Abstract

A theorem is proved which relates the number of degrees of freedom in the image space of any lens to the light-gathering effectiveness of the lens. Under appropriate approximations this theorem is equivalent to Gabor’s expansion theorem. If the number of degrees of freedom is small, these approximations are not warranted and true superresolution becomes a possibility.

© 1967 Optical Society of America

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References

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  1. D. Gabor, in Progress in Optics, Vol. I, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1961).
  2. K. Miyamoto, J. Opt. Soc. Am. 50, 865 (1960); J. Opt. Soc. Am. 51, 910 (1961).
    [CrossRef]
  3. H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).
    [CrossRef]
  4. L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965). At the request of the editor of J. Opt. Soc. Am., “light-gathering effectiveness” is used in this paper in place of the term “photometric throughput” used by Mertz.
  5. A. Walther, Am. J. Phys. 35, June (1967).
    [CrossRef]
  6. E. Schmidt, Math Ann. 63, 433 (1907).
    [CrossRef]
  7. Note that this also involves a Gabor-type theorem, which has been firmly established, however, by Landau and Pollak.3
  8. C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).
    [CrossRef]
  9. D. Slepian, J. Math. Phys. 44, 99 (1965).
  10. G. Lansraux, J. Opt. Soc. Am. 55, 595A (1965).
  11. M. Kline and E. Kay, Electromagnetic Theory and Geometrical Optics (Interscience Publishers, John Wiley & Sons, Inc., New York, 1965).

1967 (1)

A. Walther, Am. J. Phys. 35, June (1967).
[CrossRef]

1966 (1)

1965 (2)

D. Slepian, J. Math. Phys. 44, 99 (1965).

G. Lansraux, J. Opt. Soc. Am. 55, 595A (1965).

1962 (1)

H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).
[CrossRef]

1960 (1)

1907 (1)

E. Schmidt, Math Ann. 63, 433 (1907).
[CrossRef]

Barnes, C. W.

Gabor, D.

D. Gabor, in Progress in Optics, Vol. I, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1961).

Kay, E.

M. Kline and E. Kay, Electromagnetic Theory and Geometrical Optics (Interscience Publishers, John Wiley & Sons, Inc., New York, 1965).

Kline, M.

M. Kline and E. Kay, Electromagnetic Theory and Geometrical Optics (Interscience Publishers, John Wiley & Sons, Inc., New York, 1965).

Landau, H. J.

H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).
[CrossRef]

Lansraux, G.

G. Lansraux, J. Opt. Soc. Am. 55, 595A (1965).

Mertz, L.

L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965). At the request of the editor of J. Opt. Soc. Am., “light-gathering effectiveness” is used in this paper in place of the term “photometric throughput” used by Mertz.

Miyamoto, K.

Pollak, H. O.

H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).
[CrossRef]

Schmidt, E.

E. Schmidt, Math Ann. 63, 433 (1907).
[CrossRef]

Slepian, D.

D. Slepian, J. Math. Phys. 44, 99 (1965).

Walther, A.

A. Walther, Am. J. Phys. 35, June (1967).
[CrossRef]

Am. J. Phys. (1)

A. Walther, Am. J. Phys. 35, June (1967).
[CrossRef]

Bell System Tech. J. (1)

H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).
[CrossRef]

J. Math. Phys. (1)

D. Slepian, J. Math. Phys. 44, 99 (1965).

J. Opt. Soc. Am. (3)

Math Ann. (1)

E. Schmidt, Math Ann. 63, 433 (1907).
[CrossRef]

Other (4)

Note that this also involves a Gabor-type theorem, which has been firmly established, however, by Landau and Pollak.3

M. Kline and E. Kay, Electromagnetic Theory and Geometrical Optics (Interscience Publishers, John Wiley & Sons, Inc., New York, 1965).

D. Gabor, in Progress in Optics, Vol. I, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1961).

L. Mertz, Transformations in Optics (John Wiley & Sons, Inc., New York, 1965). At the request of the editor of J. Opt. Soc. Am., “light-gathering effectiveness” is used in this paper in place of the term “photometric throughput” used by Mertz.

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Figures (2)

Fig. 1
Fig. 1

Coordinate systems used to discuss optical instruments.

Fig. 2
Fig. 2

Energy transmittance as a function of mode number.

Equations (57)

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N G = ( 1 / λ 2 ) A Ω
u ( x , y , z ) = 1 N G C n α n ( x , y , z ) ,
ϕ = B A Ω ,
T = ϕ / B = A Ω = λ 2 N G .
u ( x , y , z ) = u ˆ ( L , M ) exp i k ( L x + M y + N z ) d L d M ,
N = ( 1 - L 2 - M 2 ) 1 2 .
u ˆ ( L , M ) = K ( L , M , L , M ) u ˆ ( L , M ) d L d M .
ϕ = N u ˆ ( L , M ) 2 d L d M ,
ϕ = N u ˆ ( L , M ) 2 d L d M .
N φ i ( L , M ) φ j * ( L , M ) d L d M = δ i j .
τ N u ˆ ( L , M ) 2 d L d L M = E ( L , M , L ¯ , M ¯ ) u ˆ ( L , M ) · u ˆ * ( L ¯ , M ¯ ) d L d M d L ¯ d M ¯ ,
E ( L , M , L ¯ , M ¯ ) = N K ( L , M , L , M ) K * ( L ¯ , M ¯ , L , M ) · d L d M ,
τ i N φ i ( L , M ) = E ( L , M , L ¯ , M ¯ ) φ i ( L ¯ , M ¯ ) d L d M .
ψ i ( L , M ) = K ( L , M , L , M ) φ i ( L , M ) d L d M .
N ψ i ( L , M ) 2 d L d M = τ i .
K ( L , M , L , M ) = i N φ i * ( L , M ) ψ i ( L , M ) .
N ψ i ( L , M ) ψ j * ( L , M ) d L d M = τ i δ i j ,
i τ i = N N K ( L , M , L , M ) 2 d L d M d L d M .
B = 2 ( 1 / λ 2 ) { h ν / [ exp ( h ν / k T ) - 1 ] } Δ ν .
h ν / [ exp ( h ν / k T ) - 1 ] ,
t Δ ν { h ν / [ exp ( h ν / k T ) - 1 ] } .
ϕ t = t Δ ν [ h ν / exp ( h ν / k T ) - 1 ] · 2 · i τ i ,
T = ϕ / B = λ 2 i τ i .
T = λ 2 N N K ( L , M , L , M ) 2 d L d M d L d M .
K = A exp i k f ( L L + M M )
L < p / f ,             M < p / f ,             L < p / f ,             M < p / f ,
E ( L , M , L ¯ , M ¯ ) = A 2 λ 2 f 2 sin k p ( L - L ¯ ) π ( L - L ¯ ) sin k p ( M - M ¯ ) π ( M - M ¯ ) ,
A = f / λ .
T = λ 2 ( f λ ) 2 d L d M d L d M ,
T = λ 2 ( f / λ ) 2 ( 2 p / f ) 2 ( 2 p / f ) 2 , = ( 2 p ) 2 ( 2 p / f ) 2 , = ( 2 p ) 2 ( 2 p / f ) 2 ,
u ˆ ( L , M ) = C i φ i ( L , M ) ,
u ˆ ( L , M ) = C i ψ i ( L , M ) .
T = λ 2 τ i ,
N e = ( 1 / τ 1 ) τ i .
N e = ( 1 / τ 1 λ 2 ) T ,
N s = 1 τ 1 i > N e τ i
τ i φ i ( L ) = - p / f p / f φ i ( L ) sin k p ( L - L ) π ( L - L ) d L .
τ n { 1 + exp [ ( n - N e ) / 2 ln 1 2 π N e ] } - 1 ,
N e = ( 1 / λ ) 2 p ( 2 p / f ) .
N e = ( 1 / λ 2 f ) 2 ( 2 p ) 2 ( 2 p ) 2 ,
Δ x = [ ( 2 p ) 2 / N e ] 1 2 ,
Δ x = f ( λ / 2 p ) .
C i C j * ,             i N e ,             j N e ,
A ( x , y , z , t ) = 2 Re A ( x , y , z ) e - i ω t = 2 Re e - i ω t  ( L , M ) e i k ( L x + M y + N z ) d L d M .
n · Â ( L , M ) = 0 ,
A ( r L , r M , r N , t ) = 2 Re ( λ / i r ) N Â ( L , M ) e i ( k r - ω t ) .
E = - A / t ,             B = × A .
P = ( 8 π 2 c / μ 0 r 2 ) N 2 Â 2 n .
d Ω = d L d M / N ,
ϕ = P · n r 2 d Ω = 8 π 2 c / μ 0 N Â ( L , M ) 2 d L d M .
u ( x , y , z , t ) = 2 Re u ( x , y , z ) e - i ω t = 2 Re e - i ω t u ˆ ( L , M ) e i k ( L x + M y + N z ) d L d M ,
P · n d S = - d d t D d V ,
D = 1 2 ( u / t ) 2 + 1 2 c 2 u 2 ,
P = - c 2 ( u / t ) u ,
P = i ω c 2 ( u u * - u * u ) ,
ϕ = 8 π 2 c 3 N u ˆ ( L , M ) 2 d L d M ,
u ˆ ( L , M ) = u ˆ ( L , M ) K ( L , M , L , M ) d L d M .