Abstract

It is shown that for a semireflecting film whose reflection and transmission factors are re and te, the conservation of energy requires that the angle ρτ must lie between cos−1(A/2rt) and cos−1(−A/2rt) where A=1−r2t2. The corresponding condition for an asymmetrical film is derived. The effect of this phase condition on multiple beam reflection fringes and filters is studied theoretically. It is possible to set an upper limit to the error incurred in the calculation of reflected intensities on the usual assumption that ρτ=π/2. Expressions are found for the maximum and minimum intensities with ρτ at its limit. It is extablished that a necessary condition for the reflected intensity distribution to become transmission-like is that r+t⩽1.

© 1957 Optical Society of America

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References

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  1. M. Hamy, J. phys. 5, 789 (1906).
  2. C. Dufour, J. phys. radium 11, 327 (1950).
    [Crossref]
  3. S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Clarendon Press, Oxford, 1948).
  4. J. Holden, Proc. Phys. Soc. (London) B62, 405 (1949).
  5. J. Holden, J. Opt. Soc. Am. 41, 504 (1951).
    [Crossref]

1951 (1)

1950 (1)

C. Dufour, J. phys. radium 11, 327 (1950).
[Crossref]

1949 (1)

J. Holden, Proc. Phys. Soc. (London) B62, 405 (1949).

1906 (1)

M. Hamy, J. phys. 5, 789 (1906).

Dufour, C.

C. Dufour, J. phys. radium 11, 327 (1950).
[Crossref]

Hamy, M.

M. Hamy, J. phys. 5, 789 (1906).

Holden, J.

J. Holden, J. Opt. Soc. Am. 41, 504 (1951).
[Crossref]

J. Holden, Proc. Phys. Soc. (London) B62, 405 (1949).

Tolansky, S.

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Clarendon Press, Oxford, 1948).

J. Opt. Soc. Am. (1)

J. phys. (1)

M. Hamy, J. phys. 5, 789 (1906).

J. phys. radium (1)

C. Dufour, J. phys. radium 11, 327 (1950).
[Crossref]

Proc. Phys. Soc. (London) (1)

J. Holden, Proc. Phys. Soc. (London) B62, 405 (1949).

Other (1)

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Clarendon Press, Oxford, 1948).

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Figures (4)

Fig. 1
Fig. 1

Reflection and transmission factors of film, and arrangement for derivation of phase condition.

Fig. 2
Fig. 2

The variation of I with ψ for admissible (A, B, C) and inadmissible (D, E) values of γ. For A, γ=180°; B, γ=200°; C, γ=γL=200°.8; D, γ=270°; E, γ=0 or 360°.

Fig. 3
Fig. 3

The variation of Imax and Imin with the phase angle γ for pairs of films for which γ is restricted (a), and γ is unrestricted (b). For a, r2=r2=r12=0.90, t2=0.05. For b, r2=0.4, r2=0.2, r12=1.0, t2=0.2.

Fig. 4
Fig. 4

Normal (γ=π) and transmission-like (γ=0) fringes for an etalon with r2=0.4, t2=0.2, r2=0.2, r12=1.

Equations (27)

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r e i ρ + t 2 e i ( τ + τ - δ ) n = 0 r n e i n ( ρ - δ ) = r e i ρ + t 2 e i ( τ + τ - δ ) - r e i ( τ + τ - ρ ) 1 + r 2 - 2 r cos ( ρ - δ ) ,
I = r 2 + t 2 t 2 + 2 r cos ( γ - ρ + δ ) - 2 r r cos γ 1 + r 2 - 2 r cos ( ρ - δ ) .
I = 1 + 4 r t 2 cos γ / 2 [ cos ( γ - 2 ρ + 2 δ ) / 2 - r cos γ / 2 ] 1 + r 2 - 2 r cos ( ρ - δ ) .
γ = ( 2 m + 1 ) π ( m an integer )
I = r 2 + t 2 t 2 - 2 r cos ( ψ - γ ) - 2 r r cos γ 1 + r 2 - 2 r cos ψ .
( 1 + r 2 - 2 r cos ψ ) r sin ( ψ - γ L ) + [ t 2 + 2 r cos ( ψ - γ L ) - 2 r r cos γ L ] r sin ψ = 0
r 2 + t 2 t 2 + 2 r cos ( ψ - γ L ) - 2 r r cos γ L 1 + r 2 - 2 r cos ψ = 1.
( 1 - r 2 ) cos γ L - ( 1 + r 2 ) sin γ L cot ψ + 2 r sin γ L csc ψ + r t 2 / r = 0 ,
r 2 - r t 2 sin ( ψ - γ L ) r sin ψ = 1 ,
cot ψ = r t 2 cos γ L + r - r r 2 r t 2 sin γ L ,
csc ψ = [ r 2 t 4 + r 2 ( 1 - r 2 ) 2 + 2 r r t 2 ( 1 - r 2 ) cos γ L ] 1 2 r t 2 sin γ L ,
4 r 2 r 2 t 4 cos 2 γ L - 4 r r t 2 [ t 4 + ( 1 - r 2 ) ( 1 - r 2 ) ] cos γ L + t 8 - 2 t 4 ( 1 + r 2 + r 2 - r 2 r 2 ) + ( 1 - r 2 ) 2 ( 1 - r 2 ) 2 = 0.
cos γ L = t 4 + ( 1 - r 2 ) ( 1 - r 2 ) 2 r r t 2 ± 1 r r ,
cos γ L = A A - ( r 2 + r 2 ) t 2 2 r r t 2 .
cos γ L / 2 = ± A / 2 r t .
I max = r 2 ( 1 + r 2 ) 2 ( 2 - A ) 2 ,
I min = r 2 ( 1 - r 2 ) 2 A 2 ,
I = r 2 + r 1 t 2 r 1 t 2 + 2 r cos ( ψ - γ ) - 2 r r 1 r cos γ 1 + r 2 r 1 2 - 2 r r 1 cos ψ ,
I max , min = { r 1 t 2 ± [ ( r 2 r 1 2 r + r r 1 2 t 2 - r ) 2 + 4 r r 1 2 r t 2 ( 1 - r 2 r 1 2 ) cos 2 γ / 2 ] 1 2 } 2 ( 1 - r 2 r 1 2 ) 2 ,
cot ψ = r t 2 cos γ + r I max , min - r r 2 r t 2 sin γ ,
csc ψ = [ r 2 t 4 + r 2 ( I max , min - r 2 ) 2 + 2 r r t 2 ( I max , min - r 2 ) cos γ ] 1 2 r t 2 sin γ ,
cos 2 γ L 2 = A A - ( r - r ) 2 t 2 4 r r t 2 ,
( I max , min ) L = ( r 1 t 2 ± { r 1 2 t 4 + ( 1 - r 2 r 1 2 ) [ r 2 - r 1 2 ( 1 - A - A ) ] } 1 2 ) 2 ( 1 - r 2 r 1 2 ) 2 ,
( I max , min ) L = r 2 { t 2 ± [ t 4 + 2 ( 1 - r 4 ) A ] 1 2 } 2 ( 1 - r 4 ) 2 .
A A ( r + r ) 2 t 2 ,
A 2 r t
r + t 1