Abstract

A quantitative theoretical analysis of the quadrant photodetector (QPD) sensitivity in position measurement is presented. The Gaussian light spot irradiance distribution on the QPD surface was assumed to meet most of the real-life applications of this sensor. As the result of the mathematical treatment of the problem, we obtained, in a closed form, the sensitivity function versus the ratio of the light spot 1/e radius and the QPD radius. The obtained result is valid for the full range of the ratios. To check the influence of the finite light spot radius on the interaxis cross talk and linearity, we also performed a mathematical analysis to quantitatively measure these types of errors. An optimal range of the ratio of light spot radius and QPD radius has been found to simultaneously achieve low interaxis cross talk and high linearity of the sensor.

© 2011 Optical Society of America

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References

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  1. B. Bhushan, Scanning Probe Microscopy in Nanoscience and Nanotechnology (Springer-Verlag, 2010).
    [CrossRef]
  2. A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Laser-radar-based three dimensional sensor for teaching robot paths,” Opt. Eng. 34, 2596–2602 (1995).
    [CrossRef]
  3. G. Marola, D. Santerini, and G. Prati, “Stability analysis of direct-detection cooperative optical beam tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 325–333 (1989).
    [CrossRef]
  4. C. G. Boisset, “Design and construction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–678 (1995).
    [CrossRef]
  5. F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
    [CrossRef]
  6. M. Toyoda, K. Araki, and Y. Suzuki, “Measurement of the characteristics of a quadrant avalanche photodiode and its application to a laser tracking system,” Opt. Eng. 41, 145–149(2002).
    [CrossRef]
  7. H. Wenzel, Health Monitoring of Bridges (Wiley, 2009).
    [CrossRef]
  8. A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Tracking laser radar for 3-D shape measurements of large industrial objects based on time-of-flight laser rangefinding and position-sensitive detection techniques,” IEEE Trans. Instrum. Meas. 43, 40–49 (1994).
    [CrossRef]
  9. R. Herteau, M. St-Amant, Y. Laperriere, and G. Chevrette, “Optical guidance system for underground mine vehicles,” in IEEE International Conference on Robotics and Automation (IEEE, 1992), pp. 639–644.
  10. L. G. Kazovsky, “Theory of tracking accuracy of laser systems,” Opt. Eng. 22, 339–347 (1983).
  11. E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys. 10, 834–837 (2010).
    [CrossRef]
  12. L. M. Manojlovic and Z. P. Barbaric, “Optimization of optical receiver parameters for pulsed laser-tracking systems,” IEEE Trans. Instrum. Meas. 58, 681–690 (2009).
    [CrossRef]

2010 (2)

B. Bhushan, Scanning Probe Microscopy in Nanoscience and Nanotechnology (Springer-Verlag, 2010).
[CrossRef]

E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys. 10, 834–837 (2010).
[CrossRef]

2009 (3)

L. M. Manojlovic and Z. P. Barbaric, “Optimization of optical receiver parameters for pulsed laser-tracking systems,” IEEE Trans. Instrum. Meas. 58, 681–690 (2009).
[CrossRef]

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

H. Wenzel, Health Monitoring of Bridges (Wiley, 2009).
[CrossRef]

2002 (1)

M. Toyoda, K. Araki, and Y. Suzuki, “Measurement of the characteristics of a quadrant avalanche photodiode and its application to a laser tracking system,” Opt. Eng. 41, 145–149(2002).
[CrossRef]

1995 (2)

A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Laser-radar-based three dimensional sensor for teaching robot paths,” Opt. Eng. 34, 2596–2602 (1995).
[CrossRef]

C. G. Boisset, “Design and construction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–678 (1995).
[CrossRef]

1994 (1)

A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Tracking laser radar for 3-D shape measurements of large industrial objects based on time-of-flight laser rangefinding and position-sensitive detection techniques,” IEEE Trans. Instrum. Meas. 43, 40–49 (1994).
[CrossRef]

1992 (1)

R. Herteau, M. St-Amant, Y. Laperriere, and G. Chevrette, “Optical guidance system for underground mine vehicles,” in IEEE International Conference on Robotics and Automation (IEEE, 1992), pp. 639–644.

1989 (1)

G. Marola, D. Santerini, and G. Prati, “Stability analysis of direct-detection cooperative optical beam tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 325–333 (1989).
[CrossRef]

1983 (1)

L. G. Kazovsky, “Theory of tracking accuracy of laser systems,” Opt. Eng. 22, 339–347 (1983).

Araki, K.

M. Toyoda, K. Araki, and Y. Suzuki, “Measurement of the characteristics of a quadrant avalanche photodiode and its application to a laser tracking system,” Opt. Eng. 41, 145–149(2002).
[CrossRef]

Barbaric, Z. P.

L. M. Manojlovic and Z. P. Barbaric, “Optimization of optical receiver parameters for pulsed laser-tracking systems,” IEEE Trans. Instrum. Meas. 58, 681–690 (2009).
[CrossRef]

Bhushan, B.

B. Bhushan, Scanning Probe Microscopy in Nanoscience and Nanotechnology (Springer-Verlag, 2010).
[CrossRef]

Boisset, C. G.

C. G. Boisset, “Design and construction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–678 (1995).
[CrossRef]

Chevrette, G.

R. Herteau, M. St-Amant, Y. Laperriere, and G. Chevrette, “Optical guidance system for underground mine vehicles,” in IEEE International Conference on Robotics and Automation (IEEE, 1992), pp. 639–644.

Herteau, R.

R. Herteau, M. St-Amant, Y. Laperriere, and G. Chevrette, “Optical guidance system for underground mine vehicles,” in IEEE International Conference on Robotics and Automation (IEEE, 1992), pp. 639–644.

Horsley, D. A.

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

Houng, D.

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

Islam, M. S.

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

Kazovsky, L. G.

L. G. Kazovsky, “Theory of tracking accuracy of laser systems,” Opt. Eng. 22, 339–347 (1983).

Kim, C. S.

E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys. 10, 834–837 (2010).
[CrossRef]

Kostamovaara, J.

A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Laser-radar-based three dimensional sensor for teaching robot paths,” Opt. Eng. 34, 2596–2602 (1995).
[CrossRef]

A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Tracking laser radar for 3-D shape measurements of large industrial objects based on time-of-flight laser rangefinding and position-sensitive detection techniques,” IEEE Trans. Instrum. Meas. 43, 40–49 (1994).
[CrossRef]

Kouh, T.

E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys. 10, 834–837 (2010).
[CrossRef]

Laperriere, Y.

R. Herteau, M. St-Amant, Y. Laperriere, and G. Chevrette, “Optical guidance system for underground mine vehicles,” in IEEE International Conference on Robotics and Automation (IEEE, 1992), pp. 639–644.

Lee, E. J.

E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys. 10, 834–837 (2010).
[CrossRef]

Mäkynen, A.

A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Laser-radar-based three dimensional sensor for teaching robot paths,” Opt. Eng. 34, 2596–2602 (1995).
[CrossRef]

A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Tracking laser radar for 3-D shape measurements of large industrial objects based on time-of-flight laser rangefinding and position-sensitive detection techniques,” IEEE Trans. Instrum. Meas. 43, 40–49 (1994).
[CrossRef]

Manojlovic, L. M.

L. M. Manojlovic and Z. P. Barbaric, “Optimization of optical receiver parameters for pulsed laser-tracking systems,” IEEE Trans. Instrum. Meas. 58, 681–690 (2009).
[CrossRef]

Marola, G.

G. Marola, D. Santerini, and G. Prati, “Stability analysis of direct-detection cooperative optical beam tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 325–333 (1989).
[CrossRef]

Mathai, S.

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

Myllylä, R.

A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Laser-radar-based three dimensional sensor for teaching robot paths,” Opt. Eng. 34, 2596–2602 (1995).
[CrossRef]

A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Tracking laser radar for 3-D shape measurements of large industrial objects based on time-of-flight laser rangefinding and position-sensitive detection techniques,” IEEE Trans. Instrum. Meas. 43, 40–49 (1994).
[CrossRef]

Park, Y.

E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys. 10, 834–837 (2010).
[CrossRef]

Prati, G.

G. Marola, D. Santerini, and G. Prati, “Stability analysis of direct-detection cooperative optical beam tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 325–333 (1989).
[CrossRef]

Santerini, D.

G. Marola, D. Santerini, and G. Prati, “Stability analysis of direct-detection cooperative optical beam tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 325–333 (1989).
[CrossRef]

St-Amant, M.

R. Herteau, M. St-Amant, Y. Laperriere, and G. Chevrette, “Optical guidance system for underground mine vehicles,” in IEEE International Conference on Robotics and Automation (IEEE, 1992), pp. 639–644.

Suzuki, Y.

M. Toyoda, K. Araki, and Y. Suzuki, “Measurement of the characteristics of a quadrant avalanche photodiode and its application to a laser tracking system,” Opt. Eng. 41, 145–149(2002).
[CrossRef]

Tan, M. R. T.

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

Toyoda, M.

M. Toyoda, K. Araki, and Y. Suzuki, “Measurement of the characteristics of a quadrant avalanche photodiode and its application to a laser tracking system,” Opt. Eng. 41, 145–149(2002).
[CrossRef]

VJ, L.

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

Walmsley, R. G.

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

Wang, S.-Y.

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

Wenzel, H.

H. Wenzel, Health Monitoring of Bridges (Wiley, 2009).
[CrossRef]

Wu, F.

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

Appl. Phys. A (1)

F. Wu, L. VJ, M. S. Islam, D. A. Horsley, R. G. Walmsley, S. Mathai, D. Houng, M. R. T. Tan, and S.-Y. Wang, “Integrated receiver architectures for board-to-board free-space optical interconnects,” Appl. Phys. A 95, 1079–1088 (2009).
[CrossRef]

Curr. Appl. Phys. (1)

E. J. Lee, Y. Park, C. S. Kim, and T. Kouh, “Detection sensitivity of the optical beam deflection method characterized with the optical spot size on the detector,” Curr. Appl. Phys. 10, 834–837 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

C. G. Boisset, “Design and construction of an active alignment demonstrator for a free-space optical interconnect,” IEEE Photon. Technol. Lett. 7, 676–678 (1995).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

G. Marola, D. Santerini, and G. Prati, “Stability analysis of direct-detection cooperative optical beam tracking,” IEEE Trans. Aerosp. Electron. Syst. 25, 325–333 (1989).
[CrossRef]

IEEE Trans. Instrum. Meas. (2)

A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Tracking laser radar for 3-D shape measurements of large industrial objects based on time-of-flight laser rangefinding and position-sensitive detection techniques,” IEEE Trans. Instrum. Meas. 43, 40–49 (1994).
[CrossRef]

L. M. Manojlovic and Z. P. Barbaric, “Optimization of optical receiver parameters for pulsed laser-tracking systems,” IEEE Trans. Instrum. Meas. 58, 681–690 (2009).
[CrossRef]

Opt. Eng. (3)

L. G. Kazovsky, “Theory of tracking accuracy of laser systems,” Opt. Eng. 22, 339–347 (1983).

M. Toyoda, K. Araki, and Y. Suzuki, “Measurement of the characteristics of a quadrant avalanche photodiode and its application to a laser tracking system,” Opt. Eng. 41, 145–149(2002).
[CrossRef]

A. Mäkynen, J. Kostamovaara, and R. Myllylä, “Laser-radar-based three dimensional sensor for teaching robot paths,” Opt. Eng. 34, 2596–2602 (1995).
[CrossRef]

Other (3)

B. Bhushan, Scanning Probe Microscopy in Nanoscience and Nanotechnology (Springer-Verlag, 2010).
[CrossRef]

H. Wenzel, Health Monitoring of Bridges (Wiley, 2009).
[CrossRef]

R. Herteau, M. St-Amant, Y. Laperriere, and G. Chevrette, “Optical guidance system for underground mine vehicles,” in IEEE International Conference on Robotics and Automation (IEEE, 1992), pp. 639–644.

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Figures (5)

Fig. 1
Fig. 1

Geometry of the light spot onto the QPD surface with all relevant geometrical parameters.

Fig. 2
Fig. 2

Geometry of the light spot onto the QPD surface that is used for the power sum calculation.

Fig. 3
Fig. 3

QPD sensitivity S x and S y versus the ratio of light spot radius and QPD radius w / R .

Fig. 4
Fig. 4

Ratio of the QPD sensitivity relative change and the light spot radius relative change versus the ratio of light spot radius and QPD radius.

Fig. 5
Fig. 5

Coefficients S y y y (line 1) and S y y x (line 2) versus the ratio w / R .

Equations (58)

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χ = ( I I + I IV ) ( I II + I III ) ( I I + I IV ) + ( I II + I III ) , ψ = ( I I + I II ) ( I III + I IV ) ( I I + I II ) + ( I III + I IV ) ,
ψ = ( P I + P II ) ( P III + P IV ) ( P I + P II ) + ( P III + P IV ) ,
I ( r ) = P π w 2 exp [ ( r w ) 2 ] ,
P Σ ( x , y ) = P I ( x , y ) + P II ( x , y ) + P III ( x , y ) + P IV ( x , y ) ,
P Σ ( x , y ) = θ = 0 2 π r = 0 l ( x , y , θ ) I ( r ) r · d r · d θ ,
l ( x , y , θ ) = R 2 ( x sin θ y cos θ ) 2 ( x cos θ + y sin θ ) ,
P Σ ( x , y ) = P { 1 1 2 π 0 2 π exp { [ l ( x , y , θ ) w ] 2 } d θ } .
ψ ( x , y ) = [ P I ( x , y ) + P II ( x , y ) ] [ P III ( x , y ) + P IV ( x , y ) ] [ P I ( x , y ) + P II ( x , y ) ] + [ P III ( x , y ) + P IV ( x , y ) ] , ψ ( x , y ) = 2 P I ( x , y ) + P II ( x , y ) P Σ ( x , y ) 1 .
P I ( x , y ) + P II ( x , y ) = P y 1 ( x , y ) + P y 2 ( x , y ) ,
ψ ( x , y ) = 2 P y 1 ( x , y ) + P y 2 ( x , y ) P Σ ( x , y ) 1 = 2 P y + ( x , y ) P Σ ( x , y ) 1 ,
P y 1 ( x , y ) = P π w 2 θ = β ( x , y ) π + α ( x , y ) r = 0 l ( x , y , θ ) exp [ ( r w ) 2 ] r · d r · d θ ,
α ( x , y ) = arctan ( y R x ) , β ( x , y ) = arctan ( y R + x ) .
P y 1 ( x , y ) = P 2 { 1 + 1 π [ arctan ( y R + x ) + arctan ( y R x ) ] 1 π β ( x , y ) π + α ( x , y ) exp { [ l ( x , y , θ ) w ] 2 } d θ } .
P y 2 ( x , y ) = P π w 2 θ = π + α ( x , y ) 2 π β ( x , y ) r = 0 d ( x , y , θ ) exp [ ( r w ) 2 ] r · d r · d θ ,
d ( x , y , θ ) = y sin θ .
P y 2 ( x , y ) = P 2 { 1 1 π [ arctan ( y R + x ) + arctan ( y R x ) ] 1 π θ = π + α ( x , y ) 2 π β ( x , y ) exp { [ d ( x , y , θ ) w ] 2 } d θ } .
P y + ( x , y ) = P { 1 1 2 π β ( x , y ) π + α ( x , y ) exp { [ l ( x , y , θ ) w ] 2 } d θ 1 2 π θ = π + α ( x , y ) 2 π β ( x , y ) exp { [ d ( x , y , θ ) w ] 2 } d θ } .
ψ ( x , y ) ψ ( 0 , 0 ) ( y / R ) · ( y R ) + ψ ( 0 , 0 ) ( x / R ) · ( x R ) .
ψ ( x , y ) ψ ( 0 , 0 ) ( y / R ) · ( y R ) = S y y R ,
S y = R ψ ( 0 , 0 ) y = 2 R P Σ ( 0 , 0 ) P y + ( 0 , 0 ) y ,
P Σ ( 0 , 0 ) = P { 1 exp [ ( R w ) 2 ] } .
P y + ( 0 , 0 ) y = P 2 π y { β ( x , y ) π + α ( x , y ) exp { [ l ( x , y , θ ) w ] 2 } d θ + θ = π + α ( x , y ) 2 π β ( x , y ) exp { [ d ( x , y , θ ) w ] 2 } d θ } | x = 0 y = 0 .
A ( x , y ) = β ( x , y ) π + α ( x , y ) exp { [ l ( x , y , θ ) w ] 2 } d θ , B ( x , y ) = θ = π + α ( x , y ) 2 π β ( x , y ) exp { [ d ( x , y , θ ) w ] 2 } d θ ,
a = β ( 0 , 0 ) π + α ( 0 , 0 ) { y exp { [ l ( x , y , θ ) w ] 2 } } | x = 0 y = 0 d θ + exp { { l [ 0 , 0 , π + α ( 0 , 0 ) ] w } 2 } α ( 0 , 0 ) y + exp { { l [ 0 , 0 , β ( 0 , 0 ) ] w } 2 } β ( 0 , 0 ) y .
a = 2 R [ 1 + 2 ( R w ) 2 ] exp [ ( R w ) 2 ] .
b = lim y 0 { θ = π + α ( 0 , y ) 2 π β ( 0 , y ) y { exp { [ d ( 0 , y , θ ) w ] 2 } } d θ exp { { d [ 0 , y , 2 π β ( 0 , y ) ] w } 2 } β ( 0 , y ) y exp { { d [ 0 , y , π + α ( 0 , y ) ] w } 2 } α ( 0 , y ) y } ,
b = 2 R c 2 R exp [ ( R w ) 2 ] , c = lim z 0 1 z z π z ( R w z sin θ ) 2 exp [ ( R w z sin θ ) 2 ] d θ .
c = π R w erf ( R w ) ,
b = 2 π 1 w erf ( R w ) 2 R exp [ ( R w ) 2 ] .
P y + ( 0 , 0 ) y = P 2 π ( a + b ) = P π R R w { π erf ( R w ) 2 R w exp [ ( R w ) 2 ] } ,
S y = 2 π R w erf ( R w ) exp [ ( R w ) 2 ] 2 π R w exp [ ( R w ) 2 ] 1 .
Δ S y = d S y d w Δ w = R w 2 d S y d ( R w ) Δ w .
Δ S y S y Δ w w = 1 2 ( R w ) 2 { 1 exp [ ( R w ) 2 ] 1 + 2 π R w erf ( R w ) exp [ ( R w ) 2 ] 2 π R w } .
ψ ( x , y ) S y ( y R ) { 1 + 1 S y [ S y y y ( y R ) 2 + S y x x ( x R ) 2 ] } ,
S y y y = 1 6 R 3 3 ψ ( 0 , 0 ) y 3 , S y x x = 1 2 R 3 3 ψ ( 0 , 0 ) y x 2 .
S y y y = 2 3 π p + q exp [ ( R w ) 2 ] 1 , p = 2 + 3 ( R w ) 2 + 4 3 ( R w ) 6 4 ( R w ) 2 exp [ ( R w ) 2 ] 1 , q = 2 π ( R w ) erf ( R w ) 1 exp [ ( R w ) 2 ] 3 π ( R w ) 2 erf ( R w ) exp [ ( R w ) 2 ] ,
S y x x = 2 3 π 3 + 5 ( R w ) 2 + 10 ( R w ) 4 2 ( R w ) 6 exp [ ( R w ) 2 ] 1 .
c = lim z 0 1 z z π z ( R w z sin θ ) 2 exp [ ( R w z sin θ ) 2 ] d θ .
c = lim z 0 d d z { z π z ( R w z sin θ ) 2 exp [ ( R w z sin θ ) 2 ] d θ } .
c = 2 ( R w ) 2 exp [ ( R w ) 2 ] + 2 c 1 , c 1 = lim z 0 1 z z π z ( R w z sin θ ) 4 exp [ ( R w z sin θ ) 2 ] d θ .
c 1 = 2 3 ( R w ) 4 exp [ ( R w ) 2 ] + 2 3 c 2 , c 2 = lim z 0 1 z z π z ( R w z sin θ ) 6 exp [ ( R w z sin θ ) 2 ] d θ .
c 2 = 2 5 ( R w ) 6 exp [ ( R w ) 2 ] + 2 5 c 3 , c 3 = lim z 0 1 z z π z ( R w z sin θ ) 8 exp [ ( R w z sin θ ) 2 ] d θ .
c n = 2 2 n + 1 ( R w ) 2 n + 2 exp [ ( R w ) 2 ] + 2 2 n + 1 c n + 1 ,
c = 2 s 2 exp ( s 2 ) k = 0 + ( 2 s 2 ) k j = 0 k ( 2 j + 1 ) + lim k + a k , s = R w , a k = 2 k + 1 1 · 3 · · ( 2 k + 1 ) c k , c k = lim z 0 1 z z π z ( s z sin θ ) 2 ( k + 1 ) exp [ ( s z sin θ ) 2 ] d θ .
lim k + a k + 1 a k = lim k + 2 2 k + 3 c k + 1 c k .
lim k + a k + 1 a k = 2 lim k + lim z 0 z π z ( s z sin θ ) 2 ( k + 2 ) exp [ ( s z sin θ ) 2 ] d θ z π z ( s z sin θ ) 2 ( k + 1 ) exp [ ( s z sin θ ) 2 ] d θ 2 k + 3 .
lim k + a k + 1 a k 2 s 2 lim k + 1 2 k + 3 = 0 .
lim k + a k + 1 a k = 0 .
lim k + a k = 0 ,
c = 2 s 2 exp ( s 2 ) f ( s ) , f ( s ) = k = 0 + ( 2 s ) 2 k j = 0 k ( 2 j + 1 ) .
u f ( u ) = k = 0 + u 2 k + 1 j = 0 k ( 2 j + 1 ) ,
f ( u ) + u f ˙ ( u ) = 1 + u 2 f ( u ) ,
f ˙ ( u ) + ( 1 u u ) f ( u ) = 1 u .
f ( s ) = 1 2 s exp ( s 2 ) [ C + 2 exp ( s 2 ) d s ] ,
f ( s ) = 1 2 s exp ( s 2 ) [ C + 2 0 s exp ( t 2 ) d t ] ,
f ( s ) = 1 2 s exp ( s 2 ) [ C + π 2 erf ( s ) ] .
f ( s ) = π 2 exp ( s 2 ) s erf ( s ) .
c = π s · erf ( s ) .

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