Abstract

We present a nondestructive experimental method to measure monotonically varying refractive-index profiles of planar waveguides. The technique is a modification of the Lloyd's mirage setup proposed by Allman et al. [Appl. Opt. 33, 1806 (1994)] in order to have a reference phase distribution for the interference pattern. The theoretical calculations have been implemented to account for multiple reflections inside the sample. An application of the method to ion-exchanged glass waveguides is reported.

© 1996 Optical Society of America

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References

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  1. R. V. Ramaswamy, R. Srivastava, “Ion-exchanged glass waveguides: a review,” J. Lightwave Technol. 6, 984–1002 (1988).
    [CrossRef]
  2. T. G. Giallorenzi, E. J. West, R. Kirk, R. Ginther, R. A. Andrews, “Optical waveguides formed by thermal migration of ions in glass,” Appl. Opt. 2, 1240–1245 (1973).
    [CrossRef]
  3. G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, R. M. DeLaRue, “Planar optical waveguides formed by silver-ion migration in glass,” IEEE J. Quantum Electron. QE-13, 192–200 (1977).
    [CrossRef]
  4. P. Chludzinski, R. V. Ramaswamy, T. J. Anderson, “Silver-sodium ion-exchange in soda-lime silicate glass,” Phys. Chem. Glasses 28, 169–173 (1987).
  5. R. Ulrich, “Theory of the prism-film coupler by plane wave analysis,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [CrossRef]
  6. D. Marcuse, “TE modes of graded index slab waveguide,” IEEE J. Quantum Electron. QE-9, 1000–1006 (1970).
  7. P. C. Jaussand, G. N. Chartier, “A quick method for the determination of refractive index profiles for different optical waveguides,” J. Phys. D 10, 645 (1977).
    [CrossRef]
  8. B. X. Chen, H. Hamanaka, K. Iwamura, “Recovery of refractive-index profiles of planar graded-index waveguides from measured mode indices: an iteration method,” J. Opt. Soc. Am. A 9, 1301–1305 (1992).
    [CrossRef]
  9. J. W. White, P. F. Heidrich, “Optical waveguide refractive index profiles determined from measurement of mode indices: a simple analysis,” Appl. Opt. 15, 151–155 (1976).
    [CrossRef] [PubMed]
  10. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), pp. 676–680.
  11. B. E. Allman, A. G. Klein, K. A. Nugent, G. I. Opat, “Refractive-index-profile determinations by using Lloyd's mirage,” Appl. Opt. 33, 1806–1811 (1994).
    [CrossRef] [PubMed]
  12. Ref. 10, pp. 13–20.
  13. K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indices,” J. Lightwave Technol. 3, 385–391 (1985).
    [CrossRef]
  14. E. Acosta, L. Gato, M. V. Perez, C. Gomez-Reino, “Fit method to determine the refractive index profile of planar surface waveguides,” Pure Appl. Opt. 4, 485–493 (1995).
    [CrossRef]

1995

E. Acosta, L. Gato, M. V. Perez, C. Gomez-Reino, “Fit method to determine the refractive index profile of planar surface waveguides,” Pure Appl. Opt. 4, 485–493 (1995).
[CrossRef]

1994

1992

1988

R. V. Ramaswamy, R. Srivastava, “Ion-exchanged glass waveguides: a review,” J. Lightwave Technol. 6, 984–1002 (1988).
[CrossRef]

1987

P. Chludzinski, R. V. Ramaswamy, T. J. Anderson, “Silver-sodium ion-exchange in soda-lime silicate glass,” Phys. Chem. Glasses 28, 169–173 (1987).

1985

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indices,” J. Lightwave Technol. 3, 385–391 (1985).
[CrossRef]

1977

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, R. M. DeLaRue, “Planar optical waveguides formed by silver-ion migration in glass,” IEEE J. Quantum Electron. QE-13, 192–200 (1977).
[CrossRef]

P. C. Jaussand, G. N. Chartier, “A quick method for the determination of refractive index profiles for different optical waveguides,” J. Phys. D 10, 645 (1977).
[CrossRef]

1976

1973

T. G. Giallorenzi, E. J. West, R. Kirk, R. Ginther, R. A. Andrews, “Optical waveguides formed by thermal migration of ions in glass,” Appl. Opt. 2, 1240–1245 (1973).
[CrossRef]

1970

D. Marcuse, “TE modes of graded index slab waveguide,” IEEE J. Quantum Electron. QE-9, 1000–1006 (1970).

R. Ulrich, “Theory of the prism-film coupler by plane wave analysis,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
[CrossRef]

Acosta, E.

E. Acosta, L. Gato, M. V. Perez, C. Gomez-Reino, “Fit method to determine the refractive index profile of planar surface waveguides,” Pure Appl. Opt. 4, 485–493 (1995).
[CrossRef]

Allman, B. E.

Anderson, T. J.

P. Chludzinski, R. V. Ramaswamy, T. J. Anderson, “Silver-sodium ion-exchange in soda-lime silicate glass,” Phys. Chem. Glasses 28, 169–173 (1987).

Andrews, R. A.

T. G. Giallorenzi, E. J. West, R. Kirk, R. Ginther, R. A. Andrews, “Optical waveguides formed by thermal migration of ions in glass,” Appl. Opt. 2, 1240–1245 (1973).
[CrossRef]

Chartier, G. N.

P. C. Jaussand, G. N. Chartier, “A quick method for the determination of refractive index profiles for different optical waveguides,” J. Phys. D 10, 645 (1977).
[CrossRef]

Chen, B. X.

Chiang, K. S.

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indices,” J. Lightwave Technol. 3, 385–391 (1985).
[CrossRef]

Chludzinski, P.

P. Chludzinski, R. V. Ramaswamy, T. J. Anderson, “Silver-sodium ion-exchange in soda-lime silicate glass,” Phys. Chem. Glasses 28, 169–173 (1987).

DeLaRue, R. M.

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, R. M. DeLaRue, “Planar optical waveguides formed by silver-ion migration in glass,” IEEE J. Quantum Electron. QE-13, 192–200 (1977).
[CrossRef]

Gato, L.

E. Acosta, L. Gato, M. V. Perez, C. Gomez-Reino, “Fit method to determine the refractive index profile of planar surface waveguides,” Pure Appl. Opt. 4, 485–493 (1995).
[CrossRef]

Giallorenzi, T. G.

T. G. Giallorenzi, E. J. West, R. Kirk, R. Ginther, R. A. Andrews, “Optical waveguides formed by thermal migration of ions in glass,” Appl. Opt. 2, 1240–1245 (1973).
[CrossRef]

Ginther, R.

T. G. Giallorenzi, E. J. West, R. Kirk, R. Ginther, R. A. Andrews, “Optical waveguides formed by thermal migration of ions in glass,” Appl. Opt. 2, 1240–1245 (1973).
[CrossRef]

Gomez-Reino, C.

E. Acosta, L. Gato, M. V. Perez, C. Gomez-Reino, “Fit method to determine the refractive index profile of planar surface waveguides,” Pure Appl. Opt. 4, 485–493 (1995).
[CrossRef]

Hamanaka, H.

Heidrich, P. F.

Iwamura, K.

Jaussand, P. C.

P. C. Jaussand, G. N. Chartier, “A quick method for the determination of refractive index profiles for different optical waveguides,” J. Phys. D 10, 645 (1977).
[CrossRef]

Kirk, R.

T. G. Giallorenzi, E. J. West, R. Kirk, R. Ginther, R. A. Andrews, “Optical waveguides formed by thermal migration of ions in glass,” Appl. Opt. 2, 1240–1245 (1973).
[CrossRef]

Klein, A. G.

Laybourn, P. J. R.

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, R. M. DeLaRue, “Planar optical waveguides formed by silver-ion migration in glass,” IEEE J. Quantum Electron. QE-13, 192–200 (1977).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), pp. 676–680.

Marcuse, D.

D. Marcuse, “TE modes of graded index slab waveguide,” IEEE J. Quantum Electron. QE-9, 1000–1006 (1970).

Millar, C. A.

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, R. M. DeLaRue, “Planar optical waveguides formed by silver-ion migration in glass,” IEEE J. Quantum Electron. QE-13, 192–200 (1977).
[CrossRef]

Nugent, K. A.

Opat, G. I.

Perez, M. V.

E. Acosta, L. Gato, M. V. Perez, C. Gomez-Reino, “Fit method to determine the refractive index profile of planar surface waveguides,” Pure Appl. Opt. 4, 485–493 (1995).
[CrossRef]

Ramaswamy, R. V.

R. V. Ramaswamy, R. Srivastava, “Ion-exchanged glass waveguides: a review,” J. Lightwave Technol. 6, 984–1002 (1988).
[CrossRef]

P. Chludzinski, R. V. Ramaswamy, T. J. Anderson, “Silver-sodium ion-exchange in soda-lime silicate glass,” Phys. Chem. Glasses 28, 169–173 (1987).

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), pp. 676–680.

Srivastava, R.

R. V. Ramaswamy, R. Srivastava, “Ion-exchanged glass waveguides: a review,” J. Lightwave Technol. 6, 984–1002 (1988).
[CrossRef]

Stewart, G.

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, R. M. DeLaRue, “Planar optical waveguides formed by silver-ion migration in glass,” IEEE J. Quantum Electron. QE-13, 192–200 (1977).
[CrossRef]

Ulrich, R.

West, E. J.

T. G. Giallorenzi, E. J. West, R. Kirk, R. Ginther, R. A. Andrews, “Optical waveguides formed by thermal migration of ions in glass,” Appl. Opt. 2, 1240–1245 (1973).
[CrossRef]

White, J. W.

Wilkinson, C. D. W.

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, R. M. DeLaRue, “Planar optical waveguides formed by silver-ion migration in glass,” IEEE J. Quantum Electron. QE-13, 192–200 (1977).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, R. M. DeLaRue, “Planar optical waveguides formed by silver-ion migration in glass,” IEEE J. Quantum Electron. QE-13, 192–200 (1977).
[CrossRef]

D. Marcuse, “TE modes of graded index slab waveguide,” IEEE J. Quantum Electron. QE-9, 1000–1006 (1970).

J. Lightwave Technol.

K. S. Chiang, “Construction of refractive-index profiles of planar dielectric waveguides from the distribution of effective indices,” J. Lightwave Technol. 3, 385–391 (1985).
[CrossRef]

R. V. Ramaswamy, R. Srivastava, “Ion-exchanged glass waveguides: a review,” J. Lightwave Technol. 6, 984–1002 (1988).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

P. C. Jaussand, G. N. Chartier, “A quick method for the determination of refractive index profiles for different optical waveguides,” J. Phys. D 10, 645 (1977).
[CrossRef]

Phys. Chem. Glasses

P. Chludzinski, R. V. Ramaswamy, T. J. Anderson, “Silver-sodium ion-exchange in soda-lime silicate glass,” Phys. Chem. Glasses 28, 169–173 (1987).

Pure Appl. Opt.

E. Acosta, L. Gato, M. V. Perez, C. Gomez-Reino, “Fit method to determine the refractive index profile of planar surface waveguides,” Pure Appl. Opt. 4, 485–493 (1995).
[CrossRef]

Other

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), pp. 676–680.

Ref. 10, pp. 13–20.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the experimental conditions of the Lloyd's interferometer.

Fig. 2
Fig. 2

Scheme of the refractive-index profile as a function of the depth inside the sample.

Fig. 3
Fig. 3

Experimental setup (see text for details).

Fig. 4
Fig. 4

Experimental interferogram produced by the double Lloyd's interferometer. In the upper part of the interferogram the fringes are constant and homogeneous because they are produced always by total reflection at the coated half of the sample surface. The lower part instead has small phase shifts because of total reflections inside the sample, according to the real RIP.

Fig. 5
Fig. 5

Because of multiple reflections inside the sample, more than two light beams interfere on the CCD camera to produce the fringe pattern. In this picture the intensity distributions of each interfering beam as functions of the position on the screen or, which are the same, as functions of the incidence angle on the screen are reported. The zero on the screen corresponds to the surface position, as shown in Fig. 2.

Fig. 6
Fig. 6

Experimental interference patter (dashed curve) and theoretical fit (solid curve) made with the Fermi profile [Eq. (8)] of the RIP.

Fig. 7
Fig. 7

Experimental interference patter (dashed curve) and theoretical fit (solid curve) made with the Gaussian profile [Eq. (8)] of the RIP.

Fig. 8
Fig. 8

Shapes of the RIP calculated with the two models: Fermi profile (dashed curve) and Gaussian profile (solid curve).

Fig. 9
Fig. 9

Calculated error in determining the RIP as a function of the depth for fitting with the Fermi model.

Equations (10)

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δ = 2 tan 1 [ ( sin 2 θ i n 2 ) 1 / 2 n 2 cos θ i ] ,
n sample ( z + δ z ) < n sample ( z ) ,
L o = 2 0 z tp n 2 ( z ) d z [ n 2 ( z ) n 2 ( z tp ) ] 1 / 2 ,
ϕ p = 2 π L o λ + π 2 ,
x = 2 n ( z tp ) 0 z tp d z [ n 2 ( z ) n 2 ( z tp ) ] 1 / 2 .
n ( z ) 2 = n substr 2 + Δ 2 1 + exp ( z h a ) ,
n ( z ) = n substr + Δ n exp [ ( z / z h ) 2 ] ,
I ( z ) = i = 2 { I i + 2 j = 1 i 1 cos ( ϕ i ϕ j ) [ ( I i I j ) ] 1 / 2 } ,
± Δ Γ ( z ) = { [ n ( χ , z , n 0 , n s ) χ χ sd sd ] 2
+ [ n ( χ , z , n 0 , n s ) n 0 Δ ] 2 + [ n ( χ , z , n 0 , n s ) n s Δ ] 2 } 1 / 2 ,

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