Abstract

In many interferometric applications the variation of the reflected light intensity due to the separation distance change between two optical systems is the raw signal from which some unknown parameters must be determined. We consider the general situation in which the signal offset and amplification, the initial separation, and the optical properties of one of the systems are unknown. Using some major results from the complex analysis we derive closed-form expressions that give the exact solution of the above inverse problem in terms of the signal’s Fourier coefficients. It is shown that the absolute reflectivity can be found unambiguously, while the initial separation and the reflectance phase are mutually correlated and one of these parameters can be found only if the other one is known.

© 2012 Optical Society of America

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References

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  1. J. Rädler and E. Sackmann, “Imaging optical thicknesses and separation distances of phospholipid vesicles at solid surfaces,” J. Phys. II France 3, 727–748 (1993).
    [CrossRef]
  2. O. Theodoly, Z. H. Huang, and M. P. Valignat, “New modeling of reflection interference contrast microscopy including polarization and numerical aperture effects: application to nanometric distance measurements and object profile reconstruction,” Langmuir 26, 1940–1948 (2010).
    [CrossRef]
  3. S. Y. El-Zaiat, “Group refractive index measurement by fringes of equal chromatic order,” Opt. Laser Technol. 37, 181–186 (2005).
    [CrossRef]
  4. R. Tadmor, N. Chen, and J. N. Israelachvili, “Thickness and refractive index measurements using multiple beam interference fringes (FECO),” J. Coll. Inter. Sci. 264, 548–553 (2003).
    [CrossRef]
  5. R. S. Ram, O. Prakash, J. Singh, and S. P. Varma, “Absolute reflectance measurement at normal incidence,” Opt. Laser Technol. 22, 51–55 (1990).
    [CrossRef]
  6. U. Gerhardt and G. W. Rubloff, “A normal incidence scanning reflectometer of high precision,” Appl. Opt. 8, 305–308(1969).
    [CrossRef]
  7. J. E. Shaw and W. R. Blevin, “Instrument for the absolute measurement of direct spectral reflectances at normal incidence,” J. Opt. Soc. Am. 54, 334–336 (1964).
    [CrossRef]
  8. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  9. L. R. Watkins, “Accurate heterodyne interferometric ellipsometer,” Opt. Lasers Eng. 48, 114–118 (2010).
    [CrossRef]
  10. C. C. Tsai, H. C. Wei, S. L. Huang, C. E. Lin, C. J. Yu, and C. Chou, “High speed interferometric ellipsometer,” Opt. Express 16, 7778–7788 (2008).
    [CrossRef]
  11. H. F. Hazebroek and A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
    [CrossRef]
  12. J. M. Schmitt, “Optical coherence tomography (OCT): a review,” J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
    [CrossRef]
  13. S. C. Russev, “Representation of the transparent layer-arbitrary substrate system parameters through the Fourier coefficients of the ellipsometric function,” J. Opt. Soc. Am. A 16, 364–370 (1999).
    [CrossRef]
  14. H. S. Jeffreys and B. Swirles, Methods of Mathematical Physics (Cambridge University, 1956).
  15. R. E. Edwards, Fourier Series: A Modern Introduction (Holt, Rinehart, and Winston, 1967).

2010 (2)

O. Theodoly, Z. H. Huang, and M. P. Valignat, “New modeling of reflection interference contrast microscopy including polarization and numerical aperture effects: application to nanometric distance measurements and object profile reconstruction,” Langmuir 26, 1940–1948 (2010).
[CrossRef]

L. R. Watkins, “Accurate heterodyne interferometric ellipsometer,” Opt. Lasers Eng. 48, 114–118 (2010).
[CrossRef]

2008 (1)

2005 (1)

S. Y. El-Zaiat, “Group refractive index measurement by fringes of equal chromatic order,” Opt. Laser Technol. 37, 181–186 (2005).
[CrossRef]

2003 (1)

R. Tadmor, N. Chen, and J. N. Israelachvili, “Thickness and refractive index measurements using multiple beam interference fringes (FECO),” J. Coll. Inter. Sci. 264, 548–553 (2003).
[CrossRef]

1999 (2)

1993 (1)

J. Rädler and E. Sackmann, “Imaging optical thicknesses and separation distances of phospholipid vesicles at solid surfaces,” J. Phys. II France 3, 727–748 (1993).
[CrossRef]

1990 (1)

R. S. Ram, O. Prakash, J. Singh, and S. P. Varma, “Absolute reflectance measurement at normal incidence,” Opt. Laser Technol. 22, 51–55 (1990).
[CrossRef]

1973 (1)

H. F. Hazebroek and A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

1969 (1)

1964 (1)

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Blevin, W. R.

Chen, N.

R. Tadmor, N. Chen, and J. N. Israelachvili, “Thickness and refractive index measurements using multiple beam interference fringes (FECO),” J. Coll. Inter. Sci. 264, 548–553 (2003).
[CrossRef]

Chou, C.

Edwards, R. E.

R. E. Edwards, Fourier Series: A Modern Introduction (Holt, Rinehart, and Winston, 1967).

El-Zaiat, S. Y.

S. Y. El-Zaiat, “Group refractive index measurement by fringes of equal chromatic order,” Opt. Laser Technol. 37, 181–186 (2005).
[CrossRef]

Gerhardt, U.

Hazebroek, H. F.

H. F. Hazebroek and A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

Holscher, A. A.

H. F. Hazebroek and A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

Huang, S. L.

Huang, Z. H.

O. Theodoly, Z. H. Huang, and M. P. Valignat, “New modeling of reflection interference contrast microscopy including polarization and numerical aperture effects: application to nanometric distance measurements and object profile reconstruction,” Langmuir 26, 1940–1948 (2010).
[CrossRef]

Israelachvili, J. N.

R. Tadmor, N. Chen, and J. N. Israelachvili, “Thickness and refractive index measurements using multiple beam interference fringes (FECO),” J. Coll. Inter. Sci. 264, 548–553 (2003).
[CrossRef]

Jeffreys, H. S.

H. S. Jeffreys and B. Swirles, Methods of Mathematical Physics (Cambridge University, 1956).

Lin, C. E.

Prakash, O.

R. S. Ram, O. Prakash, J. Singh, and S. P. Varma, “Absolute reflectance measurement at normal incidence,” Opt. Laser Technol. 22, 51–55 (1990).
[CrossRef]

Rädler, J.

J. Rädler and E. Sackmann, “Imaging optical thicknesses and separation distances of phospholipid vesicles at solid surfaces,” J. Phys. II France 3, 727–748 (1993).
[CrossRef]

Ram, R. S.

R. S. Ram, O. Prakash, J. Singh, and S. P. Varma, “Absolute reflectance measurement at normal incidence,” Opt. Laser Technol. 22, 51–55 (1990).
[CrossRef]

Rubloff, G. W.

Russev, S. C.

Sackmann, E.

J. Rädler and E. Sackmann, “Imaging optical thicknesses and separation distances of phospholipid vesicles at solid surfaces,” J. Phys. II France 3, 727–748 (1993).
[CrossRef]

Schmitt, J. M.

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

Shaw, J. E.

Singh, J.

R. S. Ram, O. Prakash, J. Singh, and S. P. Varma, “Absolute reflectance measurement at normal incidence,” Opt. Laser Technol. 22, 51–55 (1990).
[CrossRef]

Swirles, B.

H. S. Jeffreys and B. Swirles, Methods of Mathematical Physics (Cambridge University, 1956).

Tadmor, R.

R. Tadmor, N. Chen, and J. N. Israelachvili, “Thickness and refractive index measurements using multiple beam interference fringes (FECO),” J. Coll. Inter. Sci. 264, 548–553 (2003).
[CrossRef]

Theodoly, O.

O. Theodoly, Z. H. Huang, and M. P. Valignat, “New modeling of reflection interference contrast microscopy including polarization and numerical aperture effects: application to nanometric distance measurements and object profile reconstruction,” Langmuir 26, 1940–1948 (2010).
[CrossRef]

Tsai, C. C.

Valignat, M. P.

O. Theodoly, Z. H. Huang, and M. P. Valignat, “New modeling of reflection interference contrast microscopy including polarization and numerical aperture effects: application to nanometric distance measurements and object profile reconstruction,” Langmuir 26, 1940–1948 (2010).
[CrossRef]

Varma, S. P.

R. S. Ram, O. Prakash, J. Singh, and S. P. Varma, “Absolute reflectance measurement at normal incidence,” Opt. Laser Technol. 22, 51–55 (1990).
[CrossRef]

Watkins, L. R.

L. R. Watkins, “Accurate heterodyne interferometric ellipsometer,” Opt. Lasers Eng. 48, 114–118 (2010).
[CrossRef]

Wei, H. C.

Yu, C. J.

Appl. Opt. (1)

J. Coll. Inter. Sci. (1)

R. Tadmor, N. Chen, and J. N. Israelachvili, “Thickness and refractive index measurements using multiple beam interference fringes (FECO),” J. Coll. Inter. Sci. 264, 548–553 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. E (1)

H. F. Hazebroek and A. A. Holscher, “Interferometric ellipsometry,” J. Phys. E 6, 822–826 (1973).
[CrossRef]

J. Phys. II France (1)

J. Rädler and E. Sackmann, “Imaging optical thicknesses and separation distances of phospholipid vesicles at solid surfaces,” J. Phys. II France 3, 727–748 (1993).
[CrossRef]

J. Sel. Top. Quantum Electron. (1)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

Langmuir (1)

O. Theodoly, Z. H. Huang, and M. P. Valignat, “New modeling of reflection interference contrast microscopy including polarization and numerical aperture effects: application to nanometric distance measurements and object profile reconstruction,” Langmuir 26, 1940–1948 (2010).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (2)

S. Y. El-Zaiat, “Group refractive index measurement by fringes of equal chromatic order,” Opt. Laser Technol. 37, 181–186 (2005).
[CrossRef]

R. S. Ram, O. Prakash, J. Singh, and S. P. Varma, “Absolute reflectance measurement at normal incidence,” Opt. Laser Technol. 22, 51–55 (1990).
[CrossRef]

Opt. Lasers Eng. (1)

L. R. Watkins, “Accurate heterodyne interferometric ellipsometer,” Opt. Lasers Eng. 48, 114–118 (2010).
[CrossRef]

Other (3)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

H. S. Jeffreys and B. Swirles, Methods of Mathematical Physics (Cambridge University, 1956).

R. E. Edwards, Fourier Series: A Modern Introduction (Holt, Rinehart, and Winston, 1967).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic of the system under study. Sx is an arbitrary optical system that is fully described by its generalized complex Fresnel coefficients Rxp(s) and t is the separation distance. I(t) is the intensity of the reflected light.

Fig. 2.
Fig. 2.

Several possible experimental configurations for measuring the intensity variation I(t) as a function of the separation distance t to the system Sx: Newton rings from the convex lens/Sx (a), semi-cylindrical lens/Sx (b), or bifurcated optical fiber’s end face/Sx (c) configurations.

Fig. 3.
Fig. 3.

Simulated “measured” points with 0.5% STD added noise for three different systems (symbols) and back-computed theoretical curves, based on the calculated parameters |Rx|, δ0, a, and b. For details see text.

Equations (18)

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R(p,s)=r(p,s)+Rx(p,s)X1+r(p,s)Rx(p,s)X,
X=exp(2πit/Dϕ),
Dϕ=λ/2n12n02sin2φ0.
Ith=|R|2I0.
I(t)=Istr+k|R(t)|2I0=a+b|R(t)|2,
R=r+|Rx|Z1+r|Rx|Z,
Z=X0X=expi(δx2πt0/Dϕ)exp(2πit/Dϕ)=exp(iδ0)exp(2πit/Dϕ).
|R|2=(r+|Rx|Z)(r+|Rx|Z*)(1+r|Rx|Z)(1+r|Rx|Z*)=(r+|Rx|Z)(rZ+|Rx|)(1+r|Rx|Z)(Z+r|Rx|),
I(Z)=a+b(r+|Rx|Z)(rZ+|Rx|)(1+r|Rx|Z)(Z+r|Rx|).
LI(Z)dZ=aLdZ+bL(r+|Rx|Z)(rZ+|Rx|)(1+r|Rx|Z)(Z+r|Rx|)dZ,
LI(Z)dZ=2πibr(1r2)1|Rx|21r2|Rx|2|Rx|.
LI(Z)ZdZ=2πibr2(1r2)1|Rx|21r2|Rx|2|Rx|2.
|Rx|=1rLI(Z)ZdZLI(Z)dZ=1rX0LI(X)XdXLI(X)dX.
δ0=arg(X0)=arg[rLI(X)dXLI(X)XdX].
|Rx|2=1r2|LI(Z)ZdZLI(Z)dZ|2=1r2|LI(X)XdXLI(X)dX|2.
δ0=arctans1c2+s2c1c1c2s1c2+(1sign(r))π/2,
|Rx|2=1r2c22+s22c12+s12,
cj=1/Dϕ0DϕI(t)cos(2πtj/Dϕ)dt,sj=1/Dϕ0DϕI(t)sin(2πtj/Dϕ)dt.

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