Abstract

Cell surgery based on ultrashort laser pulses is a fast evolving field in biophotonics. Noninvasive intra cellular dissection at sub-diffraction resolution can be performed within vital cells with very little hazardous effects to adjacent cell organelles. Microscope objectives of high numerical aperture (NA) are used to focus ultrashort pulses to a small spot. Due to the high order of nonlinearity, plasma formation and thus material manipulation is limited to the very focus. Nonetheless nonlinear plasma formation is generally accompanied by a number of additional nonlinear effects like self-focusing and filamentation. These parasitic effects limit the achievable precision and reproducibility of applications. Experimentally it is known that the intensity of these effects decreases with increasing NA of the focusing optics, but the process of nonlinear plasma formation at high NA has not been studied numerically in detail yet. To simulate the interaction of ultrashort laser pulses with transparent materials at high NA a novel nonlinear Schrödinger equation is derived; the multiple rate equation (MRE) model is used to simultaneously calculate the generation of free electrons. Nonparaxial and vectorial effects are taken into account to accurately include tight focusing conditions. Parasitic effects are shown to get stronger and increasingly distortive for NA <0.9, using water as a model substance for biological soft tissue and cellular constituents.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. A. Dubietis, A. Couairon, E. Kučinskas, G. Tamošaukas, E. Gaižauskas, D. Faccio, and P. Di Trapani, "Measurement and calculation of nonlinear absorption associated with femtosecond filaments in water," Appl. Phys. B 84, 439-446 (2006).
    [CrossRef]
  3. C. L. Arnold, A. Heisterkamp, W. Ertmer and H. Lubatschowski, "Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses," Appl. Phys. B 80, 247-253 (2005).
    [CrossRef]
  4. C. B. Schaffer, A. O. Jamison, and E. Mazur, "Morphology of femtosecond laser-induced structural changes in bulk transparent materials," Appl. Phys. Lett. 84, 1441 - 1443 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  10. W. Liu, O. Kosareva, L. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl. Phys. B 76, 215-229 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore and M. D. Perry, "Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses," Phys. Rev. Lett. 74, 2248-2251 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  32. A. Couairon, L. Sudrie, M. Franco, B. Prade and A. Mysyrowicz, "Filamentation and damage in fused silica induced by tightly focused femtosecond laser pulses," Phys. Rev. B 71, 125435-1-11 (2005).
    [CrossRef]
  33. A. Kaiser, B. Rethfeld, M. Vicanek and G. Simon, "Microscopic processes in dielectrics under irradiation by subpicosecond laser pulses," Phys. Rev. B. 61, 11437-11450 (2000).
    [CrossRef]
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    [CrossRef]
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2006

A. Dubietis, A. Couairon, E. Kučinskas, G. Tamošaukas, E. Gaižauskas, D. Faccio, and P. Di Trapani, "Measurement and calculation of nonlinear absorption associated with femtosecond filaments in water," Appl. Phys. B 84, 439-446 (2006).
[CrossRef]

B. Rethfeld, "Free-electron generation in laser-irradiated dielectrics," Phys. Rev. B 73, 035101-6 (2006).
[CrossRef]

2005

A. Vogel, J. Noack, G. Hüttman and G. Paltauf, "Mechanisms of femtosecond laser nanosurgery of cells and tissue," Appl. Phys. B 81, 1015-1047 (2005).
[CrossRef]

C. L. Arnold, A. Heisterkamp, W. Ertmer and H. Lubatschowski, "Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses," Appl. Phys. B 80, 247-253 (2005).
[CrossRef]

A. Heisterkamp, I. Z. Maxwell, E. Mazur, J. M. Underwood, J. A. Nickerson, S. Kumar and D. E. Ingber, "Pulse energy dependence of subcellular dissection by femtosecond laser pulses," Opt. Express 13, 3690-3696 (2005).
[CrossRef] [PubMed]

2004

M. F. Yanik, H. Cinar, A. D. Chisholm, Y. Jin, and A. Ben-Yakar, "Neurosurgery: Functional regeneration after laser axotomy," Nature 432, 822 (2004).
[CrossRef] [PubMed]

C. B. Schaffer, A. O. Jamison, and E. Mazur, "Morphology of femtosecond laser-induced structural changes in bulk transparent materials," Appl. Phys. Lett. 84, 1441 - 1443 (2004).
[CrossRef]

Y. M. Engelberg and S. Ruschin, "Fast method for physical optics propagation of high-numerical-aperture beams," J. Opt. Soc. Am. A 21, 2135-2145 (2004).
[CrossRef]

2003

N. T. Nguyen, A. Saliminia, W. Liu, S. L. Chin, and R. Valle, "Optical breakdown versus filamentation in fused silica by use of femtosecond infrared laser pulses," Opt. Lett. 28, 1591-1593 (2003).
[CrossRef] [PubMed]

W. Liu, O. Kosareva, L. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl. Phys. B 76, 215-229 (2003).
[CrossRef]

2002

A. Heisterkamp, T. Ripken, T. Mamom, W. Drommer, H. Welling, W. Ertmer, and H. Lubatschowski, "Nonlinear side effects of fs pulses inside corneal tissue during photodisruption," Appl. Phys. B 74, 419-425 (2002).
[CrossRef]

2001

K. König, I. Riemann, and W. Fritzsche, "Nanodissection of human chromosomes with near-infrared femtosecond laser pulses," Opt. Lett. 26, 819 (2001).
[CrossRef]

G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112-146 (2001).
[CrossRef]

2000

A. Kaiser, B. Rethfeld, M. Vicanek and G. Simon, "Microscopic processes in dielectrics under irradiation by subpicosecond laser pulses," Phys. Rev. B. 61, 11437-11450 (2000).
[CrossRef]

1997

T. Brabec and F. Krausz, "Nonlinear Optical Pulse Propagation in the Single-Cycle Regime," Phys. Rev. Lett. 78,3282-3285 (1997).
[CrossRef]

Q. Feng, J. V. Moloney, A. C. Newell, E. M. Wright, K. Cook, P. K. Kennedy, D. X. Hammer, B. A. Rockwell and C. R. Thomson, "Theory and Simulation on the threshold of water breakdown induced by Focused Ultrashort Laser Pulses," IEEE J. Quantum. Electron. 33, 127-137 (1997).
[CrossRef]

1995

S. Chi and Q. Guo, "Vector theory of self-focusing of an optical beam in Kerr media," Opt. Lett. 20, 1598-1600 (1995).
[CrossRef] [PubMed]

P. K. Kennedy, "A First-Order Model for Computation of Laser-Induced Breakdown Thresholds in Ocular and Aqueous Media: Part I-Theory," IEEE J. Quantum. Electron. 31, 2241-2249 (1995).
[CrossRef]

B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore and M. D. Perry, "Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses," Phys. Rev. Lett. 74, 2248-2251 (1995).
[CrossRef] [PubMed]

1992

1976

F. Williams, S. P. Varma and S. Hillenius, "Liquid water as a lone-pair amorphous semiconductor," J. Chem. Phys. 64, 1549-1554 (1976).
[CrossRef]

1974

N. Bloembergen, "Laser-induced electric breakdown in solids," IEEE J. Quantum. Electron. 10, 375-386 (1974).
[CrossRef]

1969

C. DeMichelis, "Laser induced gas breakdown: A bibliographical review," IEEE J. Quantum. Electron. 5, 188-202 (1969).
[CrossRef]

1965

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307 (1965).

1960

L. V. Keldysh, "Kinetic theory of impact ionization in semiconductors," Sov. Phys. JETP 37, 509-518 (1960).

1959

B. Richards and E. Wolf, "Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System," Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Appl. Phys. B

A. Dubietis, A. Couairon, E. Kučinskas, G. Tamošaukas, E. Gaižauskas, D. Faccio, and P. Di Trapani, "Measurement and calculation of nonlinear absorption associated with femtosecond filaments in water," Appl. Phys. B 84, 439-446 (2006).
[CrossRef]

C. L. Arnold, A. Heisterkamp, W. Ertmer and H. Lubatschowski, "Streak formation as side effect of optical breakdown during processing the bulk of transparent Kerr media with ultra-short laser pulses," Appl. Phys. B 80, 247-253 (2005).
[CrossRef]

A. Heisterkamp, T. Ripken, T. Mamom, W. Drommer, H. Welling, W. Ertmer, and H. Lubatschowski, "Nonlinear side effects of fs pulses inside corneal tissue during photodisruption," Appl. Phys. B 74, 419-425 (2002).
[CrossRef]

W. Liu, O. Kosareva, L. S. Golubtsov, A. Iwasaki, A. Becker, V. P. Kandidov and S. L. Chin, "Femtosecond laser pulse filamentation versus optical breakdown in H2O," Appl. Phys. B 76, 215-229 (2003).
[CrossRef]

A. Vogel, J. Noack, G. Hüttman and G. Paltauf, "Mechanisms of femtosecond laser nanosurgery of cells and tissue," Appl. Phys. B 81, 1015-1047 (2005).
[CrossRef]

Appl. Phys. Lett.

C. B. Schaffer, A. O. Jamison, and E. Mazur, "Morphology of femtosecond laser-induced structural changes in bulk transparent materials," Appl. Phys. Lett. 84, 1441 - 1443 (2004).
[CrossRef]

IEEE J. Quantum. Electron.

Q. Feng, J. V. Moloney, A. C. Newell, E. M. Wright, K. Cook, P. K. Kennedy, D. X. Hammer, B. A. Rockwell and C. R. Thomson, "Theory and Simulation on the threshold of water breakdown induced by Focused Ultrashort Laser Pulses," IEEE J. Quantum. Electron. 33, 127-137 (1997).
[CrossRef]

C. DeMichelis, "Laser induced gas breakdown: A bibliographical review," IEEE J. Quantum. Electron. 5, 188-202 (1969).
[CrossRef]

P. K. Kennedy, "A First-Order Model for Computation of Laser-Induced Breakdown Thresholds in Ocular and Aqueous Media: Part I-Theory," IEEE J. Quantum. Electron. 31, 2241-2249 (1995).
[CrossRef]

N. Bloembergen, "Laser-induced electric breakdown in solids," IEEE J. Quantum. Electron. 10, 375-386 (1974).
[CrossRef]

J. Chem. Phys.

F. Williams, S. P. Varma and S. Hillenius, "Liquid water as a lone-pair amorphous semiconductor," J. Chem. Phys. 64, 1549-1554 (1976).
[CrossRef]

J. Opt. Soc. Am. A

Nature

M. F. Yanik, H. Cinar, A. D. Chisholm, Y. Jin, and A. Ben-Yakar, "Neurosurgery: Functional regeneration after laser axotomy," Nature 432, 822 (2004).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. B

B. Rethfeld, "Free-electron generation in laser-irradiated dielectrics," Phys. Rev. B 73, 035101-6 (2006).
[CrossRef]

Phys. Rev. B.

A. Kaiser, B. Rethfeld, M. Vicanek and G. Simon, "Microscopic processes in dielectrics under irradiation by subpicosecond laser pulses," Phys. Rev. B. 61, 11437-11450 (2000).
[CrossRef]

Phys. Rev. Lett.

T. Brabec and F. Krausz, "Nonlinear Optical Pulse Propagation in the Single-Cycle Regime," Phys. Rev. Lett. 78,3282-3285 (1997).
[CrossRef]

B. C. Stuart, M. D. Feit, A. M. Rubenchik, B. W. Shore and M. D. Perry, "Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses," Phys. Rev. Lett. 74, 2248-2251 (1995).
[CrossRef] [PubMed]

Physica D

G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112-146 (2001).
[CrossRef]

Proc. R. Soc. London, Ser. A

B. Richards and E. Wolf, "Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System," Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Sov. Phys. JETP

L. V. Keldysh, "Ionization in the field of a strong electromagnetic wave," Sov. Phys. JETP 20, 1307 (1965).

L. V. Keldysh, "Kinetic theory of impact ionization in semiconductors," Sov. Phys. JETP 37, 509-518 (1960).

Other

L. Sudrie, A. Couairon, M. Franco, B. Lammouroux, B. Prade, S. Tzortzakis and A. Mysyrowicz, "Femtosecond laser-induced damage and filamentary propagation in fused silica," Phys. Rev. Lett. 89, 186601-1- 4 (2002).
[CrossRef]

A. Couairon, L. Sudrie, M. Franco, B. Prade and A. Mysyrowicz, "Filamentation and damage in fused silica induced by tightly focused femtosecond laser pulses," Phys. Rev. B 71, 125435-1-11 (2005).
[CrossRef]

The International Association for the Properties of Water and Steam, "Release on the Refractive Index of Ordinary Water Substance as a Function of Wavelength, Temperature and Pressure," (1977) http://www.iapws.org/relguide/rindex.pdf.

G. P. Agraval, Nonlinear Fiber Optics, Academic Press, (San Diego, London, Boston, New York, Sydney, Tokyo, Toronto, 1995).

M. Gu, Advanced Optical Imaging Theory, Springer Series in Optical Sciences, (Springer Berlin, Heidelberg, New York 2000).

B. Rethfeld, "Unified model for the free-electron avalanche in laser-irradiated dielectrics," Phys. Rev. Lett. 92, 187401-1-4 (2004).

M. Kolesik, J. V. Moloney and M. Mlejnek, "Unidirectional Optical Pulse Propagation Equation," Phys. Rev. Lett. 89, 283902-1-4 (2002).
[CrossRef]

M. Kolesik and J.V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036604-1-11 (2004).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Initial free electrons are generated by multiphoton or tunnel ionization. Once in the conduction band, electrons can gain energy from the field by sequential one-photon absorption. When their kinetic energy exceeds the critical bandgap potential εcrit , additional free electrons are generated by impact ionization.

Fig. 2.
Fig. 2.

Incident plane waves are transformed into converging spherical waves by the focusing optics. Thereby the linearly x-polarized beam is partially transferred into the z- and y-direction on the reference sphere. A diffraction integral can be performed for each polarization direction to propagate the field component into the focus.

Fig. 3.
Fig. 3.

Focal Fluence distributions for numerical aperture NA=1.2 in water. Homogenous plane wave illumination of the focusing optics and sine-condition as apodization function were assumed. The asymmetry along the x- and y-axis is clearly observable. At these focusing conditions about 20.4% of the total power are z-polarized. The plots are normalized to the maximum fluence at the origin, the color scaling is logarithmic and the contour lines are at the levels: 0.9,0.7,0.5,0.3,0.1,0.05,0.01,0.005,0.001.

Fig. 4.
Fig. 4.

Contour plots for generated breakdown plasmas in zx-view at focusing conditions of various numerical apertures. Pulses were incident from the left. The white areas correspond to a plasma density exceeding the breakdown density ρBd=1021 cm-3. The contour lines are at the levels ρ=(1018,1019,1020,5×1020,1021)cm-3. The size of the numerical simulation window scales with the NA. Areas where no simulation was performed are hatched.

Fig. 5.
Fig. 5.

Threshold energy for different NAs obtained from the simulations in comparison to the assumption of constant fluence in the focal plane. The points in the graph are connected by lines as a guide to the eye. Since the simulations are very time-consuming, there is an uncertainty of about 5% in the plotted breakdown thresholds.

Fig. 6.
Fig. 6.

The left graph shows the normalized pulse energy as the pulse moves through the focus for different NAs at the threshold for optical breakdown. The right displays the relative absorption by the generation of a breakdown plasma as a function of the numerical aperture.

Equations (36)

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

2 E ( x , y , z ) + 1 ε 0 n 2 ( P NL ( x , y , z ) ) + k 2 E ( x , y , z ) + ω 2 c 2 P NL ( x , y , z ) ε 0 = 0
P NL ( E ) = P Kerr ( E ) + P fe ( ρ , E )
P Kerr ( E ) = c ε 0 2 n 2 n 2 1 + γ [ ( E E * ) E + γ ( E E ) E * ]
P fe ( ρ , E ) = ε 0 χ fe ( ρ ) E
χ fe ( ρ ) = ε 0 ( n 2 ω P 2 τ P 2 1 + ω 2 τ P 2 + i n 2 ω P 2 τ P ω ( 1 + ω 2 τ P 2 ) ω ε crit ( 2 1 K 1 ) 1 + i n Δ c ω W PI I )
E = 1 k 0 x 0 c ε 0 n 2 A e i k 0 z , x ˜ = x x 0 , y ˜ = y x 0 ,
˜ T 2 = x ˜ 2 + y ˜ 2 , z ˜ = z 2 L Df , L Df = k 0 x 0 2 , f = 1 k 0 x 0 ,
P ˜ Kerr = 2 3 [ ( A A * ) A + 1 2 ( A A ) A * ] , P ˜ fe = 1 f 2 χ fe n 0 2 A , P ˜ NL = P ˜ Kerr + P ˜ fe
z ˜ A x = i ˜ T 2 A x + i x 0 2 ( k 2 k 0 2 ) A x + i k 2 k 0 2 ( P ˜ NI ) x + i f 2 4 z ˜ 2 A x
+ i f 2 x 2 ( P ˜ NI ) x + i f 2 x ˜ y ˜ ( P ˜ NI ) y + i f 3 2 x ˜ z ˜ ( P ˜ NI ) z f x ˜ ( P ˜ NI ) z
z ˜ A y = i ˜ T 2 A y + i x 0 2 ( k 2 k 0 2 ) A y + i k 2 k 0 2 ( P ˜ NI ) y + i f 2 4 z ˜ 2 A y
+ i f 2 x ˜ y ˜ ( P ˜ NI ) x + i f 2 y 2 ( P ˜ NI ) y + i f 3 2 y ˜ z ˜ ( P ˜ NI ) z f y ˜ ( P ˜ NI ) z
z ˜ A z = i ˜ T 2 A z + i x 0 2 ( k 2 k 0 2 ) A z + i f 2 4 z ˜ 2 A z f ( x ˜ ( P ˜ NI ) x + y ˜ ( P ˜ NI ) y )
f 2 z ˜ ( P ˜ NI ) z + i f 3 2 ( x ˜ z ˜ ( P ˜ NI ) x + y ˜ z ˜ ( P ˜ NI ) y ) + i f 4 4 z ˜ 4 ( P ˜ NI ) z
A x = O ( 1 ) A y = O ( f 2 ) A z = i f x ˜ A x + O ( f 3 )
z ˜ A x = i ˜ T 2 A x + i x 0 2 ( k 2 k 0 2 ) A x + i k 2 k 0 2 ( P ˜ NI ) x Nonlinearity + i f 2 4 z ˜ 2 A x Nonparaxiality
f x ˜ ( P ˜ NI ) z Vector Coupling + i f 2 x 2 ( P ˜ NI ) x Nonlinear Diffraction + O ( f 3 )
( z ˜ + 2 i k 0 2 x 0 2 ) 2 A x = 4 k 0 2 x 0 4 ( k 2 + ˜ T 2 x 0 2 + k 2 k 0 2 x 0 2 A x 2 + k 2 χ fe n 2 ) A x
z ˜ A x = i 2 L Df ( k 0 + k 2 + ˜ T 2 x 0 2 + f 2 k 2 A x 2 + k 2 χ fe n 2 ) A x
k 2 + ˜ T 2 x 0 2 FFT k 2 k x 2 k y 2
z ˜ A x = i 2 L Df ( k 0 + k ) A x + i ( 1 + i t ω 0 ) 1 ˜ T 2 A x + i ( 1 + i t ω 0 ) ( A x 2 + f 2 χ fe n 0 2 ) A x
z ˜ A x = i 2 L Df ( k 0 + k 2 + ˜ T 2 x 0 2 ) A x + i 2 L Df k ( 1 + 1 + f 2 A x 2 + χ fe n 2 ) A x
i f 2 4 k 0 k [ ˜ T 2 ( A x 2 + f 2 χ fe n 2 ) + ( A x 2 + f 2 χ fe n 2 ) ˜ T 2 ] A x
+ i k 2 k 0 2 ( 2 3 A z 2 A x + 1 3 A z 2 A x * ) + i f 2 x ˜ 2 ( A x 2 + f 2 χ fe n 2 ) A x
f x ˜ ( 2 3 A x 2 A z + 1 3 A x 2 A z * + f 2 χ fe n 2 A z ) + O ( f 3 )
z ˜ A x = i 2 L Df ( k 0 + k 2 + ˜ T 2 x 0 2 ) A x i 2 L D f v g t A x
+ i 2 L Df k 0 ( 1 + 1 + f 2 A x 2 + χ fe n 0 2 ) A x
i f 2 4 [ ( ˜ T 2 4 x ˜ 2 ) ( A x 2 + f 2 χ fe n 0 2 ) + ( A x 2 + f 2 χ fe n 0 2 ) ˜ T 2 ] A x
i f 2 x ˜ ( f 2 χ fe n 0 2 x ˜ A x ) + O ( f 3 )
ρ 0 ( t ) t = W PI ( I ( t ) ) W 1 p t ( I ( t ) ) ρ 0 ( t ) + 2 W 1 pt ( I ( t ) ) ρ K 1 ( t )
ρ 1 ( t ) t = W 1 pt ( I ( t ) ) ρ 0 ( t ) W 1 pt ( I ( t ) ) ρ 1 ( t )
ρ K 1 ( t ) t = W 1 pt ( I ( t ) ) ρ K 2 ( t ) W 1 pt ( I ( t ) ) ρ K 1 ( t )
W 1 pt ( I ) = η I
η = σ ε crit ( 2 1 K 1 ) 1
ε crit = ( 1 + m e * m VB ) ( Δ + ε osc )

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