Abstract

A new model for the blaze function of the two in-plane mountings of echelle gratings is described and extended to account for defects in the edges of the grooves. Expressions for the relative efficiency are given. The model shows that the wavelength for the maximum blaze intensity differs slightly from the wavelength λ0 satisfying mλ0 = d[sinα + sin(2θα)], the difference being a function of the spectral order. The off-blaze orders of the wavelengths diffracted at the central blaze for the mounting α < θ (α = angle of incidence, θ = blaze angle) lie at the minima of the blaze function for a perfect echelle. Results are given from measurements of the blaze angle and the groove edge defects of three echelles.

© 1982 Optical Society of America

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References

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  1. G. R. Harrison, J. Opt. Soc. Am. 39, 522 (1949).
    [CrossRef]
  2. G. R. Harrison, E. G. Loewen, R. S. Wiley, Appl. Opt. 15, 971 (1976).
    [CrossRef] [PubMed]
  3. S. Engman, P. Lindblom, B. Sandberg, Phys. Scr. 24, 965 (1981).
    [CrossRef]
  4. D. J. Schroeder, R. L. Hilliard, Appl. Opt. 19, 2833 (1980).
    [CrossRef] [PubMed]
  5. M. Bottema, Appl. Opt. 20, 528 (1981).
    [CrossRef] [PubMed]
  6. D. J. Schroeder, Appl. Opt. 20, 530 (1981).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 378–382.

1981 (3)

1980 (1)

1976 (1)

1949 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 378–382.

Bottema, M.

Engman, S.

S. Engman, P. Lindblom, B. Sandberg, Phys. Scr. 24, 965 (1981).
[CrossRef]

Harrison, G. R.

Hilliard, R. L.

Lindblom, P.

S. Engman, P. Lindblom, B. Sandberg, Phys. Scr. 24, 965 (1981).
[CrossRef]

Loewen, E. G.

Sandberg, B.

S. Engman, P. Lindblom, B. Sandberg, Phys. Scr. 24, 965 (1981).
[CrossRef]

Schroeder, D. J.

Wiley, R. S.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 378–382.

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Figures (6)

Fig. 1
Fig. 1

Principal outline of the two mountings of echelle gratings (α > θ and α < θ) together with their associated models for diffraction at the groove edges. (A) and (B) reflected in the horizontal groove side give (C) and (D), respectively. For the mounting with α > θ the diffraction takes place in a slit of width d formed by the edges A and B prior to reflection from the horizontal groove side (C). For the mounting with α > θ the diffraction takes place in the edges after reflection from the horizontal groove side (D). The notations used in the text are defined in this figure.

Fig. 2
Fig. 2

Diffracted orders of the laser beam for grating 1, recorded at such angles of incidence that order 60 is diffracted in the central blaze for the two mountings [α < θ (A); α > θ (B)]. This grating is the best of the three gratings tested. As seen from (A) the off-blaze orders are close to the diffraction minima, indicating a small groove edge defect. The diagrams under the photographs show the relative intensities of the orders as calculated from the round edge model using the fitted values for the blaze angle θ and the groove edge correction parameter R. The calculated intensities seem to describe the intensity variations of the orders fairly well. Note the very low calculated intensities of the off-blaze orders for the mounting with α < θ (A). This grating has opposite blazes at θ ≈ 25.3 and θ ≈ 22.6 deg.

Fig. 3
Fig. 3

Principal outline of the two models used for the correction of groove edge defects through one parameter R. The parameter R represents the radius of curvature of the edge for the round edge model and the depth of a 45-deg cut of the edge for the cut edge model. In both cases the width of the reflecting groove side is reduced by R. In the blaze functions the quantities α, β0, β, θ, and d have been replaced by α′ = α + p, β0 = β0 + p,β′ = β + p,θ′ = θ + p. and d′, where d’ is the distance AB. The quantities d′ and p depend on R according to Eqs. (5)(7).

Fig. 4
Fig. 4

Same as Fig. 2, but for grating 2, recorded at such angles of incidence that order 39 is diffracted in the central blaze. This grating is not as good as grating 1 as can be seen by comparing the photograph of (A) with that of Fig. 2 (A). Compared to grating 1 the larger groove edge defect of this grating (see Table I) results in higher relative intensities of the off-blaze orders. The model cannot be expected to give correct intensities for the orders 34 and 35 in (A) as these orders are diffracted at an angle β close to the blaze angle. This grating has no distinct opposite blaze.

Fig. 5
Fig. 5

Same as Fig. 2, but for grating 3, recorded at such angles of incidence that order 91 is diffracted in the central blaze. This grating has a large groove edge defect (Table I) resulting in a broad blaze. The off-blaze orders of (A) lie far from the diffraction minima and have thus high relative intensities. This grating also has several other blazes, the strongest at θ = 16.3 deg, indicating a poor groove shape.

Fig. 6
Fig. 6

Plots of the blaze function for grating 2 with d = 18.339 μm and θ = 44.74 deg. (a) shows the case α < θ with α = 27.672 deg and (b) the case α > θ with α = 61.808 deg. In both cases the angles of incidence have been chosen so that the He–Ne laser line satisfies the grating equation at β = β0 = 2θα. The curves represent the blaze functions calculated with R = 0, 1, and 2 μm from the round edge model. In the expanded diagrams of the top region the shift of the maximum intensities are illustrated.

Tables (1)

Tables Icon

Table I Grating Specifications

Equations (40)

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I ~ ( sin ) 2 ,
= π d cos θ λ [ sin ( θ α ) + sin ( θ β ) ] .
I ~ [ sin ( cos α + cos ( 2 θ β ) 2 cos α ) ] 2
= π d λ [ sin ( 2 θ β ) sin α ] .
= π d cos α λ [ sin Δ β + 2 tan α sin 2 ( Δ β 2 ) ]  .
= π ( cos α cos β ) ( m λ m 0 λ 0 λ ) { 1 + ( tan β 0 + tan α 2 cos β 0 ) × ( m λ m 0 λ 0 d ) + O [ ( m λ m 0 λ 0 d ) 2 ] } ,
I ~ { sin [ cos β + cos ( 2 θ α ) 2 cos β 0 ] } 2 ,
= π d λ [ sin ( 2 θ α ) sin β ]  .
= π d cos β 0 λ [ sin Δ β 2 tan β 0 sin 2 ( Δ β 2 ) ]  .
= π ( m 0 λ 0 m λ λ )  .
= k π
= π k ( cos α cos β ) { 1 + tan β 0 + tan α 2 cos β 0 ( k λ 0 d ) + O [ ( k λ 0 d ) 2 ] } .
= π d λ [ sin ( 2 θ β + p ) sin ( α + p ) ]
= π d λ [ sin ( 2 θ α + p ) sin ( β + p ) ]
d = d [ 1 R d ( cos α + sin θ ) ] , p = R d ( sin α cos θ )
d = d [ 1 R d ( cos β 0 + sin θ ) ]  , p = R d ( sin β 0 cos θ )
d = d [ 1 R d ( cos θ + sin θ ) ]  , p = R d ( sin θ cos θ )
= l π ( l = ± 1 , ± 2 , ) ,
λ p λ 0 = 1 + 3 2 π 2 m 0 2 ( sin α cos 2 α ) ( cos β 0 cos α ) ( sin α + sin β 0 )
I p = 1 + 3 2 π 2 ( sin α cos 2 α ) 2 ( λ 0 d ) 2 .
λ p λ 0 = 1 3 2 π 2 m 0 2 ( sin β 0 cos 2 β 0 ) ( sin α + sin β 0 ) ,
I p = 1 + 3 2 π 2 ( sin β 0 cos 2 β 0 ) 2 ( λ 0 d ) 2 .
λ p λ 0 = 1 + 3 2 ( d π d m 0 ) 2 ( sin α + sin β 0 cos α ) × cos β 0 cos α [ tan α + p ( 1 + 3 tan 2 α ) ] ,
I p = 1 + 3 2 π 2 ( λ 0 d ) 2 sin α cos 3 α [ tan α + p ( 1 + 4 tan 2 α ) ]  ,
λ p λ 0 = 1 3 2 ( d π d m 0 ) 2 ( sin α + sin β 0 cos β 0 ) × [ tan β 0 + p ( 1 + 3 tan 2 β 0 ) ]  ,
I p = 1 + 3 2 π 2 ( λ 0 d ) 2 sin β 0 cos 3 β 0 [ tan β 0 + p ( 1 + 4 tan 2 β 0 ) ]  .
E r ( β ) = 1 cos α cos β { sin [ cos α + cos ( 2 θ β ) 2 ] } 2 ,
E r ( β ) = 1 cos α cos β [ sin ( cos β + cos ( 2 θ α ) 2 ) ] 2 .
E r ( β 0 ) = cos α cos β 0
E r ( β 0 ) = cos β 0 cos α
E r ( β ) = 1 cos α cos β ( d d ) 2 { sin 2 × [ cos ( α + p ) + cos ( 2 θ β + p ) ] } 2 ,
E r ( β 0 ) = cos 2 ( α + p ) cos α cos β 0 ( d d ) 2 .
E r ( β 0 ) = cos α cos β 0 ( 1 2 R d q ) ,
q = tan α ( sin α cos θ ) + cos α + sin θ
q = tan α ( sin θ cos θ ) + cos θ + sin θ
E r ( β ) = 1 cos α cos β ( d d ) 2 { sin 2 × [ cos ( β + p ) + cos ( 2 θ α + p ) ] } 2 .
E r ( β 0 ) = cos 2 ( β 0 + p ) cos α cos β 0 ( d d ) 2
E r ( β 0 ) = cos β 0 cos α ( 1 2 R d q ) ,
q = tan β 0 ( sin β 0 cos θ ) + cos β 0 + sin θ
q = tan β 0 ( sin θ cos θ ) + cos θ + sin θ

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