Abstract

The Beer–Lambert–Bouguer absorption law, known as Beer’s law for absorption in an optical medium, is precise only at power densities lower than a few kW. At higher power densities this law fails because it neglects the processes of stimulated emission and spontaneous emission. In previous models that considered those processes, an analytical expression for the absorption law could not be obtained. We show here that by utilizing the Lambert W-function, the two-level energy rate equation model is solved analytically, and this leads into a general absorption law that is exact because it accounts for absorption as well as stimulated and spontaneous emission. The general absorption law reduces to Beer’s law at low power densities. A criterion for its application is given along with experimental examples.

© 2008 Optical Society of America

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References

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  1. Wikipedia, “Beer-Lambert law,” http://en.wikipedia.org/wiki/Beer-Lambert_law.
  2. W. Koechner, “Optical Pump Systems,” in Solid-State Laser Engineering (Springer, 1995), pp. 308-326.
  3. T. Y. Fan, A. Sanchez, and W. E. Defeo, “Scalable, end-pumped, diode-laser-pumped laser,” Opt. Lett. 14, 1057-1059(1989).
    [CrossRef] [PubMed]
  4. O. Axner, F. M. Schmidt, A. Foltynowicz, J. Gustafsson, N. Omenetto, and J. D. Winefordner, “Absorption spectrometry by narrowband light in optically saturated and optically pumped collision and Doppler broadened gaseous media under arbitrary optical thickness conditions,” Appl. Spectrosc. 60, 1217-1251 (2006).
    [CrossRef] [PubMed]
  5. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329-359 (1996).
    [CrossRef]
  6. A. Sennaroglu “Continuous wave thermal loading in saturable absorbers: theory and experiment,” Appl. Opt. 36, 9528-9535(1997).
    [CrossRef]
  7. G. Xiao and M. Bass, “A generalized model for passively Q-switched laser including excited state absorption in the saturable absorber,” IEEE J. Quantum Electron. 33, 41-44 (1997).
    [CrossRef]
  8. B. E. A. Saleh and M. C. Teich, “Laser amplifiers,” in Fundamentals of Photonics (Wiley-Interscience, 1991), pp. 482-484.

2006 (1)

1997 (2)

A. Sennaroglu “Continuous wave thermal loading in saturable absorbers: theory and experiment,” Appl. Opt. 36, 9528-9535(1997).
[CrossRef]

G. Xiao and M. Bass, “A generalized model for passively Q-switched laser including excited state absorption in the saturable absorber,” IEEE J. Quantum Electron. 33, 41-44 (1997).
[CrossRef]

1996 (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329-359 (1996).
[CrossRef]

1995 (1)

W. Koechner, “Optical Pump Systems,” in Solid-State Laser Engineering (Springer, 1995), pp. 308-326.

1991 (1)

B. E. A. Saleh and M. C. Teich, “Laser amplifiers,” in Fundamentals of Photonics (Wiley-Interscience, 1991), pp. 482-484.

1989 (1)

Axner, O.

Bass, M.

G. Xiao and M. Bass, “A generalized model for passively Q-switched laser including excited state absorption in the saturable absorber,” IEEE J. Quantum Electron. 33, 41-44 (1997).
[CrossRef]

Corless, R. M.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329-359 (1996).
[CrossRef]

Defeo, W. E.

Fan, T. Y.

Foltynowicz, A.

Gonnet, G. H.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329-359 (1996).
[CrossRef]

Gustafsson, J.

Hare, D. E. G.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329-359 (1996).
[CrossRef]

Jeffrey, D. J.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329-359 (1996).
[CrossRef]

Knuth, D. E.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329-359 (1996).
[CrossRef]

Koechner, W.

W. Koechner, “Optical Pump Systems,” in Solid-State Laser Engineering (Springer, 1995), pp. 308-326.

Omenetto, N.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, “Laser amplifiers,” in Fundamentals of Photonics (Wiley-Interscience, 1991), pp. 482-484.

Sanchez, A.

Schmidt, F. M.

Sennaroglu, A.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, “Laser amplifiers,” in Fundamentals of Photonics (Wiley-Interscience, 1991), pp. 482-484.

Winefordner, J. D.

Xiao, G.

G. Xiao and M. Bass, “A generalized model for passively Q-switched laser including excited state absorption in the saturable absorber,” IEEE J. Quantum Electron. 33, 41-44 (1997).
[CrossRef]

Adv. Comput. Math. (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329-359 (1996).
[CrossRef]

Appl. Opt. (1)

Appl. Spectrosc. (1)

IEEE J. Quantum Electron. (1)

G. Xiao and M. Bass, “A generalized model for passively Q-switched laser including excited state absorption in the saturable absorber,” IEEE J. Quantum Electron. 33, 41-44 (1997).
[CrossRef]

Opt. Lett. (1)

Other (3)

Wikipedia, “Beer-Lambert law,” http://en.wikipedia.org/wiki/Beer-Lambert_law.

W. Koechner, “Optical Pump Systems,” in Solid-State Laser Engineering (Springer, 1995), pp. 308-326.

B. E. A. Saleh and M. C. Teich, “Laser amplifiers,” in Fundamentals of Photonics (Wiley-Interscience, 1991), pp. 482-484.

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Figures (4)

Fig. 1
Fig. 1

A photon flux ϕ is impinging on a thin slab of thickness d z and area A. The density of the atoms is given by N t , therefore, there are N t d z A atoms in the thin slab. If the effective area of an atom is σ, the probability of a collision (i.e., absorption) is σ N t A d z / A = σ N t d z = α d z . Therefore the attenuation of the photon flux ϕ would be αϕ d z . This model neglects the effect of stimulated emission and spontaneous emission.

Fig. 2
Fig. 2

Diagram that represents a system (an atom or a molecule) with two energy levels E 0 , E 1 : in a unit volume there would be N 0 atoms in energy state E 0 and N 1 atoms in energy state E 1 . There are three basic transitions between the energy levels of the system: absorption (upward arrow), stimulated emission (downward arrow), and spontaneous emission (dashed arrow).

Fig. 3
Fig. 3

An Nd : YVO 4 laser ( 1064 nm , 0.5 mm waist) was focused by a 40 mm lens into a 20 μm waist inside a 1 mm thick Cr + 4 : YAG saturable absorber crystal. The Cr + 4 : YAG has 84% low signal transmission, and it is coated on both sides for antireflection ( < 0.1 % at 1064 nm ). A Melles Griot powermeter (13PEM001) was used to measure the laser power at the entrance and exit surfaces of the Cr + 4 : YAG , and a beam profiler (Thorlabs BP109-IR) was used to measure the waist at the location of the Cr + 4 : YAG crystal. The waist changed slightly with increasing power of the Nd : YVO 4 laser due to thermal lensing in that laser. This affects the power density and therefore introduced some inaccuracy to the measurements ( < 10 % ).

Fig. 4
Fig. 4

Transmission of an Nd : YVO 4 laser ( 20 μm beam waist) in a 1 mm Cr + 4 : YAG saturable absorber according to Beer’s law (dashed line), Lambert W-function (continuous line), and the experimental measurements. The Lambert W-function accounts satisfactorily for the saturation of the Cr + 4 : YAG medium. The required parameters for the plots were taken from Refs. [6, 7]. The difference between the experimental results and the Lambert W-function is due to experimental errors (see text) and excited state absorption in Cr + 4 : YAG .

Equations (15)

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d ϕ d z = α ϕ .
ϕ ( z ) = ϕ ( 0 ) e α z .
d N 0 d t = σ ϕ N 0 + σ ϕ N 1 + τ 1 N 1 ,
d N 1 d t = σ ϕ N 0 σ ϕ N 1 τ 1 N 1 .
N 0 + N 1 = N t .
σ ϕ N 0 + ( σ ϕ + τ 1 ) N 1 = 0.
( σ ϕ σ ϕ + τ 1 1 1 ) ( N 0 N 1 ) = ( 0 N t ) .
d ϕ d z = σ ϕ ( N 1 N 0 ) .
d ϕ d z = τ 1 N t σ ϕ τ 1 + 2 σ ϕ .
2 σ τ ϕ + d ϕ ϕ = N t σ d z 2 σ τ ϕ ( z ) 2 σ τ ϕ ( 0 ) + ln [ ϕ ( z ) ϕ ( 0 ) ] = N t σ z exp [ 2 σ τ ϕ ( z ) ] 2 σ τ ϕ ( z ) = exp [ 2 σ τ ϕ ( 0 ) ] 2 σ τ ϕ ( 0 ) exp [ N t σ z ] .
ϕ ( z ) = 1 2 σ τ W ( 2 σ τ ϕ ( 0 ) exp [ 2 σ τ ϕ ( 0 ) N t σ z ] ) .
d ϕ = N t σ ϕ 1 1 + 2 σ τ ϕ d z .
1 1 + 2 σ τ ϕ 1 2 σ τ ϕ + ( 2 σ τ ϕ ) 2 higher orders in   2 σ τ ϕ .
d ϕ = N t σϕ d z + N t σ ϕ ( 2 σ τ ϕ ) d z higher orders in   2 σ τ ϕ .
2 σ τ ϕ 1.

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