Abstract

A diode-laser-to-waveguide butt-coupling model is described. The model takes into consideration the Fabry–Perot reflection and transmission of the étalon formed by the laser front facet and the waveguide entrance facet. The model predicts coupling efficiency and the coupled-power fluctuations that occur versus the separation between the laser and the waveguide. Calculations performed for Fabry–Perot-laser-to-KTP waveguide coupling show that the transverse and the angular alignment tolerances of the waveguide can be increased when the waveguide entrance facet is antireflection coated. The longitudinal alignment tolerance of the waveguide can be increased by use of an index-matching gel between the laser and the waveguide.

© 1996 Optical Society of America

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References

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  1. H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Microwave Research Institute Symposia Series 14 (Polytechnic Institute of Brooklyn, New York, 1964), pp. 333–347.
  2. W. B. Joyce, “Alignment of Gaussian beams,” Appl. Opt. 23, 4187–4196(1984).
    [CrossRef] [PubMed]
  3. D. G. Hall, R. R. Rice, J. D. Zino, “Simple Gaussian-beam model for GaAlAs double-heterostructure laser-diode-to-diffused-waveguide coupling calculations,” Opt. Lett. 4, 292– 294 (1979).
    [CrossRef] [PubMed]
  4. W. L. Emkey, “Optical coupling between single-mode semiconductor lasers and strip waveguides,” J. Lightwave Technol. 1, 436–443 (1983).
    [CrossRef]
  5. A. Yariv, Optical Electronics, 4th ed. (Holt, Rinehart & Winston, New York, 1991), Chap. 2, p. 46.
  6. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Hilger, London, 1986), Chap. 3, p. 71.
  7. J. Major, SDL Inc., 80 Rose Orchard Way, San Jose, Calif. 95134 (personal communication, 1994).
  8. H. Kressel, “Semiconductor laser devices,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, pp. 441–495.
  9. W. van Etten, J. van der Plaats, Fundamentals of Optical Fiber Communications (Prentice-Hall, New York, 1991).
  10. M. Ettenberg, H. S. Sommers, H. Kressel, H. F. Lockwood, “Control of facet damage in GaAs laser diodes,” Appl. Phys. Lett. 18, 571–573 (1971).
    [CrossRef]
  11. T. D. Milster, E. P. Walker, “Figures of merit for laser beam quality,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1834, 79–85 (1993).
  12. P. Karioja, K. Tukkiniemi, V. Heikkinen, I. Kaisto, “Inexpensive packaging techniques of fiber pigtailed laser diodes,” in Processing and Packaging of Semiconductor Lasers and Optoelectronic Devices, H. Temkin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1851, 48–53 (1993).
  13. W. P. Risk, IBM Almaden Research Center, 650 Harry Rd. San Jose, Calif. 95120-6099 (personal communication, 1994).
  14. M. Wale, M. Goodwin, “Flip-chip bonding optimizes opto-ICs,” IEEE Circuits Devices 8, 25–31 (1992).
    [CrossRef]
  15. C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
    [CrossRef]
  16. M. Wale, C. Edge, “Self-aligned flip-chip assembly of photonic devices with electrical and optical connections,” IEEE Trans. Components, Hybrids, Manuf. Technol. 13, 780–786 (1990).
    [CrossRef]
  17. C. Edge, R. M. Ash, C. G. Jones, M. J. Goodwin, “Flip-chip solder bond mounting of laser diodes,” Electron. Lett. 27, 499–500(1991).
    [CrossRef]

1992 (1)

M. Wale, M. Goodwin, “Flip-chip bonding optimizes opto-ICs,” IEEE Circuits Devices 8, 25–31 (1992).
[CrossRef]

1991 (2)

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

C. Edge, R. M. Ash, C. G. Jones, M. J. Goodwin, “Flip-chip solder bond mounting of laser diodes,” Electron. Lett. 27, 499–500(1991).
[CrossRef]

1990 (1)

M. Wale, C. Edge, “Self-aligned flip-chip assembly of photonic devices with electrical and optical connections,” IEEE Trans. Components, Hybrids, Manuf. Technol. 13, 780–786 (1990).
[CrossRef]

1984 (1)

1983 (1)

W. L. Emkey, “Optical coupling between single-mode semiconductor lasers and strip waveguides,” J. Lightwave Technol. 1, 436–443 (1983).
[CrossRef]

1979 (1)

1971 (1)

M. Ettenberg, H. S. Sommers, H. Kressel, H. F. Lockwood, “Control of facet damage in GaAs laser diodes,” Appl. Phys. Lett. 18, 571–573 (1971).
[CrossRef]

Armiento, C. A.

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

Ash, R. M.

C. Edge, R. M. Ash, C. G. Jones, M. J. Goodwin, “Flip-chip solder bond mounting of laser diodes,” Electron. Lett. 27, 499–500(1991).
[CrossRef]

Barry, V.

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

Edge, C.

C. Edge, R. M. Ash, C. G. Jones, M. J. Goodwin, “Flip-chip solder bond mounting of laser diodes,” Electron. Lett. 27, 499–500(1991).
[CrossRef]

M. Wale, C. Edge, “Self-aligned flip-chip assembly of photonic devices with electrical and optical connections,” IEEE Trans. Components, Hybrids, Manuf. Technol. 13, 780–786 (1990).
[CrossRef]

Emkey, W. L.

W. L. Emkey, “Optical coupling between single-mode semiconductor lasers and strip waveguides,” J. Lightwave Technol. 1, 436–443 (1983).
[CrossRef]

Ettenberg, M.

M. Ettenberg, H. S. Sommers, H. Kressel, H. F. Lockwood, “Control of facet damage in GaAs laser diodes,” Appl. Phys. Lett. 18, 571–573 (1971).
[CrossRef]

Fitzgerald, T. W.

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

Goodwin, M.

M. Wale, M. Goodwin, “Flip-chip bonding optimizes opto-ICs,” IEEE Circuits Devices 8, 25–31 (1992).
[CrossRef]

Goodwin, M. J.

C. Edge, R. M. Ash, C. G. Jones, M. J. Goodwin, “Flip-chip solder bond mounting of laser diodes,” Electron. Lett. 27, 499–500(1991).
[CrossRef]

Hall, D. G.

Haugsjaa, P. O.

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

Heikkinen, V.

P. Karioja, K. Tukkiniemi, V. Heikkinen, I. Kaisto, “Inexpensive packaging techniques of fiber pigtailed laser diodes,” in Processing and Packaging of Semiconductor Lasers and Optoelectronic Devices, H. Temkin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1851, 48–53 (1993).

Jagannath, C.

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

Jones, C. G.

C. Edge, R. M. Ash, C. G. Jones, M. J. Goodwin, “Flip-chip solder bond mounting of laser diodes,” Electron. Lett. 27, 499–500(1991).
[CrossRef]

Joyce, W. B.

Kaisto, I.

P. Karioja, K. Tukkiniemi, V. Heikkinen, I. Kaisto, “Inexpensive packaging techniques of fiber pigtailed laser diodes,” in Processing and Packaging of Semiconductor Lasers and Optoelectronic Devices, H. Temkin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1851, 48–53 (1993).

Karioja, P.

P. Karioja, K. Tukkiniemi, V. Heikkinen, I. Kaisto, “Inexpensive packaging techniques of fiber pigtailed laser diodes,” in Processing and Packaging of Semiconductor Lasers and Optoelectronic Devices, H. Temkin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1851, 48–53 (1993).

Kogelnik, H.

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Microwave Research Institute Symposia Series 14 (Polytechnic Institute of Brooklyn, New York, 1964), pp. 333–347.

Kressel, H.

M. Ettenberg, H. S. Sommers, H. Kressel, H. F. Lockwood, “Control of facet damage in GaAs laser diodes,” Appl. Phys. Lett. 18, 571–573 (1971).
[CrossRef]

H. Kressel, “Semiconductor laser devices,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, pp. 441–495.

Lockwood, H. F.

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

M. Ettenberg, H. S. Sommers, H. Kressel, H. F. Lockwood, “Control of facet damage in GaAs laser diodes,” Appl. Phys. Lett. 18, 571–573 (1971).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Hilger, London, 1986), Chap. 3, p. 71.

Major, J.

J. Major, SDL Inc., 80 Rose Orchard Way, San Jose, Calif. 95134 (personal communication, 1994).

Milster, T. D.

T. D. Milster, E. P. Walker, “Figures of merit for laser beam quality,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1834, 79–85 (1993).

Negri, A.

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

Rice, R. R.

Risk, W. P.

W. P. Risk, IBM Almaden Research Center, 650 Harry Rd. San Jose, Calif. 95120-6099 (personal communication, 1994).

Rothman, M.

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

Shieh, C. L.

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

Sommers, H. S.

M. Ettenberg, H. S. Sommers, H. Kressel, H. F. Lockwood, “Control of facet damage in GaAs laser diodes,” Appl. Phys. Lett. 18, 571–573 (1971).
[CrossRef]

Tabasky, M.

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

Tukkiniemi, K.

P. Karioja, K. Tukkiniemi, V. Heikkinen, I. Kaisto, “Inexpensive packaging techniques of fiber pigtailed laser diodes,” in Processing and Packaging of Semiconductor Lasers and Optoelectronic Devices, H. Temkin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1851, 48–53 (1993).

van der Plaats, J.

W. van Etten, J. van der Plaats, Fundamentals of Optical Fiber Communications (Prentice-Hall, New York, 1991).

van Etten, W.

W. van Etten, J. van der Plaats, Fundamentals of Optical Fiber Communications (Prentice-Hall, New York, 1991).

Wale, M.

M. Wale, M. Goodwin, “Flip-chip bonding optimizes opto-ICs,” IEEE Circuits Devices 8, 25–31 (1992).
[CrossRef]

M. Wale, C. Edge, “Self-aligned flip-chip assembly of photonic devices with electrical and optical connections,” IEEE Trans. Components, Hybrids, Manuf. Technol. 13, 780–786 (1990).
[CrossRef]

Walker, E. P.

T. D. Milster, E. P. Walker, “Figures of merit for laser beam quality,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1834, 79–85 (1993).

Yariv, A.

A. Yariv, Optical Electronics, 4th ed. (Holt, Rinehart & Winston, New York, 1991), Chap. 2, p. 46.

Zino, J. D.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. Ettenberg, H. S. Sommers, H. Kressel, H. F. Lockwood, “Control of facet damage in GaAs laser diodes,” Appl. Phys. Lett. 18, 571–573 (1971).
[CrossRef]

Electron. Lett. (2)

C. A. Armiento, M. Tabasky, C. Jagannath, T. W. Fitzgerald, C. L. Shieh, V. Barry, M. Rothman, A. Negri, P. O. Haugsjaa, H. F. Lockwood, “Passive coupling of InGaAsP/lnP laser array and singlemode fibres using silicon waferboard,” Electron. Lett. 27, 1109–1111 (1991).
[CrossRef]

C. Edge, R. M. Ash, C. G. Jones, M. J. Goodwin, “Flip-chip solder bond mounting of laser diodes,” Electron. Lett. 27, 499–500(1991).
[CrossRef]

IEEE Circuits Devices (1)

M. Wale, M. Goodwin, “Flip-chip bonding optimizes opto-ICs,” IEEE Circuits Devices 8, 25–31 (1992).
[CrossRef]

IEEE Trans. Components, Hybrids, Manuf. Technol. (1)

M. Wale, C. Edge, “Self-aligned flip-chip assembly of photonic devices with electrical and optical connections,” IEEE Trans. Components, Hybrids, Manuf. Technol. 13, 780–786 (1990).
[CrossRef]

J. Lightwave Technol. (1)

W. L. Emkey, “Optical coupling between single-mode semiconductor lasers and strip waveguides,” J. Lightwave Technol. 1, 436–443 (1983).
[CrossRef]

Opt. Lett. (1)

Other (9)

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Microwave Research Institute Symposia Series 14 (Polytechnic Institute of Brooklyn, New York, 1964), pp. 333–347.

A. Yariv, Optical Electronics, 4th ed. (Holt, Rinehart & Winston, New York, 1991), Chap. 2, p. 46.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Hilger, London, 1986), Chap. 3, p. 71.

J. Major, SDL Inc., 80 Rose Orchard Way, San Jose, Calif. 95134 (personal communication, 1994).

H. Kressel, “Semiconductor laser devices,” in Laser Handbook, F. T. Arecchi, E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 1, pp. 441–495.

W. van Etten, J. van der Plaats, Fundamentals of Optical Fiber Communications (Prentice-Hall, New York, 1991).

T. D. Milster, E. P. Walker, “Figures of merit for laser beam quality,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1834, 79–85 (1993).

P. Karioja, K. Tukkiniemi, V. Heikkinen, I. Kaisto, “Inexpensive packaging techniques of fiber pigtailed laser diodes,” in Processing and Packaging of Semiconductor Lasers and Optoelectronic Devices, H. Temkin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1851, 48–53 (1993).

W. P. Risk, IBM Almaden Research Center, 650 Harry Rd. San Jose, Calif. 95120-6099 (personal communication, 1994).

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Figures (17)

Fig. 1
Fig. 1

Field overlap geometry.

Fig. 2
Fig. 2

Power coupled to a single-mode fiber given by the Gaussian beam-overlap model (dashed curve) and the butt-coupling model (solid curve).

Fig. 3
Fig. 3

Laser output power versus the front-facet power-reflection coefficient.

Fig. 4
Fig. 4

Optical output power versus separation between the laser and the fiber facets.

Fig. 5
Fig. 5

étalon reflection, étalon transmission, and external quantum efficiency of the laser versus separation between the laser and the fiber facets.

Fig. 6
Fig. 6

Threshold current versus separation between the laser and the fiber facets.

Fig. 7
Fig. 7

Measured coupled power (solid curve with asterisks), maximum and minimum coupled power given by the butt-coupling model (solid curves), and coupled power given by the Gaussian beam-overlap model (dashed curve) versus parallel misalignment between the laser and the fiber.

Fig. 8
Fig. 8

Measured coupled power (solid curve with asterisks), maximum and minimum coupled power given by the butt-coupling model (solid curves), and coupled power given by the Gaussian beam-overlap model (dashed curve) versus perpendicular misalignment between the laser and the fiber.

Fig. 9
Fig. 9

Measured (solid curve) and calculated (dashed curve) butt-coupled power versus separation between the laser and the fiber.

Fig. 10
Fig. 10

Measured (solid curve) and calculated (dashed curve) butt-coupled power versus separation between the laser and the fiber.

Fig. 11
Fig. 11

Measured (solid curve) and calculated (dashed curve) butt-coupled power versus separation between the laser and the fiber.

Fig. 12
Fig. 12

Calculated coupling loss versus transverse and longitudinal misalignment (AR coating).

Fig. 13
Fig. 13

Calculated coupling loss versus angular and longitudinal misalignment (AR coating).

Fig. 14
Fig. 14

Calculated coupling loss versus transverse and longitudinal misalignment (no AR coating).

Fig. 15
Fig. 15

Calculated coupling loss versus angular and longitudinal misalignment (no AR coating).

Fig. 16
Fig. 16

Calculated coupling loss versus transverse and longitudinal misalignment (index matching, no AR coating).

Fig. 17
Fig. 17

Calculated coupling loss versus angular and longitudinal misalignment (index matching, no AR coating).

Tables (1)

Tables Icon

Table 1 Parameters Used for the Calculated Graphs in Figs. 9 11

Equations (23)

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ψ ( x , z ) = A ( z ) exp { [ x / w ( z ) ] 2 } exp [ i k x 2 / 2 R ( z ) ] × exp [ i ( k z Φ ) ] .
w ( z ) = w 0 [ 1 + ( 2 z / k w 0 2 ) 2 ] 1 / 2 .
R ( z ) = z [ 1 + ( k w 0 2 / 2 z ) 2 ] .
ψ ̂ ( x , z ) = ( 2 / π ) 1 / 4 w ( z ) 1 / 2 exp { [ x / w ( z ) ] 2 } × exp [ i k x 2 / 2 R ( z ) ] .
ψ ̂ ( x , z ) = ( 2 / π ) 1 / 4 w ( z ) 1 / 2 exp { [ ( x Δ ) / w ( z ) ] 2 } × exp [ i k ( x Δ ) 2 / 2 R ( z ) ] exp [ i k θ ( x Δ ) ] .
Γ = τ τ .
τ = | ψ ̂ L ( x , z ) ψ ̂ G * ( x , z ) d x | 2 .
τ = | ( 2 / w L w G ) ( 1 / a ) exp [ 2 ( b 2 / a c ) ] | ,
r = r L + t L t L r G Γ L ( 2 d ) 1 / 2 exp ( i k 2 d ) + t L t L r L r G 2 Γ L ( 4 d ) 1 / 2 × exp ( i k 4 d ) + ,
r = R L 1 / 2 ( 1 R L ) / R L 1 / 2 p = 1 Γ L ( 2 p d ) 1 / 2 × [ ( R L R G ) 1 / 2 exp ( i k 2 d ) ] p .
R = r r * .
t = t L t G Γ L G ( d ) 1 / 2 + t L t G r L r G Γ L G ( 3 d ) 1 / 2 × exp ( i k 2 d ) + t L t G ( r L r G ) 2 Γ L G ( 5 d ) 1 / 2 × exp ( i k 4 d ) + ,
t = t L t G Γ G L ( d ) 1 / 2 + t L t G r L r G Γ G L ( 3 d ) 1 / 2 exp ( i k 2 d ) + t L t G ( r L r G ) 2 Γ G L ( 5 d ) 1 / 2 exp ( i k 4 d ) + ,
T = t t * .
T = ( 1 R L ) ( 1 R G ) p = 1 Γ L G [ ( 2 p 1 ) d ] 1 / 2 × [ ( R L R G ) 1 / 2 exp ( i k 2 d ) ] p 1 × p = 1 Γ L G [ ( 2 p 1 ) d ] 1 / 2 [ ( R L R G ) 1 / 2 exp ( i k 2 d ) ] p 1
R = [ ( n 0 Y ) / ( n 0 + Y ) ] 2 ,
I th = C th { α L + ln [ ( R back R front ) 1 / 2 ] } ,
P total = ( h c / e λ ) q ext I th ( I / I th 1 ) ,
q ext = ( e λ / h c ) d P / d I .
q int = q ext { α L + ln [ ( R back R front ) 1 / 2 ] } / [ ln ( R back R front ) 1 / 2 ] .
r = P back / P front = ( R front / R back ) 1 / 2 ( 1 R back ) / ( 1 R front ) .
P front = ( 1 + r ) 1 P total ,
P back = r ( 1 + r ) 1 P total .

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