Abstract

This paper formulates a theory of noncontact point thermal sensing by fiber-optic radiometry. This theory covers the field of mid- and far-infrared fibers that are suitable for low-temperature radiometry. However, new problems arise in the infrared range, the emission of thermal radiation from the fiber itself due to infrared absorption introduces perturbations into the radiometry, and this must be taken into consideration. The model presented is based on three-dimensional optical geometry of bounded and tunneling skew rays and yields an analytical expression for the inclination and the skewness angle distribution of the guided power collected by the fiber from various layers of a thermal body. The effective field of view, the surface resolution, and the temperature resolution of fiber-optic radiometry are discussed. Thermal sensing by direct coupling is shown to have an advantage over the coupling of a focusing lens located behind the fiber tip. A formulation of fiber emissivity is presented that quantifies the suppression of radiometric perturbations in fiber-optic thermal sensing. Bulk and surface absorption in the fiber core and cladding absorption are all taken into consideration deriving emissivity. Combining the transmissivity and emissivity of the fiber, we propose a measurable criterion, a figure of merit, for fiber-optic radiometry.

© 1992 Optical Society of America

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References

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  1. S. R. Mordon, A. H. Cornil, J. M. Brunetaud, “Temperature measurement with zirconium fluoride glass fiber,” Appl. Opt. 26, 607–609 (1987).
    [CrossRef] [PubMed]
  2. A. Zur, A. Katzir, “Infrared fibers for low temperature radiometric measurements,” Appl. Phys. Lett. 48, 499–500 (1986).
    [CrossRef]
  3. V. G. Artjushenko, V. V. Voitsekhovsky, V. J. Masychev, J. V. Zubov, V. K. Sysoev, “Fiberoptic device for simultaneous laser power transmission and temperature measurement of irradiated object,” Electron. Lett. 20, 983–984 (1984).
    [CrossRef]
  4. L. M. Hobrock, J. D. Sneed, “Radiometric applications of infrared fibers,” in Advances in Infrared Fibers, L. G. De-Shazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.320, 140–144 (1982).
  5. M. Shimizu, S. Kachi, “Low temperature radiometer using infrared fiber,” presented at the Third Sensor Symposium, Tsukaba, Japan, 1983.
  6. E. Sinofsky, M. G. Dumont, “Temperature measurement using silica and fluoride based optical fibers for biological applications,” in Laser Surgery: Characterization and Therapeutics, K. Atsumi, S. N. Joffe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.907, 131–136 (1988).
  7. R. J. Burger, D. A. Greenberg, P. Kirkitelos, “Radiometry, thermometry, and minimum resolvable temperature with IR fiber optics,” in Infrared Fiber Optics II, V. P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1228, 206–215 (1990).
  8. A. Katzir, H. F. Bowman, Y. Asfour, A. Zur, C. R. Valeri, “Infrared fibers for radiometer thermometry in hypothermia and hyperthermia treatment,” IEEE Trans. Biomed Eng. 6, 634–637 (1989).
    [CrossRef]
  9. A. Zur, A. Katzir, “Theory of fiber optic radiometry, emissivity of fibers, and distributed thermal sensors,” Appl. Opt. 30, 660–673 (1991).
    [CrossRef] [PubMed]
  10. D. A. Christensen, “Thermal dosimetry and temperature measurements,” Cancer Res. 39, 2325–2331 (1979).
    [PubMed]
  11. C. T. Cetas, W. Connor, “Thermometry considerations in localized hyperthermia,” Med. Phys. 5, 79–91 (1978).
    [CrossRef] [PubMed]
  12. A. Sa’ar, A. Katzir, “Scattering effects in crystalline infrared fibers,” J. Opt. Soc. Am. A 5, 823–833 (1988).
    [CrossRef]
  13. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Secs. 35-13, 7-3.
  14. A. Sa’ar, N. Barkay, F. Moser, I. Shnitzer, A. Katzir, “Optical and mechanical properties of silver halide fibers,” in Infrared Optical Materials and Fibers V, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 98–104 (1987).

1991 (1)

1989 (1)

A. Katzir, H. F. Bowman, Y. Asfour, A. Zur, C. R. Valeri, “Infrared fibers for radiometer thermometry in hypothermia and hyperthermia treatment,” IEEE Trans. Biomed Eng. 6, 634–637 (1989).
[CrossRef]

1988 (1)

1987 (1)

1986 (1)

A. Zur, A. Katzir, “Infrared fibers for low temperature radiometric measurements,” Appl. Phys. Lett. 48, 499–500 (1986).
[CrossRef]

1984 (1)

V. G. Artjushenko, V. V. Voitsekhovsky, V. J. Masychev, J. V. Zubov, V. K. Sysoev, “Fiberoptic device for simultaneous laser power transmission and temperature measurement of irradiated object,” Electron. Lett. 20, 983–984 (1984).
[CrossRef]

1979 (1)

D. A. Christensen, “Thermal dosimetry and temperature measurements,” Cancer Res. 39, 2325–2331 (1979).
[PubMed]

1978 (1)

C. T. Cetas, W. Connor, “Thermometry considerations in localized hyperthermia,” Med. Phys. 5, 79–91 (1978).
[CrossRef] [PubMed]

Artjushenko, V. G.

V. G. Artjushenko, V. V. Voitsekhovsky, V. J. Masychev, J. V. Zubov, V. K. Sysoev, “Fiberoptic device for simultaneous laser power transmission and temperature measurement of irradiated object,” Electron. Lett. 20, 983–984 (1984).
[CrossRef]

Asfour, Y.

A. Katzir, H. F. Bowman, Y. Asfour, A. Zur, C. R. Valeri, “Infrared fibers for radiometer thermometry in hypothermia and hyperthermia treatment,” IEEE Trans. Biomed Eng. 6, 634–637 (1989).
[CrossRef]

Barkay, N.

A. Sa’ar, N. Barkay, F. Moser, I. Shnitzer, A. Katzir, “Optical and mechanical properties of silver halide fibers,” in Infrared Optical Materials and Fibers V, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 98–104 (1987).

Bowman, H. F.

A. Katzir, H. F. Bowman, Y. Asfour, A. Zur, C. R. Valeri, “Infrared fibers for radiometer thermometry in hypothermia and hyperthermia treatment,” IEEE Trans. Biomed Eng. 6, 634–637 (1989).
[CrossRef]

Brunetaud, J. M.

Burger, R. J.

R. J. Burger, D. A. Greenberg, P. Kirkitelos, “Radiometry, thermometry, and minimum resolvable temperature with IR fiber optics,” in Infrared Fiber Optics II, V. P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1228, 206–215 (1990).

Cetas, C. T.

C. T. Cetas, W. Connor, “Thermometry considerations in localized hyperthermia,” Med. Phys. 5, 79–91 (1978).
[CrossRef] [PubMed]

Christensen, D. A.

D. A. Christensen, “Thermal dosimetry and temperature measurements,” Cancer Res. 39, 2325–2331 (1979).
[PubMed]

Connor, W.

C. T. Cetas, W. Connor, “Thermometry considerations in localized hyperthermia,” Med. Phys. 5, 79–91 (1978).
[CrossRef] [PubMed]

Cornil, A. H.

Dumont, M. G.

E. Sinofsky, M. G. Dumont, “Temperature measurement using silica and fluoride based optical fibers for biological applications,” in Laser Surgery: Characterization and Therapeutics, K. Atsumi, S. N. Joffe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.907, 131–136 (1988).

Greenberg, D. A.

R. J. Burger, D. A. Greenberg, P. Kirkitelos, “Radiometry, thermometry, and minimum resolvable temperature with IR fiber optics,” in Infrared Fiber Optics II, V. P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1228, 206–215 (1990).

Hobrock, L. M.

L. M. Hobrock, J. D. Sneed, “Radiometric applications of infrared fibers,” in Advances in Infrared Fibers, L. G. De-Shazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.320, 140–144 (1982).

Kachi, S.

M. Shimizu, S. Kachi, “Low temperature radiometer using infrared fiber,” presented at the Third Sensor Symposium, Tsukaba, Japan, 1983.

Katzir, A.

A. Zur, A. Katzir, “Theory of fiber optic radiometry, emissivity of fibers, and distributed thermal sensors,” Appl. Opt. 30, 660–673 (1991).
[CrossRef] [PubMed]

A. Katzir, H. F. Bowman, Y. Asfour, A. Zur, C. R. Valeri, “Infrared fibers for radiometer thermometry in hypothermia and hyperthermia treatment,” IEEE Trans. Biomed Eng. 6, 634–637 (1989).
[CrossRef]

A. Sa’ar, A. Katzir, “Scattering effects in crystalline infrared fibers,” J. Opt. Soc. Am. A 5, 823–833 (1988).
[CrossRef]

A. Zur, A. Katzir, “Infrared fibers for low temperature radiometric measurements,” Appl. Phys. Lett. 48, 499–500 (1986).
[CrossRef]

A. Sa’ar, N. Barkay, F. Moser, I. Shnitzer, A. Katzir, “Optical and mechanical properties of silver halide fibers,” in Infrared Optical Materials and Fibers V, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 98–104 (1987).

Kirkitelos, P.

R. J. Burger, D. A. Greenberg, P. Kirkitelos, “Radiometry, thermometry, and minimum resolvable temperature with IR fiber optics,” in Infrared Fiber Optics II, V. P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1228, 206–215 (1990).

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Secs. 35-13, 7-3.

Masychev, V. J.

V. G. Artjushenko, V. V. Voitsekhovsky, V. J. Masychev, J. V. Zubov, V. K. Sysoev, “Fiberoptic device for simultaneous laser power transmission and temperature measurement of irradiated object,” Electron. Lett. 20, 983–984 (1984).
[CrossRef]

Mordon, S. R.

Moser, F.

A. Sa’ar, N. Barkay, F. Moser, I. Shnitzer, A. Katzir, “Optical and mechanical properties of silver halide fibers,” in Infrared Optical Materials and Fibers V, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 98–104 (1987).

Sa’ar, A.

A. Sa’ar, A. Katzir, “Scattering effects in crystalline infrared fibers,” J. Opt. Soc. Am. A 5, 823–833 (1988).
[CrossRef]

A. Sa’ar, N. Barkay, F. Moser, I. Shnitzer, A. Katzir, “Optical and mechanical properties of silver halide fibers,” in Infrared Optical Materials and Fibers V, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 98–104 (1987).

Shimizu, M.

M. Shimizu, S. Kachi, “Low temperature radiometer using infrared fiber,” presented at the Third Sensor Symposium, Tsukaba, Japan, 1983.

Shnitzer, I.

A. Sa’ar, N. Barkay, F. Moser, I. Shnitzer, A. Katzir, “Optical and mechanical properties of silver halide fibers,” in Infrared Optical Materials and Fibers V, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 98–104 (1987).

Sinofsky, E.

E. Sinofsky, M. G. Dumont, “Temperature measurement using silica and fluoride based optical fibers for biological applications,” in Laser Surgery: Characterization and Therapeutics, K. Atsumi, S. N. Joffe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.907, 131–136 (1988).

Sneed, J. D.

L. M. Hobrock, J. D. Sneed, “Radiometric applications of infrared fibers,” in Advances in Infrared Fibers, L. G. De-Shazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.320, 140–144 (1982).

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Secs. 35-13, 7-3.

Sysoev, V. K.

V. G. Artjushenko, V. V. Voitsekhovsky, V. J. Masychev, J. V. Zubov, V. K. Sysoev, “Fiberoptic device for simultaneous laser power transmission and temperature measurement of irradiated object,” Electron. Lett. 20, 983–984 (1984).
[CrossRef]

Valeri, C. R.

A. Katzir, H. F. Bowman, Y. Asfour, A. Zur, C. R. Valeri, “Infrared fibers for radiometer thermometry in hypothermia and hyperthermia treatment,” IEEE Trans. Biomed Eng. 6, 634–637 (1989).
[CrossRef]

Voitsekhovsky, V. V.

V. G. Artjushenko, V. V. Voitsekhovsky, V. J. Masychev, J. V. Zubov, V. K. Sysoev, “Fiberoptic device for simultaneous laser power transmission and temperature measurement of irradiated object,” Electron. Lett. 20, 983–984 (1984).
[CrossRef]

Zubov, J. V.

V. G. Artjushenko, V. V. Voitsekhovsky, V. J. Masychev, J. V. Zubov, V. K. Sysoev, “Fiberoptic device for simultaneous laser power transmission and temperature measurement of irradiated object,” Electron. Lett. 20, 983–984 (1984).
[CrossRef]

Zur, A.

A. Zur, A. Katzir, “Theory of fiber optic radiometry, emissivity of fibers, and distributed thermal sensors,” Appl. Opt. 30, 660–673 (1991).
[CrossRef] [PubMed]

A. Katzir, H. F. Bowman, Y. Asfour, A. Zur, C. R. Valeri, “Infrared fibers for radiometer thermometry in hypothermia and hyperthermia treatment,” IEEE Trans. Biomed Eng. 6, 634–637 (1989).
[CrossRef]

A. Zur, A. Katzir, “Infrared fibers for low temperature radiometric measurements,” Appl. Phys. Lett. 48, 499–500 (1986).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

A. Zur, A. Katzir, “Infrared fibers for low temperature radiometric measurements,” Appl. Phys. Lett. 48, 499–500 (1986).
[CrossRef]

Cancer Res. (1)

D. A. Christensen, “Thermal dosimetry and temperature measurements,” Cancer Res. 39, 2325–2331 (1979).
[PubMed]

Electron. Lett. (1)

V. G. Artjushenko, V. V. Voitsekhovsky, V. J. Masychev, J. V. Zubov, V. K. Sysoev, “Fiberoptic device for simultaneous laser power transmission and temperature measurement of irradiated object,” Electron. Lett. 20, 983–984 (1984).
[CrossRef]

IEEE Trans. Biomed Eng. (1)

A. Katzir, H. F. Bowman, Y. Asfour, A. Zur, C. R. Valeri, “Infrared fibers for radiometer thermometry in hypothermia and hyperthermia treatment,” IEEE Trans. Biomed Eng. 6, 634–637 (1989).
[CrossRef]

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

Med. Phys. (1)

C. T. Cetas, W. Connor, “Thermometry considerations in localized hyperthermia,” Med. Phys. 5, 79–91 (1978).
[CrossRef] [PubMed]

Other (6)

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Secs. 35-13, 7-3.

A. Sa’ar, N. Barkay, F. Moser, I. Shnitzer, A. Katzir, “Optical and mechanical properties of silver halide fibers,” in Infrared Optical Materials and Fibers V, P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.843, 98–104 (1987).

L. M. Hobrock, J. D. Sneed, “Radiometric applications of infrared fibers,” in Advances in Infrared Fibers, L. G. De-Shazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.320, 140–144 (1982).

M. Shimizu, S. Kachi, “Low temperature radiometer using infrared fiber,” presented at the Third Sensor Symposium, Tsukaba, Japan, 1983.

E. Sinofsky, M. G. Dumont, “Temperature measurement using silica and fluoride based optical fibers for biological applications,” in Laser Surgery: Characterization and Therapeutics, K. Atsumi, S. N. Joffe, eds., Proc. Soc. Photo-Opt. Instrum. Eng.907, 131–136 (1988).

R. J. Burger, D. A. Greenberg, P. Kirkitelos, “Radiometry, thermometry, and minimum resolvable temperature with IR fiber optics,” in Infrared Fiber Optics II, V. P. Klocek, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1228, 206–215 (1990).

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Figures (8)

Fig. 1
Fig. 1

(a) Inclination angle θ and the skewness angle γ of a guided ray. (b) Point thermal sensing by fiber-optic radiometry.

Fig. 2
Fig. 2

Geometry of direct power coupling from the thermal surface to the fiber endface.

Fig. 3
Fig. 3

(a) Skewness distribution of power at the fiber’s input endface for various ratios of inclination i, Eq. (3.5c). (b) Inclination distribution of guided power at the fiber’s input endface for various fiber critical angles, Eq. (3.5b).

Fig. 4
Fig. 4

(a) Maximum incident angle θ1 m in direct coupling as a function of the fiber’s critical angle θ1 c , Eq. (3.7). The linear dashed curve (θ1 c ) represents the maximum incident angle for bounded power. The upper, solid curve is the θ1 m of the total guided power, including both bounded and tunneling rays (τ = 1). The upper dashed curves also take into account the power losses that are due to reflection at the fiber endfaces. In the bottom solid curve, θ1 m is reduced by limiting the field of detection of the fiber’s output power (θ1 υ = 20°). (b) Input power collected by direct coupling (top solid curve) as a function of the fiber’s critical angle θ1 c , Eq. (3.5). Bounded and tunneling rays are taken into account. The long dashed curve shows the reduction (%) in the power collected if only bounded rays (θ1 < θ1 c ) are considered. The medium dashed curves illustrate the reduction (%) in the power collected (relative to the top, solid curve) that is due to Fresnel reflection losses at the fiber endfaces. The bottom solid curve shows the power collected by a high NA fiber with a limited FOV.

Fig. 5
Fig. 5

Normalized input power collected by direct coupling as a function of the distance between the fiber tip and the thermal surface for various fiber critical angles. The curves are based on Eq. (3.5), with Eq. (3.9) as the upper integration limit and normalized as follows: Pin(h)/Pin(h = 0). The medium dashed curve represents a high NA fiber whose FOV has been limited (θ1 υ = 10°) by the field of detection of the fiber’s radiation output.

Fig. 6
Fig. 6

(a) Configuration of power coupling with a focusing lens behind the fiber tip (b) another view of (a).

Fig. 7
Fig. 7

(a) Input power coupled into the fiber as a function of the radius of the resolved area on the thermal surface. Both direct coupling, Eq. (3.10), and lens coupling, Eq. (3.13), with various f/numbers (f/2a L ) are illustrated (a = 350 μm, a L = 1.5 cm, a T = 1 cm, θ1 m = 50°). (b) As in (a) but for a smaller fiber FOV (θ1 m = 20°).

Fig. 8
Fig. 8

(a) Normalized layer emissivity, ∊λ1, Δx, x)/∊λ(0, Δx, x), as a function of the inclination angle (θ1) for layers at various optical depths in the bulk, Eq. (3.14a), with n b = 1.2. The dashed curve represents the surface layer (x = 0) of a highly absorbing material (αΔx > 10). (b) Normalized power, P d (λ, Δx, x)/P d (λ, Δx, 0), collected by the fiber from each layer of the bulk (solid curves) as a function of the layer’s optical depth, Eq. (3.15). The FOV (2θ1 m ) of the radiometry (dashed curves) associated with each layer of the bulk is also illustrated. Only the effect of coupling is considered; the critical angle is θ1 c = 90°, and fiber power losses have been ignored (τ = 1). Curves denoted by L represent a fiber whose FOV has been limited to 20° by the field of detection at the fiber’s output endface.

Equations (79)

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P out ( θ, γ, λ ) = P in ( θ, γ, λ ) τ ( θ, γ, λ ) .
τ ( θ, γ, λ ) ( 1 r 1 ) ( 1 r 2 ) exp ( α t L ) 1 r 1 r 2 exp ( 2 α t L ) ,
α t = α t ( θ, γ, λ ) , r 1 2 [ r ( θ, λ ) + r ( θ, λ ) ] .
n co sin θ = n 1 sin θ 1 = n 2 sin θ 2 ,
P ( θ 2 , γ, λ ) out = P ( θ 1 , γ, λ ) in J ( θ 1 , θ 2 ) τ ( θ, γ, λ ) .
P λ ( d A T d a S ) = I λ cos θ T d A T d Ω S ,
I λ = λ ( T ) W bb λ ( T ) π ( W cm 2 sr μ m ) ,
W bb λ ( T ) = c 1 λ 5 [ exp ( c 2 λ T ) 1 ] 1 , c 1 = 37 , 413 W μ 4 cm 2 , c 2 = 14 , 388 μ K .
P λ ( A T A S ) = I λ 0 π/ 2 d θ 1 sin θ 1 cos θ 1 × 0 2 π d Φ 4 T Φ T Φ T + π / 2 d Φ s 0 a R S d R S .
P λ ( A T A S ) = 4 I λ 0 π/ 2 d θ 1 sin θ 1 cos θ 1 0 π/ 2 d λ × 0 2 π d Φ T a sin γ a d R S a cos γ R S R s 2 a 2 sin 2 γ .
Θ = { 1 cos γ sin θ sin θ c 0 cos γ sin θ > sin θ c ,
P ( λ ) in = P a ( λ, T ) θ 1 = 0 π / 2 F in ( θ 1 , θ 1 c ) d θ 1 ,
P a ( λ, T ) π 2 a 2 I λ ( T ) ( W ) .
F in ( θ 1 , θ 1 c ) 8 π 0 π/ 2 cos θ 1 sin θ 1 f ( γ, θ 1 ) d γ,
f ( γ, θ 1 ) cos 2 γΘ ( sin θ 1 c sin θ 1 cos γ ) ,
sin θ 1 c = min [ ( n co / n 1 ) sin θ c , 1 ] .
P ( θ 1 , γ, λ ) in = 8 π a 2 I λ sin θ 1 cos θ 1 f ( γ, θ 1 ) .
0 θ 1 m d θ 1 0 π/ 2 d γ P ( θ 1 , γ, λ ) in τ ( θ , γ, λ ) = P d ( λ ) ,
A r = 2 0 π d Φ T 0 R T ( Φ T ) R T d R T .
R ( Φ T ) T = a cos θ 1 m + h sin θ 1 m cos ( θ 1 m + β ) ,
A r = π a r 2 ,
θ ( h ) 1 min [ θ 1 υ , tan 1 ( a T / h ) ] .
P ( λ ) in = P a ( λ, T ) sin 2 θ 1 m for h < a T cot θ 1 m , P ( λ ) in = P a ( λ, T ) 1 1 + ( h / a T ) 2 for h a T cot θ 1 m .
m = υ u , u = f ( 1 + 1 / m ) , υ = f ( 1 + m ) .
P λ ( d A T d A L ) = I λ cos 2 θ T R 2 d A T d A L , I λ = λ ( T ) W b b λ ( T ) π ,
P ( λ ) in = π 2 2 I λ [ X ( X 2 4 a r f 2 tan 2 θ 1 m ) 1 / 2 ] for a a r a L a f tan θ 1 m 1 ,
P ( λ ) in = π 2 2 I λ [ Z ( Z 2 4 a r a L 2 ) 1 / 2 ] for a a r a L + a f tan θ 1 m 1 ,
P ( θ 1 , γ, λ, Δ x , x ) in = P ( θ 1 , γ, λ ) in λ ( θ 1 , Δ x , x ) .
λ ( θ 1 , Δ x , x ) = { 1 exp [ α ( λ ) Δ x cos θ b ] } exp [ α ( λ ) x cos θ b ] .
θ b = sin 1 ( n 1 sin θ 1 / n b ) ,
0 θ 1 m ( Δ x , x ) d θ 1 0 π/ 2 P ( θ 1 , γ, λ, Δ x , x ) in × τ ( θ , γ, λ ) d λ = P d ( λ, Δ x , x ) .
a r ( Δ x , x ) = a + h tan θ 1 m ( Δ x , x ) + x tan θ b m ( Δ x , x ) .
f ( λ ) = 0 θ υ d θ 0 π/ 2 d γ g s ( α ab / α t ) [ 1 exp ( α t L ) ] ( R ev + R od ) ,
g s 8 sin θ cos θ f ( γ, θ ) / π sin 2 θ υ ,
R ev ( θ, γ, λ ) 1 r 1 r 2 exp ( 2 α t L ) , R od ( θ, γ, λ ) R ev r exp ( α t L ) .
f α co ( a ) L ( a 2 b 2 a 2 ) η + α co ( b ) L ( b 2 a 2 ) η for θ υ θ c ,
f α cl L η ,
η' = 4 λ π 2 a n cl r sin 2 θ υ sin 2 θ c 0 θ υ d θ sin 3 θ 0 π / 2 d γ × cos 2 γ ( sin 2 θ c sin 2 θ cos 2 γ ) 1 / 2 .
FOM | Δ T f ( max ) / Δ T b b ( min ) | T 0 ,
Δ S = λ a λ b res ( λ ) Δ P out 2 ( λ ) d λ = N ,
π a 2 sin 2 θ υ λ a λ b res ( λ ) f ( λ ) Δ W b b λ [ Δ T f ( max ) ] d λ = N ,
π a 2 sin 2 θ υ λ a λ b res ( λ ) τ b b ( λ ) Δ W b b λ [ Δ T b b ( min ) ] d λ = N ,
τ b b ( λ ) = n co 2 n 1 2 0 θ υ d θ 0 π/ 2 d γ g s ( 1 r ) R ev exp ( α t L ) .
FOM = n co 2 τ b b / n 1 2 f .
FOM = 16 n co 4 ( n co + 1 ) 4 f .
r ( n co , n t , θ ) = | n co cos θ Δ n co cos θ + Δ | 2 , r ( n co , n t , θ ) = | n t 2 cos θ n co Δ n t 2 cos θ + n co Δ | 2 ,
Δ ( n t 2 n co 2 sin 2 θ ) 1 / 2 ,
n co ( r ) = n co r i n co i ( r ) , n co i ( r ) = λ 4 π α co ( r ) .
α ab = ν α co ( r ) d s , ν = tan θ 2 a cos γ ,
s = a sin γ ( r 2 a 2 sin γ ) 1 / 2 sin θ .
α ab ( θ, γ ) = 1 a cos γ cos θ a sin γ a α co ( r ) r ( r 2 a 2 sin 2 γ ) 1 / 2 d r .
α ab ( θ ) = α co / cos θ ,
α co ( r ) = { α co ( a ) b r a α co ( b ) r < b ,
α ab ( θ, γ ) = 1 cos θ × { α co ( a ) sin γ b a α co ( a ) [ 1 Γ ( γ ) ] + α co ( b ) Γ ( γ ) sin γ b a ,
Γ ( γ ) ( b 2 / a 2 sin 2 γ ) 1 / 2 cos γ .
α ab = ν ( θ, γ ) [ 1 r ( θ, γ, λ ) ] .
α ab = λ 2 π a sin θ tan θ sin 2 θ c ( sin 2 θ c sin 2 θ cos 2 γ ) 1 / 2 ( α cl n cl r α co n co r ) .
α t ( θ , γ , λ ) = α sc + α ab ( core ) + α ab ( clad ) .
P in ( θ, γ ) d θ = P ( θ 1 , γ ) in d θ 1 , P out ( θ, γ ) = P ( θ 2 , γ ) out d θ 2 ,
d θ = J 1 , 2 d θ 1 , 2 , J 1 , 2 = n 1 , 2 cos θ 1 , 2 ( n co 2 n 1 , 2 2 sin 2 θ 1 , 2 ) 1 / 2 .
P in ( θ, γ ) = 1 J 1 P ( θ 1 , γ ) in , P out ( θ, γ ) = 1 J 2 P ( θ 2 , γ ) out .
J ( θ 1 , θ 2 ) J 2 / J 1 .
sin β = cos Φ T sin ψ, if Φ T = 0 , then β = ψ,
sin Φ T = sin Φ T / cos β , cos Φ T = cos Φ T cos ψ / cos β ,
tan θ 1 = R T cos β / ( h 0 + R T sin β ) ,
R T = h 0 sin θ 1 sin ( π / 2 β θ 1 ) = h 0 sin θ 1 ( cos β cos θ 1 sin β sin θ 1 ) ,
h 0 = h + R s cos Φ s tan ψ
R 2 = R T 2 + h 0 2 2 R T h 0 cos ( π 2 + β ) = h 0 2 cos 2 β ( cos β cos θ 1 sin β sin θ 1 ) 2 ,
cos θ T = cos ψ ( cos β cos θ 1 sin β sin θ 1 ) / cos β,
d A T = R T d R T d Φ T , d A s = R s d R s d Φ s .
d R T d Φ T = ( cos 2 β / cos ψ ) h 0 cos β ( cos β cos θ 1 sin β sin θ 1 ) 2 d θ 1 d Φ T .
sin γ = R s a | sin ( Φ s Φ T ) | .
d R s d Φ s = a cos γ ( R s 2 a 2 sin 2 γ ) 1 / 2 d R s d γ .
cos θ T = u R ,
R 2 = [ ( 1 + m ) 2 R T 2 + R L 2 2 ( 1 + m ) R T R L cos ( π Φ L ) ] + u 2 ,
R L d R L = υ 2 tan θ 1 cos 2 θ 1 d θ 1 .
P ( λ ) in = I λ 0 θ 1 m d θ 1 0 a r R T d R T 0 2 π d Φ L 0 2 π d Φ T u 2 υ 2 tan θ 1 cos 2 θ 1 R 4 ,
R 2 = R T 2 + R L 2 2 R T R L cos ( π Φ L ) + u 2 .
P ( λ ) in = I λ 0 a r R T d R T 0 a L R L d R L 0 2 π d Φ L 0 2 π d Φ T u 2 R 4 .

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