m+11,N+1=m0S¯2m+11,N+1.
S01,N+1=1,
S11,N+1=j=1NRj,j+1=R1,2+R2,3++RN,N+1,
S21,N+1=1j<kNNRj,j+1R¯k,k+1=R1,2R¯2,3+R1,2R¯3,4++R1,2R¯N,N+1,+R2,3R¯3,4+R2,3R¯4,5++R2,3R¯N,N+1,++RN-1,NR¯N,N+1,
S31,N+1=1j<k<lNNRj,j+1R¯k,k+1Rl,l+1=R1,2R¯2,3R3,4+R1,2R¯2,3R4,5++RN-2,N-1R¯N-1,NRN,N+1,
R1,N+1=E1-E1+=M21M11=D21D11=S2m+11,N+1S¯2m1,N+1,
T1,N+1=EN+1+E1+=1M11=exp[i(β1+β2++βN)]D11
=j=1Ntj,j+1 exp(iβj)S¯2m1,N+1.
[M]=[D1,M+1][DM+1,N+1][β1+β2++βN],
[D1,M+1]=[R1,2][R2,3]  [RM,M+1],
[DM+1,N+1]=[RM+1,M+2][RM+2,M+3]  [RN,N+1].
S2m1,N+1=S2m1,M+1S2mM+1,N+1+S2m+11,M+1S¯2m+1M+1,N+1
S2m+11,N+1=S2m1,M+1S2m+1M+1,N+1+S2m+11,M+1S¯2mM+1,N+1.
R1,N+1T1,N+1=R1,M+1+RM+1,N+1Ө1,M+11+R¯1,M+1RM+1,N+1Ө1,M+1T1,M+1TM+1,N+11+R¯1,M+1RM+1,N+1Ө1,M+1
=R1,M+1T1,M+1RM+1,N+1Ө1,M+1TM+1,N+1,
Ө1,M+1=S2m1,M+1S¯2m1,M+1.
Ө1,3=1+R1,2R¯2,31+R¯1,2R2,3,
Ө1,4=1+R1,2R¯2,3+R1,2R¯3,4+R2,3R¯3,41+R¯1,2R2,3+R¯1,2R3,4+R¯2,3R3,4.

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