Abstract

Conoscopic holography is an incoherent holographic technique based on the properties of crystal optics. More precisely, for each point of the object, the interference pattern between the ordinary and the extraordinary rays is presented. The pattern is a Gabor-zone-lens pattern, with a scale parameter that is a function of the distance of the point. The superposition of the Gabor zone lens from each point of the object is the hologram, which contains information on the shape of the object through the scale-parameter dependence of each point is presented. I present a simplified version of the theory of conoscopic holography. The point-spread function and the transfer function of the conoscopic system are presented by using simple arguments, and the conoscopic hologram is defined. The basic schemes for reconstruction, i.e., retrieving, optically or numerically, this three-dimensional information about the object from the recorded hologram, are presented. Finally, the resolution of the system is quantified.

© 1992 Optical Society of America

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References

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  1. G. Sirat, D. Psaltis, “Conoscopic holography,” Opt. Lett. 10, 4–6 (1985).
    [CrossRef] [PubMed]
  2. G. Sirat, D. Psaltis, “Monochromatic incoherent light holography,” U.S. patent4,602,844 (July29, 1986).
  3. D. Charlot, “Holographie conoscopique: principe et reconstructions numériques,” thése DDI (Ecole Nationale Supérieure des Télécommunications, 46 rue Barrault, Paris, 1987).
  4. G. Y. Sirat, D. Psaltis, “Conoscopic holograms,” Opt. Commun. 65, 243–249 (1988).
    [CrossRef]
  5. D. Charlot, G. Y. Sirat, A. D. Maruani, E. Dufresne, “Holographie conoscopique: reconstructions numériques,” Ann. Télécommun. 43, 460–466 (1988).
  6. G. Y. Sirat, D. Charlot, E. Dufresne, A. D. Maruani, “Procédé et dispositif holographique perfectionné en lumière incohérente,” French patent88-1725 (December22, 1988).
  7. E. Dufresne, “Holographie conoscopique,” nouvelle thése, (Ecole Nationale Supérieure des Télécommunications, 46 rue Barrault, Paris, 1990).
  8. G. Y. Sirat, “Conoscopic holography. II. Rigorous derivation,” J. Opt. Soc. Am. A 9, 84–90 (1990).
    [CrossRef]
  9. G. Y. Sirat, “Conoscopic holography. III. Exponential holograms,” to be submitted to J. Opt. Soc. Am. A.
  10. J. W. Goodman, Statistical Optics (Interscience, New York, 1985), Chap. 2.
  11. E. N. Leith, B. J. Chang, “Space-invariant holography with quasi-coherent light,” Appl. Opt. 12, 1957–1963 (1973).
    [CrossRef] [PubMed]
  12. Ref. 10, Subsec. 5-1.
  13. Courtesy of L. Mugnier, Department of Images, Telecom Paris.
  14. Ref. 10, App. A.
  15. P. Chavel, E. Dufresne, G. Y. Sirat, “Dispositif holographique perfectionné en lumière incohérente,” French patent89-05344 (April21, 1989).
  16. L. M. Soroko, Holography and Coherent Optics (Plenum, New York, 1978), Chap. 2-20.
  17. L. Bergstein, T. Zachos, “A Huygens principle for uniaxially anisotropic media,”J. Opt. Soc. Am. 56, 931–937 (1966).
    [CrossRef]
  18. N. Streibl, “Fundamental restriction for 3-D light distributions,” Optik 66, 341–354 (1984).
  19. N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
    [CrossRef]
  20. D. N. Sitter, W. T. Rhodes, “Three-dimensional imaging: a space invariant model for space variant systems,” Appl. Opt. 29, 3789–3794 (1990).
    [CrossRef] [PubMed]
  21. Ref. 10, Subsec. 9-4.

1990 (2)

1988 (2)

G. Y. Sirat, D. Psaltis, “Conoscopic holograms,” Opt. Commun. 65, 243–249 (1988).
[CrossRef]

D. Charlot, G. Y. Sirat, A. D. Maruani, E. Dufresne, “Holographie conoscopique: reconstructions numériques,” Ann. Télécommun. 43, 460–466 (1988).

1985 (1)

1984 (2)

N. Streibl, “Fundamental restriction for 3-D light distributions,” Optik 66, 341–354 (1984).

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

1973 (1)

1966 (1)

Bergstein, L.

Chang, B. J.

Charlot, D.

D. Charlot, G. Y. Sirat, A. D. Maruani, E. Dufresne, “Holographie conoscopique: reconstructions numériques,” Ann. Télécommun. 43, 460–466 (1988).

D. Charlot, “Holographie conoscopique: principe et reconstructions numériques,” thése DDI (Ecole Nationale Supérieure des Télécommunications, 46 rue Barrault, Paris, 1987).

G. Y. Sirat, D. Charlot, E. Dufresne, A. D. Maruani, “Procédé et dispositif holographique perfectionné en lumière incohérente,” French patent88-1725 (December22, 1988).

Chavel, P.

P. Chavel, E. Dufresne, G. Y. Sirat, “Dispositif holographique perfectionné en lumière incohérente,” French patent89-05344 (April21, 1989).

Dufresne, E.

D. Charlot, G. Y. Sirat, A. D. Maruani, E. Dufresne, “Holographie conoscopique: reconstructions numériques,” Ann. Télécommun. 43, 460–466 (1988).

G. Y. Sirat, D. Charlot, E. Dufresne, A. D. Maruani, “Procédé et dispositif holographique perfectionné en lumière incohérente,” French patent88-1725 (December22, 1988).

P. Chavel, E. Dufresne, G. Y. Sirat, “Dispositif holographique perfectionné en lumière incohérente,” French patent89-05344 (April21, 1989).

E. Dufresne, “Holographie conoscopique,” nouvelle thése, (Ecole Nationale Supérieure des Télécommunications, 46 rue Barrault, Paris, 1990).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Interscience, New York, 1985), Chap. 2.

Leith, E. N.

Maruani, A. D.

D. Charlot, G. Y. Sirat, A. D. Maruani, E. Dufresne, “Holographie conoscopique: reconstructions numériques,” Ann. Télécommun. 43, 460–466 (1988).

G. Y. Sirat, D. Charlot, E. Dufresne, A. D. Maruani, “Procédé et dispositif holographique perfectionné en lumière incohérente,” French patent88-1725 (December22, 1988).

Psaltis, D.

G. Y. Sirat, D. Psaltis, “Conoscopic holograms,” Opt. Commun. 65, 243–249 (1988).
[CrossRef]

G. Sirat, D. Psaltis, “Conoscopic holography,” Opt. Lett. 10, 4–6 (1985).
[CrossRef] [PubMed]

G. Sirat, D. Psaltis, “Monochromatic incoherent light holography,” U.S. patent4,602,844 (July29, 1986).

Rhodes, W. T.

Sirat, G.

G. Sirat, D. Psaltis, “Conoscopic holography,” Opt. Lett. 10, 4–6 (1985).
[CrossRef] [PubMed]

G. Sirat, D. Psaltis, “Monochromatic incoherent light holography,” U.S. patent4,602,844 (July29, 1986).

Sirat, G. Y.

G. Y. Sirat, “Conoscopic holography. II. Rigorous derivation,” J. Opt. Soc. Am. A 9, 84–90 (1990).
[CrossRef]

G. Y. Sirat, D. Psaltis, “Conoscopic holograms,” Opt. Commun. 65, 243–249 (1988).
[CrossRef]

D. Charlot, G. Y. Sirat, A. D. Maruani, E. Dufresne, “Holographie conoscopique: reconstructions numériques,” Ann. Télécommun. 43, 460–466 (1988).

G. Y. Sirat, D. Charlot, E. Dufresne, A. D. Maruani, “Procédé et dispositif holographique perfectionné en lumière incohérente,” French patent88-1725 (December22, 1988).

P. Chavel, E. Dufresne, G. Y. Sirat, “Dispositif holographique perfectionné en lumière incohérente,” French patent89-05344 (April21, 1989).

G. Y. Sirat, “Conoscopic holography. III. Exponential holograms,” to be submitted to J. Opt. Soc. Am. A.

Sitter, D. N.

Soroko, L. M.

L. M. Soroko, Holography and Coherent Optics (Plenum, New York, 1978), Chap. 2-20.

Streibl, N.

N. Streibl, “Fundamental restriction for 3-D light distributions,” Optik 66, 341–354 (1984).

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

Zachos, T.

Ann. Télécommun. (1)

D. Charlot, G. Y. Sirat, A. D. Maruani, E. Dufresne, “Holographie conoscopique: reconstructions numériques,” Ann. Télécommun. 43, 460–466 (1988).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

Opt. Commun. (1)

G. Y. Sirat, D. Psaltis, “Conoscopic holograms,” Opt. Commun. 65, 243–249 (1988).
[CrossRef]

Opt. Lett. (1)

Optik (1)

N. Streibl, “Fundamental restriction for 3-D light distributions,” Optik 66, 341–354 (1984).

Other (12)

G. Y. Sirat, “Conoscopic holography. III. Exponential holograms,” to be submitted to J. Opt. Soc. Am. A.

J. W. Goodman, Statistical Optics (Interscience, New York, 1985), Chap. 2.

Ref. 10, Subsec. 9-4.

G. Sirat, D. Psaltis, “Monochromatic incoherent light holography,” U.S. patent4,602,844 (July29, 1986).

D. Charlot, “Holographie conoscopique: principe et reconstructions numériques,” thése DDI (Ecole Nationale Supérieure des Télécommunications, 46 rue Barrault, Paris, 1987).

G. Y. Sirat, D. Charlot, E. Dufresne, A. D. Maruani, “Procédé et dispositif holographique perfectionné en lumière incohérente,” French patent88-1725 (December22, 1988).

E. Dufresne, “Holographie conoscopique,” nouvelle thése, (Ecole Nationale Supérieure des Télécommunications, 46 rue Barrault, Paris, 1990).

Ref. 10, Subsec. 5-1.

Courtesy of L. Mugnier, Department of Images, Telecom Paris.

Ref. 10, App. A.

P. Chavel, E. Dufresne, G. Y. Sirat, “Dispositif holographique perfectionné en lumière incohérente,” French patent89-05344 (April21, 1989).

L. M. Soroko, Holography and Coherent Optics (Plenum, New York, 1978), Chap. 2-20.

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Figures (5)

Fig. 1
Fig. 1

Simplified representation of the on-axis conoscope.

Fig. 2
Fig. 2

Experimental results. (a) Positive conoscopic on-axis figure recorded with the photographic system developed in Ref. 3. (b) Positive conoscopic on-axis figure (a negative one) with numerical substraction (the hologram) as described in Eqs. (16), (19), and (20) and the contrast image defined as (positive − negative)/(positive + negative). Note that all the defects present in the positive and the negative holograms disappear in the hologram and in the contrast image.13

Fig. 3
Fig. 3

Off-axis conoscopic systems7: (a) aperture-displaced system, (b) optical-axis-tilted crystal, (c) geometrically tilted crystal.

Fig. 4
Fig. 4

General (with respect to polarization) scheme of a conoscopic system3: (a) general case, (b) linear-polarizer-circular-analyzer case.

Fig. 5
Fig. 5

Definition of a fringe.

Equations (103)

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Γ ( P , τ ) = u ( P , t + τ ) u * ( P , t ) ,
γ ( P , τ ) = u ( P , t + τ ) u * ( P , t ) u ( P , t ) u * ( P , t ) = Γ ( P , τ ) Γ ( P , 0 ) ,
I ( P ) = ( 0 c / 2 ) u ( P , t ) 2 = ( 0 c / 2 ) Γ ( P , 0 ) .
A = A o ( Q , P ) = A e ( Q , P ) .
I o ( Q , P ) = ( 0 c / 2 ) ( 2 A ) 2 .
I o ( Q , P ) = ( T / z c 2 ) I ( P ) ,
A 2 = 2 I o 4 0 c = T 0 c z c 2 I ( P ) ,
Δ ϕ = 2 π ( l e - l o ) / λ .
l o = P A + n o A B + B Q ,
l e = P A + n e ( θ ) A B + B Q .
l e - l o = n e ( θ ) A B - n o A B = Δ n e ( θ ) A B Δ n e ( θ ) L ,
Δ n e ( θ ) Δ n n o 2 n c 2 θ 2 ,
θ r / z c ,
Δ ϕ = 2 π l o - l e λ 2 π l o - l e λ = 2 π Δ n L r 2 λ n c 2 z c 2 = π κ o z c 2 r 2 ,
I + ( Q ) = ( 0 c / 2 ) [ A o ( Q , P ) + A e ( Q , P ) ] × [ A o ( Q , P ) + A e ( Q , P ) ] * ,
I + ( Q ) = ( 0 c / 2 ) 2 A 2 [ 1 + cos ( Δ ϕ ) ] = [ T z c 2 + h 2 ( M ) ] I ( P ) ,
H ( Q ) = h 2 ( M ) I ( P ) = T z c 2 cos ( Δ ϕ ) I ( P ) ,
h 2 ( M ) = T z c 2 cos ( Δ ϕ ) = T z c 2 cos [ 2 π Δ n L r 2 λ n c 2 z c 2 ] = T z c 2 cos [ π κ o r 2 z c 2 ] ,
L ( Q ) = ( 0 c / 2 ) [ A o ( Q , P ) - A e ( Q , P ) ] [ A o ( Q , P ) - A e ( Q , P ) ] * = ( 0 c / 2 ) 2 A 2 [ 1 - cos ( Δ ϕ ) ] = [ ( T / z c 2 ) - h 2 ( M ) ] I ( P ) .
H ( Q ) = I + ( Q ) - I - ( Q ) 2 = h 2 ( M ) I ( P ) = T z c 2 cos ( Δ ϕ ) I ( P ) = T z c 2 cos [ π κ o r 2 z c 2 ] I ( P ) .
f 2 ( M ) = T z c 2 exp ( j Δ ϕ ) = T z c 2 exp ( j 2 π Δ n L r 2 λ n c 2 z c 2 ) = T z c 2 exp ( j π κ o r 2 z c 2 ) ,
h 2 ( M ) = [ f 2 ( M ) + f 2 * ( M ) ] / 2.
f ˜ 2 ( u , v ) = FT x y [ 1 z c 2 exp ( j π κ o r 2 z c 2 ) ] = - j κ o exp ( j π z c 2 ρ 2 κ o ) = - j f R z c 2 exp ( j π ρ 2 f R ) ,
h ˜ 2 ( u , v ) = FT x y [ f 2 ( x , y ) + f 2 * ( x , y ) 2 ] = - 1 κ o sin ( π z c 2 ρ 2 κ o ) .
f ˜ ˜ S ( u , v , ξ ) = f ( x , y , z c ) exp { j [ 2 π ( x u + y v + z c ξ ) ] } d x d y d z c .
f ˜ ˜ 2 ( u , v , ξ ) = FT z c [ - j κ o exp ( j π z c 2 ρ 2 κ o ) ] = - ( 1 + j ) ( 2 κ o ) 1 / 2 ρ 2 exp ( - j π κ o ξ 2 ρ 2 ) ,
h ˜ ˜ 2 ( u , v , ξ ) = FT z c [ h ˜ 2 ( u , v , z c ) ] .
h 2 ( M ) = hol ( π κ o r 2 z c 2 ) .
H ( Q ) = all points P I ( P ) hol [ j π κ o ( x - x ) 2 + ( y - y ) 2 ( z c - z ) 2 ] ,
H ( x , y , z ) = V I ( x , y , z c ) × hol [ j π κ o ( x - x ) 2 + ( y - y ) 2 ( z c - z ) 2 ] d x d y d z c .
H ( x , y , z ) = Z c I ( x , y , z c ) * { hol [ j π κ o x 2 + y 2 ( z c - z ) 2 ] Win [ X , Y ] } d z c ,
H ( x , y , z ) = Z c I ( x , y , z c ) * hol [ j π κ o x 2 + y 2 ( z c - z ) 2 ] d z c .
H lin ( M , ψ ) = [ t b 2 ( ψ ) + t lin 2 ( ψ ) sin ( 2 ψ ) sin ( Δ ϕ ) ] I ( P ) z c 2 ,
f ˜ qsin ( u , v ) = [ f ˜ lin ( 0 ) - f ˜ lin ( π / 2 ) ] cos ( 2 Ψ ) + [ f ˜ lin ( π / 4 ) - f ˜ lin ( - π / 4 ) ] sin ( 2 Ψ ) ,
H ( x , y ) = S I ( x , y ) z c ( x , y ) 2 × hol { π κ o [ ( x - x ) 2 + ( y - y ) ] 2 z c ( x , y ) 2 } d x d y .
H ( x , y , z ) = V I ( x , y , z c ) ( z c - z ) 2 × hol { [ π κ o ( x - x ) 2 + ( y - y ) 2 ( z c - z ) 2 ] } d x d y d z c .
Δ x R = 0.61 R F ,
Δ z R = 2 R 2 λ F 2 ,
Δ t = F λ / 2.
Δ β = β o - β e = Δ ϕ π r n o λ .
Δ β = n o F λ / R .
Δ λ λ = 1 F .
Δ n = n e - n o .
n c 2 = 2 n o n e 2 n o + n e ,
2 Δ n n c 2 = n e 2 - n o 2 n o n e 2 = 1 n o - n o n e 2 .
z o = ( z - z ) - L + ( L / n o ) .
sin β o = Q Q / z o .
z e = ( z - z ) - L + ( L n o / n e 2 ) .
sin β e = Q Q / z e .
z c = ( z o z e ) 1 / 2 .
d z c d z = 1 2 ( z o z e ) 1 / 2 ( z o d z e d z + z e d z o d z ) = ( z o + z e ) / 2 ( z o z e ) 1 / 2 .
z l = ( z - z ) + ( n o - l ) L ,
l o z l cos β .
l o z c .
κ o = 2 Δ n L n c 2 λ .
f r = 2 Δ n L n c 2 λ 1 z c 2 = κ o z c 2 ,
F = 2 Δ n L n c 2 λ R 2 z c 2 = κ o R 2 z c 2 = f r R 2 ,
x G 2 = G z c 2 κ o
Δ x F = x F - x F - 1 = x F 2 - x F - 1 2 x F + x F - 1 [ F - ( F - 1 ) ] z c 2 κ o ( 2 R )
Δ x F = z c 2 κ o ( 2 R ) = R 2 2 F R = R 2 F .
u o ( x , y , z ) = - j n o λ R u ( ξ , η , z A ) 1 R o exp ( j k n o R o ) d ξ d η
u e ( x , y , z ) = - j n o λ R u ( ξ , η , z A ) 1 R e exp ( j k n e R e ) d ξ d η ,
R o 2 = ( x - ξ ) 2 + ( y - η ) 2 + ( z - z A ) 2 .
R e 2 = R o 2 - n e 2 - n o 2 n e 2 ( z - z A ) 2 = ( x = ξ ) 2 + ( y - η ) 2 + n o 2 n e 2 ( z - z A ) 2 ,
u ( ξ , η , z A ) = exp ( j k ξ 2 + η 2 2 z A ) ,
u ( x , y , z ) = - j n o λ R exp ( j k ξ 2 + η 2 2 z A ) 1 R o exp ( j k n o R o ) d ξ d η = - j n o λ R 1 R o exp [ j k ( ξ 2 + η 2 2 z A + n o R o ) ] d ξ d η ,
u e ( x , y , z ) = - j n o λ R exp ( j k ξ 2 + η 2 2 z A ) 1 R e exp ( j k n e R e ) d ξ d η = - j n o λ R 1 R e exp [ j k ( ξ 2 + η 2 2 z A + n e R e ) ] d ξ d η .
1 n e 2 ( θ e ) = sin 2 θ n e 2 + cos 2 θ e n o 2 ,
1 n e 2 ( θ e ) = θ e 2 n e 2 + 1 - θ e 2 n o 2 = 1 n o 2 + θ e 2 ( 1 n e 2 - 1 n o 2 ) .
n e ( θ e ) = n o + Δ n e ( θ e ) ,
1 [ n o + Δ n e ( θ e ) ] = 1 n o 2 + θ e 2 ( 1 n e 2 - 1 n o 2 )
1 [ n o + Δ n e ( θ e ) ] 2 - 1 n o 2 θ e 2 ( 1 n e 2 - 1 n o 2 ) 1 n o 2 { 1 + [ 2 Δ n e ( θ e ) / n o ] } - 1 n o 2 1 - [ 2 Δ n e ( θ e ) / n o ] n o 2 - 1 n o 2 - 2 Δ n e ( θ e ) n o 3 .
- 2 Δ n e ( θ e ) n o 3 θ e 2 ( 1 n e 2 - 1 n o 2 )
Δ n e ( θ e ) θ e 2 ( n o 3 n e 2 - n o 3 n o 2 ) n o θ e 2 ( n o 2 n e 2 - 1 ) θ e 2 Δ n n o 2 n c 2
θ r / z c .
I + ( x , y ) = I o ( 1 + cos { π κ o [ ( x o - x ) 2 + ( y o - y ) 2 ] z c 2 } ) .
I + ( l , m ) = I o ( 1 + cos { π F R p 2 [ ( l - l o ) 2 + ( m - m o ) 2 ] } ) ,
x o = l o Δ x ,             y o = m o Δ y ,
F = κ o R 2 z c 2 ,
I - ( l , m ) = I o ( 1 - cos { π F R p 2 [ ( l - l o ) 2 + ( m - m o ) 2 ] } ) .
H ( l , m ) = I + ( l , m ) - I - ( l , m ) 2 = I o cos { π F R p 2 [ ( l - l o ) 2 + ( m o - m ) 2 ] } .
c 0 = I 0 η τ h ν A p
C T = I 0 η τ h ν π R 2 = c 0 π R 2 A p ,
K ( l , m ) = α Q H ( l , m ) = c o cos { π F R p 2 [ ( l o - l ) 2 + ( m - m o ) 2 ] } ,
α Q = c 0 I 0 = η τ h ν A p .
F r ( F 1 , l o , m o ) = l , m R K ( l , m ) exp { - j π F 1 R p 2 [ ( l - l o ) 2 + ( m - m o ) 2 ] } ,
F r ( F 1 , l o , m o ) = K r + j K i ,
F r ( F 1 ) = c o 1 2 ( l , m R { exp [ - j π ( F 1 - F ) R p 2 ( l 2 + m 2 ) ] + exp [ - j π ( F 1 + F ) R p 2 ( l 2 + m 2 ) ] } ) .
Δ F = F 1 - F .
F r ( F 1 ) c o 2 l m R exp [ - j π Δ F R p 2 ( l 2 + m 2 ) ] .
d x = Δ l , d y = Δ d m , r d r d θ = d x d y = Δ 2 d l d m = A p d l d m
F r ( F 1 ) c o 2 A p θ = 0 2 π d θ r = 0 R exp ( - j π Δ F r 2 R 2 ) r d r 2 π c o 2 A p r = 0 R exp ( - j π Δ F r 2 R 2 ) r d r C T R 2 r = 0 R exp ( - j π Δ F r 2 R 2 ) r d r .
F r ( F 1 ) C T 2 R 2 u = 0 R 2 exp ( - j π Δ F u R 2 ) d u ,
F r ( F 1 ) j C T 2 π Δ F [ exp ( - j π Δ F ) - 1 ] = C T π Δ F exp ( j π Δ F 2 ) sin π Δ F 2 = C T 2 exp ( j π Δ F 2 ) [ ( sin π Δ F 2 ) ( π Δ F 2 ) ] = C T 2 sinc ( Δ F 2 ) exp ( - j π Δ F 2 ) ,
K r = C T 2 sinc ( Δ F 2 ) cos ( π Δ F 2 ) ,
K i = - C T 2 sinc ( Δ F 2 ) sin ( π Δ F 2 ) .
K = C T 2 sinc ( Δ F 2 ) ,
ϕ K = π Δ F 2 ,
Δ F = 2 ϕ K π .
Δ F rms = 2 σ i π K r 2 σ i π K ¯ r .
σ i 2 = C T 2 ,
K ¯ r = C T 2 .
Δ F rms = 2 σ i π K ¯ r = 2 2 π C T .

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