Abstract

Focusing from a plane grating can be accomplished by using convergent radiation incident on the grating in such a manner that any incident angle αn, and the resulting diffraction angle βn, will be on the same side of the grating normal. The theory for the focal properties is developed by applying Fermat’s principle of least time to selected terms resulting from a finite series expansion of the system’s distance function. Derivations are given for finding the focal curve equation, astigmatism, and coma, of the most usable configuration of the optical components. Discussions of the aberrations disclose methods for eliminating the astigmatism and reducing the coma.

© 1966 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. S. Monk, J. Opt. Soc. Am. 17, 358 (1928).
    [CrossRef]
  2. H. T. Smyth, J. Opt. Soc. Am. 45, 312 (1935).
    [CrossRef]
  3. A. H. C. P. Gillieson, J. Sci. Instr. 26, 335 (1949).
    [CrossRef]
  4. H. C. Beutler, J. Opt. Soc. Am. 35, 311 (1945).
    [CrossRef]
  5. D. J. Schroeder, Appl. Opt. 5, 545 (1966).
    [CrossRef] [PubMed]
  6. M. V. R. K. Murty, J. Opt. Soc. Am. 52, 768 (1962).
    [CrossRef]

1966

1962

1949

A. H. C. P. Gillieson, J. Sci. Instr. 26, 335 (1949).
[CrossRef]

1945

1935

H. T. Smyth, J. Opt. Soc. Am. 45, 312 (1935).
[CrossRef]

1928

Beutler, H. C.

Gillieson, A. H. C. P.

A. H. C. P. Gillieson, J. Sci. Instr. 26, 335 (1949).
[CrossRef]

Monk, G. S.

Murty, M. V. R. K.

Schroeder, D. J.

Smyth, H. T.

H. T. Smyth, J. Opt. Soc. Am. 45, 312 (1935).
[CrossRef]

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 (2)

Fig. 1
Fig. 1

Geometry of convergent radiation diffracted by a plane grating.

Fig. 2
Fig. 2

Coordinate system for a general ray path.

Equations (28)

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

F = A ( x , y , z ) P ( , w , l ) + B ( x , y , z ) P ( , w , l ) ± ( w / a ) m λ .
A P = [ ( x - ) 2 + ( y - w ) 2 + ( z - l ) 2 ] 1 2
B P = [ ( x - ) 2 + ( y - w ) 2 + ( z - l ) 2 ] 1 2
( A P ) 2 = r 2 - 2 r sin α × w + w 2 + z 2 - 2 z l + l 2
( B P ) 2 = r 2 - 2 r sin β × w + w 2 + z 2 - 2 z l + l 2 .
A P r - w sin α + w 2 2 r + z 2 2 r - z l r + l 2 2 r - sin 2 α 2 r w 2 + sin α 2 r 2 w 3 + sin α 2 r 2 w z 2 - sin α r 2 w z l + sin α 2 r 2 w l 2 - w 4 8 r 3 - w 2 z 2 4 r 3 + w 2 z l 2 r 3 - w 2 l 2 4 r 3 - ( z - l ) 4 8 r 3 + r - w sin α + w 2 2 ( 1 r - sin 2 α r ) + ( z - l ) 2 2 r + w sin α 2 r 2 ( z - l ) 2 + w 3 2 r 2 sin α - w 4 8 r 3 - w 2 4 r 3 ( z - l ) 2 - ( z - l ) 4 8 r 3 + .
B P r - w sin β + w 2 2 ( 1 r - sin 2 β r ) + ( z - l ) 2 2 r + w sin β 2 r 2 ( z - l ) 2 + w 3 2 r 2 sin β + .
F = T 1 + T 1 ± ( w / a ) m λ + T 2 + T 2 + T 3 + T 3 + ,
T 1 + T 1 + ( w / a ) m λ = r - w sin α + r - w sin β ± ( w / a ) m λ = T 1 0 ,
± m λ / a = sin β h + sin α h .
T 2 0 = ( T 2 + T 2 ) = w 2 2 ( 1 r - sin 2 α r ) + w 2 2 ( 1 r - sin 2 β r ) = w 2 2 r ( cos 2 α ) + w 2 2 r ( cos 2 β ) T 2 0 w = w ( cos 2 α h r + cos 2 β h r ) = 0.
r n = - r h n ( cos 2 β h cos 2 α h ) .
r n = - r n [ ( cos 2 β h ) / ( cos 2 α h ) ] + [ f m / ( cos α m ) - f m ] × [ ( cos 2 β h ) / ( cos 2 α h ) ]
r n = [ r n 0 - 2 f m + ( f m / cos α m ) ] [ cos 2 β h ) / ( cos 2 α h ) ] .
( T 3 + T 3 ) = ( l - z ) 2 2 r + ( l - z ) 2 2 r = T 3 0 , T 3 0 = l 2 2 ( 1 r + 1 r ) - l ( z r + z r ) + z 2 2 r + z 2 2 r , T 3 0 l = l ( 1 r + 1 r ) - ( z r + z r ) .
T 3 0 l = l ( 1 r n - cos 2 α h r h n cos 2 β h ) + z cos 2 α h r h n cos 2 β h
z = - l [ ( cos 2 β h ) / ( cos 2 α h ) - 1 ] ,
z = - l ( cos 2 β h / cos 2 α h - 1 ) + z ( cos 2 β h / cos 2 α h ) .
z = l { [ r n 0 - 2 f m + ( f m / cos α m ) ] ( f m - r n 0 ) cos 2 α h cos 2 β h + 1 } - z { [ r n 0 - 2 f m + ( f m / cos α m ) ] ( f m - r m 0 ) cos 2 α h cos 2 β h } .
T 4 0 = ( T 4 + T 4 ) = w sin α 2 r n 2 ( z - l ) 2 + w 3 2 r 2 n s in α + w sin β 2 r n 2 ( z - l ) 2 + w 3 2 r m 2 sin β T 4 0 w = sin α h 2 r n 2 ( z - l ) 2 + sin β h 2 r n 2 ( z - l ) 2 + 3 w 2 2 r n 2 sin α h + 3 w 2 2 r n 2 sin β h .
T 4 0 w = sin α h 2 r n 2 ( z - l ) 2 + sin β h [ r n 0 - 2 f m + ( f m / cos α m ) ] 2 cos 4 β h ( z - l ) 2 2 r n 2 ( f m - r n 0 ) 2 cos 4 α h + 3 w 2 2 r n 2 sin α h + 3 w 2 2 r n 2 sin β h .
T 4 0 w = ( z - l ) 2 2 ( f m - r n 0 ) 2 ( sin β h + sin α h ) + 3 w 2 { sin β h cos 4 α h 2 [ r n 0 - 2 f m + ( f m / cos α m ) ] 2 cos 4 β h + sin α h 2 ( f m - r n 0 ) 2 } .
T 4 0 l = w sin α v r n 2 ( z - l ) + w sin β v r n 2 ( z - l ) .
T 4 0 l = w sin α v r n 2 ( z - l ) - w sin β v [ r n 0 - 2 f m + ( f m / cos α m ) ] r n 2 ( f m - r n 0 ) cos 2 α h cos 2 β h ( z - l ) .
T 4 0 l = w sin α v r n 2 ( z - l ) - w sin β v × cos 2 α h ( z - l ) [ r n 0 - 2 f m + ( f m / cos α m ) ] ( f m - r n 0 ) cos 2 β h .
T 4 0 l = w l ( z - l ) ( f m - r n 0 ) 2 { - cos 2 α h cos 2 β h [ r n 0 - 2 f m + ( f m / cos α m ) ] + 1 ( f m - r n 0 ) } .
T 4 0 w = ( z - l ) 2 2 r n 2 ( sin β h + sin α h ) + 3 w 2 2 r n 2 ( sin β h + sin α h )
T 4 0 / l = 2 w l ( z - l ) / r n 3 .

Metrics