Abstract

A class of coherent optical spectrum analyzers for one-dimensional signals is described that is characterized by a frequency variant response to the spectral components of the input signal. Operations performed exploit the second degree of freedom inherent in the optical systems. One example considered is a constant proportional bandwidth, log-frequency spectrum analyzer. Experimental and analytical results are presented.

© 1976 Optical Society of America

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References

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  1. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
    [CrossRef]
  2. L. Cutrona, in Optical and Electro-Optical Information Processing, J. T. Tippet et al., Eds. (MIT Press, Cambridge, 1965), pp. 83–123.
  3. C. E. Thomas, Appl. Opt. 5, 1782 (1966).
    [CrossRef] [PubMed]
  4. M. R. Mueller, F. P. Carlson, Appl. Opt. 14, 2207 (1975).
    [CrossRef] [PubMed]
  5. W. T. Rhodes, J. Opt. Soc. Am. 64, 545(A) (1974).
  6. J. L. Flanagan, Speech Analysis, Synthesis, and Perception (Springer-Verlag, New York, 1972), p. 149.
  7. O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974).
    [CrossRef]
  8. W. T. Rhodes, to be published in Opt. Commun.
  9. J. W. Goodman, Proc. IEEE. 64, No. 11 (1976).
  10. A. Papoulis, Systems and Transforms in Applications in Optics (McGraw-Hill, New York, 1968), p. 95.

1976 (1)

J. W. Goodman, Proc. IEEE. 64, No. 11 (1976).

1975 (1)

1974 (2)

W. T. Rhodes, J. Opt. Soc. Am. 64, 545(A) (1974).

O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974).
[CrossRef]

1966 (1)

1960 (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Bryngdahl, O.

Carlson, F. P.

Cutrona, L.

L. Cutrona, in Optical and Electro-Optical Information Processing, J. T. Tippet et al., Eds. (MIT Press, Cambridge, 1965), pp. 83–123.

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Flanagan, J. L.

J. L. Flanagan, Speech Analysis, Synthesis, and Perception (Springer-Verlag, New York, 1972), p. 149.

Goodman, J. W.

J. W. Goodman, Proc. IEEE. 64, No. 11 (1976).

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Mueller, M. R.

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms in Applications in Optics (McGraw-Hill, New York, 1968), p. 95.

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

Rhodes, W. T.

W. T. Rhodes, J. Opt. Soc. Am. 64, 545(A) (1974).

W. T. Rhodes, to be published in Opt. Commun.

Thomas, C. E.

Appl. Opt. (2)

IRE Trans. Inf. Theory (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
[CrossRef]

J. Opt. Soc. Am. (2)

O. Bryngdahl, J. Opt. Soc. Am. 64, 1092 (1974).
[CrossRef]

W. T. Rhodes, J. Opt. Soc. Am. 64, 545(A) (1974).

Proc. IEEE (1)

J. W. Goodman, Proc. IEEE. 64, No. 11 (1976).

Other (4)

A. Papoulis, Systems and Transforms in Applications in Optics (McGraw-Hill, New York, 1968), p. 95.

J. L. Flanagan, Speech Analysis, Synthesis, and Perception (Springer-Verlag, New York, 1972), p. 149.

W. T. Rhodes, to be published in Opt. Commun.

L. Cutrona, in Optical and Electro-Optical Information Processing, J. T. Tippet et al., Eds. (MIT Press, Cambridge, 1965), pp. 83–123.

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Figures (9)

Fig. 1
Fig. 1

Optical spectrum analyzer with continuously varying time frequency resolution. Lens pair L2, L3 images input recording in vertical direction (with top-to-bottom inversion); Fourier transforms it in horizontal direction.

Fig. 2
Fig. 2

Output of variable resolution spectrum analyzer. Single spectral component shown (a) results from hyperbolic input window (b). Multiple spectral lines (c) result with Ronchi ruling input (d).

Fig. 3
Fig. 3

Frequency mapping analyzer. Astigmatic lenses L1 and L2 image in vertical direction and Fourier transform in horizontal direction. Mask M contains slit that performs mapping operation. Mask in output plane is opaque except along vertical axis.

Fig. 4
Fig. 4

Frequency mapping spectrum analyzer: intermediate stages of operation. Mapping performed is assumed to be logarithmic; input signal consists of sinusoids with frequencies in the proportions 1:2:4:8.

Fig. 5
Fig. 5

Output of frequency mapping spectrum analyzer: (a) with straight line mapping slit (display is linear with frequency), (b) with curved mapping slit (approximately logarithmic).

Fig. 6
Fig. 6

Frequency variant spectrum analyzer. Mask in plane P1 determines the form of the system response to each frequency component of input signal. Mask in plane P2 performs frequency mapping operation.

Fig. 7
Fig. 7

Mapping plane resolution cell diagram showing limitations on mapping slit width.

Fig. 8
Fig. 8

Output of frequency variant spectrum analyzer set up for constant Q, log frequency analysis. (This particular result was obtained with inexpensive astigmatic opthalmic lenses.)

Fig. 9
Fig. 9

Example of a frequency variant spectrum analysis, showing input plane window (negative) (a), mapping plane distribution (b), mapping slit (c), and the output spectral distribution (d). The input was a Ronchi ruling.

Equations (30)

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u 2 ( ν , y ) = exp ( i α ν 2 ) F x [ u 1 ( x , y ) ] ,
u 1 ( x , y ) = f ( x ) w t ( x , y ) ,
u 2 ( ν , y ) = - F ( ξ ) W t ( ν - ξ , y ) d ξ ,
F ( ν ) = F x [ f ( x ) ]
W t ( ν , y ) = F x [ w t ( x , y ) ] .
f ( x ) = exp ( i 2 π ν o x )
u 2 ( ν , y ) = - δ ( ξ - ν o ) W t ( ν - ξ , y ) d ξ = W t ( ν - ν o , y ) ,
w t ( x , y ) = w t ( x y ) .
u 2 ( ν , y ) = 1 y W t ( ν - ν o y ) .
u 2 ( ν , y ) = F ( ν ) * W t ( ν , y ) ,
y = g ( ν ) ,
ν = h ( y ) = g - 1 ( y ) .
t m ( ν , y ) = m ( y ) δ [ ν - h ( y ) ] ,
m ( y ) = [ 1 + ( d h d y ) 2 ] 1 / 2 .
u 2 ( ν , y ) = [ F ( ν ) * W t ( ν , y ) ] m ( y ) δ [ ν - h ( y ) ] .
u 3 ( x , y ) = F ν - 1 { [ F ( ν ) * W t ( ν , y ) ] m ( y ) δ [ ν - h ( y ) ] } = [ f ( x ) w t ( x , y ) ] * { m ( y ) exp [ i 2 π h ( y ) x ] } ,
u 3 ( 0 , y ) = - f ( ξ ) w t ( ξ , y ) m ( y ) exp [ - i 2 π h ( y ) ξ ] d ξ .
w t ( x , y ) = w t [ x Δ ( y ) ] .
y = g ( ν ) = ln ( ν )
ν = h ( y ) = exp ( y ) .
1 Δ ( y ) exp ( y ) ,
Δ ( y ) exp ( - y ) .
u 2 ( ν , y ) = F x [ g ( x ) w t ( x , y ) ] ,
t m ( ν , y ) = W f [ ν - h ( y ) , y ] .
u 2 ( ν , y ) = F x [ f ( x ) w t ( x , y ) ] W f [ ν - h ( y ) , y ] ;
u 3 ( x , y ) = F ν - 1 { F x [ f ( x ) w t ( x , y ) ] W f [ ν - h ( y ) , y ] } = [ f ( x ) w t ( x , y ) ] * { w f ( x , y ) exp [ i 2 π h ( y ) x ] } ,
w f ( x , y ) = F ν - 1 [ W f ( ν , y ) ]
u 3 ( 0 , y ) = - f ( ξ ) w t ( ξ , y ) w f ( - ξ , y ) exp [ - i 2 π h ( y ) ξ ] d ξ .
w ( x , y ) = w t ( x , y ) w f ( - x , y ) .
u 3 ( x , y ) - f ( ξ ) w t ( ξ , y ) δ ( x - ξ ) exp [ i 2 π h ( y ) ( x - ξ ) ] d ξ f ( x ) w t ( x , y ) ;

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