Abstract

Optical sensors applied to practical devices often encounter beam steering: the wander and/or diffusion of laser light. Here we provide a framework for minimizing the sensitivity of transmission-based sensors to beam steering without quantitative prediction of the severity of the beam-steering field. Typical goals are increased transmission and/or minimized fluctuations in transmission; such features can improve optical sensor performance (e.g., improved signal-to-noise ratio, response time, or spectral resolution). In our framework, we introduce a parameter for characterizing beam-steering severity. We then compare two approaches for absorption spectroscopy and show that the preferred approach depends on the total spectral range monitored, the spectral resolution desired, and the severity of the beam steering.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. T. Sanders, “Diode-laser sensors for harsh environments with application to pulse detonation. engines,” in Mechanical Engineering (Stanford University, 2001).
  2. P. P. Radi, B. Mischler, A. Schlegel, A. Tzannis, P. Beaud, T. Gerber, “Absolute concentration measurements using DFWM and modeling of OH and S2 in a fuel-rich H2/air/SO2 flame,” Combust. Flame 118, 301–307 (1999).
    [CrossRef]
  3. S. T. Sanders, D. W. Mattison, L. Ma, J. B. Jeffries, R. K. Hanson, “Wavelength-agile diode-laser sensing strategies for monitoring gas properties in optically harsh flows: application in cesium-seeded pulse detonation engine,” Opt. Express 10, 505–514 (2002).
    [CrossRef] [PubMed]
  4. S. D. Wehe, D. S. Baer, R. K. Hanson, “Diode-laser sensor for velocity measurements in hypervelocity flows,” AIAA J. 37, 1013–1015 (1999).
    [CrossRef]
  5. L. A. Kranendonk, A. W. Caswell, A. N. Myers, S. T. Sanders, “Wavelength-agile laser sensors for measuring gas properties in engines,” , (Society of Automotive Engineers, 2002).
  6. J. E. Supplee, E. A. Whittaker, W. Lenth, “Theoretical description of frequency modulation and wavelength modulation spectroscopy,” Appl. Opt. 33, 6294–6302 (1994).
    [CrossRef] [PubMed]
  7. E. L. Peterson, “A shock tube and diagnostic for chemistry measurements at elevated pressures with application to methane ignition,” in Mechanical Engineering (Stanford University, 1998).
  8. L. A. Kranendonk, J. W. Walewski, T. Kim, S. T. Sanders, “Wavelength-agile sensor applied for HCCI Engine Measurements,” Proc. Combust. Symp. 30, 1619–1627 (2005).
    [CrossRef]
  9. E. J. Jumper, E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–399 (2001).
    [CrossRef]
  10. Landolt- Bornstein, Zahlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik, und Technik (Springer-Verlag, 1962).
  11. M. Born, E. Wolf, Principles of Optics (MacMillan, 1964).
  12. M. P. B. Musculus, L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in high-pressure transient combustion environments,” Combust. Flame 141, 371–391 (2005).
    [CrossRef]
  13. G. W. Sutton, “Effect of turbulent fluctuations in an optically active fluid medium,” AIAA J. 7, 1737–1743 (1969).
    [CrossRef]
  14. I. Celik, Y. Ibrahin, “An assessment of turbulence scales relevant to IC engines,” in ASME Spring Technical Conference 28 97-ICE-S (ASME, 1997), pp. 35–44.
  15. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).
  16. W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, 1978).
  17. M. Young, Optics and Laser (Springer-Verlag, 1986).
    [CrossRef]
  18. W. H. Steel, “Luminosity, throughput, or etendue?” Appl. Opt. 13, 704 (1974).
    [CrossRef] [PubMed]
  19. S. S. Kee, H. Mohammadi, Y. Hirano, Y. Kidoguchi, K. Miwa, “Experimental study on combustion characteristics and emissions reduction of emulsified fuels in diesel combustion using rapid compression,” (Society of Automotive Engineers, 2003).
    [CrossRef]
  20. F. E. Corcione, S. S. Merola, B. M. Vaglieco, “Nanometric particle formation in optically accessible diesel engine,” (Society of Automotive Engineers, 2001).
    [CrossRef]
  21. M. Stoner, T. Litzinger, “Effects of structure and boiling point of oxygenated blending compounds in reducing diesel emissions,” (Society of Automotive Engineers, 1999).
    [CrossRef]
  22. D. R. Tree, J. E. Dec, “Extinction measurements of in-cylinder soot deposition in a heavy-duty DI diesel engine,” (Society of Automotive Engineers, 2001).
    [CrossRef]
  23. J. W. Walewski, S. T. Sanders, “High-resolution wavelength-agile laser source based on pulsed super-continua,” Appl. Phys. B 79, 415–418 (2004).
    [CrossRef]

2005 (2)

L. A. Kranendonk, J. W. Walewski, T. Kim, S. T. Sanders, “Wavelength-agile sensor applied for HCCI Engine Measurements,” Proc. Combust. Symp. 30, 1619–1627 (2005).
[CrossRef]

M. P. B. Musculus, L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in high-pressure transient combustion environments,” Combust. Flame 141, 371–391 (2005).
[CrossRef]

2004 (1)

J. W. Walewski, S. T. Sanders, “High-resolution wavelength-agile laser source based on pulsed super-continua,” Appl. Phys. B 79, 415–418 (2004).
[CrossRef]

2002 (1)

2001 (1)

E. J. Jumper, E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–399 (2001).
[CrossRef]

1999 (2)

P. P. Radi, B. Mischler, A. Schlegel, A. Tzannis, P. Beaud, T. Gerber, “Absolute concentration measurements using DFWM and modeling of OH and S2 in a fuel-rich H2/air/SO2 flame,” Combust. Flame 118, 301–307 (1999).
[CrossRef]

S. D. Wehe, D. S. Baer, R. K. Hanson, “Diode-laser sensor for velocity measurements in hypervelocity flows,” AIAA J. 37, 1013–1015 (1999).
[CrossRef]

1994 (1)

1974 (1)

1969 (1)

G. W. Sutton, “Effect of turbulent fluctuations in an optically active fluid medium,” AIAA J. 7, 1737–1743 (1969).
[CrossRef]

Baer, D. S.

S. D. Wehe, D. S. Baer, R. K. Hanson, “Diode-laser sensor for velocity measurements in hypervelocity flows,” AIAA J. 37, 1013–1015 (1999).
[CrossRef]

Beaud, P.

P. P. Radi, B. Mischler, A. Schlegel, A. Tzannis, P. Beaud, T. Gerber, “Absolute concentration measurements using DFWM and modeling of OH and S2 in a fuel-rich H2/air/SO2 flame,” Combust. Flame 118, 301–307 (1999).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (MacMillan, 1964).

Caswell, A. W.

L. A. Kranendonk, A. W. Caswell, A. N. Myers, S. T. Sanders, “Wavelength-agile laser sensors for measuring gas properties in engines,” , (Society of Automotive Engineers, 2002).

Celik, I.

I. Celik, Y. Ibrahin, “An assessment of turbulence scales relevant to IC engines,” in ASME Spring Technical Conference 28 97-ICE-S (ASME, 1997), pp. 35–44.

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).

Corcione, F. E.

F. E. Corcione, S. S. Merola, B. M. Vaglieco, “Nanometric particle formation in optically accessible diesel engine,” (Society of Automotive Engineers, 2001).
[CrossRef]

Dec, J. E.

D. R. Tree, J. E. Dec, “Extinction measurements of in-cylinder soot deposition in a heavy-duty DI diesel engine,” (Society of Automotive Engineers, 2001).
[CrossRef]

Fitzgerald, E. J.

E. J. Jumper, E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–399 (2001).
[CrossRef]

Gerber, T.

P. P. Radi, B. Mischler, A. Schlegel, A. Tzannis, P. Beaud, T. Gerber, “Absolute concentration measurements using DFWM and modeling of OH and S2 in a fuel-rich H2/air/SO2 flame,” Combust. Flame 118, 301–307 (1999).
[CrossRef]

Hanson, R. K.

Hirano, Y.

S. S. Kee, H. Mohammadi, Y. Hirano, Y. Kidoguchi, K. Miwa, “Experimental study on combustion characteristics and emissions reduction of emulsified fuels in diesel combustion using rapid compression,” (Society of Automotive Engineers, 2003).
[CrossRef]

Ibrahin, Y.

I. Celik, Y. Ibrahin, “An assessment of turbulence scales relevant to IC engines,” in ASME Spring Technical Conference 28 97-ICE-S (ASME, 1997), pp. 35–44.

Jeffries, J. B.

Jumper, E. J.

E. J. Jumper, E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–399 (2001).
[CrossRef]

Kee, S. S.

S. S. Kee, H. Mohammadi, Y. Hirano, Y. Kidoguchi, K. Miwa, “Experimental study on combustion characteristics and emissions reduction of emulsified fuels in diesel combustion using rapid compression,” (Society of Automotive Engineers, 2003).
[CrossRef]

Kidoguchi, Y.

S. S. Kee, H. Mohammadi, Y. Hirano, Y. Kidoguchi, K. Miwa, “Experimental study on combustion characteristics and emissions reduction of emulsified fuels in diesel combustion using rapid compression,” (Society of Automotive Engineers, 2003).
[CrossRef]

Kim, T.

L. A. Kranendonk, J. W. Walewski, T. Kim, S. T. Sanders, “Wavelength-agile sensor applied for HCCI Engine Measurements,” Proc. Combust. Symp. 30, 1619–1627 (2005).
[CrossRef]

Kranendonk, L. A.

L. A. Kranendonk, J. W. Walewski, T. Kim, S. T. Sanders, “Wavelength-agile sensor applied for HCCI Engine Measurements,” Proc. Combust. Symp. 30, 1619–1627 (2005).
[CrossRef]

L. A. Kranendonk, A. W. Caswell, A. N. Myers, S. T. Sanders, “Wavelength-agile laser sensors for measuring gas properties in engines,” , (Society of Automotive Engineers, 2002).

Lenth, W.

Litzinger, T.

M. Stoner, T. Litzinger, “Effects of structure and boiling point of oxygenated blending compounds in reducing diesel emissions,” (Society of Automotive Engineers, 1999).
[CrossRef]

Ma, L.

Mattison, D. W.

Merola, S. S.

F. E. Corcione, S. S. Merola, B. M. Vaglieco, “Nanometric particle formation in optically accessible diesel engine,” (Society of Automotive Engineers, 2001).
[CrossRef]

Mischler, B.

P. P. Radi, B. Mischler, A. Schlegel, A. Tzannis, P. Beaud, T. Gerber, “Absolute concentration measurements using DFWM and modeling of OH and S2 in a fuel-rich H2/air/SO2 flame,” Combust. Flame 118, 301–307 (1999).
[CrossRef]

Miwa, K.

S. S. Kee, H. Mohammadi, Y. Hirano, Y. Kidoguchi, K. Miwa, “Experimental study on combustion characteristics and emissions reduction of emulsified fuels in diesel combustion using rapid compression,” (Society of Automotive Engineers, 2003).
[CrossRef]

Mohammadi, H.

S. S. Kee, H. Mohammadi, Y. Hirano, Y. Kidoguchi, K. Miwa, “Experimental study on combustion characteristics and emissions reduction of emulsified fuels in diesel combustion using rapid compression,” (Society of Automotive Engineers, 2003).
[CrossRef]

Musculus, M. P. B.

M. P. B. Musculus, L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in high-pressure transient combustion environments,” Combust. Flame 141, 371–391 (2005).
[CrossRef]

Myers, A. N.

L. A. Kranendonk, A. W. Caswell, A. N. Myers, S. T. Sanders, “Wavelength-agile laser sensors for measuring gas properties in engines,” , (Society of Automotive Engineers, 2002).

Peterson, E. L.

E. L. Peterson, “A shock tube and diagnostic for chemistry measurements at elevated pressures with application to methane ignition,” in Mechanical Engineering (Stanford University, 1998).

Pickett, L.

M. P. B. Musculus, L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in high-pressure transient combustion environments,” Combust. Flame 141, 371–391 (2005).
[CrossRef]

Radi, P. P.

P. P. Radi, B. Mischler, A. Schlegel, A. Tzannis, P. Beaud, T. Gerber, “Absolute concentration measurements using DFWM and modeling of OH and S2 in a fuel-rich H2/air/SO2 flame,” Combust. Flame 118, 301–307 (1999).
[CrossRef]

Sanders, S. T.

L. A. Kranendonk, J. W. Walewski, T. Kim, S. T. Sanders, “Wavelength-agile sensor applied for HCCI Engine Measurements,” Proc. Combust. Symp. 30, 1619–1627 (2005).
[CrossRef]

J. W. Walewski, S. T. Sanders, “High-resolution wavelength-agile laser source based on pulsed super-continua,” Appl. Phys. B 79, 415–418 (2004).
[CrossRef]

S. T. Sanders, D. W. Mattison, L. Ma, J. B. Jeffries, R. K. Hanson, “Wavelength-agile diode-laser sensing strategies for monitoring gas properties in optically harsh flows: application in cesium-seeded pulse detonation engine,” Opt. Express 10, 505–514 (2002).
[CrossRef] [PubMed]

S. T. Sanders, “Diode-laser sensors for harsh environments with application to pulse detonation. engines,” in Mechanical Engineering (Stanford University, 2001).

L. A. Kranendonk, A. W. Caswell, A. N. Myers, S. T. Sanders, “Wavelength-agile laser sensors for measuring gas properties in engines,” , (Society of Automotive Engineers, 2002).

Schlegel, A.

P. P. Radi, B. Mischler, A. Schlegel, A. Tzannis, P. Beaud, T. Gerber, “Absolute concentration measurements using DFWM and modeling of OH and S2 in a fuel-rich H2/air/SO2 flame,” Combust. Flame 118, 301–307 (1999).
[CrossRef]

Steel, W. H.

Stoner, M.

M. Stoner, T. Litzinger, “Effects of structure and boiling point of oxygenated blending compounds in reducing diesel emissions,” (Society of Automotive Engineers, 1999).
[CrossRef]

Supplee, J. E.

Sutton, G. W.

G. W. Sutton, “Effect of turbulent fluctuations in an optically active fluid medium,” AIAA J. 7, 1737–1743 (1969).
[CrossRef]

Tree, D. R.

D. R. Tree, J. E. Dec, “Extinction measurements of in-cylinder soot deposition in a heavy-duty DI diesel engine,” (Society of Automotive Engineers, 2001).
[CrossRef]

Tzannis, A.

P. P. Radi, B. Mischler, A. Schlegel, A. Tzannis, P. Beaud, T. Gerber, “Absolute concentration measurements using DFWM and modeling of OH and S2 in a fuel-rich H2/air/SO2 flame,” Combust. Flame 118, 301–307 (1999).
[CrossRef]

Vaglieco, B. M.

F. E. Corcione, S. S. Merola, B. M. Vaglieco, “Nanometric particle formation in optically accessible diesel engine,” (Society of Automotive Engineers, 2001).
[CrossRef]

Walewski, J. W.

L. A. Kranendonk, J. W. Walewski, T. Kim, S. T. Sanders, “Wavelength-agile sensor applied for HCCI Engine Measurements,” Proc. Combust. Symp. 30, 1619–1627 (2005).
[CrossRef]

J. W. Walewski, S. T. Sanders, “High-resolution wavelength-agile laser source based on pulsed super-continua,” Appl. Phys. B 79, 415–418 (2004).
[CrossRef]

Wehe, S. D.

S. D. Wehe, D. S. Baer, R. K. Hanson, “Diode-laser sensor for velocity measurements in hypervelocity flows,” AIAA J. 37, 1013–1015 (1999).
[CrossRef]

Welford, W. T.

W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, 1978).

Whittaker, E. A.

Winston, R.

W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, 1978).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (MacMillan, 1964).

Young, M.

M. Young, Optics and Laser (Springer-Verlag, 1986).
[CrossRef]

AIAA J. (2)

G. W. Sutton, “Effect of turbulent fluctuations in an optically active fluid medium,” AIAA J. 7, 1737–1743 (1969).
[CrossRef]

S. D. Wehe, D. S. Baer, R. K. Hanson, “Diode-laser sensor for velocity measurements in hypervelocity flows,” AIAA J. 37, 1013–1015 (1999).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

J. W. Walewski, S. T. Sanders, “High-resolution wavelength-agile laser source based on pulsed super-continua,” Appl. Phys. B 79, 415–418 (2004).
[CrossRef]

Combust. Flame (2)

P. P. Radi, B. Mischler, A. Schlegel, A. Tzannis, P. Beaud, T. Gerber, “Absolute concentration measurements using DFWM and modeling of OH and S2 in a fuel-rich H2/air/SO2 flame,” Combust. Flame 118, 301–307 (1999).
[CrossRef]

M. P. B. Musculus, L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in high-pressure transient combustion environments,” Combust. Flame 141, 371–391 (2005).
[CrossRef]

Opt. Express (1)

Proc. Combust. Symp. (1)

L. A. Kranendonk, J. W. Walewski, T. Kim, S. T. Sanders, “Wavelength-agile sensor applied for HCCI Engine Measurements,” Proc. Combust. Symp. 30, 1619–1627 (2005).
[CrossRef]

Prog. Aerosp. Sci. (1)

E. J. Jumper, E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–399 (2001).
[CrossRef]

Other (13)

Landolt- Bornstein, Zahlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik, und Technik (Springer-Verlag, 1962).

M. Born, E. Wolf, Principles of Optics (MacMillan, 1964).

I. Celik, Y. Ibrahin, “An assessment of turbulence scales relevant to IC engines,” in ASME Spring Technical Conference 28 97-ICE-S (ASME, 1997), pp. 35–44.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).

W. T. Welford, R. Winston, The Optics of Non-Imaging Concentrators (Academic, 1978).

M. Young, Optics and Laser (Springer-Verlag, 1986).
[CrossRef]

S. T. Sanders, “Diode-laser sensors for harsh environments with application to pulse detonation. engines,” in Mechanical Engineering (Stanford University, 2001).

E. L. Peterson, “A shock tube and diagnostic for chemistry measurements at elevated pressures with application to methane ignition,” in Mechanical Engineering (Stanford University, 1998).

L. A. Kranendonk, A. W. Caswell, A. N. Myers, S. T. Sanders, “Wavelength-agile laser sensors for measuring gas properties in engines,” , (Society of Automotive Engineers, 2002).

S. S. Kee, H. Mohammadi, Y. Hirano, Y. Kidoguchi, K. Miwa, “Experimental study on combustion characteristics and emissions reduction of emulsified fuels in diesel combustion using rapid compression,” (Society of Automotive Engineers, 2003).
[CrossRef]

F. E. Corcione, S. S. Merola, B. M. Vaglieco, “Nanometric particle formation in optically accessible diesel engine,” (Society of Automotive Engineers, 2001).
[CrossRef]

M. Stoner, T. Litzinger, “Effects of structure and boiling point of oxygenated blending compounds in reducing diesel emissions,” (Society of Automotive Engineers, 1999).
[CrossRef]

D. R. Tree, J. E. Dec, “Extinction measurements of in-cylinder soot deposition in a heavy-duty DI diesel engine,” (Society of Automotive Engineers, 2001).
[CrossRef]

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 (10)

Fig. 1
Fig. 1

Scanner: wavelength-agile absorption spectroscopy schematic.

Fig. 2
Fig. 2

OMA: absorption spectroscopy schematic.

Fig. 3
Fig. 3

Schematic for line-of-sight optical access to a turbulent fluid flow test section of length L, with the integral length scale Λ and Kolmogorov length scale d.

Fig. 4
Fig. 4

Diameter and divergence angle of laser light as a function of distance through a turbulent flow field. Assumptions: K = Kmeas, f1 = 4.5 mm, 9 μm diameter, 0.09 NA pitching fiber.

Fig. 5
Fig. 5

Example of optical extent conservation. The lens can reduce the diameter of the light (from d3 to d4) at the expense of the dispersion expansion half-angle (from θ3 to θ4).

Fig. 6
Fig. 6

Growth of optical extent in a turbulent flow field. Extent values set by common fibers are shown. Assumptions: K = Kmeas, 9 μm diameter, 0.09 NA pitching fiber.

Fig. 7
Fig. 7

Optimum pitching and collection lens selection map. Assumptions: 9 μm diameter, 0.09 NA, L = 10 cm.

Fig. 8
Fig. 8

Measured transmission during a single compression stroke of a firing HCCI engine. Data from two lens sets are shown: Both have a 4.5 mm focal length pitching lens (f1) but different collection lens focal lengths (f2).

Fig. 9
Fig. 9

Calculated transmission through a representative turbulent flow field. The collection fiber in the OMA case is 9 μm diameter with a 0.09 NA. The collection fiber in the scanner cases is 62.5 μm with a 0.27 NA. Assumptions: K = Kmeas, pitching fiber is 9 μm diameter, 0.09 NA.

Fig. 10
Fig. 10

Design guide for choosing between the scanner and OMA methods, showing which technique is preferred for maximizing transmission in beam-steering environments. Assumptions: 9 μm diameter, 0.09 NA pitching fiber.

Tables (2)

Tables Icon

Table 1 Sample Commercial Photodetectorsa

Tables Icon

Table 2 Assuming a Known Pitching Fiber (d1, θ1), Collection Fiber (d5, θ5), and Test Section Length (L), the Lenses Can Be Chosen by Solving the Following Equations, and the Resulting K Is the Maximum K That the Lenses Can Tolerate Without a Loss of Signal

Equations (19)

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

( n o - 1 ) = ( n S T P - 1 ) ρ o ρ S T P ,
Δ n = ( n o - 1 ) Δ ρ ρ o .
f 1 = d 2 - d 1 2 tan ( θ 1 ) .
f 2 = d 5 2 tan ( θ 3 ) .
T = d 5 2 sin 2 ( θ 5 ) d 4 2 sin 2 ( θ 4 ) { d 5 d 4 θ 5 θ 4 } , T = d 5 2 d 4 2 { d 5 d 4 θ 5 > θ 4 } , T = sin 2 ( θ 5 ) sin 2 ( θ 4 ) { d 5 > d 4 θ 5 θ 4 } .
d θ d s = n ,
θ y x = n y ,             θ z x = n z .
Δ θ y 2 = Δ θ 2 2 = K x .
K = 3.8 Δ n 2 Λ ( Λ d ) 1 / 3 .
Δ θ max Δ θ 2 = K x ,
θ θ 2 + Δ θ max = θ 2 + K x ,
d d x ( d ) = 2 θ .
d = d 2 + 2 [ θ 2 x + K ( 2 3 x 1.5 ) ] .
Extent 2 π 2 d 2 sin 2 ( θ ) = 2 π 2 d 2 NA 2 .
Extent = 2 π 2 [ ( d 2 + 2 θ 2 x + 4 3 K x 3 / 2 ) ( θ 2 + Kx ) ] 2 ,
Extent = 2 π 2 { [ d 1 ( 1 + x f 1 ) + 2 f 1 θ 1 + 4 3 K x 3 / 2 ] × ( d 1 2 f 1 + K x ) } 2
d λ = ( 6.2 E - 6 ) d 5 .
0 = ( d 1 2 L f 1 3 + 5 L d 1 K L 3 f 1 2 - 2 θ 1 K L ) .
T = d λ Δ λ

Metrics