Abstract

As the technology for manufacturing radial diffraction gratings improves, there will be increasing interest in using high-groove-density radial gratings as scales for precision interferometer-type rotary encoders. If efficiency in optical designs is to be optimized, the focusing properties of these gratings must be understood. We use analytical geometry to investigate the focusing properties of a radial diffraction grating illuminated by laser light diverging from a pointlike source. We compare the results with experiments that we performed with a state-of-the-art custom radial grating, and we obtain excellent agreement with our calculations, which improve on earlier analytical work.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. H. Stevenson, “Optical frequency shifting by means of a rotating diffraction grating,” Appl. Opt. 9, 649–652 (1970).
    [CrossRef] [PubMed]
  2. K. Matsumoto, “Method for optical detection and/or measurement of movement of a diffraction grating,” U.S. patent3,726,595 (April17, 1973).
  3. A. Kozlowska, “Fibre optic grating interferometer for in-plane displacement measurement,” in Interferometry ’94: Interferometric Fiber Sensing, E. Udd, R. P. Tatam, eds., Proc. SPIE2341, 124–131 (1994).
    [CrossRef]
  4. Linear grating interferometers are available from several companies, including Heidenhain Corporation, Schaumburg, Ill.
  5. A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14, 147–154 (1992).
    [CrossRef]
  6. Rotary grating interferometers are available from Canon U.S.A. Inc., Lake Success, N.Y.
  7. Conventional optical rotary encoders are commonly used for many applications and are a continuing focus for research efforts; see, for example, K. Engelhardt, P. Seitz, “Absolute, high-resolution optical position encoder,” Appl. Opt. 35, 201–208 (1996).
    [CrossRef] [PubMed]
  8. Yu Hong-Lin, “Research on the small-sized high-resolution radial grating,” in Measurement Technology and Intelligent Instruments, L. Zhu, ed., Proc. SPIE2101, 1168–1171 (1993).
    [CrossRef]
  9. S. V. Gordeev, B. G. Turukhano, “Investigation of the interference field of two spherical waves for holographic recording of precision radial gratings,” Opt. Laser Technol. 28, 225–261 (1996).
    [CrossRef]
  10. R. Sawada, H. Tanaka, O. Ohguchi, J. Shimada, S. Hara, “Fabrication of active integrated optical micro-encoder,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1991 (IEEE, New York, 1991), pp. 233–238.
  11. R. Sawada, O. Ohguchi, K. Mise, M. Tsubamoto, “Fabrication of advanced integrated optical micro-encoder chip,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1994 (IEEE, New York, 1994), pp. 337–342.
  12. M. Dobosz, “Application of a divergent laser beam in a grating interferometer for high-resolution displacement measurements,” Opt. Eng. 33, 897–901 (1994).
    [CrossRef]
  13. J. A. Koch, “Compact high-resolution soft x-ray spectrograph design using two matched grazing-incidence gratings,” Appl. Opt. 34, 3693–3701 (1995).
    [CrossRef] [PubMed]
  14. J. H. Underwood, J. A. Koch, “High resolution tunable spectrograph for x-ray laser line width measurements using a plane varied line spacing grating,” Appl. Opt. 36, 4913–4921 (1997).
    [CrossRef] [PubMed]
  15. J. Alonso, E. Bernabeu, “Spatial evolution of Gaussian beams diffracted by radial gratings,” Opt. Commun. 98, 323–330 (1993).
    [CrossRef]
  16. J. Alonso, E. Bernabeu, “Use of effective focal lengths to describe laser-beam evolution after diffraction in radial gratings,” J. Opt. Soc. Am. A 10, 1963–1970 (1993).
    [CrossRef]
  17. J. F. James, R. S. Sternberg, The Design of Optical Spectrometers (Chapman & Hall, London, 1969).
  18. M. C. Hettrick, S. Bowyer, “Variable line-space gratings: new designs for use in grazing incidence spectrometers,” Appl. Opt. 22, 3921–3924 (1983).
    [CrossRef] [PubMed]

1997 (1)

1996 (2)

Conventional optical rotary encoders are commonly used for many applications and are a continuing focus for research efforts; see, for example, K. Engelhardt, P. Seitz, “Absolute, high-resolution optical position encoder,” Appl. Opt. 35, 201–208 (1996).
[CrossRef] [PubMed]

S. V. Gordeev, B. G. Turukhano, “Investigation of the interference field of two spherical waves for holographic recording of precision radial gratings,” Opt. Laser Technol. 28, 225–261 (1996).
[CrossRef]

1995 (1)

1994 (1)

M. Dobosz, “Application of a divergent laser beam in a grating interferometer for high-resolution displacement measurements,” Opt. Eng. 33, 897–901 (1994).
[CrossRef]

1993 (2)

J. Alonso, E. Bernabeu, “Spatial evolution of Gaussian beams diffracted by radial gratings,” Opt. Commun. 98, 323–330 (1993).
[CrossRef]

J. Alonso, E. Bernabeu, “Use of effective focal lengths to describe laser-beam evolution after diffraction in radial gratings,” J. Opt. Soc. Am. A 10, 1963–1970 (1993).
[CrossRef]

1992 (1)

A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14, 147–154 (1992).
[CrossRef]

1983 (1)

1970 (1)

Alonso, J.

J. Alonso, E. Bernabeu, “Use of effective focal lengths to describe laser-beam evolution after diffraction in radial gratings,” J. Opt. Soc. Am. A 10, 1963–1970 (1993).
[CrossRef]

J. Alonso, E. Bernabeu, “Spatial evolution of Gaussian beams diffracted by radial gratings,” Opt. Commun. 98, 323–330 (1993).
[CrossRef]

Bernabeu, E.

J. Alonso, E. Bernabeu, “Spatial evolution of Gaussian beams diffracted by radial gratings,” Opt. Commun. 98, 323–330 (1993).
[CrossRef]

J. Alonso, E. Bernabeu, “Use of effective focal lengths to describe laser-beam evolution after diffraction in radial gratings,” J. Opt. Soc. Am. A 10, 1963–1970 (1993).
[CrossRef]

Bowyer, S.

Dobosz, M.

M. Dobosz, “Application of a divergent laser beam in a grating interferometer for high-resolution displacement measurements,” Opt. Eng. 33, 897–901 (1994).
[CrossRef]

Engelhardt, K.

Gordeev, S. V.

S. V. Gordeev, B. G. Turukhano, “Investigation of the interference field of two spherical waves for holographic recording of precision radial gratings,” Opt. Laser Technol. 28, 225–261 (1996).
[CrossRef]

Hara, S.

R. Sawada, H. Tanaka, O. Ohguchi, J. Shimada, S. Hara, “Fabrication of active integrated optical micro-encoder,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1991 (IEEE, New York, 1991), pp. 233–238.

Hettrick, M. C.

Hong-Lin, Yu

Yu Hong-Lin, “Research on the small-sized high-resolution radial grating,” in Measurement Technology and Intelligent Instruments, L. Zhu, ed., Proc. SPIE2101, 1168–1171 (1993).
[CrossRef]

James, J. F.

J. F. James, R. S. Sternberg, The Design of Optical Spectrometers (Chapman & Hall, London, 1969).

Koch, J. A.

Kozlowska, A.

A. Kozlowska, “Fibre optic grating interferometer for in-plane displacement measurement,” in Interferometry ’94: Interferometric Fiber Sensing, E. Udd, R. P. Tatam, eds., Proc. SPIE2341, 124–131 (1994).
[CrossRef]

Matsumoto, K.

K. Matsumoto, “Method for optical detection and/or measurement of movement of a diffraction grating,” U.S. patent3,726,595 (April17, 1973).

Mise, K.

R. Sawada, O. Ohguchi, K. Mise, M. Tsubamoto, “Fabrication of advanced integrated optical micro-encoder chip,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1994 (IEEE, New York, 1994), pp. 337–342.

Ohguchi, O.

R. Sawada, O. Ohguchi, K. Mise, M. Tsubamoto, “Fabrication of advanced integrated optical micro-encoder chip,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1994 (IEEE, New York, 1994), pp. 337–342.

R. Sawada, H. Tanaka, O. Ohguchi, J. Shimada, S. Hara, “Fabrication of active integrated optical micro-encoder,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1991 (IEEE, New York, 1991), pp. 233–238.

Sawada, R.

R. Sawada, H. Tanaka, O. Ohguchi, J. Shimada, S. Hara, “Fabrication of active integrated optical micro-encoder,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1991 (IEEE, New York, 1991), pp. 233–238.

R. Sawada, O. Ohguchi, K. Mise, M. Tsubamoto, “Fabrication of advanced integrated optical micro-encoder chip,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1994 (IEEE, New York, 1994), pp. 337–342.

Seitz, P.

Shimada, J.

R. Sawada, H. Tanaka, O. Ohguchi, J. Shimada, S. Hara, “Fabrication of active integrated optical micro-encoder,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1991 (IEEE, New York, 1991), pp. 233–238.

Sternberg, R. S.

J. F. James, R. S. Sternberg, The Design of Optical Spectrometers (Chapman & Hall, London, 1969).

Stevenson, W. H.

Tanaka, H.

R. Sawada, H. Tanaka, O. Ohguchi, J. Shimada, S. Hara, “Fabrication of active integrated optical micro-encoder,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1991 (IEEE, New York, 1991), pp. 233–238.

Teimel, A.

A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14, 147–154 (1992).
[CrossRef]

Tsubamoto, M.

R. Sawada, O. Ohguchi, K. Mise, M. Tsubamoto, “Fabrication of advanced integrated optical micro-encoder chip,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1994 (IEEE, New York, 1994), pp. 337–342.

Turukhano, B. G.

S. V. Gordeev, B. G. Turukhano, “Investigation of the interference field of two spherical waves for holographic recording of precision radial gratings,” Opt. Laser Technol. 28, 225–261 (1996).
[CrossRef]

Underwood, J. H.

Appl. Opt. (5)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

J. Alonso, E. Bernabeu, “Spatial evolution of Gaussian beams diffracted by radial gratings,” Opt. Commun. 98, 323–330 (1993).
[CrossRef]

Opt. Eng. (1)

M. Dobosz, “Application of a divergent laser beam in a grating interferometer for high-resolution displacement measurements,” Opt. Eng. 33, 897–901 (1994).
[CrossRef]

Opt. Laser Technol. (1)

S. V. Gordeev, B. G. Turukhano, “Investigation of the interference field of two spherical waves for holographic recording of precision radial gratings,” Opt. Laser Technol. 28, 225–261 (1996).
[CrossRef]

Precis. Eng. (1)

A. Teimel, “Technology and applications of grating interferometers in high-precision measurement,” Precis. Eng. 14, 147–154 (1992).
[CrossRef]

Other (8)

Rotary grating interferometers are available from Canon U.S.A. Inc., Lake Success, N.Y.

Yu Hong-Lin, “Research on the small-sized high-resolution radial grating,” in Measurement Technology and Intelligent Instruments, L. Zhu, ed., Proc. SPIE2101, 1168–1171 (1993).
[CrossRef]

K. Matsumoto, “Method for optical detection and/or measurement of movement of a diffraction grating,” U.S. patent3,726,595 (April17, 1973).

A. Kozlowska, “Fibre optic grating interferometer for in-plane displacement measurement,” in Interferometry ’94: Interferometric Fiber Sensing, E. Udd, R. P. Tatam, eds., Proc. SPIE2341, 124–131 (1994).
[CrossRef]

Linear grating interferometers are available from several companies, including Heidenhain Corporation, Schaumburg, Ill.

R. Sawada, H. Tanaka, O. Ohguchi, J. Shimada, S. Hara, “Fabrication of active integrated optical micro-encoder,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1991 (IEEE, New York, 1991), pp. 233–238.

R. Sawada, O. Ohguchi, K. Mise, M. Tsubamoto, “Fabrication of advanced integrated optical micro-encoder chip,” in Proceedings of the Micro Electro-Mechanical Systems Conference 1994 (IEEE, New York, 1994), pp. 337–342.

J. F. James, R. S. Sternberg, The Design of Optical Spectrometers (Chapman & Hall, London, 1969).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Geometry used for effective-focal-length calculations. The pointlike source is at P0 and diffracts off the grating annular track with central radius R and central groove spacing d0. Geometrical relations and conical-diffraction formulas can then be used to investigate the constraints necessary for the grating to produce a focus at point P2. The center of rotation of the grating is at point C, and the zero-order beam reflects to point Pzero.

Fig. 2
Fig. 2

Experimental setup used to test the analytical focal-length calculations. The diffracted focus measurements were made above and in the plane of the grating, which is the xy plane of Fig. 1.

Fig. 3
Fig. 3

Photograph of the diffracted beams from the experiments, taken at best focus in the xy plane of Fig. 1 with χ0=0. The source distance p was 12 mm, and the image distance q was 58 mm; the distance scale is approximate. At best focus the beams are sharply focused arcs, with depths of focus given approximately by the error bars.

Fig. 4
Fig. 4

Experimentally measured image distance q versus source distance p for χ0=0 (points) plotted against the calculated results from Eq. (18) (solid curve). The calculated results from Ref. 16 are also shown for comparison (dashed curve).

Fig. 5
Fig. 5

Experimentally measured image distance q versus source distance p for χ0=9.8° (points) plotted against the calculated results from Eq. (18) for χ0=9.8° (solid curve). The dashed curve shows the calculated results for χ0=0 from Eq. (18).

Equations (21)

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

P0=(p sin χ0, 0, p cos χ0),
P1=(0, 0, 0),
P2=(-q cos θ sin χd, q sin θ, q cos θ cos χd),
ri=(-sin χ0, 0, -cos χ0),
rd=-cos θ sin χd, sin θ, cos θ cos χd.
rd=rix, riy+nλd0, riz2-2nλd0 riy-n2λ2d021/2,
sin θ=nλd0,
sin χd=sin χ01-n2λ2d021/2=sin χ0cos θ.
P3=(-R, Rδ, 0),
ri=-sin χ0-R cos2 χ0p, Rδp, -cos χ0+R sin χ0 cos χ0p,
rd=-sin χ0-Rδ sin θ sin χ0q+R cos2 χ0q, sin θ+R sin θ sin χ0q-Rδ cos2 θq,1+Rq ( sin χ0+δ sin θ)(1-sin2 χ0-sin2 θ)1/2.
rd=rix+δ sin θ, riy+(sin θ)(1-),{riz2-2(sin θ)[rixδ+riy(1-)]-(sin2 θ)(1-2)}1/2,
R(cos2 χ0)1p+1q=(sin θ)1+R sin χ0qδ,
(sin θ)1+R sin χ0q=R1p+cos2 θqδ.
1p+1q|θ|R=|n|λd0R.
p=R cos χ0|sin θ|=Rd0 cos χ0|n|λ.
q=Rcos χ0|tan θ|-sin χ0.
(sin2 θ)1+R sin χ0q2
=R2(cos2 χ0)1p+cos2 θq1p+1q.
1p+1q=tan2 θ2p ±1+4p2R2 tan2 θ1/2-11f,
f=2p±1+4p2R2 tan2 θ1/2-1tan2 θ.

Metrics