Abstract

Theoretical calculations are presented for the performance of a fiber-optic radiometer which makes use of infrared transmitting fibers to measure low temperatures (near room temperature). We calculate the radiometer spatial resolution, the dependence of the radiometer signal on the surface temperature, and the minimum resolvable temperature difference (MRΔT) of the radiometer. The performance of the fiber-optic radiometer is compared with a conventional optical (thin lens) radiometer.

© 1987 Optical Society of America

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References

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  1. L. M. Hobrock, J. D. Sneed, “Radiometric Applications of Infrared Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 320, 140 (1982).
  2. M. Shimizu, S. Kachi, “Low Temperature Radiometer Using Infrared Fiber,” in Proceedings, Third Sensor Symposium, Tsukaba, Japan (1983), p. 275.
  3. S. Kachi, K. Nakamura, M. Kimura, K. Shiroyama, “Reduction of the Scattering Loss of Polycrystalline Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 484, 128 (1984).
  4. T. Ueda, A. Hosokawa, A. Yamamoto, “Temperature Measurement Method by Infrared Radiation Pyrometer Using Optical Fiber,” J. Jpn. Soc. Precis. Eng. 48, 629 (1982).
    [CrossRef]
  5. A. Zur, A. Katzir, “Use of Infrared Fibers for Low-Temperature Radiometric Measurments,” Appl. Phys. Lett. 48, 499 (1986).
    [CrossRef]
  6. A. W. Snyder, D. J. Mitchell, “Leaky Rays on Circular Optical Fibers,” J. Opt. Soc. Am. 64, 599 (1974).
    [CrossRef]
  7. C. Pask, A. W. Snyder, “Multimode Optical Fibers: Interplay of Absorption and Radiation Losses,” Appl. Opt. 15, 1295 (1976).
    [CrossRef] [PubMed]
  8. Transmission theory of polycrystalline fibers. A full discussion will appear in a subsequent paper.
  9. R. J. Potter, “Transmission Properties of Optical Fibers,” J. Opt. Soc. Am. 51, 1079 (1961).
    [CrossRef]

1986 (1)

A. Zur, A. Katzir, “Use of Infrared Fibers for Low-Temperature Radiometric Measurments,” Appl. Phys. Lett. 48, 499 (1986).
[CrossRef]

1984 (1)

S. Kachi, K. Nakamura, M. Kimura, K. Shiroyama, “Reduction of the Scattering Loss of Polycrystalline Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 484, 128 (1984).

1982 (2)

T. Ueda, A. Hosokawa, A. Yamamoto, “Temperature Measurement Method by Infrared Radiation Pyrometer Using Optical Fiber,” J. Jpn. Soc. Precis. Eng. 48, 629 (1982).
[CrossRef]

L. M. Hobrock, J. D. Sneed, “Radiometric Applications of Infrared Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 320, 140 (1982).

1976 (1)

1974 (1)

1961 (1)

Hobrock, L. M.

L. M. Hobrock, J. D. Sneed, “Radiometric Applications of Infrared Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 320, 140 (1982).

Hosokawa, A.

T. Ueda, A. Hosokawa, A. Yamamoto, “Temperature Measurement Method by Infrared Radiation Pyrometer Using Optical Fiber,” J. Jpn. Soc. Precis. Eng. 48, 629 (1982).
[CrossRef]

Kachi, S.

S. Kachi, K. Nakamura, M. Kimura, K. Shiroyama, “Reduction of the Scattering Loss of Polycrystalline Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 484, 128 (1984).

M. Shimizu, S. Kachi, “Low Temperature Radiometer Using Infrared Fiber,” in Proceedings, Third Sensor Symposium, Tsukaba, Japan (1983), p. 275.

Katzir, A.

A. Zur, A. Katzir, “Use of Infrared Fibers for Low-Temperature Radiometric Measurments,” Appl. Phys. Lett. 48, 499 (1986).
[CrossRef]

Kimura, M.

S. Kachi, K. Nakamura, M. Kimura, K. Shiroyama, “Reduction of the Scattering Loss of Polycrystalline Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 484, 128 (1984).

Mitchell, D. J.

Nakamura, K.

S. Kachi, K. Nakamura, M. Kimura, K. Shiroyama, “Reduction of the Scattering Loss of Polycrystalline Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 484, 128 (1984).

Pask, C.

Potter, R. J.

Shimizu, M.

M. Shimizu, S. Kachi, “Low Temperature Radiometer Using Infrared Fiber,” in Proceedings, Third Sensor Symposium, Tsukaba, Japan (1983), p. 275.

Shiroyama, K.

S. Kachi, K. Nakamura, M. Kimura, K. Shiroyama, “Reduction of the Scattering Loss of Polycrystalline Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 484, 128 (1984).

Sneed, J. D.

L. M. Hobrock, J. D. Sneed, “Radiometric Applications of Infrared Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 320, 140 (1982).

Snyder, A. W.

Ueda, T.

T. Ueda, A. Hosokawa, A. Yamamoto, “Temperature Measurement Method by Infrared Radiation Pyrometer Using Optical Fiber,” J. Jpn. Soc. Precis. Eng. 48, 629 (1982).
[CrossRef]

Yamamoto, A.

T. Ueda, A. Hosokawa, A. Yamamoto, “Temperature Measurement Method by Infrared Radiation Pyrometer Using Optical Fiber,” J. Jpn. Soc. Precis. Eng. 48, 629 (1982).
[CrossRef]

Zur, A.

A. Zur, A. Katzir, “Use of Infrared Fibers for Low-Temperature Radiometric Measurments,” Appl. Phys. Lett. 48, 499 (1986).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Zur, A. Katzir, “Use of Infrared Fibers for Low-Temperature Radiometric Measurments,” Appl. Phys. Lett. 48, 499 (1986).
[CrossRef]

J. Jpn. Soc. Precis. Eng. (1)

T. Ueda, A. Hosokawa, A. Yamamoto, “Temperature Measurement Method by Infrared Radiation Pyrometer Using Optical Fiber,” J. Jpn. Soc. Precis. Eng. 48, 629 (1982).
[CrossRef]

J. Opt. Soc. Am. (2)

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

S. Kachi, K. Nakamura, M. Kimura, K. Shiroyama, “Reduction of the Scattering Loss of Polycrystalline Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 484, 128 (1984).

L. M. Hobrock, J. D. Sneed, “Radiometric Applications of Infrared Fibers,” Proc. Soc. Photo-Opt. Instrum. Eng. 320, 140 (1982).

Other (2)

M. Shimizu, S. Kachi, “Low Temperature Radiometer Using Infrared Fiber,” in Proceedings, Third Sensor Symposium, Tsukaba, Japan (1983), p. 275.

Transmission theory of polycrystalline fibers. A full discussion will appear in a subsequent paper.

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Figures (15)

Fig. 1
Fig. 1

Infrared fiber radiometer.

Fig. 2
Fig. 2

Guided skew ray at the fiber. A ray is characterized by the angles θ and γ.

Fig. 3
Fig. 3

Fiber is held at a distance h from the thermal surface (ϕ = 0).

Fig. 4
Fig. 4

Fiber axis generates an angle ϕ with the surface normal.

Fig. 5
Fig. 5

Fiber is held at a distance h (R s s ) from the thermal surface (ϕ ≠ 0).

Fig. 6
Fig. 6

Relative guided power vs the alignment angle ϕ based on Eq. (10): a = 0.45 mm; n1 = 2; θ M = 90°.

Fig. 7
Fig. 7

Thin lens radiometer, first view.

Fig. 8
Fig. 8

Thin lens radiometer, second view.

Fig. 9
Fig. 9

Relative transmitted power vs the sampled area diameter d T based on Eq. (15): d L = 2.5 cm; d D = 0.1 cm; f/# ≡ f/d L .

Fig. 10
Fig. 10

Ratio of the power guided by a fiber to the power transmitted by a thin lens vs the spatial resolution d T : f/# = 1.5; d D = 0.1 cm; a = 0.45 mm; n1 = 2; θ M = 90°; L = 1 m; α = ½(db)/m.

Fig. 11
Fig. 11

Direct coupling of the fiber to the detector, view 1.

Fig. 12
Fig. 12

Direct coupling of the fiber to the detector, view 2.

Fig. 13
Fig. 13

Radiometer signal (dashed line) vs the thermal surface temperature based on Eq. (27). The solid line represents a signal which depends on the temperature as −T4: λ a = 6.5 μm; λ b = 14 μm; resλ = const; λ = const; α(db/m) = (36.2/λ2) + 2.36. Graph for temperatures from 0 to 55°C.

Fig. 14
Fig. 14

Same as Fig. 13 but for temperatures from 40 to 700°C.

Fig. 15
Fig. 15

MRΔT vs time constant τ at different surface temperatures (37°C and 46°C): measured values of MRΔT at 37°C; measured values of MRΔT at 46°C; λ a = 6.5 μm; λ b = 14 μm; L = 1 m; = 0.9; α(db/m) = (36.2/λ2) + 2.36; f r = 5 Hz; NEP = 7 × 10 - 10 W / Hz ; n1 = 2; a = 0.45 mm; θeff = 55°; h D = 0.4 cm.

Equations (39)

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P λ out = 4 λ ( T ) W b b λ ( T ) θ = 0 π / 2 M ( θ ) d θ ;
W b b λ = C 1 λ - 5 [ exp ( c 2 / λ T ) - 1 ] , c 1 = 3.74 × 10 4 W μ 4 cm - 2 ,             c 2 = 14 , 388 μ K ;
M ( θ ) = R s = 0 a d R s R s Φ T = 0 π cos θ sin θ T 2 R η exp ( - α l ) S d Φ T .
T = 1 - 1 2 [ r ( n 0 , n 1 , θ ) + r ( n 0 , n 1 , θ ) ] , R = 1 2 [ r ( n 1 , n 2 , α 1 ) + r ( n 1 , n 2 , α 1 ) ] ,
r ( n , n , α ) = | n cos α - ( n 2 - n 2 sin 2 α ) 1 / 2 n cos α + ( n 2 - n 2 sin 2 α ) 1 / 2 | 2 , r ( n , n , α ) = | n 2 cos α - n ( n 2 - n 2 sin 2 α ) 1 / 2 n 2 cos α + n ( n 2 - n 2 sin 2 α ) 1 / 2 | 2 .
η = L tan θ 1 2 a cos γ = L n 0 sin θ 2 ( n 1 2 - n 0 2 sin 2 θ ) 1 / 2 ( a 2 - R s 2 sin 2 Φ T ) 1 / 2 ;
l = L cos θ 1 = L n 1 ( n 1 2 - n 0 2 sin 2 θ ) 1 / 2 ;
cos γ sin θ sin θ M ( n 1 2 - n 2 2 ) 1 / 2 n 0 ,
S = { 1 for [ 1 - ( R s sin Φ T / a ) 2 ] 1 / 2 sin θ sin θ M , 0 for [ 1 - ( R s sin Φ T / a ) 2 ] 1 / 2 sin θ > sin θ M .
d T = 2 ( h tan θ eff + a ) .
h < ( D T / 2 ) - a tan θ eff .
P λ out = λ ( T ) π W b b λ ( T ) θ = 0 π / 2 M ( θ , ϕ ) d θ ,
M ( θ , ϕ ) = 0 2 π d Φ s 0 a d R s R s 0 2 π sin θ cos θ T 2 exp ( - α l ) R η s ( 1 - cos 2 Φ T sin 2 ϕ ) 1 / 2 d Φ T ,
s = { 1 for { 1 - [ R s sin ( Φ s - Φ T ) / a ] 2 } 1 / 2 sin θ sin θ M , 0 for { 1 - [ R s sin ( Φ s - Φ T ) / a ] 2 } 1 / 2 sin θ > sin θ M .
d T = d D V D U T .
P λ ( T ) = π λ ( T ) W b b λ ( T ) L P ( d T ) ,
L P ( d T ) = 0 d T / 2 R T { 1 + ( d L / 2 ) 2 - f 2 ( 1 + d T / d D ) 2 - R T 2 [ ( d L / 2 ) 2 + f 2 ( 1 + d T / d D ) 2 + R T 2 ] 2 - d L 2 R T 2 } d R T ;
P S λ = 0 a d R s R s 0 2 π d Φ s 0 θ D d θ 0 2 π d Φ D W ω λ ( θ ) sin θ s D ;
S D = { 1 for R D 2 b 2 , 0 for R D 2 > b 2 ,
R D 2 R s 2 + h D 2 tan 2 θ + 2 R s h D tan θ cos Φ D .
W ω λ ( θ ) = λ ( T ) π W b b λ ( T ) M ( θ , ϕ ) / 2 π · π a 2 .
P S λ ( T ) = λ ( T ) W b b λ ( T ) g s λ ;
g s λ = 1 π 2 a 2 0 θ D d θ 0 a d R s R s 0 2 π d Φ D sin θ M ( θ , ϕ ) S D .
P λ ( t ) = { P L λ ( T L ) + P S λ ( T ) , 0 < ω r t < π , P r λ ( T r ) , - π < ω r t < 0 ,
P λ ( t ) = P 0 + 2 P 1 λ sin ( ω r t + ϕ r ) + ,
P 1 λ = [ P L λ ( T L ) + P S λ ( T ) - P r λ ( T r ) ] 2 π .
S 1 λ = 2 P 1 λ sin ( ω r t + ϕ r ) res λ ( f r ) ;
S λ = 2 2 π A G A L ( 0 ) P 1 λ res λ ( f r ) ,
S ( T ) = K G λ a λ b [ res λ ( f r ) λ ( T ) W b b λ ( T ) g s λ ] d λ + V r ,
K G 4 π 2 A L ( 0 ) A G ,
V r K G λ a λ b [ P L λ ( T L ) - P r λ ( T r ) ] res λ ( f r ) d λ - V dc .
Δ S = [ K G λ a λ b λ ( T ) c 2 λ T 2 W b b λ ( T ) res λ ( f r ) g s λ d λ ] Δ T .
Δ S N = Δ S [ 8 π 2 A G 2 0 B in V n 2 ( f ) A L 2 ( f - f r ) d f ] 1 / 2 = 2.
Δ T min = 2 π A L ( 0 ) [ 0 B in V n 2 ( f ) A L 2 ( f - f r ) d f ] 1 / 2 λ a λ b λ ( T ) c 2 λ T 2 W b b λ ( T ) res λ ( f r ) g s λ d λ .
M R Δ T Δ T min = π 2 [ τ λ a λ b λ ( T ) W b b λ ( T ) c 2 g s λ NEP λ ( f r ) λ T 2 d λ ] - 1 ;
NEP λ ( f r ) = V n ( f r ) res λ ( f r ) ,
0 B in A L 2 ( f - f r ) A L 2 ( 0 ) d f = 0 5 / τ 1 { 1 + [ 2 π τ ( f - f r ) ] 2 } 1 / 2 d f = 1 4 τ .
Δ S = [ K G λ a λ b res λ ( f r ) W b b λ ( T ) g s λ d λ ] Δ .
M R Δ = π 2 [ τ λ a λ b W b b λ ( T ) g s λ NEP λ ( f r ) d λ ] - 1 .

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