Abstract

For lack of alternatives, echelle-grating diffraction behavior has in the past been modeled on scalar theory, despite observations that indicate significant deviations. To resolve this difficulty a detailed experimental, theoretical, and numerical study is performed for several echelles that work at low (8–13), medium (35–55), high (84–140), and very-high (to 660) diffraction orders. Noticeable deviations from the scalar model were detected both experimentally and numerically, on the basis of electromagnetic theory: (1) the shift of the observed blaze position was shown to decrease with the wavelength-to-period ratio, and it tends to zero more rapidly than the decrease of the maximum width, so that the TE- and TM-plane responses tend to merge into each other; (2) cut-off effects (Rayleigh anomalies) were found to play a significant role for high groove angles, where passing-off orders are close to the blaze order. A possibility for evaluation of the blaze angle from angular, rather than from spectral, measurements is discussed. Several reasons for the differences between real and ideal echelles (material-index deviations, profile deformations, and groove-angle errors) are analyzed, and their effects on the performance of echelles is studied.

© 1995 Optical Society of America

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References

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  1. G. R. Harrison, “The production of diffraction gratings: II. The design of echelle gratings and spectrographs,” J. Opt. Soc. Am. 39, 522–528 (1949).
    [CrossRef]
  2. G. R. Harrison, E. G. Loewen, R. S. Wiley, “Echelle gratings: their testing and improvement,” Appl. Opt. 15, 971–976 (1976).
    [CrossRef] [PubMed]
  3. W. M. Burton, N. K. Reay, “Echelle efficiency measurements in the ultraviolet,” Appl. Opt. 9, 1227–1229 (1970).
    [CrossRef] [PubMed]
  4. D. J. Schroeder, R. L. Hillard, “Echelle efficiencies: theory and experiment,” Appl. Opt. 19, 2833–2841 (1980).
    [CrossRef] [PubMed]
  5. R. G. Tull, “A comparison of holographic and echelle gratings in astronomical spectrometry,” in Proceedings of the Ninth Workshop on Instrumentation of Ground-Based Optical Astronomy, L. B. Robinson, ed. (Springer-Verlag, Berlin, 1988), pp. 104–117.
    [CrossRef]
  6. D. J. Schroeder, “An echelle spectrometer–spectrograph for astronomical use,” Appl. Opt. 6, 1976–1980 (1967).
    [CrossRef] [PubMed]
  7. R. C. M. Learner, “Spectrograph design 1918–68,” J. Phys. E 1, 589–594 (1968).
    [CrossRef]
  8. J. Kielkopf, “Echelle and holographic gratings compared for scattering and spectral resolution,” Appl. Opt. 20, 3327–3331 (1981).
    [CrossRef] [PubMed]
  9. S. Engman, P. Lindblom, “Multiechelle grating mountings with high spectral resolution and dispersion,” Appl. Opt. 21, 4363–4371 (1982).
    [CrossRef] [PubMed]
  10. P. Lindblom, F. Stenman, “Resolving power of multigrating spectrometers,” Appl. Opt. 28, 2542–2549 (1989).
    [CrossRef] [PubMed]
  11. M. Bottema, “Echelle efficiencies: theory and experiment; comment,” Appl. Opt. 20, 528–530 (1981).
    [CrossRef] [PubMed]
  12. D. J. Schroeder, “Echelle efficiencies: theory and experiment; author’s reply to comment,” Appl. Opt. 20, 530–531 (1981).
    [CrossRef] [PubMed]
  13. S. Engman, P. Lindblom, “Blaze characteristics of echelle gratings,” Appl. Opt. 21, 4356–4362 (1982).
    [CrossRef] [PubMed]
  14. F. Zhao, “A diffraction model for echelle gratings,” J. Mod. Opt. 38, 2241–2246 (1991).
    [CrossRef]
  15. J. W. Strutt, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
    [CrossRef]
  16. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  17. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1984), Vol. 21, pp. 1–67.
    [CrossRef]
  18. R. A. Brown, R. L. Hilliard, A. L. Phillips, “Actual blaze angle of the Bausch & Lomb R4 echelle grating,” Appl. Opt. 21, 167–168 (1982).
    [CrossRef] [PubMed]
  19. S. Engman, P. Lindblom, “Blaze angle of the Bausch & Lomb R4 echelle grating,” Appl. Opt. 22, 2512–2513 (1983).
    [CrossRef] [PubMed]
  20. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).
  21. A. Marechal, G. W. Stroke, “Sur l’origine des effets de polarisation et de diffraction dans les réseaux optiques,” C. R. Acad. Sci. 29, 2042–2044 (1959).
  22. L. Mashev, E. Popov, “Reflection gratings in conical diffraction mounting,” J. Opt. (Paris) 18, 3–8 (1987).
    [CrossRef]

1991 (1)

F. Zhao, “A diffraction model for echelle gratings,” J. Mod. Opt. 38, 2241–2246 (1991).
[CrossRef]

1989 (1)

1987 (1)

L. Mashev, E. Popov, “Reflection gratings in conical diffraction mounting,” J. Opt. (Paris) 18, 3–8 (1987).
[CrossRef]

1983 (1)

1982 (3)

1981 (3)

1980 (1)

1976 (1)

1970 (1)

1968 (1)

R. C. M. Learner, “Spectrograph design 1918–68,” J. Phys. E 1, 589–594 (1968).
[CrossRef]

1967 (1)

1959 (1)

A. Marechal, G. W. Stroke, “Sur l’origine des effets de polarisation et de diffraction dans les réseaux optiques,” C. R. Acad. Sci. 29, 2042–2044 (1959).

1949 (1)

1907 (1)

J. W. Strutt, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Bottema, M.

Brown, R. A.

Burton, W. M.

Engman, S.

Harrison, G. R.

Hillard, R. L.

Hilliard, R. L.

Kielkopf, J.

Learner, R. C. M.

R. C. M. Learner, “Spectrograph design 1918–68,” J. Phys. E 1, 589–594 (1968).
[CrossRef]

Lindblom, P.

Loewen, E. G.

Marechal, A.

A. Marechal, G. W. Stroke, “Sur l’origine des effets de polarisation et de diffraction dans les réseaux optiques,” C. R. Acad. Sci. 29, 2042–2044 (1959).

Mashev, L.

L. Mashev, E. Popov, “Reflection gratings in conical diffraction mounting,” J. Opt. (Paris) 18, 3–8 (1987).
[CrossRef]

Maystre, D.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1984), Vol. 21, pp. 1–67.
[CrossRef]

Phillips, A. L.

Popov, E.

L. Mashev, E. Popov, “Reflection gratings in conical diffraction mounting,” J. Opt. (Paris) 18, 3–8 (1987).
[CrossRef]

Reay, N. K.

Schroeder, D. J.

Stenman, F.

Stroke, G. W.

A. Marechal, G. W. Stroke, “Sur l’origine des effets de polarisation et de diffraction dans les réseaux optiques,” C. R. Acad. Sci. 29, 2042–2044 (1959).

Strutt, J. W.

J. W. Strutt, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Tull, R. G.

R. G. Tull, “A comparison of holographic and echelle gratings in astronomical spectrometry,” in Proceedings of the Ninth Workshop on Instrumentation of Ground-Based Optical Astronomy, L. B. Robinson, ed. (Springer-Verlag, Berlin, 1988), pp. 104–117.
[CrossRef]

Wiley, R. S.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Zhao, F.

F. Zhao, “A diffraction model for echelle gratings,” J. Mod. Opt. 38, 2241–2246 (1991).
[CrossRef]

Appl. Opt. (12)

J. Kielkopf, “Echelle and holographic gratings compared for scattering and spectral resolution,” Appl. Opt. 20, 3327–3331 (1981).
[CrossRef] [PubMed]

S. Engman, P. Lindblom, “Multiechelle grating mountings with high spectral resolution and dispersion,” Appl. Opt. 21, 4363–4371 (1982).
[CrossRef] [PubMed]

P. Lindblom, F. Stenman, “Resolving power of multigrating spectrometers,” Appl. Opt. 28, 2542–2549 (1989).
[CrossRef] [PubMed]

M. Bottema, “Echelle efficiencies: theory and experiment; comment,” Appl. Opt. 20, 528–530 (1981).
[CrossRef] [PubMed]

D. J. Schroeder, “Echelle efficiencies: theory and experiment; author’s reply to comment,” Appl. Opt. 20, 530–531 (1981).
[CrossRef] [PubMed]

S. Engman, P. Lindblom, “Blaze characteristics of echelle gratings,” Appl. Opt. 21, 4356–4362 (1982).
[CrossRef] [PubMed]

G. R. Harrison, E. G. Loewen, R. S. Wiley, “Echelle gratings: their testing and improvement,” Appl. Opt. 15, 971–976 (1976).
[CrossRef] [PubMed]

W. M. Burton, N. K. Reay, “Echelle efficiency measurements in the ultraviolet,” Appl. Opt. 9, 1227–1229 (1970).
[CrossRef] [PubMed]

D. J. Schroeder, R. L. Hillard, “Echelle efficiencies: theory and experiment,” Appl. Opt. 19, 2833–2841 (1980).
[CrossRef] [PubMed]

D. J. Schroeder, “An echelle spectrometer–spectrograph for astronomical use,” Appl. Opt. 6, 1976–1980 (1967).
[CrossRef] [PubMed]

R. A. Brown, R. L. Hilliard, A. L. Phillips, “Actual blaze angle of the Bausch & Lomb R4 echelle grating,” Appl. Opt. 21, 167–168 (1982).
[CrossRef] [PubMed]

S. Engman, P. Lindblom, “Blaze angle of the Bausch & Lomb R4 echelle grating,” Appl. Opt. 22, 2512–2513 (1983).
[CrossRef] [PubMed]

C. R. Acad. Sci. (1)

A. Marechal, G. W. Stroke, “Sur l’origine des effets de polarisation et de diffraction dans les réseaux optiques,” C. R. Acad. Sci. 29, 2042–2044 (1959).

J. Mod. Opt. (1)

F. Zhao, “A diffraction model for echelle gratings,” J. Mod. Opt. 38, 2241–2246 (1991).
[CrossRef]

J. Opt. (Paris) (1)

L. Mashev, E. Popov, “Reflection gratings in conical diffraction mounting,” J. Opt. (Paris) 18, 3–8 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. E (1)

R. C. M. Learner, “Spectrograph design 1918–68,” J. Phys. E 1, 589–594 (1968).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. W. Strutt, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Other (4)

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1984), Vol. 21, pp. 1–67.
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

R. G. Tull, “A comparison of holographic and echelle gratings in astronomical spectrometry,” in Proceedings of the Ninth Workshop on Instrumentation of Ground-Based Optical Astronomy, L. B. Robinson, ed. (Springer-Verlag, Berlin, 1988), pp. 104–117.
[CrossRef]

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Figures (22)

Fig. 1
Fig. 1

Schematic representation of the wave vectors and electric-field vectors of incident and diffracted-backward plane waves along the two facets of a triangular grating with a 90° apex angle. The two planes of polarization are TM and TE. E represents the electric field; the open circles indicate that the direction of the field is perpendicular to the paper in the TE case, and the circles with center dots represent that there is no center within these circles.

Fig. 2
Fig. 2

Numerical results showing the spectral dependence of the diffraction efficiency in the TM plane of a perfectly conducting r-2 echelle with a 90° apex angle and 31.6 grooves/mm. The angle of incidence θ i is equal to the facet angle ϕ B , which is equal to 64.48°. The vertical axis is the efficiency.

Fig. 3
Fig. 3

Numerical results showing the spectral dependence of the diffraction efficiency of an r-4 echelle with 79 grooves/mm. θ i = ϕ B = 76°. The solid curves represent the TE case, and the dashed curves represent the TM case for (a) a perfectly conducting substrate, (b) nAl = 1.2 + i7, (c) nAl = 1.09 + i5.31. The vertical axes are in arbitrary units.

Fig. 4
Fig. 4

Numerical results showing the spectral dependence of a large spectral interval on the diffraction efficiency of an r-4 echelle with 79 grooves/mm. θ i = ϕ B = 76°. The refractive index corresponds to the bulk values of aluminum, and its dispersion is taken into account. (a) TE and (b) TM polarization. The vertical axes are in arbitrary units.

Fig. 5
Fig. 5

Numerical results showing the spectral shift of the position of the maximum from the scalar position given by Eq. (3). A double-logarithmic scale is used. Squares represent an r-4 aluminum echelle with 79 grooves/mm and TE polarization; triangles represent the same but for the TM plane. Circles and crosses represent the TE- and TM-plane results, respectively, for an r-2 aluminum echelle with 79 grooves/mm. Diamonds represent the TE case for a perfectly conducting echelle with 43 grooves/mm and a 70° facet angle. The central part of (a) is shown enlarged in (b).

Fig. 6
Fig. 6

Record of numerical investigation: Efficiency in order 660 for the TE case with a perfectly conducting echelle with 42.713 grooves/mm. θ i = ϕ B = 70°. The position of the maximum (labeled real) is shifted from the ideal position (labeled expected) as determined from Eq. (3).

Fig. 7
Fig. 7

Numerical results showing the TE efficiency in orders 120 and 58 for the TE plane of an r-4 echelle with 79 grooves/mm. θ i = ϕ B = 76°. The passing-off positions of the adjacent orders are indicated at the tops of the graphs. Note the magnification of the ordinate for order 120, which was necessary to reveal the cut-off anomaly.

Fig. 8
Fig. 8

Dye-laser measurements of (a) the diffraction efficiency in several consecutive orders for an r-4 echelle with 79 grooves/mm, where θ i = ϕ B = 76°, and (b) the corresponding numerical results with n A 1 = 1.9 + i5.31. The vertical axes are in arbitrary units.

Fig. 9
Fig. 9

Numerically determined efficiency of the 84th order for an r-2 echelle with 31.6 grooves/mm and a 64.48° facet angle. (a) The angular dependence for several wavelength values, shown in nanometers in the figures. (b) A three-dimensional view of the efficiency peak in the wavelength and the angle of incidence planes.

Fig. 10
Fig. 10

Experimental angular dependencies of the efficiency of an aluminum r-4 echelle with 31.6 grooves/mm and a 76° facet angle. Wavelengths are shown in nanometers in the upper right-hand corner of each figure. Solid lines represent TE polarization and dashed lines represent TM.

Fig. 11
Fig. 11

Experimental angular dependencies of the efficiency of an aluminum r-2 echelle with 31.6 grooves/mm and a 63.5° facet angle. Wavelengths are shown in nanometers in the upper right-hand corner of each figure.

Fig. 12
Fig. 12

Numerical (theoretical) results that correspond to the experimental results from Figs. 11.

Fig. 13
Fig. 13

Experimental angular dependencies of the efficiency of an aluminum r-4 echelle with 79 grooves/mm and a 76° facet angle. Wavelengths are shown in the upper right-hand corners of each figure.

Fig. 14
Fig. 14

Numerical (theoretical) results in (a) and (b) correspond to the experimental results from Figs. 13. (c) Refractive index n = 1.09 + i5.31, and (d) n = 1.3 + i7.11.

Fig. 15
Fig. 15

Numerical investigation of an r-4 echelle with 79 grooves/mm: Influence of the passing-off effect on the −1 order and on the orders adjacent to peak one for several different wavelengths (shown in nanometers). The region in which the −1 order propagates is shown in the rectangular area in the upper part of each graph and the region in which the order next to peak one propagates is marked at the bottom of each graph.

Fig. 16
Fig. 16

Numerical results for n A 1 = 1.09 + i5.31. Two-dimensional views of (a) the TE and (b) the TM planes with corresponding isolines and their levels of the diffraction efficiency in the 38th order as a function of the wavelength and angle of incidence for an r-4 echelle with 79 grooves/mm and a 76° facet angle [corresponding to Figs. 13(k) and 14(c)].

Fig. 17
Fig. 17

Experimental angular dependencies of the diffraction efficiency of an r-2 echelle with 79 grooves/mm.

Fig. 18
Fig. 18

The same as for Figs. 17, except that a 316-grooves/mm echelle was used.

Fig. 19
Fig. 19

Numerical values that correspond to the experimental results in Figs. 18.

Fig. 20
Fig. 20

Numerical results that correspond to Fig. 14(d) (n = 1.3 + i7.11) but with a refractive-index value of n = 1.09 + i5.31.

Fig. 21
Fig. 21

Numerical comparison of the angular dependencies of the efficiency for a 31.6-grooves/mm r-2 echelle with two slightly different facet angles: 64.481° in (a) and (c), and 64.400° in (b) and (d), for two different wavelength values.

Fig. 22
Fig. 22

Representations of an r-2 echelle with 79 grooves/mm and a 63.5° facet angle: (a) SEM pictures of a thin-film replica cut in liquid nitrogen: left-hand side, top view; right-hand side, lateral view. (b) Drawings of the reconstructed (2), inverted (3), and ideal (1) profiles of the echelle; the ideal profile has a facet angle of 62.4° and apex angle of 84.2°. (c) Numerically determined spectral dependence of order 51 for profiles 1, 2, and 3 from (b). Solid curves represent TE polarization, and dashed curves represent TM.

Tables (2)

Tables Icon

Table 1 Experimental and Theoretical Values of the Apparent Facet Angle a ϕ ˜ B

Tables Icon

Table 2 Experimental Values of the Apparent Facet Angle a ϕ ˜ B

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

1 M 2 I g ( p ) I f B f = [ sin ( M k d p 2 ) M sin ( k d p 2 ) ] 2 [ sin ( k s p 2 ) k s p 2 ] 2 ,
p sin θ d + sin θ i = N M λ d .
2 sin ϕ B = N λ d ,
sin θ N = sin θ i + N λ d ,
λ B N 1 N .
λ B N λ B N + 1 1 N 1 N + 1 1 N 2 .
Δ max N 1 N 2 + δ .
λ B N TE λ B N TM λ B N TM λ B N + 1 TM = 1 N δ λ 0 { δ < 0 . 0 δ > 0
2 sin θ i = n λ d .
θ N = ϕ B Δ θ ,
θ i = ϕ B + Δ θ .
cos Δ θ = λ Δ λ B .
sin ( θ N C ) = N λ d 1 ,
sin [ θ ( 1 ) C ] = 1 λ d ,

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