Abstract

The possibilities for using the pulsed time-of-flight (TOF) laser radar technique for hot refractory lining measurements are examined, and formulas are presented for calculating the background radiation collected, the achievable signal-to-noise ratio (SNR), and the measurement resolution. An experimental laser radar device is presented based on the use of a laser diode as a transmitter. Results obtained under real industrial conditions show that a SNR of 10 can be achieved at measurement distances of up to 15–20 m if the temperature of the converter is 1400 °C and the peak power of the laser diode used is 10 W. The single-shot resolution is about 60 mm (sigma value), but it can be improved to millimeter range by averaging techniques over a measurement time of 0.5 s. A commercial laser radar profiler based on the experimental laser radar device is also presented, and results obtained with it in real measurement situations are shown. These measurements indicate that it is possible to use the pulsed TOF laser radar technique in demanding measurement applications of this kind to obtain reliable data on the lining wear rate of a hot converter in a steel works.

© 1993 Optical Society of America

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References

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  1. M. Manninen, “Task-oriented approach to interactive control of heavy-duty manipulators based on coarse scene description,” Acta Polytech. Scand. Math. Comput. Sci. Ser. 42, 1–81 (1984).
  2. J. Kostamovaara, K. Määttä, M. Koskinen, R. Myllylä, “Pulsed laser radars with high-modulation-frequency in industrial applications,” in Laser Radar VII: Advanced Technology for Applications, Richard J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1633, 114–127 (1992).
  3. T. A. Clarke, K. T. V. Grattan, N. E. Lindsey, “Laser-based triangulation techniques in optical inspection of industrial structures,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 474–485 (1990).
  4. D. E. Smith, “Electronic distance measurement for industrial and scientific applications,” Hewlett-Packard J. 31, 3–10 (1980).
  5. M. Manninen, J. Jaatinen, “Productive method and system to control dimensional uncertainties at final assembling stages in ship production,” J. Ship Product. 8, 244–248 (1992).
  6. K. Määttä, J. Kostamovaara, R. Myllylä, “On the measurement of hot surfaces by pulsed time-of-flight laser radar techniques,” in Industrial Inspection II, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1265, 179–191 (1990).
  7. L. Green, “Condition monitoring—its impact on plant performance at British Steel, Scunthorpe Works,” Ironmak. Steelmak. 17, 355–362 (1990).
  8. K. Määttä, J. Kostamovaara, R. Myllylä, “A laser range-finder for hot surface profiling measurements, in Laser Technologies in Industry, O. D. D. Soares, ed., Proc. Soc. Photo-Opt. Instrum. Eng.952, 356–364 (1988).
  9. J. F. Ready, Industrial Applications of Lasers (Academic, New York, 1978).
  10. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, New York, 1981).
  11. G. Bertolini, “Pulse shape and time resolution,” in Semiconductor Detectors, G. Bertolini, A. Coche, eds. (North-Holland, Amsterdam, 1968), pp. 243–276.
  12. J. Kostamovaara, “Techniques and devices for positron lifetime measurement and time-of-flight laser rangefinding,” Acta Univ. Oulu 37, 1–69 (1986).
  13. C30902E, C30902S, C30921E, and C30921S Data sheet (RCA, Inc., Electro Optics, Vaudreuil, Canada, 1988).
  14. H. Kressel, Semiconductor Devices for Optical Communication (Springer-Verlag, Berlin, 1980).
  15. R. G. Meyer, R. A. Blauschild, “A wide-band low-noise monolithic transimpedance amplifier,” IEEE J. Solid-State Circuits SSC-21, 530–533 (1986).
    [CrossRef]
  16. RCA Corporation, RCA-Electro-Optics Handbook, tech. ser. EOH-11 (RCA, Inc., Lancaster, Pa., 1974).
  17. J. Wang, K. Määttä, J. Kostamovaara, “Signal power estimation in short range laser radars,” in ICALEO’91: Optical Sensing and Measurement Symposium (Laser Institute of America, Toledo, Ohio, 1992), pp. 16–26.
  18. K. Määttä, J. Kostamovaara, R. Myllylä, “Time-to-digital converter for fast, accurate laser rangefinding, in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 60–67 (1988).
  19. P. P. Webb, R. J. McIntyre, J. Conradi, “Properties of avalanche photodiodes,” RCA Rev. 35, 234–278 (1974).
  20. M. Karppinen, J. Kostamovaara, R. Myllylä, M. Seppänen, K. Määttä, “A novel laser rangefinder system for the profile measurements of refractory linings,” in Proceedings of the UNITECR’89 Meeting, L. J. Trostel, ed. (American Ceramic Society, Westerville, Ohio, 1989), 1340–1353.
  21. G. Klages, “Measurement of refractory lining in metallurgical vessels,” Metall. Plant Technol. 12, 32–41 (1989).

1992 (1)

M. Manninen, J. Jaatinen, “Productive method and system to control dimensional uncertainties at final assembling stages in ship production,” J. Ship Product. 8, 244–248 (1992).

1990 (1)

L. Green, “Condition monitoring—its impact on plant performance at British Steel, Scunthorpe Works,” Ironmak. Steelmak. 17, 355–362 (1990).

1989 (1)

G. Klages, “Measurement of refractory lining in metallurgical vessels,” Metall. Plant Technol. 12, 32–41 (1989).

1986 (2)

J. Kostamovaara, “Techniques and devices for positron lifetime measurement and time-of-flight laser rangefinding,” Acta Univ. Oulu 37, 1–69 (1986).

R. G. Meyer, R. A. Blauschild, “A wide-band low-noise monolithic transimpedance amplifier,” IEEE J. Solid-State Circuits SSC-21, 530–533 (1986).
[CrossRef]

1984 (1)

M. Manninen, “Task-oriented approach to interactive control of heavy-duty manipulators based on coarse scene description,” Acta Polytech. Scand. Math. Comput. Sci. Ser. 42, 1–81 (1984).

1980 (1)

D. E. Smith, “Electronic distance measurement for industrial and scientific applications,” Hewlett-Packard J. 31, 3–10 (1980).

1974 (1)

P. P. Webb, R. J. McIntyre, J. Conradi, “Properties of avalanche photodiodes,” RCA Rev. 35, 234–278 (1974).

Bertolini, G.

G. Bertolini, “Pulse shape and time resolution,” in Semiconductor Detectors, G. Bertolini, A. Coche, eds. (North-Holland, Amsterdam, 1968), pp. 243–276.

Blauschild, R. A.

R. G. Meyer, R. A. Blauschild, “A wide-band low-noise monolithic transimpedance amplifier,” IEEE J. Solid-State Circuits SSC-21, 530–533 (1986).
[CrossRef]

Clarke, T. A.

T. A. Clarke, K. T. V. Grattan, N. E. Lindsey, “Laser-based triangulation techniques in optical inspection of industrial structures,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 474–485 (1990).

Conradi, J.

P. P. Webb, R. J. McIntyre, J. Conradi, “Properties of avalanche photodiodes,” RCA Rev. 35, 234–278 (1974).

Grattan, K. T. V.

T. A. Clarke, K. T. V. Grattan, N. E. Lindsey, “Laser-based triangulation techniques in optical inspection of industrial structures,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 474–485 (1990).

Green, L.

L. Green, “Condition monitoring—its impact on plant performance at British Steel, Scunthorpe Works,” Ironmak. Steelmak. 17, 355–362 (1990).

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, New York, 1981).

Jaatinen, J.

M. Manninen, J. Jaatinen, “Productive method and system to control dimensional uncertainties at final assembling stages in ship production,” J. Ship Product. 8, 244–248 (1992).

Karppinen, M.

M. Karppinen, J. Kostamovaara, R. Myllylä, M. Seppänen, K. Määttä, “A novel laser rangefinder system for the profile measurements of refractory linings,” in Proceedings of the UNITECR’89 Meeting, L. J. Trostel, ed. (American Ceramic Society, Westerville, Ohio, 1989), 1340–1353.

Klages, G.

G. Klages, “Measurement of refractory lining in metallurgical vessels,” Metall. Plant Technol. 12, 32–41 (1989).

Koskinen, M.

J. Kostamovaara, K. Määttä, M. Koskinen, R. Myllylä, “Pulsed laser radars with high-modulation-frequency in industrial applications,” in Laser Radar VII: Advanced Technology for Applications, Richard J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1633, 114–127 (1992).

Kostamovaara, J.

J. Kostamovaara, “Techniques and devices for positron lifetime measurement and time-of-flight laser rangefinding,” Acta Univ. Oulu 37, 1–69 (1986).

J. Kostamovaara, K. Määttä, M. Koskinen, R. Myllylä, “Pulsed laser radars with high-modulation-frequency in industrial applications,” in Laser Radar VII: Advanced Technology for Applications, Richard J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1633, 114–127 (1992).

M. Karppinen, J. Kostamovaara, R. Myllylä, M. Seppänen, K. Määttä, “A novel laser rangefinder system for the profile measurements of refractory linings,” in Proceedings of the UNITECR’89 Meeting, L. J. Trostel, ed. (American Ceramic Society, Westerville, Ohio, 1989), 1340–1353.

J. Wang, K. Määttä, J. Kostamovaara, “Signal power estimation in short range laser radars,” in ICALEO’91: Optical Sensing and Measurement Symposium (Laser Institute of America, Toledo, Ohio, 1992), pp. 16–26.

K. Määttä, J. Kostamovaara, R. Myllylä, “Time-to-digital converter for fast, accurate laser rangefinding, in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 60–67 (1988).

K. Määttä, J. Kostamovaara, R. Myllylä, “A laser range-finder for hot surface profiling measurements, in Laser Technologies in Industry, O. D. D. Soares, ed., Proc. Soc. Photo-Opt. Instrum. Eng.952, 356–364 (1988).

K. Määttä, J. Kostamovaara, R. Myllylä, “On the measurement of hot surfaces by pulsed time-of-flight laser radar techniques,” in Industrial Inspection II, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1265, 179–191 (1990).

Kressel, H.

H. Kressel, Semiconductor Devices for Optical Communication (Springer-Verlag, Berlin, 1980).

Lindsey, N. E.

T. A. Clarke, K. T. V. Grattan, N. E. Lindsey, “Laser-based triangulation techniques in optical inspection of industrial structures,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 474–485 (1990).

Määttä, K.

K. Määttä, J. Kostamovaara, R. Myllylä, “Time-to-digital converter for fast, accurate laser rangefinding, in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 60–67 (1988).

J. Wang, K. Määttä, J. Kostamovaara, “Signal power estimation in short range laser radars,” in ICALEO’91: Optical Sensing and Measurement Symposium (Laser Institute of America, Toledo, Ohio, 1992), pp. 16–26.

K. Määttä, J. Kostamovaara, R. Myllylä, “On the measurement of hot surfaces by pulsed time-of-flight laser radar techniques,” in Industrial Inspection II, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1265, 179–191 (1990).

K. Määttä, J. Kostamovaara, R. Myllylä, “A laser range-finder for hot surface profiling measurements, in Laser Technologies in Industry, O. D. D. Soares, ed., Proc. Soc. Photo-Opt. Instrum. Eng.952, 356–364 (1988).

M. Karppinen, J. Kostamovaara, R. Myllylä, M. Seppänen, K. Määttä, “A novel laser rangefinder system for the profile measurements of refractory linings,” in Proceedings of the UNITECR’89 Meeting, L. J. Trostel, ed. (American Ceramic Society, Westerville, Ohio, 1989), 1340–1353.

J. Kostamovaara, K. Määttä, M. Koskinen, R. Myllylä, “Pulsed laser radars with high-modulation-frequency in industrial applications,” in Laser Radar VII: Advanced Technology for Applications, Richard J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1633, 114–127 (1992).

Manninen, M.

M. Manninen, J. Jaatinen, “Productive method and system to control dimensional uncertainties at final assembling stages in ship production,” J. Ship Product. 8, 244–248 (1992).

M. Manninen, “Task-oriented approach to interactive control of heavy-duty manipulators based on coarse scene description,” Acta Polytech. Scand. Math. Comput. Sci. Ser. 42, 1–81 (1984).

McIntyre, R. J.

P. P. Webb, R. J. McIntyre, J. Conradi, “Properties of avalanche photodiodes,” RCA Rev. 35, 234–278 (1974).

Meyer, R. G.

R. G. Meyer, R. A. Blauschild, “A wide-band low-noise monolithic transimpedance amplifier,” IEEE J. Solid-State Circuits SSC-21, 530–533 (1986).
[CrossRef]

Myllylä, R.

J. Kostamovaara, K. Määttä, M. Koskinen, R. Myllylä, “Pulsed laser radars with high-modulation-frequency in industrial applications,” in Laser Radar VII: Advanced Technology for Applications, Richard J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1633, 114–127 (1992).

M. Karppinen, J. Kostamovaara, R. Myllylä, M. Seppänen, K. Määttä, “A novel laser rangefinder system for the profile measurements of refractory linings,” in Proceedings of the UNITECR’89 Meeting, L. J. Trostel, ed. (American Ceramic Society, Westerville, Ohio, 1989), 1340–1353.

K. Määttä, J. Kostamovaara, R. Myllylä, “A laser range-finder for hot surface profiling measurements, in Laser Technologies in Industry, O. D. D. Soares, ed., Proc. Soc. Photo-Opt. Instrum. Eng.952, 356–364 (1988).

K. Määttä, J. Kostamovaara, R. Myllylä, “On the measurement of hot surfaces by pulsed time-of-flight laser radar techniques,” in Industrial Inspection II, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1265, 179–191 (1990).

K. Määttä, J. Kostamovaara, R. Myllylä, “Time-to-digital converter for fast, accurate laser rangefinding, in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 60–67 (1988).

Ready, J. F.

J. F. Ready, Industrial Applications of Lasers (Academic, New York, 1978).

Seppänen, M.

M. Karppinen, J. Kostamovaara, R. Myllylä, M. Seppänen, K. Määttä, “A novel laser rangefinder system for the profile measurements of refractory linings,” in Proceedings of the UNITECR’89 Meeting, L. J. Trostel, ed. (American Ceramic Society, Westerville, Ohio, 1989), 1340–1353.

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, New York, 1981).

Smith, D. E.

D. E. Smith, “Electronic distance measurement for industrial and scientific applications,” Hewlett-Packard J. 31, 3–10 (1980).

Wang, J.

J. Wang, K. Määttä, J. Kostamovaara, “Signal power estimation in short range laser radars,” in ICALEO’91: Optical Sensing and Measurement Symposium (Laser Institute of America, Toledo, Ohio, 1992), pp. 16–26.

Webb, P. P.

P. P. Webb, R. J. McIntyre, J. Conradi, “Properties of avalanche photodiodes,” RCA Rev. 35, 234–278 (1974).

Acta Polytech. Scand. Math. Comput. Sci. Ser. (1)

M. Manninen, “Task-oriented approach to interactive control of heavy-duty manipulators based on coarse scene description,” Acta Polytech. Scand. Math. Comput. Sci. Ser. 42, 1–81 (1984).

Acta Univ. Oulu (1)

J. Kostamovaara, “Techniques and devices for positron lifetime measurement and time-of-flight laser rangefinding,” Acta Univ. Oulu 37, 1–69 (1986).

Hewlett-Packard J. (1)

D. E. Smith, “Electronic distance measurement for industrial and scientific applications,” Hewlett-Packard J. 31, 3–10 (1980).

IEEE J. Solid-State Circuits (1)

R. G. Meyer, R. A. Blauschild, “A wide-band low-noise monolithic transimpedance amplifier,” IEEE J. Solid-State Circuits SSC-21, 530–533 (1986).
[CrossRef]

Ironmak. Steelmak. (1)

L. Green, “Condition monitoring—its impact on plant performance at British Steel, Scunthorpe Works,” Ironmak. Steelmak. 17, 355–362 (1990).

J. Ship Product. (1)

M. Manninen, J. Jaatinen, “Productive method and system to control dimensional uncertainties at final assembling stages in ship production,” J. Ship Product. 8, 244–248 (1992).

Metall. Plant Technol. (1)

G. Klages, “Measurement of refractory lining in metallurgical vessels,” Metall. Plant Technol. 12, 32–41 (1989).

RCA Rev. (1)

P. P. Webb, R. J. McIntyre, J. Conradi, “Properties of avalanche photodiodes,” RCA Rev. 35, 234–278 (1974).

Other (13)

M. Karppinen, J. Kostamovaara, R. Myllylä, M. Seppänen, K. Määttä, “A novel laser rangefinder system for the profile measurements of refractory linings,” in Proceedings of the UNITECR’89 Meeting, L. J. Trostel, ed. (American Ceramic Society, Westerville, Ohio, 1989), 1340–1353.

C30902E, C30902S, C30921E, and C30921S Data sheet (RCA, Inc., Electro Optics, Vaudreuil, Canada, 1988).

H. Kressel, Semiconductor Devices for Optical Communication (Springer-Verlag, Berlin, 1980).

RCA Corporation, RCA-Electro-Optics Handbook, tech. ser. EOH-11 (RCA, Inc., Lancaster, Pa., 1974).

J. Wang, K. Määttä, J. Kostamovaara, “Signal power estimation in short range laser radars,” in ICALEO’91: Optical Sensing and Measurement Symposium (Laser Institute of America, Toledo, Ohio, 1992), pp. 16–26.

K. Määttä, J. Kostamovaara, R. Myllylä, “Time-to-digital converter for fast, accurate laser rangefinding, in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 60–67 (1988).

K. Määttä, J. Kostamovaara, R. Myllylä, “On the measurement of hot surfaces by pulsed time-of-flight laser radar techniques,” in Industrial Inspection II, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1265, 179–191 (1990).

J. Kostamovaara, K. Määttä, M. Koskinen, R. Myllylä, “Pulsed laser radars with high-modulation-frequency in industrial applications,” in Laser Radar VII: Advanced Technology for Applications, Richard J. Becherer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1633, 114–127 (1992).

T. A. Clarke, K. T. V. Grattan, N. E. Lindsey, “Laser-based triangulation techniques in optical inspection of industrial structures,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 474–485 (1990).

K. Määttä, J. Kostamovaara, R. Myllylä, “A laser range-finder for hot surface profiling measurements, in Laser Technologies in Industry, O. D. D. Soares, ed., Proc. Soc. Photo-Opt. Instrum. Eng.952, 356–364 (1988).

J. F. Ready, Industrial Applications of Lasers (Academic, New York, 1978).

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, New York, 1981).

G. Bertolini, “Pulse shape and time resolution,” in Semiconductor Detectors, G. Bertolini, A. Coche, eds. (North-Holland, Amsterdam, 1968), pp. 243–276.

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Figures (20)

Fig. 1
Fig. 1

Schematic diagram of the receiver optics.

Fig. 2
Fig. 2

Measured background radiation in a hot oven as a function of distance at focus distances of the receiver optics of 6 m (□), 8 m (⋄), 10 m (△), and 12 m (×).

Fig. 3
Fig. 3

Calculated beam overlap function with three adjustments of the transmitter and receiver optics: A, The transmitter and receiver focus distance is 10 m, and the optical axes cross at 10 m. B, The transmitter and receiver focus distance is 24 m, and the optical axes cross at 10 m. C, The transmitter and receiver focus distance is 24 m, and the optical axes cross at 24 m. D, The received signal observes the 1/x2 rule when it is fitted to curve A at 10 m or to curve C at 24 m.

Fig. 4
Fig. 4

Block diagram of the distance measurement electronics.

Fig. 5
Fig. 5

Schematic diagram of the optomechanical measuring head.

Fig. 6
Fig. 6

Calculated (△) and measured (□) single shot resolution of the electronics as a function of SNR (ratio of the peak value of the signal to the rms value of the electronics noise).

Fig. 7
Fig. 7

Resolution of the electronics as a function of the number of measurements averaged, with SNR values of 10 (⋄) and 270 (□).

Fig. 8
Fig. 8

Nonlinearity of the electronics.

Fig. 9
Fig. 9

Temperature stability of the electronics with temperature compensation (⋄) and no temperature compensation (□).

Fig. 10
Fig. 10

Stabilization of the electronics after power on. The temperature drift of the distance is compensated for by the processor.

Fig. 11
Fig. 11

Measured (□) and calculated background radiation in the hot converter. The emissivity of the surface is 0.6 (×), 0.7 (△), and 0.8 (⋄).

Fig. 12
Fig. 12

Measured (□) and calculated (⋄) bias current of the AP photodiode as a function of the temperature of the converter.

Fig. 13
Fig. 13

Gain of the AP photodiode as a function of its bias current.

Fig. 14
Fig. 14

Measured (□) and calculated noise scaled to the input of the preamplifier as a function of target temperature. Calculations are based on the measured background radiation (△) or the calculated background radiation (⋄).

Fig. 15
Fig. 15

Measured (□) and calculated (⋄) signal-to-noise ratio as a function of the temperature of the converter.

Fig. 16
Fig. 16

Stability of actual measurements of a hot converter. Each measurement is a average of 1000 successive single-shot results.

Fig. 17
Fig. 17

Measured signal and single shot resolution of measurements of a hot converter.

Fig. 18
Fig. 18

LR-2000 laser profiler. Its height, width and depth are 160 cm, 55 cm, and 73 cm. The weight is about 120 kg. Thermal shields are not shown.

Fig. 19
Fig. 19

Tuyere brick lengths during campaigns with two brick types.

Fig. 20
Fig. 20

Typical converter profiling results. The grid scale in both pictures is 500 mm.

Tables (5)

Tables Icon

Table 1 Radiance of the Graybody L B and Peak Wavelength λ P at Different Temperatures (λ = 906 nm, Δλ = 886 nm … 926 nm = 40 nm and ∊ = 0.8)

Tables Icon

Table 2 Noise Sources of the TOF-Laser Radar

Tables Icon

Table 3 Typical Noise Levels ( pA / Hz )

Tables Icon

Table 4 Typical Values of the Electronics Parameters

Tables Icon

Table 5 Parameters of the Optical Measurement Head

Equations (11)

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

P B = τ R L B A R Ω I = τ R L B A R A I s 1 2 = τ R L B A I Ω R ,
A R Ω I = A R A I s 1 2 = A R A D ( s 1 - f R ) 2 s 1 2 f R 2 = A R A D f R 2 ( 1 - f R s 1 ) 2 .
σ RCL = c 2 σ n d u / d t 0.35 c 2 B SNR ,
σ RCF c 2 σ L 2 + σ T 2 ( d u L / d t + d u T / d t ) ,
σ R = 0.35 c 2 2 B [ 1 ( SNR START ) 2 + 1 ( SNR STOP ) 2 ] 1 / 2 ,
F ( M ) = 0.98 ( 2 - 1 M ) + 0.02 M .
i n B n = [ 2 q ( k T P P S + P B ) R 0 M 2 F ( M ) + i n a ] 1 / 2 .
P S ( x ) = ρ P T τ T τ R A R π x 2 T ( x ) e - 2 γ x ,
i S ( x ) = P S ( x ) M R 0 .
SNR ( x ) = i S ( x ) i n = M R 0 P S ( x ) [ 2 q ( k T P P S ( x ) + P B ) R 0 M 2 F ( M ) B n + i n a 2 B n ] 1 / 2 .
M = I B τ F D P B R 0 .

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