Abstract

An analytical expression for the response of a complete Fabry–Perot photoelectric spectrometer to a line profile is derived. The case of a doppler-broadened Lorentzian profile examined by a spectrometer comprised of an etalon of variable thickness and reflectivity, with surface defects consisting of spherical curvature, as well as microscopic flatness imperfections, a finite exploring slit in front of the photodetector, limiting circuits is considered. Some of the properties and uses of the expression obtained and bandwidth are discussed.

© 1966 Optical Society of America

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References

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  1. K. Krebs, A. Sauer, Ann. Physik 13, 359 (1953).
    [CrossRef]
  2. E. A. Ballik, Appl. Opt. 5, 170 (1966).
    [CrossRef] [PubMed]
  3. R. M. Hill, Opt. Acta 10, 141 (1963).
    [CrossRef]
  4. J. O. Stoner, J. Opt. Soc. Am. 36, 370 (1966).
    [CrossRef]
  5. F. Bayer-Helms, Z. Angew. Physik 15, 330 (1963) a; Z. Angew. Physik 15, 416 (1963) b; Z. Angew. Physik 16, 44 (1963) c.
  6. R. J. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 24, 138 (1953).
  7. A. H. Jarret, M. J. Hoey, J. Atmospheric Terrest. Phys. 28, 175 (1966).
    [CrossRef]
  8. E. C. Turgeon, G. G. Shepherd, Planetary Space Sci. 9, 925 (1962).
    [CrossRef]
  9. S. G. Rautian, Usp. Fiz. Nauk 66, 475 (1958); Soviet Phys.-Usp. 1, 245 (1958).
  10. P. Jacquinot, Ch. Dufour, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 6, 91 (1948).
  11. R. M. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill Book Co., Inc., New York, 1965).
  12. M. Born, E. Wolf, Principles of Optics, (The Macmillan Company, New York, 2nd ed., 1964), p. 323ff.
  13. J. E. Mack, D. P. McNutt, F. L. Roesler, R. Chabbal, Appl. Opt. 2, 873 (1963).
  14. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Pergamon Press, Inc., New York, 1965), p. 40.
  15. T. Tako, M. Ohi, Japan J. Appl. Phys. 4, Suppl. 1, 451 (1965).
  16. P. Giacomo, Compt. Rend. 235, 1627 (1952).

1966

E. A. Ballik, Appl. Opt. 5, 170 (1966).
[CrossRef] [PubMed]

J. O. Stoner, J. Opt. Soc. Am. 36, 370 (1966).
[CrossRef]

A. H. Jarret, M. J. Hoey, J. Atmospheric Terrest. Phys. 28, 175 (1966).
[CrossRef]

1965

T. Tako, M. Ohi, Japan J. Appl. Phys. 4, Suppl. 1, 451 (1965).

1963

J. E. Mack, D. P. McNutt, F. L. Roesler, R. Chabbal, Appl. Opt. 2, 873 (1963).

F. Bayer-Helms, Z. Angew. Physik 15, 330 (1963) a; Z. Angew. Physik 15, 416 (1963) b; Z. Angew. Physik 16, 44 (1963) c.

R. M. Hill, Opt. Acta 10, 141 (1963).
[CrossRef]

1962

E. C. Turgeon, G. G. Shepherd, Planetary Space Sci. 9, 925 (1962).
[CrossRef]

1958

S. G. Rautian, Usp. Fiz. Nauk 66, 475 (1958); Soviet Phys.-Usp. 1, 245 (1958).

1953

R. J. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 24, 138 (1953).

K. Krebs, A. Sauer, Ann. Physik 13, 359 (1953).
[CrossRef]

1952

P. Giacomo, Compt. Rend. 235, 1627 (1952).

1948

P. Jacquinot, Ch. Dufour, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 6, 91 (1948).

Ballik, E. A.

Bayer-Helms, F.

F. Bayer-Helms, Z. Angew. Physik 15, 330 (1963) a; Z. Angew. Physik 15, 416 (1963) b; Z. Angew. Physik 16, 44 (1963) c.

Born, M.

M. Born, E. Wolf, Principles of Optics, (The Macmillan Company, New York, 2nd ed., 1964), p. 323ff.

Bracewell, R. M.

R. M. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill Book Co., Inc., New York, 1965).

Chabbal, R.

Chabbal, R. J.

R. J. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 24, 138 (1953).

Dufour, Ch.

P. Jacquinot, Ch. Dufour, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 6, 91 (1948).

Giacomo, P.

P. Giacomo, Compt. Rend. 235, 1627 (1952).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Pergamon Press, Inc., New York, 1965), p. 40.

Hill, R. M.

R. M. Hill, Opt. Acta 10, 141 (1963).
[CrossRef]

Hoey, M. J.

A. H. Jarret, M. J. Hoey, J. Atmospheric Terrest. Phys. 28, 175 (1966).
[CrossRef]

Jacquinot, P.

P. Jacquinot, Ch. Dufour, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 6, 91 (1948).

Jarret, A. H.

A. H. Jarret, M. J. Hoey, J. Atmospheric Terrest. Phys. 28, 175 (1966).
[CrossRef]

Krebs, K.

K. Krebs, A. Sauer, Ann. Physik 13, 359 (1953).
[CrossRef]

Mack, J. E.

McNutt, D. P.

Ohi, M.

T. Tako, M. Ohi, Japan J. Appl. Phys. 4, Suppl. 1, 451 (1965).

Rautian, S. G.

S. G. Rautian, Usp. Fiz. Nauk 66, 475 (1958); Soviet Phys.-Usp. 1, 245 (1958).

Roesler, F. L.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Pergamon Press, Inc., New York, 1965), p. 40.

Sauer, A.

K. Krebs, A. Sauer, Ann. Physik 13, 359 (1953).
[CrossRef]

Shepherd, G. G.

E. C. Turgeon, G. G. Shepherd, Planetary Space Sci. 9, 925 (1962).
[CrossRef]

Stoner, J. O.

J. O. Stoner, J. Opt. Soc. Am. 36, 370 (1966).
[CrossRef]

Tako, T.

T. Tako, M. Ohi, Japan J. Appl. Phys. 4, Suppl. 1, 451 (1965).

Turgeon, E. C.

E. C. Turgeon, G. G. Shepherd, Planetary Space Sci. 9, 925 (1962).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, (The Macmillan Company, New York, 2nd ed., 1964), p. 323ff.

Ann. Physik

K. Krebs, A. Sauer, Ann. Physik 13, 359 (1953).
[CrossRef]

Appl. Opt.

Compt. Rend.

P. Giacomo, Compt. Rend. 235, 1627 (1952).

J. Atmospheric Terrest. Phys.

A. H. Jarret, M. J. Hoey, J. Atmospheric Terrest. Phys. 28, 175 (1966).
[CrossRef]

J. Opt. Soc. Am.

J. O. Stoner, J. Opt. Soc. Am. 36, 370 (1966).
[CrossRef]

J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris)

R. J. Chabbal, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 24, 138 (1953).

P. Jacquinot, Ch. Dufour, J. Rech. Centre Nat. Rech. Sci. Lab. Bellevue (Paris) 6, 91 (1948).

Japan J. Appl. Phys.

T. Tako, M. Ohi, Japan J. Appl. Phys. 4, Suppl. 1, 451 (1965).

Opt. Acta

R. M. Hill, Opt. Acta 10, 141 (1963).
[CrossRef]

Planetary Space Sci.

E. C. Turgeon, G. G. Shepherd, Planetary Space Sci. 9, 925 (1962).
[CrossRef]

Usp. Fiz. Nauk

S. G. Rautian, Usp. Fiz. Nauk 66, 475 (1958); Soviet Phys.-Usp. 1, 245 (1958).

Z. Angew. Physik

F. Bayer-Helms, Z. Angew. Physik 15, 330 (1963) a; Z. Angew. Physik 15, 416 (1963) b; Z. Angew. Physik 16, 44 (1963) c.

Other

R. M. Bracewell, The Fourier Transform and Its Applications (McGraw–Hill Book Co., Inc., New York, 1965).

M. Born, E. Wolf, Principles of Optics, (The Macmillan Company, New York, 2nd ed., 1964), p. 323ff.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Pergamon Press, Inc., New York, 1965), p. 40.

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Figures (1)

Fig. 1
Fig. 1

Broadening of a doppler line profile by a Fabry–Perot spectrometer with arbitrary separation, reflectivity, and surface defects, as well as an exploring diaphragm (see text for details). A: ϕ (x) = G. B: ϕ (x) = G*A*DF. C: ϕ (x) = G*A*DF*F.

Equations (23)

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Y = B * I = Y ( x ) = - B ( y ) I ( x - y ) d y ,
Y = B * I = L * G * A * D f * D g * F .
a ( x ) = τ A [ 1 + 4 R ( 1 - R ) - 2 sin 2 ( x / 2 ) ] - 1 = τ A ( 1 - R ) 2 ( 1 - 2 R cos x + R 2 ) - 1 ,
A ( x ) = ( 1 - R 2 ) ( 2 π ) - 1 ( 1 - 2 R cos x + R 2 ) - 1 .
D g = D - 1 π - 1 2 exp - ( x D - 1 ) 2 ,
D f = Δ σ ( 4 π d f ) - 1 Π             ( x ) Π ( x ) = 1 2 π d f ( Δ σ ) - 1 > x 1 2 2 π d f ( Δ σ ) - 1 = x 0 2 π d f ( Δ σ ) - 1 < x
L ( x ) = L ( π - 1 ) ( L 2 + x 2 ) - 1 ,
( 1 - R 2 ) ( 2 π ) - 1 ( 1 - 2 R cos x + R 2 ) - 1 = ( 2 π ) - 1 × [ 1 + 2 n = 1 R n cos n x ] .
L * A = ( 2 π ) - 1 [ 1 + 2 n = 1 ( R e - L ) n cos n x ]
= ( 2 π ) - 1 [ 1 - ( R e - L ) 2 ] [ 1 - 2 ( R e - L ) cos x + ( R e - L ) 2 ] - 1 .
L * A * G = ( 2 π ) - 1 { 1 + 2 n = 1 ( R e - L ) n [ exp - ( n G 2 - 1 ) 2 ] cos n x } .
L * A * G * D G = ( 2 π ) - 1 { { 1 + 2 n = 1 ( R e - L ) n × { exp - [ ( n 2 - 1 ) 2 ( G 2 + D 2 ) ] } cos n x } } .
L * A * G * D G * D F = ( 2 π ) - 1 { { 1 + 2 n = 1 ( R e - L ) n × { exp - [ ( n 2 - 1 ) 2 ( G 2 + D 2 ) ] } sinc [ 2 n d F ( Δ σ ) - 1 ] cos n x } } ,
Y ( x ) = L * A * G * D G * D F * F = ( 2 π ) - 1 × { { 1 + 2 n = 1 ( R e - L ) n { exp - [ ( n 2 - 1 ) 2 ( G 2 + D 2 ) ] { × sinc [ 2 n d F ( Δ σ ) - 1 ] sinc [ 2 n f ( Δ σ ) - 1 ] cos n x } }
Y ( x , t ) = ( 2 π ) - 1 { { 1 + 2 n = 1 ( R e - L ) n { exp - [ ( 2 - 1 n ) 2 ( G 2 + D 2 ) ] } sinc [ 2 n d F ( Δ σ ) - 1 ] sinc [ ( 2 n f ( Δ σ ) - 1 ] × ( 1 + t 2 n 2 ) - 1 [ cos n x + n t sin n x ] } } .
ψ ( x , t ) = Y ( x , t ) Y ( x max , t ) = { { 1 + 2 n = 1 ( R e - L ) n × { exp - [ ( 2 - 1 n ) 2 ( G 2 + D 2 ) ] } sinc [ 2 n d F ( Δ σ ) - 1 ] × sinc [ 2 n f ( Δ σ ) - 1 ] ( 1 + t 2 n 2 ) - 1 [ cos n x + n t sin n x ] } } { { 1 + 2 n = 1 ( R e - L ) n × { exp - [ ( 2 - 1 n ) 2 ( G 2 + D 2 ) ] } sinc [ 2 n d F × ( Δ σ ) - 1 ] sinc [ 2 n f ( Δ σ ) - 1 ] ( 1 + t 2 n 2 ) - 1 × [ cos ( t g - 1 n t ) + n t sin ( t g - 1 n t ) } } - 1 .
Y = A * G * F * T .
C = A ( 0 ) [ A ( π ) ] - 1 = ( 1 + R ) 2 ( 1 - R ) - 2 ,
τ = [ ( 2 π ) A ( 0 ) ] - 1 = ( 1 - R ) ( 1 + R ) - 1 .
I ^ T ( I ^ min ) - 1 = ( 1 + R ) ( 1 - R ) - 1 .
C = Y ( 0 ) [ Y ( π ) ] - 1
τ = [ 2 π Y ( 0 ) ] - 1
I ^ T ( I min ) - 1 = [ Y ( π ) ] - 1 .

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