Abstract

Fiber-optic radiance microprobes, increasingly applied for measurements of internal light fields in living tissues, provide three-dimensional radiance distribution solids and radiant energy fluence rates at different depths of turbid samples. These data are, however, distorted because of an inherent feature of optical fibers: nonuniform angular sensitivity. Because of this property a radiance microprobe during a single measurement partly underestimates light from the envisaged direction and partly senses light from other directions. A theory of three-dimensional equidistant radiance measurements has been developed that provides correction for this instrumental error using the independently obtained function of the angular sensitivity of the microprobe. For the first time, as far as we know, the measurements performed with different radiance microprobes are comparable. An example of application is presented. The limitations of this theory and the prospects for this approach are discussed.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. C. Vogelmann, L. O. Björn, “Measurements of light gradients and spectral regime in plant tissue with a fiber optic probe,” Physiol. Plant. 60, 361–368 (1984).
    [CrossRef]
  2. T. C. Vogelmann, G. Martin, G. Chen, D. Buttry, “Fiber optic microprobes and measurement of the light microenvironment within plant tissues,” Adv. Bot. Res. 18, 255–295 (1991).
    [CrossRef]
  3. L. Fukshansky, N. Fukshansky-Kazarinova, A. M. von Remisowsky, “Estimation of optical parameters in a living tissue by solving the inverse problem of the multiflux radiative transfer,” Appl. Opt. 30, 3145–3153 (1991).
    [CrossRef] [PubMed]
  4. L. Lilge, T. Haw, B. C. Wilson, “Miniature isotropic optical fibre probes for quantitative light dosimetry in tissues,” Phys. Med. Biol. 38, 215–230 (1987).
    [CrossRef]
  5. B. B. Jørgensen, D. J. Des Marais, “A simple fiber-optic microprobe for high resolution light measurements: application in marine sediment,” Limnol. Oceanogr. 31, 1376– 1383 (1986).
    [CrossRef] [PubMed]
  6. M. Kühl, B. B. Jørgensen, “Spectral light measurements in microbenthic communities with a fiber-optic microprobe coupled to a sensitive diode array detector system,” Limnol. Oceanogr. 37, 1813–1823 (1992).
    [CrossRef]
  7. M. Kühl, B. B. Jørgensen, “The light field of microbenthic communities: radiance distribution and microscale optics of sandy coastal sediments,” Limnol. Oceanogr. 39, 1368–1398 (1994).
    [CrossRef]
  8. M. Kühl, C. Lassen, B. B. Jørgensen, “Optical properties of microbial mats: light measurements with fiber-optic microprobes,” in Microbial Mats: Structure, Development and Environmental Significance, L. J. Stal, P. Caumette, eds., NATO Advanced Studies Institute Series G (Springer-Verlag, Berlin, 1994), pp. 149–166.
  9. J. M. Senior, Optical Fiber Communications: Principles and Practice (Prentice-Hall, New York, 1985).
  10. P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1501 (1971).
    [CrossRef] [PubMed]

1994 (1)

M. Kühl, B. B. Jørgensen, “The light field of microbenthic communities: radiance distribution and microscale optics of sandy coastal sediments,” Limnol. Oceanogr. 39, 1368–1398 (1994).
[CrossRef]

1992 (1)

M. Kühl, B. B. Jørgensen, “Spectral light measurements in microbenthic communities with a fiber-optic microprobe coupled to a sensitive diode array detector system,” Limnol. Oceanogr. 37, 1813–1823 (1992).
[CrossRef]

1991 (2)

T. C. Vogelmann, G. Martin, G. Chen, D. Buttry, “Fiber optic microprobes and measurement of the light microenvironment within plant tissues,” Adv. Bot. Res. 18, 255–295 (1991).
[CrossRef]

L. Fukshansky, N. Fukshansky-Kazarinova, A. M. von Remisowsky, “Estimation of optical parameters in a living tissue by solving the inverse problem of the multiflux radiative transfer,” Appl. Opt. 30, 3145–3153 (1991).
[CrossRef] [PubMed]

1987 (1)

L. Lilge, T. Haw, B. C. Wilson, “Miniature isotropic optical fibre probes for quantitative light dosimetry in tissues,” Phys. Med. Biol. 38, 215–230 (1987).
[CrossRef]

1986 (1)

B. B. Jørgensen, D. J. Des Marais, “A simple fiber-optic microprobe for high resolution light measurements: application in marine sediment,” Limnol. Oceanogr. 31, 1376– 1383 (1986).
[CrossRef] [PubMed]

1984 (1)

T. C. Vogelmann, L. O. Björn, “Measurements of light gradients and spectral regime in plant tissue with a fiber optic probe,” Physiol. Plant. 60, 361–368 (1984).
[CrossRef]

1971 (1)

Björn, L. O.

T. C. Vogelmann, L. O. Björn, “Measurements of light gradients and spectral regime in plant tissue with a fiber optic probe,” Physiol. Plant. 60, 361–368 (1984).
[CrossRef]

Buttry, D.

T. C. Vogelmann, G. Martin, G. Chen, D. Buttry, “Fiber optic microprobes and measurement of the light microenvironment within plant tissues,” Adv. Bot. Res. 18, 255–295 (1991).
[CrossRef]

Chen, G.

T. C. Vogelmann, G. Martin, G. Chen, D. Buttry, “Fiber optic microprobes and measurement of the light microenvironment within plant tissues,” Adv. Bot. Res. 18, 255–295 (1991).
[CrossRef]

Des Marais, D. J.

B. B. Jørgensen, D. J. Des Marais, “A simple fiber-optic microprobe for high resolution light measurements: application in marine sediment,” Limnol. Oceanogr. 31, 1376– 1383 (1986).
[CrossRef] [PubMed]

Fukshansky, L.

Fukshansky-Kazarinova, N.

Haw, T.

L. Lilge, T. Haw, B. C. Wilson, “Miniature isotropic optical fibre probes for quantitative light dosimetry in tissues,” Phys. Med. Biol. 38, 215–230 (1987).
[CrossRef]

Jørgensen, B. B.

M. Kühl, B. B. Jørgensen, “The light field of microbenthic communities: radiance distribution and microscale optics of sandy coastal sediments,” Limnol. Oceanogr. 39, 1368–1398 (1994).
[CrossRef]

M. Kühl, B. B. Jørgensen, “Spectral light measurements in microbenthic communities with a fiber-optic microprobe coupled to a sensitive diode array detector system,” Limnol. Oceanogr. 37, 1813–1823 (1992).
[CrossRef]

B. B. Jørgensen, D. J. Des Marais, “A simple fiber-optic microprobe for high resolution light measurements: application in marine sediment,” Limnol. Oceanogr. 31, 1376– 1383 (1986).
[CrossRef] [PubMed]

M. Kühl, C. Lassen, B. B. Jørgensen, “Optical properties of microbial mats: light measurements with fiber-optic microprobes,” in Microbial Mats: Structure, Development and Environmental Significance, L. J. Stal, P. Caumette, eds., NATO Advanced Studies Institute Series G (Springer-Verlag, Berlin, 1994), pp. 149–166.

Kühl, M.

M. Kühl, B. B. Jørgensen, “The light field of microbenthic communities: radiance distribution and microscale optics of sandy coastal sediments,” Limnol. Oceanogr. 39, 1368–1398 (1994).
[CrossRef]

M. Kühl, B. B. Jørgensen, “Spectral light measurements in microbenthic communities with a fiber-optic microprobe coupled to a sensitive diode array detector system,” Limnol. Oceanogr. 37, 1813–1823 (1992).
[CrossRef]

M. Kühl, C. Lassen, B. B. Jørgensen, “Optical properties of microbial mats: light measurements with fiber-optic microprobes,” in Microbial Mats: Structure, Development and Environmental Significance, L. J. Stal, P. Caumette, eds., NATO Advanced Studies Institute Series G (Springer-Verlag, Berlin, 1994), pp. 149–166.

Lassen, C.

M. Kühl, C. Lassen, B. B. Jørgensen, “Optical properties of microbial mats: light measurements with fiber-optic microprobes,” in Microbial Mats: Structure, Development and Environmental Significance, L. J. Stal, P. Caumette, eds., NATO Advanced Studies Institute Series G (Springer-Verlag, Berlin, 1994), pp. 149–166.

Lilge, L.

L. Lilge, T. Haw, B. C. Wilson, “Miniature isotropic optical fibre probes for quantitative light dosimetry in tissues,” Phys. Med. Biol. 38, 215–230 (1987).
[CrossRef]

Martin, G.

T. C. Vogelmann, G. Martin, G. Chen, D. Buttry, “Fiber optic microprobes and measurement of the light microenvironment within plant tissues,” Adv. Bot. Res. 18, 255–295 (1991).
[CrossRef]

Mudgett, P. S.

Richards, L. W.

Senior, J. M.

J. M. Senior, Optical Fiber Communications: Principles and Practice (Prentice-Hall, New York, 1985).

Vogelmann, T. C.

T. C. Vogelmann, G. Martin, G. Chen, D. Buttry, “Fiber optic microprobes and measurement of the light microenvironment within plant tissues,” Adv. Bot. Res. 18, 255–295 (1991).
[CrossRef]

T. C. Vogelmann, L. O. Björn, “Measurements of light gradients and spectral regime in plant tissue with a fiber optic probe,” Physiol. Plant. 60, 361–368 (1984).
[CrossRef]

von Remisowsky, A. M.

Wilson, B. C.

L. Lilge, T. Haw, B. C. Wilson, “Miniature isotropic optical fibre probes for quantitative light dosimetry in tissues,” Phys. Med. Biol. 38, 215–230 (1987).
[CrossRef]

Adv. Bot. Res. (1)

T. C. Vogelmann, G. Martin, G. Chen, D. Buttry, “Fiber optic microprobes and measurement of the light microenvironment within plant tissues,” Adv. Bot. Res. 18, 255–295 (1991).
[CrossRef]

Appl. Opt. (2)

Limnol. Oceanogr. (3)

B. B. Jørgensen, D. J. Des Marais, “A simple fiber-optic microprobe for high resolution light measurements: application in marine sediment,” Limnol. Oceanogr. 31, 1376– 1383 (1986).
[CrossRef] [PubMed]

M. Kühl, B. B. Jørgensen, “Spectral light measurements in microbenthic communities with a fiber-optic microprobe coupled to a sensitive diode array detector system,” Limnol. Oceanogr. 37, 1813–1823 (1992).
[CrossRef]

M. Kühl, B. B. Jørgensen, “The light field of microbenthic communities: radiance distribution and microscale optics of sandy coastal sediments,” Limnol. Oceanogr. 39, 1368–1398 (1994).
[CrossRef]

Phys. Med. Biol. (1)

L. Lilge, T. Haw, B. C. Wilson, “Miniature isotropic optical fibre probes for quantitative light dosimetry in tissues,” Phys. Med. Biol. 38, 215–230 (1987).
[CrossRef]

Physiol. Plant. (1)

T. C. Vogelmann, L. O. Björn, “Measurements of light gradients and spectral regime in plant tissue with a fiber optic probe,” Physiol. Plant. 60, 361–368 (1984).
[CrossRef]

Other (2)

M. Kühl, C. Lassen, B. B. Jørgensen, “Optical properties of microbial mats: light measurements with fiber-optic microprobes,” in Microbial Mats: Structure, Development and Environmental Significance, L. J. Stal, P. Caumette, eds., NATO Advanced Studies Institute Series G (Springer-Verlag, Berlin, 1994), pp. 149–166.

J. M. Senior, Optical Fiber Communications: Principles and Practice (Prentice-Hall, New York, 1985).

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 (9)

Fig. 1
Fig. 1

Fiber-optic radiance microprobe: 1, fiber cable; 2, syringe; 3, hypodermic needle; 4, optical fiber. θ a is the apex angle of the circular cone of collection also known as the acceptance angle of the probe.

Fig. 2
Fig. 2

Spatial design of three-dimensional radiance measurements: θ0 = 0, θ1, … θ i , … θ k −1 = π represents the direction of measurements. Each measurement represents the radiance within the corresponding spherical band. The radiance value within a band is assumed to be constant.

Fig. 3
Fig. 3

Angular sensitivity distribution of a microprobe oriented in the θ direction: ϑ is the internal angular coordinate, ϑ = 0 coincides with θ, h(ϑ) is the relative sensitivity, h(ϑ) is maximal in the direction of the microprobe axis (ϑ = 0) whereas light that deviates from ϑ = 0 is captured with lower sensitivity. Moreover, the probe senses light from neighboring spheric bands (at ϑ > + θ a and ϑ < − θ a ). The dashed line shows the ideal angular sensitivity that is required to ensure that each measurement provides an undistorted value of the constant radiance within the corresponding spherical band.

Fig. 4
Fig. 4

Spatial and sensitivity relations between different components of radiation sensed in the course of a single measurement at point P under zenith angle θ = θ i . x, y, and z are local Cartesian coordinates and ϑ is the local spherical coordinate associated with this measurement. ϑ is zero in the θ i direction. h(ϑ) is the angular sensitivity of the microprobe. Li −1 Li and Li +1 are the unknown radiances within spherical bands i − 1, i, i + 1, respectively. α is the half-angle of the zenithal angular interval that corresponds to a single measurement; thus α is the apex angle of the circular cone whose cross section with the unit sphere—spherical surface S 0—matches the zenithal angular interval of the measured spherical band (in this case band number i). The entire radiation sensed in the course of this measurement is spread over spherical surface S, which includes parts of spherical bands i − 1, i, i + 1 and is the cross section of the unit sphere with a circular cone having apex angle 3α. Circle C is a mapping of spherical surface S onto the xPy plane.

Fig. 5
Fig. 5

Structure of sphercial surface S, which perceives the entire radiation sensed in measurement number i. Sii −1, Sii , and Sii +1 are parts of S that belong to sphercial bands i −1, i, i + 1 and, therefore, perceive radiances Li −1, L 1,Li +1, respectively. S 0Sii is the spherical surface that would perceive the entire radiation sensed in measurement number i if the angular sensitivity of the microprobe had an ideal rectangular shape with acceptance angle α as presented by the dashed line in Fig. 3. For the position of surface S on the unit sphere see Fig. 4.

Fig. 6
Fig. 6

Illustration of the transition from the integral over a curvilinear surface S to the integral over a plane surface σ.n and n 1 are the normals to surfaces S and σ, respectively. M and N are the current points on surfaces S and σ, respectively.

Fig. 7
Fig. 7

Mapping of the surface of a unit sphere onto the xPy plane that is normal to the θ i direction of measurement number i (see Fig. 4). Circle C with diameter R = sin(3α) is the mapping of spherical surface S. Ellipses Cii −1 and Cii +1 are mappings of circles that delimit the spherical bands i − 1, i, i + 1 and, in particular, areas Sii −1, Si , and Sii +1 inside surface S, respectively (see Fig. 5). Areas σ ii −1, σ i , and σ ii +1 are mappings of spherical surfaces Sii −1, Si , and Sii +1, respectively. φii±1 and R are polar coordinates of points of intersection of ellipses Cii ±1 with circle C. φ i 0 and Ri 0 are polar coordinates of the point of contact of the tangent to ellipse Cii −1 from point P.

Fig. 8
Fig. 8

Derivation of the parametric description of ellipses Cii ±1 presented in Fig. 7 and given in Appendix A. x and y are Cartesian coordinates, and φ and r are polar coordinates on the xPy plane. a and b are the axes of the ellipse with center P 1. M is the point of contact of the tangent to the ellipse from point P, which is outside the ellipse because P 1 P = q > a.

Fig. 9
Fig. 9

Diagram of the corrected (dashed line) and uncorrected (solid line) radiant fluxes over single spherical bands. The bands are specified according to the directions of microprobe θ i at successive measurements as presented in Fig. 2. The diagram presents measurements at depth d = 0.5 mm. The corresponding radiances are also shown in Table 1. The flux values are normalized against the incident flux.

Tables (1)

Tables Icon

Table 1 Comparison of Corrected and Uncorrected Radiance Values Obtained from Measurements at Different Depths of a Coastal Sediment with Diatoms (λ = 650 nm) a d

Equations (28)

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

K = π 2 α + 1 ,
M i = S 0 f ( ϑ ) h ( ϑ ) d S = L i S 0 ,
M i = S f ( ϑ ) h ( ϑ ) d S = L i 1 S i i 1 h ( ϑ ) d S + L i Sii h ( ϑ ) d S + L i + 1 S i i + 1 h ( ϑ ) d S = L i 1 J i i 1 + L i J i i + L i + 1 J i i + 1 .
S f ( M ) d S = σ f ( N ) cos ( n , n 1 ) d σ ,
a i i ± 1 = sin ( θ i ± α ) cos ( θ i ) , b i i ± 1 = sin ( θ i ± α ) , q i i ± 1 = cos ( θ i ± α ) sin ( θ i ) , a 0 , 1 = b 0 , 1 = a k 1 , k = b k 1 , k = 0 .
for C , r = R [ R = sin ( 3 α ) as desribed above ] ; for C i i ± 1 , r = f i i ± 1 ( φ ) .
f i j ± ( φ ) = b i j a i j 2 + c i j 2 cos 2 ( φ ) { q i j b i j cos ( φ ) ± a i j [ a i j 2 q i j 2 + ( q i j 2 + c i j 2 ) cos 2 ( φ ) 1 / 2 } , g i j = 1 R c i j 2 { q i j 2 b i j 2 a i j 2 × [ q i j 2 b i j 2 c i j 2 ( R 2 b i j 2 ) ] 1 / 2 } ,
r = f i i 1 ± ( φ ) , cos ( φ i i 1 ) = g i i 1 , cos ( φ i 0 ) = ( q i i 1 2 a i i 1 2 q i i 1 2 + c i i 1 2 ) 1 / 2 , R i 0 = 1 q i i 1 [ ( q i i 1 2 a i i 1 2 ) ( q i i 1 2 c i i 1 2 ) ] 1 / 2 ,
r = f i i + 1 + ( φ ) , cos ( φ i i + 1 ) = g i i + 1 .
J i i 1 = σ i i 1 h ( ϑ ) cos ( ϑ ) d σ ,
J i i 1 = 2 0 φ i i 1 d φ f i i 1 ( φ ) R h ( ϑ ) cos ( ϑ ) r d r .
J ii 1 = 2 0 φ i i 1 d φ f i i 1 ( φ ) R h ( ϑ ) cos ( ϑ ) r d r + 2 φ i i 1 φ i 0 d φ f i i 1 ( φ ) f i i 1 + ( φ ) h ( ϑ ) cos ( ϑ ) r d r .
J i i 1 = 2 0 φ i 0 d φ f i i + 1 ( φ ) f i i 1 + ( φ ) h ( ϑ ) cos ( ϑ ) r d r .
J i i + 1 = 2 0 φ i i + 1 d φ f i i + 1 ( φ ) R h ( ϑ ) cos ( ϑ ) r d r ,
J i i = 2 0 π d φ 0 R h ( ϑ ) cos ( ϑ ) r d r J i i 1 J i i 1 J i i + 1 .
h ( ϑ ) = exp [ m sin 2 ( ϑ ) ] cos ( ϑ ) ,
f 1 ( φ ) f 2 ( φ ) exp ( m r 2 ) r d r = 1 2 m { exp [ m f 1 2 ( φ ) ] exp [ m f 2 2 ( φ ) ] } .
J i i = J K 1 i , K 1 i , J i i 1 = J K 1 i , K i , J i i + 1 = J K 1 i , K i 2 .
L i 1 J i i 1 + L i J i i + L i + 1 J i i + 1 = M i , J 0 , 1 = J K 1 , K = 0 .
( q x ) 2 a 2 + y 2 b 2 = 1 ,
y 2 = b 2 b 2 a 2 ( q x ) 2 .
r 2 = x 2 + y 2 = x 2 + b 2 b 2 a 2 ( q x ) 2 .
r 2 [ 1 + c 2 a 2 cos 2 ( φ ) ] 2 q b 2 a 2 cos ( φ ) + q 2 b 2 a 2 b 2 = 0 ,
r = q b 2 cos ( φ ) ± a b [ a 2 q 2 + ( q 2 + c 2 ) cos 2 ( φ ) ] 1 / 2 a 2 + c 2 cos 2 ( φ ) .
a b [ a 2 q 2 + ( q 2 + c 2 ) cos 2 ( φ ) ] 1 / 2 | q b 2 cos ( φ ) | .
cos ( φ 0 ) = ± ( q 2 a 2 q 2 + c 2 ) 1 / 2 ,
R 0 = r ( φ 0 ) = [ ( q 2 a 2 ) ( q 2 + c 2 ) ] 1 / 2 | q | .
cos ( φ 1 ) = 1 R c 2 { q b 2 ± a [ q 2 b 2 c 2 ( R 2 b 2 ) ] 1 / 2 } .

Metrics