Abstract

A simple graphical method is developed for calculating modulation and groove-to-period ratio of a shallow lamellar grating from its diffraction spectra and for simultaneously checking its assumed shape. It applies to both reflection nd transmission and to the conducting or dielectric nature of both the grating and its substrate. The effect of noise and distortion is briefly discussed. The method is illustrated for the limiting case of a reflecting metallic grating. Uses will be presented in subsequent papers.

© 1984 Optical Society of America

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Corrections

Geraldo F. Mendes, Lucila Cescato, and Jaime Frejlich, "Gratings for metrology and process control. 1: a simple parameter optimization problem; errata," Appl. Opt. 23, 3517-3517 (1984)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-23-20-3517

References

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  1. H. P. Kleinknecht, H. Meier, J. Electrochem. Soc. 125, 798(1978).
    [CrossRef]
  2. E. S. Braga, G. F. Mendes, J. Frejlich, A. P. Mammana, Thin Solid Films 109, 4, 363 (1983).
    [CrossRef]
  3. J. Frejlich, J. J. Clair, J. Opt. Soc. Am. 67, 1644 (1977), Appendix A.
    [CrossRef]
  4. H. P. Kleinknecht, H. Meier, Appl. Opt. 19, 525 (1980).
    [CrossRef] [PubMed]
  5. H. P. Baltes, Ed. Inverse Source Problems in Optics (Springer, Berlin, 1978), p. 42.
  6. R. Petit, Nouv. Rev. Opt. 6, 129 (1975).
    [CrossRef]
  7. M. G. Moharam, T. K. Gaylord, J. Opt. Soc. Am. 72, 1385 (1982).
    [CrossRef]
  8. A. Roger, D. Maystre, J. Opt. Soc. Am. 70, 1483 (1980).
    [CrossRef]
  9. A. M. J. Huiser, A. Quattropani, H. P. Baltes, Opt. Commun. 41, 149 (1982).
    [CrossRef]
  10. E. G. Loewen, M. Neviere, D. Maystre, Appl. Opt. 15, 2937 (1976).
    [CrossRef] [PubMed]
  11. Reference 8 claims from experience that scalar theory may be accurately considered for λ/d ratios < 0.2.
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), 630.

1983 (1)

E. S. Braga, G. F. Mendes, J. Frejlich, A. P. Mammana, Thin Solid Films 109, 4, 363 (1983).
[CrossRef]

1982 (2)

A. M. J. Huiser, A. Quattropani, H. P. Baltes, Opt. Commun. 41, 149 (1982).
[CrossRef]

M. G. Moharam, T. K. Gaylord, J. Opt. Soc. Am. 72, 1385 (1982).
[CrossRef]

1980 (2)

1978 (1)

H. P. Kleinknecht, H. Meier, J. Electrochem. Soc. 125, 798(1978).
[CrossRef]

1977 (1)

1976 (1)

1975 (1)

R. Petit, Nouv. Rev. Opt. 6, 129 (1975).
[CrossRef]

Baltes, H. P.

A. M. J. Huiser, A. Quattropani, H. P. Baltes, Opt. Commun. 41, 149 (1982).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), 630.

Braga, E. S.

E. S. Braga, G. F. Mendes, J. Frejlich, A. P. Mammana, Thin Solid Films 109, 4, 363 (1983).
[CrossRef]

Clair, J. J.

Frejlich, J.

E. S. Braga, G. F. Mendes, J. Frejlich, A. P. Mammana, Thin Solid Films 109, 4, 363 (1983).
[CrossRef]

J. Frejlich, J. J. Clair, J. Opt. Soc. Am. 67, 1644 (1977), Appendix A.
[CrossRef]

Gaylord, T. K.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Huiser, A. M. J.

A. M. J. Huiser, A. Quattropani, H. P. Baltes, Opt. Commun. 41, 149 (1982).
[CrossRef]

Kleinknecht, H. P.

H. P. Kleinknecht, H. Meier, Appl. Opt. 19, 525 (1980).
[CrossRef] [PubMed]

H. P. Kleinknecht, H. Meier, J. Electrochem. Soc. 125, 798(1978).
[CrossRef]

Loewen, E. G.

Mammana, A. P.

E. S. Braga, G. F. Mendes, J. Frejlich, A. P. Mammana, Thin Solid Films 109, 4, 363 (1983).
[CrossRef]

Maystre, D.

Meier, H.

H. P. Kleinknecht, H. Meier, Appl. Opt. 19, 525 (1980).
[CrossRef] [PubMed]

H. P. Kleinknecht, H. Meier, J. Electrochem. Soc. 125, 798(1978).
[CrossRef]

Mendes, G. F.

E. S. Braga, G. F. Mendes, J. Frejlich, A. P. Mammana, Thin Solid Films 109, 4, 363 (1983).
[CrossRef]

Moharam, M. G.

Neviere, M.

Petit, R.

R. Petit, Nouv. Rev. Opt. 6, 129 (1975).
[CrossRef]

Quattropani, A.

A. M. J. Huiser, A. Quattropani, H. P. Baltes, Opt. Commun. 41, 149 (1982).
[CrossRef]

Roger, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), 630.

Appl. Opt. (2)

J. Electrochem. Soc. (1)

H. P. Kleinknecht, H. Meier, J. Electrochem. Soc. 125, 798(1978).
[CrossRef]

J. Opt. Soc. Am. (3)

Nouv. Rev. Opt. (1)

R. Petit, Nouv. Rev. Opt. 6, 129 (1975).
[CrossRef]

Opt. Commun. (1)

A. M. J. Huiser, A. Quattropani, H. P. Baltes, Opt. Commun. 41, 149 (1982).
[CrossRef]

Thin Solid Films (1)

E. S. Braga, G. F. Mendes, J. Frejlich, A. P. Mammana, Thin Solid Films 109, 4, 363 (1983).
[CrossRef]

Other (4)

H. P. Baltes, Ed. Inverse Source Problems in Optics (Springer, Berlin, 1978), p. 42.

Reference 8 claims from experience that scalar theory may be accurately considered for λ/d ratios < 0.2.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), 630.

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Figures (5)

Fig. 1
Fig. 1

Schematic description of a lamellar grating.

Fig. 2
Fig. 2

Nomograms for a metallic lamellar reflecting grating calculated for the first [I1/I0 in (a)], second [I2/I0 in (b)], and third [I3/I0 in (c)] diffracted orders using Eqs. (3)(6) with n0 = 1, r ^ 1 = - 1 , r ^ 2 = 0 , r ^ 3 = - 1 , and λ = 6328 Å. The bar width-to-period (a/d) and the geometrical modulation (h) of the grating are plotted as ordinates and abscissas, respectively. The IN/I0 ratios are shown as parameters.

Fig. 3
Fig. 3

Compound nomogram for a metallic lamellar reflecting grating. Superposition of figures 2(a), (b), and (c), are shown. The measured diffracted intensity ratios (I1/I0 = 0.0651, I2/I0 = 0.0007, and I3/I0 = 0.0066) for the sample shown in the top picture are plotted in bold lines. The accuracy of their intersection confirms the assumed rectangular grating shape (as in the picture). From the nomogram we obtain h = 380 Å (an independent elipsometric method gives h = 400 Å) and a/d = 0.535. The period of the grating is 20 μm.

Fig. 4
Fig. 4

Nomogram for a metallic lamellar reflecting grating is the same as Fig. 3 but for another sample as shown in the top picture. The measured diffracted orders (I1/I0 = 0.0592, I2/I0 = 0.0021, I3/I0 = 0.0036) (bold lines) do not simultaneously intersect indicating that the sample does not fit the assumed rectangular shape (see the picture). From the first-order ratio we read h = 375 Å on the nomogram but this result must be considered with less confidence than for the same in Fig. 3.

Fig. 5
Fig. 5

On ordinates is plotted the percent precision of the calculated h grating geometrical modulation, assuming 1% in both I1 and I0 as the only error sources [see Eq. (8)].

Equations (18)

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T ( f ) FT [ r ^ ( x ) ] ,
r ^ ( x ) = [ r ^ a rect ( x / a ) + r ^ b rect ( x / b ) * δ ( x - d / 2 ) ] * L L L ( x / d ) d ,
I N / I 0 = r ^ a - r ^ b 2 ( a / d ) 2 r ^ b + ( r ^ a - r ^ b ) ( a / d ) 2 sinc 2 ( N a / d ) ,
r ^ a = r ^ 1 + r ^ 2 exp ( i 2 β ) 1 + r ^ 1 r ^ 2 exp ( i 2 β )             where 2 β 4 π h n ^ 1 / λ , r ^ b = r ^ 3 exp ( i 2 γ )             where 2 γ 4 π h n 0 / λ , }
r ^ 1 = ( n 0 - n ^ 1 ) / ( n 0 + n ^ 1 ) , r ^ 2 = ( n ^ 1 - n ^ 2 ) / ( n ^ 1 + n ^ 2 ) , r ^ 3 = ( n 0 - n ^ 2 ) / ( n 0 + n ^ 2 ) . }
F ( I N / I 0 , h , a / d ) = 0 ] n 0 , n ^ 1 , n ^ 2 .
| Δ h h | | h ( I 1 / I 0 ) | ( Δ I 1 / I 1 + Δ I 0 / I 0 ) I 1 / I 0 h .
| Δ h h | 0.02 | h ( I 1 / I 0 ) | ( I 1 / I 0 h ) .
r ^ n ( x ) = r ^ ( x ) exp [ i ϕ ( x ) ] ,
T n ( f ) = [ r ^ b + ( r ^ a - r ^ b ) ( a d ) ] δ ( f ) + i [ r ^ b + ( r ^ a - r ^ b ) ( a d ) ] Φ ( f ) + [ ( r ^ a - r ^ b ) a d sinc ( a d N ) ] δ ( f - N d ) + i [ ( r ^ a - r ^ b ) a d sinc ( a d N ) ] Φ ( f - N d ) ,
( I N ) n = N l d - ɛ N / d + ɛ T n ( f ) 2 d f ,
( I 0 ) n = - ɛ + ɛ T n ( f ) 2 d f ;
( I N ) n = | ( r ^ a - r ^ b ) a d sinc ( a d N ) | 2 [ 1 - 2 Im { Φ ( 0 ) } ] + | ( r ^ a - r ^ b ) a d sinc ( a d N ) | 2 N / d - ɛ N / d + ɛ | Φ ( f - N d ) | 2 d f ,
( I 0 ) n = | r ^ b + ( r ^ a - r ^ b ) a d | 2 [ 1 - 2 Im { Φ ( 0 ) } ] + | r ^ b + ( r ^ a - r ^ b ) a d | 2 - ɛ + ɛ Φ ( 0 ) 2 d f ,
( I N ) n | ( r ^ a - r ^ b ) a d sinc ( a d N ) | 2 [ 1 - 2 Im { Φ ( 0 ) } ] , ( I 0 ) n | r ^ b + ( r ^ a - r ^ b ) a d | 2 [ 1 - 2 Im { Φ ( 0 ) } ] , ( I N I 0 ) n | r ^ a - r ^ b | 2 ( a d ) 2 | r ^ b + ( r ^ a - r ^ b ) a d | 2 sinc 2 ( a d N ) .
r ^ a d ( x ) = r ^ b + ( r ^ a - r ^ b ) exp [ i ϕ ( x ) ] rect ( x a ) * L L L ( x d ) d .
T o d ( f ) = r ^ b δ ( f ) + { ( r ^ a - r ^ b ) [ δ ( f ) + i Φ ( f ) ] * a s i n c ( a f ) } L L L ( d f ) .
R N ( I N I 0 ) o d = ( I N I 0 ) R N , 1 + [ C ( N d ) / sinc ( a d N ) ] 2 1 + ( a d ) 2 | ( r ^ a - r ^ b ) r ^ b + ( r ^ a - r ^ b ) a d | 2 [ C ( 0 ) ] 2 - 2 C ( 0 ) ( a d ) Im { r ^ a r ^ b } | r ^ b + ( r ^ a - r ^ b ) a d | 2 ,

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