Abstract

The surface-plasmon resonance (SPR) effect in metals is highly sensitive to fluctuations in the optical properties of the interface and has been frequently employed in the Kretschmann configuration for optical sensing. The operating conditions required for using the SPR effect for probing nonabsorbing media under maximum sensitivity are derived analytically under the Lorentzian approximation. It is found that the film thickness that maximizes sensitivity occurs when the radiation damping of the oscillation is half the intrinsic damping. Numerical results are presented for the spectral dependence of the optimum thickness as well as of the SPR parameters of gold, copper, silver, and aluminum films, useful for the design of optical sensors for both gaseous and aqueous environments.

© 2006 Optical Society of America

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  1. E. Kretschmann, "Determination of optical constants of metals through the stimulation of surface plasma oscillations," Z. Phys. 241, 313-324 (1971), in German.
    [CrossRef]
  2. G. P. Anderson, E. C. Merrick, S. A. Trammell, T. M. Chinowsky, and D. K. Shenoy, "Simplified avidin-biotin mediated antibody attachment for a surface plasmon resonance biosensor," Sens. Lett. 3, 151-156 (2005).
    [CrossRef]
  3. J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
    [CrossRef]
  4. S. H. Zaidi, M. Yousaf, and S. R. J. Brueck, "Grating coupling to surface plasma waves. I. First-order coupling," J. Opt. Soc. Am. 8, 770-779 (1991).
    [CrossRef]
  5. M. J. Jory, P. S. Vukusic, and J. R. Sambles, "Development of a prototype gas sensor using surface-plasmon resonance on gratings," Sens. Actuators B 17, 203-209 (1994).
    [CrossRef]
  6. E. Fontana, "Theoretical and experimental study of the surface plasmon resonance effect on a recordable optical disk," Appl. Opt. 43, 79-87 (2004).
    [CrossRef] [PubMed]
  7. R. C. Jorgenson and S. S. Yee, "A fiber-optic chemical sensor based on surface plasmon resonance," Sens. Actuators B 12, 213-220 (1993).
    [CrossRef]
  8. L. DeMaria, M. Martinelli, and G. Vegetti, "Fiber-optic sensor based on surface plasmon interrogation," Sens. Actuators B 12, 221-223 (1993).
    [CrossRef]
  9. A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, "Surface-plasmon resonance spectrometry and characterization of absorbing liquids," Appl. Opt. 39, 3314-3320 (2000).
    [CrossRef]
  10. S. Wang, S. Boussaad, and N. J. Tao, "Surface plasmon resonance enhanced optical absorption spectroscopy for studying molecular adsorbates," Rev. Sci. Instrum. 72, 3055-3060 (2001).
    [CrossRef]
  11. J. van Gent, P. V. Lambeck, H. J. M. Kreuwel, G. J. Gerritsma, E. J. R. Sudhölter, D. N. Reinhoudt, and T. J. A. Popma, "Optimization of a chemooptical surface plasmon resonance based sensor," Appl. Opt. 29, 2843-2849 (1990).
    [CrossRef] [PubMed]
  12. S. Ekgasit, C. Thammacharoen, F. Yu, and W. Knoll, "Influence of the metal film thickness on the sensitivity of surface plasmon resonance bionsensors," Appl. Spectrosc. 59, 661-667 (2005).
    [CrossRef] [PubMed]
  13. C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005), Chap. 14, pp. 393-426.
  14. W. P. Chen and J. M. Chen, "Use of surface plasma waves for determination of the thickness and optical constants of thin metallic films," J. Opt. Soc. Am. 71, 189-191 (1981).
    [CrossRef]
  15. D. R. Lide, ed., Handbook of Chemistry and Physics, 85th ed. (CRC, 2005), pp. 12-133-12-156.
  16. M. Bass, ed., Handbook of Optics, 2nd ed. (McGraw-Hill, 1995), Vol. II, Chap. 33, pp. 33.3-33.101.
  17. I. H. Malitson, "Interspecimen comparison of the refractive index of fused silica," J. Opt. Soc. Am. 55, 1205-1209 (1965).
    [CrossRef]
  18. P. Schiebener, J. Straub, J. M. H. Levelt Sengers, and J. S. Gallagher, "Refractive index of water and steam as functions of wavelength, temperature and density," J. Phys. Chem. Ref. Data 19, 677-717 (1990).
    [CrossRef]
  19. E. Fontana, R. H. Pantell, and M. Moslehi, "Characterization of dielectric-coated, metal mirrors using surface plasmon spectroscopy," Appl. Opt. 27, 3334-3340 (1988).
    [CrossRef] [PubMed]

2005 (2)

G. P. Anderson, E. C. Merrick, S. A. Trammell, T. M. Chinowsky, and D. K. Shenoy, "Simplified avidin-biotin mediated antibody attachment for a surface plasmon resonance biosensor," Sens. Lett. 3, 151-156 (2005).
[CrossRef]

S. Ekgasit, C. Thammacharoen, F. Yu, and W. Knoll, "Influence of the metal film thickness on the sensitivity of surface plasmon resonance bionsensors," Appl. Spectrosc. 59, 661-667 (2005).
[CrossRef] [PubMed]

2004 (1)

2001 (1)

S. Wang, S. Boussaad, and N. J. Tao, "Surface plasmon resonance enhanced optical absorption spectroscopy for studying molecular adsorbates," Rev. Sci. Instrum. 72, 3055-3060 (2001).
[CrossRef]

2000 (1)

1999 (1)

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

1994 (1)

M. J. Jory, P. S. Vukusic, and J. R. Sambles, "Development of a prototype gas sensor using surface-plasmon resonance on gratings," Sens. Actuators B 17, 203-209 (1994).
[CrossRef]

1993 (2)

R. C. Jorgenson and S. S. Yee, "A fiber-optic chemical sensor based on surface plasmon resonance," Sens. Actuators B 12, 213-220 (1993).
[CrossRef]

L. DeMaria, M. Martinelli, and G. Vegetti, "Fiber-optic sensor based on surface plasmon interrogation," Sens. Actuators B 12, 221-223 (1993).
[CrossRef]

1991 (1)

S. H. Zaidi, M. Yousaf, and S. R. J. Brueck, "Grating coupling to surface plasma waves. I. First-order coupling," J. Opt. Soc. Am. 8, 770-779 (1991).
[CrossRef]

1990 (2)

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, and J. S. Gallagher, "Refractive index of water and steam as functions of wavelength, temperature and density," J. Phys. Chem. Ref. Data 19, 677-717 (1990).
[CrossRef]

J. van Gent, P. V. Lambeck, H. J. M. Kreuwel, G. J. Gerritsma, E. J. R. Sudhölter, D. N. Reinhoudt, and T. J. A. Popma, "Optimization of a chemooptical surface plasmon resonance based sensor," Appl. Opt. 29, 2843-2849 (1990).
[CrossRef] [PubMed]

1988 (1)

1981 (1)

1971 (1)

E. Kretschmann, "Determination of optical constants of metals through the stimulation of surface plasma oscillations," Z. Phys. 241, 313-324 (1971), in German.
[CrossRef]

1965 (1)

Anderson, G. P.

G. P. Anderson, E. C. Merrick, S. A. Trammell, T. M. Chinowsky, and D. K. Shenoy, "Simplified avidin-biotin mediated antibody attachment for a surface plasmon resonance biosensor," Sens. Lett. 3, 151-156 (2005).
[CrossRef]

Bass, M.

M. Bass, ed., Handbook of Optics, 2nd ed. (McGraw-Hill, 1995), Vol. II, Chap. 33, pp. 33.3-33.101.

Boussaad, S.

S. Wang, S. Boussaad, and N. J. Tao, "Surface plasmon resonance enhanced optical absorption spectroscopy for studying molecular adsorbates," Rev. Sci. Instrum. 72, 3055-3060 (2001).
[CrossRef]

Brueck, S. R. J.

S. H. Zaidi, M. Yousaf, and S. R. J. Brueck, "Grating coupling to surface plasma waves. I. First-order coupling," J. Opt. Soc. Am. 8, 770-779 (1991).
[CrossRef]

Chen, J. M.

Chen, W. P.

Chinowsky, T. M.

G. P. Anderson, E. C. Merrick, S. A. Trammell, T. M. Chinowsky, and D. K. Shenoy, "Simplified avidin-biotin mediated antibody attachment for a surface plasmon resonance biosensor," Sens. Lett. 3, 151-156 (2005).
[CrossRef]

DeMaria, L.

L. DeMaria, M. Martinelli, and G. Vegetti, "Fiber-optic sensor based on surface plasmon interrogation," Sens. Actuators B 12, 221-223 (1993).
[CrossRef]

Ekgasit, S.

Fontana, E.

Gallagher, J. S.

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, and J. S. Gallagher, "Refractive index of water and steam as functions of wavelength, temperature and density," J. Phys. Chem. Ref. Data 19, 677-717 (1990).
[CrossRef]

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Gerritsma, G. J.

Gershon, P. D.

Homola, J.

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Jorgenson, R. C.

R. C. Jorgenson and S. S. Yee, "A fiber-optic chemical sensor based on surface plasmon resonance," Sens. Actuators B 12, 213-220 (1993).
[CrossRef]

Jory, M. J.

M. J. Jory, P. S. Vukusic, and J. R. Sambles, "Development of a prototype gas sensor using surface-plasmon resonance on gratings," Sens. Actuators B 17, 203-209 (1994).
[CrossRef]

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005), Chap. 14, pp. 393-426.

Knoll, W.

Kolomenskii, A. A.

Kretschmann, E.

E. Kretschmann, "Determination of optical constants of metals through the stimulation of surface plasma oscillations," Z. Phys. 241, 313-324 (1971), in German.
[CrossRef]

Kreuwel, H. J. M.

Lambeck, P. V.

Levelt Sengers, J. M. H.

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, and J. S. Gallagher, "Refractive index of water and steam as functions of wavelength, temperature and density," J. Phys. Chem. Ref. Data 19, 677-717 (1990).
[CrossRef]

Lide, D. R.

D. R. Lide, ed., Handbook of Chemistry and Physics, 85th ed. (CRC, 2005), pp. 12-133-12-156.

Malitson, I. H.

Martinelli, M.

L. DeMaria, M. Martinelli, and G. Vegetti, "Fiber-optic sensor based on surface plasmon interrogation," Sens. Actuators B 12, 221-223 (1993).
[CrossRef]

Merrick, E. C.

G. P. Anderson, E. C. Merrick, S. A. Trammell, T. M. Chinowsky, and D. K. Shenoy, "Simplified avidin-biotin mediated antibody attachment for a surface plasmon resonance biosensor," Sens. Lett. 3, 151-156 (2005).
[CrossRef]

Moslehi, M.

Pantell, R. H.

Popma, T. J. A.

Reinhoudt, D. N.

Sambles, J. R.

M. J. Jory, P. S. Vukusic, and J. R. Sambles, "Development of a prototype gas sensor using surface-plasmon resonance on gratings," Sens. Actuators B 17, 203-209 (1994).
[CrossRef]

Schiebener, P.

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, and J. S. Gallagher, "Refractive index of water and steam as functions of wavelength, temperature and density," J. Phys. Chem. Ref. Data 19, 677-717 (1990).
[CrossRef]

Schuessler, H. A.

Shenoy, D. K.

G. P. Anderson, E. C. Merrick, S. A. Trammell, T. M. Chinowsky, and D. K. Shenoy, "Simplified avidin-biotin mediated antibody attachment for a surface plasmon resonance biosensor," Sens. Lett. 3, 151-156 (2005).
[CrossRef]

Straub, J.

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, and J. S. Gallagher, "Refractive index of water and steam as functions of wavelength, temperature and density," J. Phys. Chem. Ref. Data 19, 677-717 (1990).
[CrossRef]

Sudhölter, E. J. R.

Tao, N. J.

S. Wang, S. Boussaad, and N. J. Tao, "Surface plasmon resonance enhanced optical absorption spectroscopy for studying molecular adsorbates," Rev. Sci. Instrum. 72, 3055-3060 (2001).
[CrossRef]

Thammacharoen, C.

Trammell, S. A.

G. P. Anderson, E. C. Merrick, S. A. Trammell, T. M. Chinowsky, and D. K. Shenoy, "Simplified avidin-biotin mediated antibody attachment for a surface plasmon resonance biosensor," Sens. Lett. 3, 151-156 (2005).
[CrossRef]

van Gent, J.

Vegetti, G.

L. DeMaria, M. Martinelli, and G. Vegetti, "Fiber-optic sensor based on surface plasmon interrogation," Sens. Actuators B 12, 221-223 (1993).
[CrossRef]

Vukusic, P. S.

M. J. Jory, P. S. Vukusic, and J. R. Sambles, "Development of a prototype gas sensor using surface-plasmon resonance on gratings," Sens. Actuators B 17, 203-209 (1994).
[CrossRef]

Wang, S.

S. Wang, S. Boussaad, and N. J. Tao, "Surface plasmon resonance enhanced optical absorption spectroscopy for studying molecular adsorbates," Rev. Sci. Instrum. 72, 3055-3060 (2001).
[CrossRef]

Yee, S. S.

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

R. C. Jorgenson and S. S. Yee, "A fiber-optic chemical sensor based on surface plasmon resonance," Sens. Actuators B 12, 213-220 (1993).
[CrossRef]

Yousaf, M.

S. H. Zaidi, M. Yousaf, and S. R. J. Brueck, "Grating coupling to surface plasma waves. I. First-order coupling," J. Opt. Soc. Am. 8, 770-779 (1991).
[CrossRef]

Yu, F.

Zaidi, S. H.

S. H. Zaidi, M. Yousaf, and S. R. J. Brueck, "Grating coupling to surface plasma waves. I. First-order coupling," J. Opt. Soc. Am. 8, 770-779 (1991).
[CrossRef]

Appl. Opt. (4)

Appl. Spectrosc. (1)

J. Opt. Soc. Am. (3)

J. Phys. Chem. Ref. Data (1)

P. Schiebener, J. Straub, J. M. H. Levelt Sengers, and J. S. Gallagher, "Refractive index of water and steam as functions of wavelength, temperature and density," J. Phys. Chem. Ref. Data 19, 677-717 (1990).
[CrossRef]

Rev. Sci. Instrum. (1)

S. Wang, S. Boussaad, and N. J. Tao, "Surface plasmon resonance enhanced optical absorption spectroscopy for studying molecular adsorbates," Rev. Sci. Instrum. 72, 3055-3060 (2001).
[CrossRef]

Sens. Actuators B (4)

M. J. Jory, P. S. Vukusic, and J. R. Sambles, "Development of a prototype gas sensor using surface-plasmon resonance on gratings," Sens. Actuators B 17, 203-209 (1994).
[CrossRef]

R. C. Jorgenson and S. S. Yee, "A fiber-optic chemical sensor based on surface plasmon resonance," Sens. Actuators B 12, 213-220 (1993).
[CrossRef]

L. DeMaria, M. Martinelli, and G. Vegetti, "Fiber-optic sensor based on surface plasmon interrogation," Sens. Actuators B 12, 221-223 (1993).
[CrossRef]

J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999).
[CrossRef]

Sens. Lett. (1)

G. P. Anderson, E. C. Merrick, S. A. Trammell, T. M. Chinowsky, and D. K. Shenoy, "Simplified avidin-biotin mediated antibody attachment for a surface plasmon resonance biosensor," Sens. Lett. 3, 151-156 (2005).
[CrossRef]

Z. Phys. (1)

E. Kretschmann, "Determination of optical constants of metals through the stimulation of surface plasma oscillations," Z. Phys. 241, 313-324 (1971), in German.
[CrossRef]

Other (3)

C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005), Chap. 14, pp. 393-426.

D. R. Lide, ed., Handbook of Chemistry and Physics, 85th ed. (CRC, 2005), pp. 12-133-12-156.

M. Bass, ed., Handbook of Optics, 2nd ed. (McGraw-Hill, 1995), Vol. II, Chap. 33, pp. 33.3-33.101.

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Figures (7)

Fig. 1
Fig. 1

(Color online) Qualitative representation of the transversal profile of the power density associated with a SP propagating along a single interface.

Fig. 2
Fig. 2

(Color online) Spectral dependences of the (a) real part and (b) imaginary part of the relative permittivities of Au, Cu, Ag, and Al.[15]

Fig. 3
Fig. 3

(Color online) Kretschmann configuration[1] for the excitation of SPs.

Fig. 4
Fig. 4

(Color online) Lorentzian curve as the approximation of the reflectance curve and parameter representation in the k x and θ space.

Fig. 5
Fig. 5

(Color online) Spectral dependences of the optimum thickness of bare and water-exposed Au, Ag, Al, and Cu films for a BK7 prism in the Kretschmann configuration.

Fig. 6
Fig. 6

(Color online) Spectral dependences of the resonance angle at optimum thickness of bare and water-exposed Au, Ag, Al, and Cu films for a BK7 prism in the Kretschmann configuration.

Fig. 7
Fig. 7

(Color online) (a) Spectral dependences of the reduced half-width at optimum thickness of bare and water-exposed Au, Ag, Al, and Cu films for a BK7 prism in the Kretschmann configuration. (b) Expanded view of the spectral dependences of reduced half-widths smaller than 0.2° in the spectral region of 0.6 μ m λ 1   μ m .

Tables (1)

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Table 1 Constants for the Definition of the Relative Permittivity Function a

Equations (74)

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ε 2 + ε 3 < 0
K 0 = K 0 + j K 0
K 0 = k 0 ( ε 2 ε 3 ε 2 + ε 3 ) 1 / 2 ,
K 0 = k 0 ( ε 2 ε 3 ε 2 + ε 3 ) 1 / 2 ,
K 0 = K 0 ε 2 ε 3 2 ε 2 ( ε 2 + ε 3 ) ,
| ε 2 | ε 3 2 ε 2 ( ε 2 + ε 3 )   ≪   1 .
R = | r 12 + r 23   exp ( - 2 j k 2 d ) 1 + r 12 r 23   exp ( - j 2 k 2 d ) | 2 ,
r i , i + 1 = ε i + 1 k i ε i k i + 1 ε i + 1 k i + ε i k i + 1 ,
k i = k 0 [ ε i ε 1 ( sin   θ ) 2 ] 1 / 2 ,
R 1 ( 1 R min ) K 2 ( k x K ) 2 + K 2 ,
k x = ε 1 k 0   sin   θ .
R ¯ = 1 2 ( 1 + R min )
K = ε 1 k 0   sin   θ SP .
| K | 2 π ε 1 λ w SP   cos   θ SP .
sin θ sin   θ SP + cos   θ SP ( θ θ SP )
R ( θ ) 1 ( 1 R min ) w SP 2 ( θ θ SP ) 2 + w SP 2 ,
R = 1 ( 1 R min ) w SP 2 u 2 + w SP 2 ,
u = θ θ SP .
R ε 3 = a u ε 3 + b w SP ε 3 + c R min ε 3 ,
a = 2 ( 1 R min ) w SP 2 u ( u 2 + w SP 2 ) 2 ,
b = 2 ( 1 R min ) w SP u 2 ( u 2 + w SP 2 ) 2 ,
c = w SP 2 u 2 + w SP 2 .
c max = 1 .
a u = 2 ( 1 R min ) w SP 2 w SP 2 3 u 2 ( u 2 + w SP 2 ) 3 ,
w r = w SP / 3 .
a ( ± w r ) a ± = ± 3 ( 1 R min ) 8 w r .
b u = 2 ( 1 R min ) w SP 2 u ( w SP 2 u 2 ) ( u 2 + w SP 2 ) 3 .
b ( u = ± w SP ) b ± = ( 1 R min ) 2 w SP ,
b ± = ( 1 R min ) 2 3 w r .
c | a ± | ,
c | b ± | .
R ε 3 a u ε 3 + b w SP ε 3 .
| u ε 3 | w SP ε 3 .
a + < | a | .
R ε 3 | u = w r R ε 3 | max a u ε 3 ,
R ε 3 | max R max θ SP ε 3 ,
R max = 3 ( 1 R min ) 8 w r
K 0 ε 3 K 0 3 2 ε 3 2 k 0 2 .
θ SP ε 3 ε 1 2 ε 3 2 ( sin θ SP ) 3 cos   θ SP .
K = K 0 + K 1 ,
K 1 = A   exp ( - α d ) ,
A = 2 Im [ k 0 ( ε 2 ε 3 ) 2 ( ε 2 + ε 3 ) 3 ] ,
α = 2 k 0 ε 2 ( | ε 2 | ε 3 ) 1 / 2 .
R min = 1 4 η ( 1 + η ) 2 ,
η = K 0 K 1
w SP = w 0 η + 1 2 η ,
w 0 = λ | K 0 | π ε 1   cos   θ SP
R max ( η ) = 3 2 1 w r 0 η 2 ( 1 + η ) 3 ,
w r ( η = 2 ) = 3 4 w r 0 ,
R min ( η = 2 ) = 1 9 ,
R max ( 2 ) R max ( 1 ) = 32 27 1.185.
ε 1 ( λ ) = 1 + A 1 λ 2 λ 2 B 1 + A 2 λ 2 λ 2 B 2 + A 3 λ 2 λ 2 B 3 ,
K 1 = K 0 ,
d 0 ,guess = 1 α  ln ( A K 0 ) ,
K 1 ( η = 2 ) K 1 ( η = 1 ) = 1 2 = exp [ - α ( d opt,guess d 0 ) ] ,
d opt,guess = d 0 + 1 α  ln   2,
K K 0 = K 0 + j K 0 = k 0 ( ε 2 ε 3 ε 2 + ε 3 ) 1 / 2 ,
K 0 = n 1 k 0   sin   θ SP ,
K 0 = n 1 k 0 w SP   cos   θ SP .
K 0 ε 3 = k 0 2 ( ε 2 ε 3 ε 2 + ε 3 ) 1 / 2 ε 2     2 ( ε 2 + ε 3 ) 2 ,
K 0 ε 3 = k 0 2 1 ε 3 2 ( K 0 k 0 ) 3 .
K 0 ε 3 = K 0 3 + 3 K 0 K 0 2 2 ε 3 2 k 0 2 ,
K 0 ε 3 = K 0 3 + 3 K 0 2 K 0 2 ε 3 2 k 0 2 .
K 0 ε 3 / K 0 ε 3 = K 0 3 + 3 K 0 K 0 2 K 0 3 + 3 K 0 2 K 0 .
K 0 ε 3 / K 0 ε 3 K 0 3 K 0 .
K 0 ε 3 = n 1 k 0   cos   θ SP θ SP ε 3 ,
θ SP ε 3 = 1 n 1 k 0   cos   θ SP K 0 ε 3 .
K 0 ε 3 = n 1 k 0 [ sin   θ SP w SP θ SP ε 3 cos   θ SP w SP ε 3 ] .
K 0 ε 3 = [ w SP sin   θ SP cos   θ SP K 0 ε 3 n 1 k 0   cos   θ SP w SP ε 3 ] ,
w SP ε 3 = 1 n 1 k 0   cos   θ SP [ w SP sin   θ SP cos   θ SP K 0 ε 3 K 0 ε 3 ] .
w SP ε 3 1 n 1 k 0   cos   θ SP [ 1 + ( tan   θ SP ) 2 3 ] K 0 ε 3 .
θ SP ε 3 / w SP ε 3 - 3 3 + ( tan   θ SP ) 2 K 0 ε 3 / K 0 ε 3 ,
θ SP ε 3 / w SP ε 3 - 1 3 + ( tan   θ SP ) 2 K 0 K 0 .
u ε 3 / w SP ε 3 1 3 + ( tan   θ SP ) 2 K 0 K 0 .

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