Abstract

An accurate method of analysis of underwater imaging systems using optical spread functions and their transforms, the modulation transfer functions, is presented. The overall system spread function is obtained from the water spread functions and the spread functions of the system components. The water point spread and beam spread functions are defined, and measurements of these are presented for clear coastal water for distances up to nine attenuation lengths. Relationships between the spread functions and their dependence on range are also given. In addition, their relationships to the conventional optical oceangraphic parameters of beam attenuation, absorption, and scattering are described. Combined optics-water spread functions and their transforms are developed for illuminator and receiver geometries typically used in underwater imaging. These are then used to determine the system (optics plus water) response to the target reflectance. An analytic technique for accurate computation of backscatter signals is developed. Computed and measured signals compare favorably. It is concluded that the use of spread functions is a convenient and viable technique for analytic computation of underwater images.

© 1977 Optical Society of America

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References

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  1. S. Q. Duntley, J. Opt. Soc. Am. 53, 214 (1963).
    [Crossref]
  2. C. J. Funk, Appl. Opt. 12, 301 (1973).
    [Crossref] [PubMed]
  3. R. E. Morrison, Ph.D. thesis (Dept. of Meteorology and Oceanography, New York University, 1967).
  4. L. E. Mertens and D. L. Phillips, “Measurements of theVolume Scattering Function of Sea Water,” Air Force Eastern Test Range, Tech. Rep. 334, Patrick AFB, Florida (1972), obtainable from author (unpublished).
  5. T. J. Petzold, “Volume Scattering Functions for Selected Ocean Waters,” Scripps Inst, of Oceanography Visibility Laboratory, San Diego, Calif. (1972).
  6. W. H. Wells, AGARD Lecture Series No. 61, Sec. 4.1, p. 1, obtainable from Report Distribution Unit, NASA, Langley, Va. (1973).
  7. Linear summation has been verified empirically for spatial frequencies below 10 000 cycles per rad—the region where salinity and thermal gradient structure in the water do not produce beam broadening (see Ref. 8 below).
  8. R. T. Hodgson and D. R. Caldwell, J. Opt. Soc. Am. 62, 1434 (1972).
    [Crossref]
  9. A receiver beam may be defined as the cylindrical volume whose umbral surface is bounded at the ends by the receiver aperture and by the image of the receiver field stop.
  10. W. H. Wells, Ref. 6, Sec. 3.4, p. 1.
  11. E. C. Jordan, Electromagnetic Waves and Radiating Systems (Prentice-Hall, New York, 1950), p. 327.
  12. W. H. Wells, J. Opt. Soc. Am. 59, 686 (1969).
    [Crossref]
  13. W. H. Wells, Ref. 6, Sec. 3.3, p. 1.
  14. This relationship has been verified up to a range of 5.7 attenuation lengths. The maximum range of its validity is not known.
  15. H. Hodara, AGARD Lecture Series No. 61, Sec. 3.4 (1973), see Ref. 6 for procurement.
  16. W. H. Wells, Ref. 6, Secs. 3.3 and 4.3.
  17. Equation (5) and others in the derivation assume that the spread functions are independent of position. This is true on spherical surfaces at a constant range. On plane surfaces, it represents an approximation useful over small fields.
  18. V. R. Muratov and R. L. Struzer, Optical Tech. Theory Exper. 39, 519 (1972).
  19. W. H. Wells, Ref. 6, Sec. 4.3, p. 7.
  20. A. Gordon, D. Cozen, C. Funk, and P. Heckman, “Design Study of Advanced Underwater Optical Imaging Systems, Appendix B,” Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).
  21. By scattering calculations or from direct visual observations at a point outside a beam, it may be shown that the direction giving the maximum contribution to the illumination of a backscattering volume follows the direct line from the transmitter to the volume [dashed line of Fig. 8(b)], and the angular distribution of the flux falling on this backscattering volume is effectively only a few degrees wide. A similar observation applies to the flux backscattered from this volume and striking the receiver. Thus the dominant (back) scattering angle ne for the incremental volume shown is approximately (π− G/r) rad over most of the scattering volume in a thin extended slab at range r, and the distribution of back-scattering angles for this range is only a few degrees wide. Since the amplitude of the volume scattering function VSF varies only fractionally in a span of a few degrees when the mean scattering angle lies in the vicinity of π rad (Ref. 5), a similar accuracy will result if a single value of backseat -ter coefficient is used in the summation of the backscattering contribution from such a slab.
  22. W. H. Wells, Ref. 6, Sec. 4.3, p. 2.

1973 (1)

1972 (2)

R. T. Hodgson and D. R. Caldwell, J. Opt. Soc. Am. 62, 1434 (1972).
[Crossref]

V. R. Muratov and R. L. Struzer, Optical Tech. Theory Exper. 39, 519 (1972).

1969 (1)

1963 (1)

Caldwell, D. R.

Cozen, D.

A. Gordon, D. Cozen, C. Funk, and P. Heckman, “Design Study of Advanced Underwater Optical Imaging Systems, Appendix B,” Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).

Duntley, S. Q.

Funk, C.

A. Gordon, D. Cozen, C. Funk, and P. Heckman, “Design Study of Advanced Underwater Optical Imaging Systems, Appendix B,” Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).

Funk, C. J.

Gordon, A.

A. Gordon, D. Cozen, C. Funk, and P. Heckman, “Design Study of Advanced Underwater Optical Imaging Systems, Appendix B,” Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).

Heckman, P.

A. Gordon, D. Cozen, C. Funk, and P. Heckman, “Design Study of Advanced Underwater Optical Imaging Systems, Appendix B,” Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).

Hodara, H.

H. Hodara, AGARD Lecture Series No. 61, Sec. 3.4 (1973), see Ref. 6 for procurement.

Hodgson, R. T.

Jordan, E. C.

E. C. Jordan, Electromagnetic Waves and Radiating Systems (Prentice-Hall, New York, 1950), p. 327.

Mertens, L. E.

L. E. Mertens and D. L. Phillips, “Measurements of theVolume Scattering Function of Sea Water,” Air Force Eastern Test Range, Tech. Rep. 334, Patrick AFB, Florida (1972), obtainable from author (unpublished).

Morrison, R. E.

R. E. Morrison, Ph.D. thesis (Dept. of Meteorology and Oceanography, New York University, 1967).

Muratov, V. R.

V. R. Muratov and R. L. Struzer, Optical Tech. Theory Exper. 39, 519 (1972).

Petzold, T. J.

T. J. Petzold, “Volume Scattering Functions for Selected Ocean Waters,” Scripps Inst, of Oceanography Visibility Laboratory, San Diego, Calif. (1972).

Phillips, D. L.

L. E. Mertens and D. L. Phillips, “Measurements of theVolume Scattering Function of Sea Water,” Air Force Eastern Test Range, Tech. Rep. 334, Patrick AFB, Florida (1972), obtainable from author (unpublished).

Struzer, R. L.

V. R. Muratov and R. L. Struzer, Optical Tech. Theory Exper. 39, 519 (1972).

Wells, W. H.

W. H. Wells, J. Opt. Soc. Am. 59, 686 (1969).
[Crossref]

W. H. Wells, Ref. 6, Sec. 3.3, p. 1.

W. H. Wells, Ref. 6, Secs. 3.3 and 4.3.

W. H. Wells, Ref. 6, Sec. 4.3, p. 7.

W. H. Wells, AGARD Lecture Series No. 61, Sec. 4.1, p. 1, obtainable from Report Distribution Unit, NASA, Langley, Va. (1973).

W. H. Wells, Ref. 6, Sec. 3.4, p. 1.

W. H. Wells, Ref. 6, Sec. 4.3, p. 2.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Optical Tech. Theory Exper. (1)

V. R. Muratov and R. L. Struzer, Optical Tech. Theory Exper. 39, 519 (1972).

Other (17)

W. H. Wells, Ref. 6, Sec. 4.3, p. 7.

A. Gordon, D. Cozen, C. Funk, and P. Heckman, “Design Study of Advanced Underwater Optical Imaging Systems, Appendix B,” Naval Undersea Research and Development Center, San Diego, Tech. Publ. 275 (1972) (unpublished).

By scattering calculations or from direct visual observations at a point outside a beam, it may be shown that the direction giving the maximum contribution to the illumination of a backscattering volume follows the direct line from the transmitter to the volume [dashed line of Fig. 8(b)], and the angular distribution of the flux falling on this backscattering volume is effectively only a few degrees wide. A similar observation applies to the flux backscattered from this volume and striking the receiver. Thus the dominant (back) scattering angle ne for the incremental volume shown is approximately (π− G/r) rad over most of the scattering volume in a thin extended slab at range r, and the distribution of back-scattering angles for this range is only a few degrees wide. Since the amplitude of the volume scattering function VSF varies only fractionally in a span of a few degrees when the mean scattering angle lies in the vicinity of π rad (Ref. 5), a similar accuracy will result if a single value of backseat -ter coefficient is used in the summation of the backscattering contribution from such a slab.

W. H. Wells, Ref. 6, Sec. 4.3, p. 2.

W. H. Wells, Ref. 6, Sec. 3.3, p. 1.

This relationship has been verified up to a range of 5.7 attenuation lengths. The maximum range of its validity is not known.

H. Hodara, AGARD Lecture Series No. 61, Sec. 3.4 (1973), see Ref. 6 for procurement.

W. H. Wells, Ref. 6, Secs. 3.3 and 4.3.

Equation (5) and others in the derivation assume that the spread functions are independent of position. This is true on spherical surfaces at a constant range. On plane surfaces, it represents an approximation useful over small fields.

A receiver beam may be defined as the cylindrical volume whose umbral surface is bounded at the ends by the receiver aperture and by the image of the receiver field stop.

W. H. Wells, Ref. 6, Sec. 3.4, p. 1.

E. C. Jordan, Electromagnetic Waves and Radiating Systems (Prentice-Hall, New York, 1950), p. 327.

R. E. Morrison, Ph.D. thesis (Dept. of Meteorology and Oceanography, New York University, 1967).

L. E. Mertens and D. L. Phillips, “Measurements of theVolume Scattering Function of Sea Water,” Air Force Eastern Test Range, Tech. Rep. 334, Patrick AFB, Florida (1972), obtainable from author (unpublished).

T. J. Petzold, “Volume Scattering Functions for Selected Ocean Waters,” Scripps Inst, of Oceanography Visibility Laboratory, San Diego, Calif. (1972).

W. H. Wells, AGARD Lecture Series No. 61, Sec. 4.1, p. 1, obtainable from Report Distribution Unit, NASA, Langley, Va. (1973).

Linear summation has been verified empirically for spatial frequencies below 10 000 cycles per rad—the region where salinity and thermal gradient structure in the water do not produce beam broadening (see Ref. 8 below).

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Figures (18)

FIG. 1
FIG. 1

Schematic diagram of separated transmitter-receiver system geometry. The system elements are RS: radiant source; TL: transmitting lens; CBV: common backscattering volume, DET: detector; RL: receiver lens; TP: target plane. Ideal ray transmission is shown. The x direction, not shown, is orthogonal to y and z. All elements are assumed immersed in a water medium.

FIG. 2
FIG. 2

Geometry for beam spread function BSF(θ, ϕ, R) measurement. The beam B travels from a source S at the origin to its focal point at the range R. Irradiance is measured on a small area a of the spherical surface at range R. If there is source polarization SP, it is in the x direction.

FIG. 3
FIG. 3

Geometry for point spread function PSF(θ, ϕ, R) measurement. The photometer PH at the origin is focused on a small area a of the spherical surface at range R. It measures the apparent radiance in the direction (θ, ϕ) resulting from scattering in the medium of radiance from an unresolved Lambertian source US at (0, 0, R).

FIG. 4
FIG. 4

Beam spread function for clear coastal water. Measurements were made at High Cay, Andros Island, Bahamas, using a 15-mm-diam collimated laser beam (λ = 488 nm) with linear polarization ( ϕ = 1 2 π ) and a 2.5-cm-diam radiometer detector with a cosine pattern diffuser. The numbers labeling the curves are ranges in attenuation lengths. Data on curves A and B were taken on 17 March 1971 with water attenuation lengths of 6.8 and 6.7 m. Data on curves C, D, and E were taken on 4 June 1971 with attenuation lengths of 6.1, 5.3, and 5.6 m. The attenuation length during runs C and E was varying rapidly.

FIG. 5
FIG. 5

Beam spread function measurements for the planes ϕ = 0 and ϕ = 1 2 π. The transmitter was linearly polarized, the receiver unpolarized. Data were taken on 31 May 1971 in the Tongue of the Ocean, Bahamas, with a water attenuation length of 9.71 m at a range of 1.0 attenuation length.

FIG. 6
FIG. 6

Point spread function for clear coastal water measured at High Cay, Andros Island, Bahamas. A 488 nm laser and small diameter diffuser provided the point source; the receiver was an unpolarized telephotometer with a 2.1 m, f/15 lens and a 1.08 mrad (in water) field of view. Numbers on the curves are ranges in attenuation lengths. The data on curves A, B, and C were taken on 18 March 1971 with a water attenuation length of 7.0 m. The data on curves D, E, and F were taken on 7 June 1971 with water attenuation lengths of 4.3, 5.1, and 5.0 m.

FIG. 7
FIG. 7

Spatial frequency decay function D(ψ) in clear coastal water. The dotted curves are Fourier transformations by Wells (Ref. 12) of curves A, B, and C of Fig. 5. The attenuation length for high frequencies is 7.0 m. The attenuation length for very low frequencies is 30 m.

FIG. 8
FIG. 8

Schematic geometries for calculating flux backscattered from the medium. The element symbols are REC: receiver; TR: transmitter; CVR: common volume region. a shows dimensions for single and double scattering; b for multiple scattering. Details are contained in the text.

FIG. 9
FIG. 9

Geometry of scattering volume in the plane z = constant. The element symbols are TBA: transmitter beam axis; RBA: receiver beam axis; SV: scattering volume.

FIG. 10
FIG. 10

Differential backscatter (backscatter power per meter thickness of a transverse water slab) with a wide transmitter and a narrow receiver beam imaging system. The curve codes are target range (attenuation lengths) and transmitter-receiver separation (m). The receiver beam width is 500 mrad; the transmitter beam width is 2.2 mrad; the water attenuation length is 6.9 m.

FIG. 11
FIG. 11

Differential backscatter (backscatter power per meter thickness of a transverse water slab) with a dual narrow beam imaging system with 1 m transmitter-receiver separation. Numbers on curves are target ranges in attenuation lengths. The transmitter beam width is 1.0 mrad; the receiver beam width is 2.2 mrad; the water attenuation length is 6.9 m.

FIG. 12
FIG. 12

Differential backscatter (backscatter power per meter thickness of a transverse water slab) with dual narrow beam imaging system with 2 m transmitter-receiver separation. Numbers on curves are target ranges in attenuation lengths. The transmitter beam width is 1.0 mrad; the receiver beam width is 2.2 mrad; the water attenuation length is 6.9 m.

FIG. 13
FIG. 13

Differential backscatter (backscatter power per meter thickness of a transverse water slab) with narrower dual beam imaging system with 1 m transmitter-receiver separation. Numbers on curves are target ranges in attenuation lengths. The transmitter beam width is 1.0 mrad; the receiver beam width is 1.1 mrad; the water attenuation length is 6.9 m.

FIG. 14
FIG. 14

Differential backscatter (backscatter power per meter thickness of a transverse water slab) with narrower dual beam imaging system with 2 m transmitter-receiver separation. Numbers on curves are target ranges in attenuation lengths. The transmitter beam width is 1.0 mrad; the receiver beamwidth is 1.1 mrad; the water attenuation length is 6.9 m.

FIG. 15
FIG. 15

Computed total backscatter. The curve code is: transmitter-receiver separation (m); transmitter beam width (mr); receiver beam width (mr); figure whose curves were integrated.

FIG. 16
FIG. 16

Comparison of computed values with measured values of total backscatter. Upper measurements were in the Tongue of the Ocean, Bahamas (attenuation lengths were 6.3 and 5.6 m). Lower measurements were at High Cay, Bahamas (attenuation length was 6.9 m). The curve code is transmitter-receiver separation (m); receiver beamwidth (mr); computed or measured value.

FIG. 17
FIG. 17

BSF and PSF geometry in the umbral region. A circular source or field stop S′ near the lens focal point is imaged in the field as the disk S. The point P′ is imaged in the field as the point P. The illustration assumes that the optical indices on both sides of the lens are equal.

FIG. 18
FIG. 18

Geometry of rays irradiating or backscattered from a point P in the medium. Symbols have the following meanings: IL: imaging lens; I: image of source or receiver field stop, DL: lens (pupil) diameter; Ae: lens area used for solid angle computation; DSo: diameter of projection of disk I through P = DSz/(Rz); ρo: radius of projection of point (0, R) through P, and DS: diameter of image of source or receiver field stop.

Equations (51)

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MTF ( ψ , R ) = 2 π 0 θ max d θ θ J 0 ( 2 π ψ θ ) [ R 2 PSF ( θ , R ) ] .
R 2 PSF ( θ , R ) = 2 π 0 d ψ ψ J 0 ( 2 π ψ θ ) MTF ( ψ , R ) ,
MTF ( ψ , R ) = exp [ D ( ψ ) R ] ,
E u ( x , y ) = A t T t R 2 L s ( x f / R , y f / R ) ,
E ( x , y ) = d x i d y i E u ( x i , y i ) BSFP ( x x i , y y i , R ) .
TSF ( x , y , R ) = E ( x , y ) / P t = E ( x , y ) ( dx dy E u ( x , y ) ) 1 ,
L t ( x , y ) = E ( x , y ) RF ( x , y ) / π = P t TSF ( x , y , R ) RF ( x , y ) / π .
L t a ( x , y ) = dx dy L t ( x , y ) PSFP ( x x , y y , R ) .
P r ( x , y ) = A r T r R 2 x Δ x / 2 x + Δ x / 2 d x y Δ y / 2 y + Δ y / 2 d y × dx dy L t ( x , y ) PSFP ( x x , y y , R ) = dx dy RSF ( x x , y y , R ) L t ( x , y ) ,
RSF ( x x , y y , R ) = A r T r R 2 x Δ x / 2 x + Δ x / 2 d x y Δ y / 2 y + Δ y / 2 d y × PSFP ( x x , y y , R ) .
P r ( x , x , y , y ) = P t π 1 dx dy RSF ( x x , y y , R ) × RF ( x , y ) TSF ( x x , y y , R ) ,
FT [ P r ( x , y ) ] = P t π 1 FT [ RSF ] FT [ ( RF ) ( TSF ) ] .
FT [ P r ( x , y ) ] = P t π 1 FT [ ( RSF ) ( RF ) ] FT [ TSF ] .
FT [ P r ( x , y ) ] = P t π 1 FT [ RF ] FT [ RSF ( x δ x , y δ y , R ) × TSF ( x , y , R ) ] .
FT [ RSF ( x δ x , y δ y ) TSF ( x , y ) ] = FT [ RSF ( x δ x , y δ y ) ] * FT [ TSF ( x , y ) ] .
BSC ( z ) VSF [ π 2 arctan ( G / 2 z ) ] in m 1 sr 1 .
d 3 ( BS ) = TSF ( ρ 1 , z ) BSC ( z ) RST ( ρ 2 , z ) d V ,
d ( BS ) = BSC ( z ) d z 0 2 π d χ 0 d ρ 1 ρ 1 TSF ( ρ 1 , z ) RSF ( ρ 2 , z ) .
d ( BS ) = 2 π BSC ( z ) d z 0 dkk TSF ( k , z ) RSF ( k , z ) J 0 ( 2 π k G q ) ,
F ( k ) = 2 π 0 d ρ ρ f ( ρ ) J 0 ( 2 π k ρ )
f ( ρ ) = 2 π 0 dkk F ( k ) J 0 ( 2 π k ρ ) .
E u ( x , y , z ) = 2 π ( 4 R q z D L t D S t ) 2 0 dkk Rect 1 t ( k , z ) × Rect 2 t ( k , z ) J 0 ( 2 π k ρ ) ,
Rect ( a , b , c ) = 1 if a 2 + b 2 c 2
Rect ( a , b , c ) = 0 if a 2 + b 2 > c 2 .
E ( x , y , z ) = TSF ( x , y , z ) = dudw E u ( u , w , z ) × BSFP ( u x , w y , z ) .
TSF ( x , y , z ) = 2 π dkk E u ( k , z ) BSFP ( k , z ) J 0 ( 2 π k ρ ) .
TSF ( k , z ) = E u ( k , z ) BSFP ( k , z ) = ( 4 R q z D L t D S t ) 2 Rect 1 t ( k , z ) Rect 2 t ( k , z ) × BSFP ( k , z ) .
d P r u ( x , y , z ) = 2 π Ldx dy q 2 z 2 0 dkk Rec t 1 r Rect 2 r J 0 ( 2 π k ρ ) ,
d P r ( x , y , z ) = du dw d P r u ( u , w , z ) PSFP ( u x , w y , z ) .
d P r ( x , y , z ) = 2 π 0 dkk FT ( d P r u ) PSFP ( k , z ) J 0 ( 2 π k ρ ) = Ldx dy 2 π q 2 z 2 0 dkk PSFP ( k , z ) Rect 1 r ( k , z ) × Rect 2 r ( k , z ) J 0 ( 2 π k ρ ) = Ldx dy RSF ( x , y , z ) .
RSF ( k , z ) = q 2 z 2 Rect 1 r ( k , z ) Rect 2 r ( k , z ) PSFP ( k , z ) .
d ( BS ) = 2 π ( 4 R D L r q 2 z 2 D L t 2 D S t 2 ) 2 BSC ( z ) d z 0 dkk Rect 1 t ( k , z ) × Rect 2 t ( k , z ) Rect 1 r ( k , z ) Rect 2 r ( k , z ) × BSFP ( k , z ) PSFP ( k , z ) J 0 ( 2 π kGq ) .
Rect ( k , ρ max ) = 2 π 0 d ρ ρ Rect ( ρ , ρ max ) J 0 ( 2 π k ρ ) = π ρ max 2 U ( k ρ max ) ,
U ( k ρ max ) = J 1 ( 2 π k ρ max ) π k ρ max .
d ( BS ) = BSC ( z ) d z 2 π K 0 dkkU ( k q D L t 2 ) U ( k q D L r 2 ) U ( k z η t 2 ) × U ( k z η r 2 ) BSFP ( k , z ) PSFP ( k , z ) J 0 ( 2 π k G q ) ,
K = ( π D L r η r / 4 ) 2 P t T r ,
BSFP ( k , z ) = PSFP ( k , z ) .
exp [ D ( ψ ) R ] = 2 π R 2 0 θ max d θ θ PSF ( θ , R ) J 0 ( 2 π ψ θ ) ,
exp [ D ( ψ ) R ] 2 π 0 d ρ ρ PSFP ( ρ , R ) J 0 ( 2 π k ρ ) = PSFP ( k , R ) .
BSFP ( k , R ) PSFP ( k , R ) [ PSFP ( k , R ) ] 2 = PSFP ( k , 2 R ) ,
d ( BS ) = BSC ( z ) d z 2 π K 0 dkkU ( 1 2 k q D L t ) U ( 1 2 k z η t ) × U ( 1 2 k q D L r ) U ( 1 2 k z η r ) PSFP ( k , 2 z ) J 0 ( 2 π k G q ) ,
d ( BS ) BSC ( z ) d z 2 π K 0 dkk PSFP ( k , 2 z ) J 0 ( 2 π k G q ) = BSC ( z ) d z K 4 PSFP ( G q , 2 z ) ,
PSFP ( k , z ) = exp ( c z b z 1 exp ( 2 π 3 k z / 100 ) 2 π 3 k z / 100 ) ,
VSF ( θ ) = [ 1.7 + θ 3 + 2.9 ( cos θ ) 2 ] × 10 4 in m 1 sr 1 ,
E u ( P ) = L T t d ω = L A e T t / z 2 ,
E u ( x , y , z ) = L T t z 2 d x d y Rect ( x , y , 1 2 D L ) × Rect ( x o x , y o y , 1 2 D S o ) ,
D S o = z D s / ( R q ) , x o = x / q , y o = y / q , ρ o 2 = x o 2 + y o 2 , q = 1 z / R .
E u ( x , y , z ) = ( L T t / q 2 z 2 ) d x d y Rect 1 t ( x , y , 1 2 q D L t ) × Rect 2 t ( x x , y y , D S t 1 2 z / R ) .
E u ( ρ , z ) = 2 π ( 4 R q z D L t D S t ) 2 0 dkk Rect 1 t ( k , z ) × Rect 2 t ( k , z ) J 0 ( 2 π k ρ ) ,
d P r u ( x , y , z ) = LdA T r A e z 2 = ( LdA T r / q 2 z 2 ) × d x d y Rect 1 r ( x , y , 1 2 q D L r ) × Rect 2 r ( x x , y y , D S r 1 2 z / R ) ,
d P r u ( x , y , z ) = ( 2 π LdA T r / q 2 z 2 ) 0 dkk Rect 1 r ( k , z ) × Rect 2 r ( k , z ) J 0 ( k r ) ,