Abstract

Simplified formulas for the intensity distribution among the orders of an echelette grating are derived from the Green’s function method, and compared with previous expressions. The application to special types of mounts and the possibility of using gratings as filters is discussed.

© 1958 Optical Society of America

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References

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  1. R. D. Hatcher and J. H. Rohrbaugh, J. Opt. Soc. Am. 46, 104 (1956), hereafter referred to as I; J. Opt. Soc. Am. 48, 704 (1958), hereafter referred to as II; Rohrbaugh, Pine, Zoellner, and Hatcher, J. Opt. Soc. Am. 48, 710 (1958), hereafter referred to as III.
    [Crossref]
  2. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), Vol. II, p. 1429.
  3. L. Zadoff, Ph.D. thesis, New York University (1957).
  4. W. C. Meecham, J. Appl. Phys. 27, 361 (1956), where references to other approaches may be found.
    [Crossref]
  5. W. C. Meecham and C. W. Peters, J. Appl. Phys. 28, 216 (1957).
    [Crossref]
  6. P. M. Morse and H. Feshabach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York), p. 806, 1544.
  7. A similar change should be made in Eq. (9) of I which should readλ2π2sin2πc′a(μ-cρ)λ(c+c′)[(1+c2)12(μ-cρ)-(1+c′2)12(μ+c′ρ)]2.We thank Dr. Masao Seya for correspondence on this point. This equation arises from Eq. (5) above by considering the scalar product (bs·n0) to be absent which can be done by having (Ψr0s) equal 0 over the surface of the grating in Eq. (2). (It must also be remarked that Fig. 3 of I was not the intended figure, the meaning being sufficiently clear in our judgment to make it unnecessary to replace the figure here.)
  8. N. Finkelstein, J. Opt. Soc. Am. 41, 179 (1951).
    [Crossref]
  9. J. H. Greig and W. F. C. Ferguson, J. Opt. Soc. Am. 40, 504 (1950).
    [Crossref]

1957 (1)

W. C. Meecham and C. W. Peters, J. Appl. Phys. 28, 216 (1957).
[Crossref]

1956 (2)

1951 (1)

1950 (1)

Ferguson, W. F. C.

Feshabach, H.

P. M. Morse and H. Feshabach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York), p. 806, 1544.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), Vol. II, p. 1429.

Finkelstein, N.

Greig, J. H.

Hatcher, R. D.

Meecham, W. C.

W. C. Meecham and C. W. Peters, J. Appl. Phys. 28, 216 (1957).
[Crossref]

W. C. Meecham, J. Appl. Phys. 27, 361 (1956), where references to other approaches may be found.
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), Vol. II, p. 1429.

P. M. Morse and H. Feshabach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York), p. 806, 1544.

Peters, C. W.

W. C. Meecham and C. W. Peters, J. Appl. Phys. 28, 216 (1957).
[Crossref]

Rohrbaugh, J. H.

Zadoff, L.

L. Zadoff, Ph.D. thesis, New York University (1957).

J. Appl. Phys. (2)

W. C. Meecham, J. Appl. Phys. 27, 361 (1956), where references to other approaches may be found.
[Crossref]

W. C. Meecham and C. W. Peters, J. Appl. Phys. 28, 216 (1957).
[Crossref]

J. Opt. Soc. Am. (3)

Other (4)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York, 1953), Vol. II, p. 1429.

L. Zadoff, Ph.D. thesis, New York University (1957).

P. M. Morse and H. Feshabach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., New York), p. 806, 1544.

A similar change should be made in Eq. (9) of I which should readλ2π2sin2πc′a(μ-cρ)λ(c+c′)[(1+c2)12(μ-cρ)-(1+c′2)12(μ+c′ρ)]2.We thank Dr. Masao Seya for correspondence on this point. This equation arises from Eq. (5) above by considering the scalar product (bs·n0) to be absent which can be done by having (Ψr0s) equal 0 over the surface of the grating in Eq. (2). (It must also be remarked that Fig. 3 of I was not the intended figure, the meaning being sufficiently clear in our judgment to make it unnecessary to replace the figure here.)

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Figures (5)

Fig. 1
Fig. 1

One groove of a grating surface plus an enclosing surface at infinity. The surfaces enclose a volume V.

Fig. 2
Fig. 2

Cross section of one groove of an echelette grating.

Fig. 3
Fig. 3

One groove of an echelette grating being irradiated so as to define u.

Fig. 4
Fig. 4

Laboratory axes xyz′ and grating axes xyz with the incident ray along the x′ axis.

Fig. 5
Fig. 5

The Finkelstein mount (showing just one groove) with a grating as a filter in place of the plane mirror.

Equations (53)

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Ψ ( r ) = C e i b i · r + 1 4 π [ G b ( r r 0 s ) 0 Ψ ( r 0 s ) - Ψ ( r 0 s ) 0 G b ( r r 0 s ) ] · d S ,
G b ( r r 0 s ) e i b r r exp ( - i b s · r 0 s ) ,
Ψ s C f ( b i b s ) e i b r r ,
f ( b i b s ) ~ [ exp ( - i b s · r 0 s ) 0 Ψ ( r 0 s ) + i b s Ψ ( r 0 s ) exp ( - i b s · r 0 s ) ] · d S .
Ψ ( r 0 s ) exp ( i b i · r 0 s ) + A exp ( i b a · r 0 s ) ,
f ( b i b s ) ~ i ( b i · n 0 ) exp ( - i b s · r 0 s ) × { exp ( i b i · r 0 s ) - A exp ( i b a · r 0 s ) } d S + i ( b s · n 0 ) exp ( - i b s · r 0 s ) × { exp ( i b i · r 0 s ) + A exp ( i b a · r 0 s ) } d S
f ( b i b s ) ~ i [ ( b i · n 0 ) + ( b s · n 0 ) ] × exp [ i r 0 s · ( b i - b s ) ] d S ,
λ 2 π 2 sin 2 π c a ( μ - c ρ ) ( c + c ) λ [ ( μ 2 + ρ 2 ) ( c + c ) ( μ - c ρ ) ( μ + c ρ ) ] 2 .
b a = b i - 2 ( b i · n 0 ) n 0 ,
f ( b i b s ) ~ exp [ i r 0 s · ( b i - b s ) ] × [ ( b i · n 0 ) ( 1 - B ) + ( b s · n 0 ) ( 1 + B ) ] d S ,
f ( b i b s ) ~ [ ( b s · n 0 ) + D ( - b i · n 0 ) ] × exp [ i r 0 s · ( b i - b s ) ] d S .
Relative intensity sin 2 π c a ( μ - c ρ ) ( c + c ) λ × [ ρ 2 + μ 2 ( μ - c ρ ) ( μ + c ρ ) ] 2
Relative intensity ~ [ ( α - c β ) + D ( α - c β ) ( μ + c ρ ) ] 2 × sin 2 π c a ( μ - c ρ ) ( c + c ) λ .
A = - exp [ 2 i ( b i · n 0 ) ( n 0 · r 0 s ) ]
A = exp [ 2 i ( b i · n 0 ) ( n 0 · r 0 s ) ] .
f ( b i b s ) ~ ( b s · n 0 ) exp [ i r 0 s · ( b i - b s ) ] d S ,
Relative intensity ( α - c β ) 2 ( μ + c ρ ) 2 sin 2 π ( a - u ) ( μ + c ρ ) λ ,
( ρ - μ c ) 2 ( μ + c ρ ) 2 sin 2 π ( a - u ) ( μ + c ρ ) λ ,
( a - u ) = a cos θ cos ψ cos ( θ - ψ ) .
β + β = - c ( α + α ) .
β + β = λ N / a .
γ + γ = 0.
α 2 + β 2 + γ 2 = 1
α 2 + β 2 + γ 2 = 1.
            γ = - γ ,             β = ( λ N / a ) - β , and                         α = ( - λ N / a c ) - α .
λ = 2 a N c ( 1 + c 2 ) ( - α + β c ) .
D = i d 1 + j d 2 + k d 3 .
α = l 1 ,             β = l 2 ,             γ = l 3 , α = l 1 d 1 + m 1 d 2 + n 1 d 3 , β = l 2 d 1 + m 2 d 2 + n 2 d 3 , γ = l 3 d 1 + m 3 d 2 + n 3 d 3 .
d 1 l 3 + d 2 m 3 + d 3 n 3 = - l 3 ,
d 1 l 2 + d 2 m 2 + d 3 n 2 = ( λ N / a ) - l 2 ,
d 1 l 1 + d 2 m 1 + d 3 n 1 = - ( λ N / a c ) - l 1
d 1 = - 1 + λ N a c ( c l 2 - l 1 ) ,
d 2 = λ N a c ( m 2 c - m 1 ) ,
d 3 = λ N a c ( n 2 c - n 1 ) .
λ = 2 a N c ( 1 + c 2 ) ( c l 2 - l 1 )
n 2 c = n 1 ,
n 1 2 + n 2 2 + n 3 2 = 1.
n 2 = 1 / ( 1 + c 2 ) 1 2 = ± cos ψ ,
n 2 = K c ± [ ( 1 + c 2 ) - K 2 ] 1 2 1 + c 2 .
λ = 2 a 1 sin ψ 1 cos θ ,
( λ / a 2 ) > 1 + sin θ
λ a 2 < 2 ( 1 + sin θ ) .
λ a 2 > 2 ( 1 - sin θ ) .
λ 2 a 1 sin ψ 1 > 2 2 3
λ 2 - 2 [ 1 - ( λ / 2 a 1 sin ψ 1 ) 2 ] 1 2 > a 2 > λ 2 + 2 [ 1 - ( λ / 2 a 1 sin ψ 1 ) 2 ] 1 2 ,
λ 2 a 1 sin ψ 1 < 2 2 3
λ 1 + [ 1 - ( λ / 2 a 1 sin ψ 1 ) 2 ] 1 2 > a 2 > λ 2 + 2 [ 1 - ( λ / 2 a 1 sin ψ 1 ) 2 ] 1 2 .
101 > a 2 > 50.
μ = - c ρ
λ 2 a 2 = 2 cos ( θ - ψ 2 ) sin ψ 2 ,
sin 4 ψ 2 + sin 2 ψ 2 [ - λ ( 1 - A 2 ) 1 2 2 a 2 - A 2 ] + λ 2 16 a 2 2 = 0 ,
a 2 sin ψ 2 a 1 sin ψ 1 2 π .
λ2π2sin2πca(μ-cρ)λ(c+c)[(1+c2)12(μ-cρ)-(1+c2)12(μ+cρ)]2.