Abstract

We have fabricated large, coarsely ruled, echelle patterns on silicon wafers by using photolithography and chemical-etching techniques. The grating patterns consist of 142-µm-wide, V-shaped grooves with an opening angle of 70.6°, blazed at 54.7°. We present a detailed description of our grating-fabrication techniques and the results of extensive testing. We have measured peak diffraction efficiencies of 70% at λ = 632.8 nm and conclude that the gratings produced by our method are of sufficient quality for use in high-resolution spectrographs in the visible and near IR (λ ≃ 500–5000 nm).

© 2000 Optical Society of America

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References

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  1. G. R. Harrison, “The production of diffraction gratings: the design of echelle gratings and spectrographs,” J. Opt. Soc. Am. 39, 522–528 (1949).
    [CrossRef]
  2. E. G. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), pp. 191–251.
  3. M. C. Hutley, Diffraction Gratings (Academic, London, UK, 1982).
  4. D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 1987), pp. 284–294.
  5. W. Tsang, S. Wang, “Preferentially etched diffraction gratings in silicon,” J. Appl. Phys. 46, 2163–2166 (1975).
    [CrossRef]
  6. Y. Fujii, “Optical demultiplexer using a silicon echelette grating,” IEEE J. Quantum Electron. QE-16, 165–169 (1980).
    [CrossRef]
  7. J. C. Greenwood, “Integrated fabrication of micromachined structures in silicon,” in Novel Silicon Based Technologies, R. A. Levy, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 123–141.
    [CrossRef]
  8. U. U. Graf, D. T. Jaffe, E. Kim, J. H. Lacy, H. Ling, J. T. Moore, G. Rebeiz, “Fabrication and evaluation of an etched infrared diffraction grating,” Appl. Opt. 33, 96–102 (1994).
    [CrossRef] [PubMed]
  9. G. Wiedemann, D. E. Jennings, “Immersion grating for infrared astronomy,” Appl. Opt. 32, 1176–1178 (1993).
    [CrossRef] [PubMed]
  10. P. J. Kuzmenko, D. R. Ciarlo, C. D. Stevens, “Fabrication and testing of a silicon immersion grating for infrared astronomy,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research, J. Wang, P. B. Hays, eds., Proc. SPIE2266, 566–577 (1994).
    [CrossRef]
  11. P. J. Kuzmenko, D. R. Ciarlo, “Improving the optical performance of etched silicon gratings,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 357–367 (1998).
    [CrossRef]
  12. D. T. Jaffe, L. D. Keller, O. A. Ershov, “Micromachined silicon diffraction gratings for infrared spectroscopy,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 201–212 (1998).
    [CrossRef]
  13. L. D. Keller, D. T. Jaffe, G. W. Doppmann, “Design for a near infrared immersion echelle spectrograph: breaking the R = 100,000 barrier from 1.5 to 5 µm,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 295–304 (1998).
  14. R. G. Tull, P. J. MacQueen, C. Sneden, D. Lambert, “The high-resolution cross-dispersed echelle white pupil spectrometer of the McDonald Observatory 2.7-m telescope,” Publ. Astron. Soc. Pac. 107, 251–264 (1995).
    [CrossRef]
  15. M. Kohketsu, S. Isome, “Effect of oxidizing ambient on oxygen precipitation in silicon crystals,” Jpn. J. Appl. Phys. 30, L1337–L1339 (1995).
    [CrossRef]
  16. D. L. Kendall, “On etching very narrow grooves in silicon,” Appl. Phys. Lett. 26, 195–198 (1974).
    [CrossRef]
  17. K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron Devices ED-25, 1185–1193 (1978).
    [CrossRef]
  18. H. A. Waggener, R. C. Kragness, A. L. Taylor, “Anisotropic etching for forming isolation slots,” Electronics 40, 274–276 (1967).
  19. T. Baum, D. J. Schiffrin, “AFM study of surface finish improvement by ultrasound in the anisotropic etching of Si (100) in KOH for micormachining applications,” J. Micromech. Microeng. 7, 338–342 (1997).
    [CrossRef]
  20. J. B. Price, “Anisotropic etching of silicon with KOH-H2O-isopropyl alcohol,” in Semiconductor Silicon, H. R. Huff, R. R. Burgess, eds. (The Electrochemical Society, Princeton, N.J., 1973), pp. 339–353.
  21. K. E. Petersen, “Silicon as a mechanical material,” Proc. IEEE 70, 420–457 (1982).
    [CrossRef]
  22. V. N. Mahajan, “Aberrated point-spread functions for rotationally symmetric aberrations,” Appl. Opt. 22, 3035–3041 (1983).
    [CrossRef] [PubMed]
  23. R. Barakat, “Total illumination of a diffraction image containing spherical aberration,” J. Opt. Soc. Am. 51, 152–157 (1961).
    [CrossRef]
  24. J. Ruze, “Antenna tolerance theory—a review,” Proc. IEEE 54, 633–640 (1966).
    [CrossRef]
  25. D. F. Gray, The Observation and Analysis of Stellar Photospheres (Cambridge U. Press, Cambridge, UK, 1992), p. 58.

1997 (1)

T. Baum, D. J. Schiffrin, “AFM study of surface finish improvement by ultrasound in the anisotropic etching of Si (100) in KOH for micormachining applications,” J. Micromech. Microeng. 7, 338–342 (1997).
[CrossRef]

1995 (2)

R. G. Tull, P. J. MacQueen, C. Sneden, D. Lambert, “The high-resolution cross-dispersed echelle white pupil spectrometer of the McDonald Observatory 2.7-m telescope,” Publ. Astron. Soc. Pac. 107, 251–264 (1995).
[CrossRef]

M. Kohketsu, S. Isome, “Effect of oxidizing ambient on oxygen precipitation in silicon crystals,” Jpn. J. Appl. Phys. 30, L1337–L1339 (1995).
[CrossRef]

1994 (1)

1993 (1)

1983 (1)

1982 (1)

K. E. Petersen, “Silicon as a mechanical material,” Proc. IEEE 70, 420–457 (1982).
[CrossRef]

1980 (1)

Y. Fujii, “Optical demultiplexer using a silicon echelette grating,” IEEE J. Quantum Electron. QE-16, 165–169 (1980).
[CrossRef]

1978 (1)

K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron Devices ED-25, 1185–1193 (1978).
[CrossRef]

1975 (1)

W. Tsang, S. Wang, “Preferentially etched diffraction gratings in silicon,” J. Appl. Phys. 46, 2163–2166 (1975).
[CrossRef]

1974 (1)

D. L. Kendall, “On etching very narrow grooves in silicon,” Appl. Phys. Lett. 26, 195–198 (1974).
[CrossRef]

1967 (1)

H. A. Waggener, R. C. Kragness, A. L. Taylor, “Anisotropic etching for forming isolation slots,” Electronics 40, 274–276 (1967).

1966 (1)

J. Ruze, “Antenna tolerance theory—a review,” Proc. IEEE 54, 633–640 (1966).
[CrossRef]

1961 (1)

1949 (1)

Barakat, R.

Baum, T.

T. Baum, D. J. Schiffrin, “AFM study of surface finish improvement by ultrasound in the anisotropic etching of Si (100) in KOH for micormachining applications,” J. Micromech. Microeng. 7, 338–342 (1997).
[CrossRef]

Bean, K. E.

K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron Devices ED-25, 1185–1193 (1978).
[CrossRef]

Ciarlo, D. R.

P. J. Kuzmenko, D. R. Ciarlo, C. D. Stevens, “Fabrication and testing of a silicon immersion grating for infrared astronomy,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research, J. Wang, P. B. Hays, eds., Proc. SPIE2266, 566–577 (1994).
[CrossRef]

P. J. Kuzmenko, D. R. Ciarlo, “Improving the optical performance of etched silicon gratings,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 357–367 (1998).
[CrossRef]

Doppmann, G. W.

L. D. Keller, D. T. Jaffe, G. W. Doppmann, “Design for a near infrared immersion echelle spectrograph: breaking the R = 100,000 barrier from 1.5 to 5 µm,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 295–304 (1998).

Ershov, O. A.

D. T. Jaffe, L. D. Keller, O. A. Ershov, “Micromachined silicon diffraction gratings for infrared spectroscopy,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 201–212 (1998).
[CrossRef]

Fujii, Y.

Y. Fujii, “Optical demultiplexer using a silicon echelette grating,” IEEE J. Quantum Electron. QE-16, 165–169 (1980).
[CrossRef]

Graf, U. U.

Gray, D. F.

D. F. Gray, The Observation and Analysis of Stellar Photospheres (Cambridge U. Press, Cambridge, UK, 1992), p. 58.

Greenwood, J. C.

J. C. Greenwood, “Integrated fabrication of micromachined structures in silicon,” in Novel Silicon Based Technologies, R. A. Levy, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 123–141.
[CrossRef]

Harrison, G. R.

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, London, UK, 1982).

Isome, S.

M. Kohketsu, S. Isome, “Effect of oxidizing ambient on oxygen precipitation in silicon crystals,” Jpn. J. Appl. Phys. 30, L1337–L1339 (1995).
[CrossRef]

Jaffe, D. T.

U. U. Graf, D. T. Jaffe, E. Kim, J. H. Lacy, H. Ling, J. T. Moore, G. Rebeiz, “Fabrication and evaluation of an etched infrared diffraction grating,” Appl. Opt. 33, 96–102 (1994).
[CrossRef] [PubMed]

L. D. Keller, D. T. Jaffe, G. W. Doppmann, “Design for a near infrared immersion echelle spectrograph: breaking the R = 100,000 barrier from 1.5 to 5 µm,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 295–304 (1998).

D. T. Jaffe, L. D. Keller, O. A. Ershov, “Micromachined silicon diffraction gratings for infrared spectroscopy,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 201–212 (1998).
[CrossRef]

Jennings, D. E.

Keller, L. D.

D. T. Jaffe, L. D. Keller, O. A. Ershov, “Micromachined silicon diffraction gratings for infrared spectroscopy,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 201–212 (1998).
[CrossRef]

L. D. Keller, D. T. Jaffe, G. W. Doppmann, “Design for a near infrared immersion echelle spectrograph: breaking the R = 100,000 barrier from 1.5 to 5 µm,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 295–304 (1998).

Kendall, D. L.

D. L. Kendall, “On etching very narrow grooves in silicon,” Appl. Phys. Lett. 26, 195–198 (1974).
[CrossRef]

Kim, E.

Kohketsu, M.

M. Kohketsu, S. Isome, “Effect of oxidizing ambient on oxygen precipitation in silicon crystals,” Jpn. J. Appl. Phys. 30, L1337–L1339 (1995).
[CrossRef]

Kragness, R. C.

H. A. Waggener, R. C. Kragness, A. L. Taylor, “Anisotropic etching for forming isolation slots,” Electronics 40, 274–276 (1967).

Kuzmenko, P. J.

P. J. Kuzmenko, D. R. Ciarlo, “Improving the optical performance of etched silicon gratings,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 357–367 (1998).
[CrossRef]

P. J. Kuzmenko, D. R. Ciarlo, C. D. Stevens, “Fabrication and testing of a silicon immersion grating for infrared astronomy,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research, J. Wang, P. B. Hays, eds., Proc. SPIE2266, 566–577 (1994).
[CrossRef]

Lacy, J. H.

Lambert, D.

R. G. Tull, P. J. MacQueen, C. Sneden, D. Lambert, “The high-resolution cross-dispersed echelle white pupil spectrometer of the McDonald Observatory 2.7-m telescope,” Publ. Astron. Soc. Pac. 107, 251–264 (1995).
[CrossRef]

Ling, H.

Loewen, E. G.

E. G. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), pp. 191–251.

MacQueen, P. J.

R. G. Tull, P. J. MacQueen, C. Sneden, D. Lambert, “The high-resolution cross-dispersed echelle white pupil spectrometer of the McDonald Observatory 2.7-m telescope,” Publ. Astron. Soc. Pac. 107, 251–264 (1995).
[CrossRef]

Mahajan, V. N.

Moore, J. T.

Petersen, K. E.

K. E. Petersen, “Silicon as a mechanical material,” Proc. IEEE 70, 420–457 (1982).
[CrossRef]

Popov, E.

E. G. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), pp. 191–251.

Price, J. B.

J. B. Price, “Anisotropic etching of silicon with KOH-H2O-isopropyl alcohol,” in Semiconductor Silicon, H. R. Huff, R. R. Burgess, eds. (The Electrochemical Society, Princeton, N.J., 1973), pp. 339–353.

Rebeiz, G.

Ruze, J.

J. Ruze, “Antenna tolerance theory—a review,” Proc. IEEE 54, 633–640 (1966).
[CrossRef]

Schiffrin, D. J.

T. Baum, D. J. Schiffrin, “AFM study of surface finish improvement by ultrasound in the anisotropic etching of Si (100) in KOH for micormachining applications,” J. Micromech. Microeng. 7, 338–342 (1997).
[CrossRef]

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 1987), pp. 284–294.

Sneden, C.

R. G. Tull, P. J. MacQueen, C. Sneden, D. Lambert, “The high-resolution cross-dispersed echelle white pupil spectrometer of the McDonald Observatory 2.7-m telescope,” Publ. Astron. Soc. Pac. 107, 251–264 (1995).
[CrossRef]

Stevens, C. D.

P. J. Kuzmenko, D. R. Ciarlo, C. D. Stevens, “Fabrication and testing of a silicon immersion grating for infrared astronomy,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research, J. Wang, P. B. Hays, eds., Proc. SPIE2266, 566–577 (1994).
[CrossRef]

Taylor, A. L.

H. A. Waggener, R. C. Kragness, A. L. Taylor, “Anisotropic etching for forming isolation slots,” Electronics 40, 274–276 (1967).

Tsang, W.

W. Tsang, S. Wang, “Preferentially etched diffraction gratings in silicon,” J. Appl. Phys. 46, 2163–2166 (1975).
[CrossRef]

Tull, R. G.

R. G. Tull, P. J. MacQueen, C. Sneden, D. Lambert, “The high-resolution cross-dispersed echelle white pupil spectrometer of the McDonald Observatory 2.7-m telescope,” Publ. Astron. Soc. Pac. 107, 251–264 (1995).
[CrossRef]

Waggener, H. A.

H. A. Waggener, R. C. Kragness, A. L. Taylor, “Anisotropic etching for forming isolation slots,” Electronics 40, 274–276 (1967).

Wang, S.

W. Tsang, S. Wang, “Preferentially etched diffraction gratings in silicon,” J. Appl. Phys. 46, 2163–2166 (1975).
[CrossRef]

Wiedemann, G.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

D. L. Kendall, “On etching very narrow grooves in silicon,” Appl. Phys. Lett. 26, 195–198 (1974).
[CrossRef]

Electronics (1)

H. A. Waggener, R. C. Kragness, A. L. Taylor, “Anisotropic etching for forming isolation slots,” Electronics 40, 274–276 (1967).

IEEE J. Quantum Electron. (1)

Y. Fujii, “Optical demultiplexer using a silicon echelette grating,” IEEE J. Quantum Electron. QE-16, 165–169 (1980).
[CrossRef]

IEEE Trans. Electron Devices (1)

K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron Devices ED-25, 1185–1193 (1978).
[CrossRef]

J. Appl. Phys. (1)

W. Tsang, S. Wang, “Preferentially etched diffraction gratings in silicon,” J. Appl. Phys. 46, 2163–2166 (1975).
[CrossRef]

J. Micromech. Microeng. (1)

T. Baum, D. J. Schiffrin, “AFM study of surface finish improvement by ultrasound in the anisotropic etching of Si (100) in KOH for micormachining applications,” J. Micromech. Microeng. 7, 338–342 (1997).
[CrossRef]

J. Opt. Soc. Am. (2)

Jpn. J. Appl. Phys. (1)

M. Kohketsu, S. Isome, “Effect of oxidizing ambient on oxygen precipitation in silicon crystals,” Jpn. J. Appl. Phys. 30, L1337–L1339 (1995).
[CrossRef]

Proc. IEEE (2)

K. E. Petersen, “Silicon as a mechanical material,” Proc. IEEE 70, 420–457 (1982).
[CrossRef]

J. Ruze, “Antenna tolerance theory—a review,” Proc. IEEE 54, 633–640 (1966).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

R. G. Tull, P. J. MacQueen, C. Sneden, D. Lambert, “The high-resolution cross-dispersed echelle white pupil spectrometer of the McDonald Observatory 2.7-m telescope,” Publ. Astron. Soc. Pac. 107, 251–264 (1995).
[CrossRef]

Other (10)

E. G. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), pp. 191–251.

M. C. Hutley, Diffraction Gratings (Academic, London, UK, 1982).

D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 1987), pp. 284–294.

J. C. Greenwood, “Integrated fabrication of micromachined structures in silicon,” in Novel Silicon Based Technologies, R. A. Levy, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1991), pp. 123–141.
[CrossRef]

P. J. Kuzmenko, D. R. Ciarlo, C. D. Stevens, “Fabrication and testing of a silicon immersion grating for infrared astronomy,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research, J. Wang, P. B. Hays, eds., Proc. SPIE2266, 566–577 (1994).
[CrossRef]

P. J. Kuzmenko, D. R. Ciarlo, “Improving the optical performance of etched silicon gratings,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 357–367 (1998).
[CrossRef]

D. T. Jaffe, L. D. Keller, O. A. Ershov, “Micromachined silicon diffraction gratings for infrared spectroscopy,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 201–212 (1998).
[CrossRef]

L. D. Keller, D. T. Jaffe, G. W. Doppmann, “Design for a near infrared immersion echelle spectrograph: breaking the R = 100,000 barrier from 1.5 to 5 µm,” in Infrared Astronomical Instrumentation, A. M. Fowler, ed., Proc. SPIE3354, 295–304 (1998).

D. F. Gray, The Observation and Analysis of Stellar Photospheres (Cambridge U. Press, Cambridge, UK, 1992), p. 58.

J. B. Price, “Anisotropic etching of silicon with KOH-H2O-isopropyl alcohol,” in Semiconductor Silicon, H. R. Huff, R. R. Burgess, eds. (The Electrochemical Society, Princeton, N.J., 1973), pp. 339–353.

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Figures (9)

Fig. 1
Fig. 1

Schematic of our silicon micromachining processing steps (Section 3). The process begins at the top of the diagram, and each step is illustrated on a cross section of a silicon wafer with the (100) crystal lattice plane perpendicular to the paper plane. The orientation of each cross section is such that the grooves are seen end-on. The thicknesses of coatings and layers and the widths of grooves are exaggerated to make the illustration more clear. Details are in the text and refer to (a)–(i).

Fig. 2
Fig. 2

Edge-on view for comparison of groove geometry for etched and ruled gratings: (a) The 70.6° opening angle groove profiles of an etched echelle compared with (b) the 90° profiles of ruled echelle grooves. For the etched grooves the crystal lattices that define the geometry are shown; note the flat intergroove spaces, an artifact of the lithography and etching process Subsection (3.C).

Fig. 3
Fig. 3

SEM images of preliminary tests fabricating gratings with 25-µm-wide grooves produced by the same process as for the 142-µm-wide grooves. We emphasize uncharacteristically flawed areas on an otherwise good grating. The σ = 142-µm gratings contain the same defects so the illustration is useful for understanding the coarser gratings as well. (a) SEM images of an etched silicon echelle with examples of groove breaks (centered in the image) at 33×; the white horizontal bar at the base of the image is 1 mm long. Note that the slightly diagonal pattern of darkened stripes is a moiré pattern in the SEM display and not intrinsic to the grating pattern. (b) SEM images of groove breaks at 300×; the white horizontal bar is 100 µm long. Note also the partially broken groove wall three grooves to the right of the breaks.

Fig. 4
Fig. 4

SEM images of the same echelle as in Fig. 3 but focused on line jogs (Subsection 4.A) at (a) 300× (the white horizontal bar is 100 µm long) and (b) 600× (the white horizontal bar is 10 µm long). The wide vertical stripes are groove tops, and you can see the SiO2 mask material (passivation layer) as very narrow white stripes on either side of the groove tops. The white SiO2 layer is much more visible over the jog where the mask material was severely undercut. Dark narrow vertical stripes are the sharp groove bottoms. In the higher magnification image (b), the jog is centered.

Fig. 5
Fig. 5

Optical test setup. Red (λ = 632.8-nm) and green (λ = 543.5-nm) He–Ne lasers project coaligned beams to a spatially filtered Keplarian beam expander through an iris and to a beam splitter. The beam transmitted to Arm 1 hits our grating test sample in Littrow and returns to the beam splitter where it reflects to a camera lens and comes to a focus on the CCD. The beam reflected to Arm 2 hits a flat reference mirror and reflects to the camera, which focuses the reference image on the CCD to one side of the spectrum. This setup is a spectrograph that we can configure as a Twymann–Green interferometer by realigning the mirror in Arm 2 and moving the camera lens to image the exit pupil of the system (details in Subsection 4.B).

Fig. 6
Fig. 6

Spectra measured with our spectrograph (Subsection 4.B and Fig. 5) by using our etched echelle sample with σ = 142 µm. The tick marks on the ordinate axis measure the spectral line intensity on an arbitrary scale. (a) Spectrum at λ = 543.5 nm, spectral orders of 425–429. (b) Spectrum at λ = 543.5 nm with the intensity scale magnified to emphasize the light spread into the blaze and to show the Gaussian fit to this signal.

Fig. 7
Fig. 7

Spectra measured with our spectrograph (Subsection 4.B and Fig. 5) by using our etched sample with 142-µm-wide grooves. (a) Spectrum at λ = 632.8 nm, spectral orders of 365–368. (b) Spectrum at λ = 632.8 nm with the intensity scale magnified and an overlay of the Gaussian fit to the signal spread within the brightest order.

Fig. 8
Fig. 8

(a) An interferogram in green (λ = 543.5-nm) He–Ne light of an unetched silicon wafer sample (see Subsection 4.B for more details). (b) An interferogram in green (λ = 543.5-nm) He–Ne light of an etched echelle grating in a Littrow configuration. The fringe patterns clearly indicate the presence of spherical aberration and astigmatism caused by warps in the silicon substrates that we used to fabricate our gratings. In both interferograms the interferometer aperture was 20 mm.

Fig. 9
Fig. 9

Computer simulations of (a) green He–Ne (λ = 543.5 nm) and (b) red He–Ne (λ = 632.8-nm) diffraction spectra produced with a model grating that simulates gratings that we have fabricated and tested. With our simulation code we produce model spectra by displaying the Fourier transform of a user-specified grating pattern. We input the grating length, groove spacing and width, wavelength, and the period and amplitude of groove-spacing errors. These model spectra simulate the observed spectra with the Rowland ghosts presented in Figs. 6 and 7. The intensity scale is arbitrary (see Subsection 5.B for more details).

Equations (8)

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

mλ=2σ sinδ,
R=λΔλ=mN=2Nσ sinδλ.
phase=22πλΔσrms,
η=η0 exp-phase2,
η=η0 exp-2=η0 exp-22 πΔσrmsλ2.
Δλghostλ=±MσmD,
Δλghost=±ΔλFSRσD.
IghostIline=πmAσ2,

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