December 15, 2014 / Vol. 39, No. 24 / OPTICS LETTERS

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Effect of multiphoton ionization on performance of crystalline lens Pradeep Kumar Gupta,1,* Ram Kishor Singh,1 D. Strickland,2 M. C. W. Campbell,2 and R. P. Sharma1 1

2

Centre for Energy Studies, Indian Institute of Technology, Delhi 110016, India Department of Physics and Astronomy, Guelph-Waterloo Physics Institute, University of Waterloo, Waterloo, Ontario N2 L 3G1, Canada *Corresponding author: [email protected] Received September 15, 2014; accepted October 24, 2014; posted October 31, 2014 (Doc. ID 222909); published December 3, 2014

This Letter presents a model for propagation of a laser pulse in a human crystalline lens. The model contains a transverse beam diffraction effect, laser-induced optical breakdown for the creation of plasma via a multiphoton ionization process, and the gradient index (GRIN) structure. Plasma introduces the nonlinearity in the crystalline lens which affects the propagation of the beam. The multiphoton ionization process generates plasma that changes the refractive index and hence leads to the defocusing of the laser beam. The Letter also points out the relevance of the present investigation to cavitation bubble formation for restoring the elasticity of the eyes. © 2014 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (190.1900) Diagnostic applications of nonlinear optics; (190.4180) Multiphoton processes; (350.5400) Plasmas; (350.7420) Waves. http://dx.doi.org/10.1364/OL.39.006775

Femtosecond laser pulses are widely used in medical science because they make noninvasive, intraocular microsurgery possible [1,2]. It is expected that the surgical effect occurs due to optical breakdown and leads to the formation of plasma in the focal volume at the focusing point [3]. Microscopic cavitation bubbles are generated due to laser plasma interactions at the focusing point [3,4]. Some studies have suggested that the formation of cavitation bubbles can restore the elasticity of a crystalline lens and hence can be used to treat presbyopia, which is an age-related loss of the accommodation power of a crystalline lens [5,6]. A study of femtosecond laser optical breakdown, with a proper safety factor, has been made, and the outcome of plasma formation on the crystalline lenses of animals was studied. A study of the eye lenses of pigs found that optical breakdown in a characteristic pattern can be used to restore the elasticity of the eye lens [7]. The effects of plasma formation in surgery by short pulse lasers, and the evaluation of plasma dimension and its dependence on laser pulse energy also has been investigated [2,8]. Creation of plasma in the crystalline lens is a very crucial phenomenon [9,10]. Experiments have verified that optical breakdown occurs in an identical fashion both in water and ocular media because of high water content in ocular media or a crystalline lens [11]. Keldysh [12] has derived the theoretical expression for probability of ionization for gases and solids by a strong field of electromagnetic wave and introduced a parameter, γ  ω∕ωt , that separates the tunneling and multiphoton ionization regime. In this case, ω is the frequency of the electromagnetic wave and 1∕ωt is tunneling time through atomic potential barrier, which is inversely proportional to strength of the electromagnetic wave. When γ ≫ 1 (for typical frequency and moderate strength of field of electromagnetic wave) the probability of multiphoton ionization is much higher than tunneling; however, for γ ≪ 1, tunneling ionization will dominate [13]. To study the phenomenon of optical breakdown, water can be treated as an amorphous solid [14]. Therefore, the 0146-9592/14/246775-04$15.00/0

process by which the plasma forms in the crystalline lens depends upon the frequency and the field strength of the laser pulse used for surgery. Besides, this plasma density can increase due to the cascade ionization process, but the contribution of the cascade ionization process is small in the femtosecond pulse duration [15]. In general, moderate field strength is used via very short pulse duration of laser pulse for laser eye surgery [2,16,17]; therefore, the multiphoton ionization process is necessary phenomenon for plasma formation in the crystalline lens for femtosecond pulse duration [17,18]. In particular, the multiphoton ionization process is a nonlinear absorption process in which transition of an electron from bound state to free state is accompanied by simultaneous absorption of several photons [15,19–21]. In the presence of a high electromagnetic field, the energy levels become lower than the zero field stable energy levels. So, at a high energy laser pulse, an electron can tunnel through the potential barrier leading to tunneling ionization. Electrons have high energy in newly formed plasma so temperatures of these electrons also are very high. Electrons transfer their energy to the ions and molecules via inelastic collisions. Therefore, after a characteristic time, a uniform temperature is set up within the plasma volume. This characteristic time is of the order of picoseconds [22]. This gives rise to a high temperature on the order of thousands of Kelvin and generates a high pressure up to gigapascal [22]. This high pressure and temperature is enough to create a supercritical state of water inside the plasma that leads to the explosive expansion of plasma. As a result, the temperature and pressure falls adiabatically [23]. Because of this pressure drop, the supercritical water turns into steam and a phase boundary called a “cavitation bubble” develops between the steam and plasma fluid [24]. Recently, the self-focusing phenomenon in the eye lens that has a gradient index (GRIN) profile has been studied [25]. But plasma formation changes the propagation properties of laser radiations in several ways. First, the density of generated electrons is maximum along the axis © 2014 Optical Society of America

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where the intensity of laser radiation is maximum, which causes the laser beam to defocus. Second, the laser energy is reduced by an amount equal to the amount of energy absorbed by the molecules in a crystalline lens to generate electron plasma via multiphoton absorption. The effect of varying preformed plasma parameters with typical lens and laser parameters on beam dynamics has been analyzed [26]. Plasma density will depend on the time duration of laser pulse so it also will change the propagation of laser pulse with time evolution. In this study, we propose a model for laser pulse propagation, under graded index (GRIN) profile along with the time evolution of plasma density. The nonlinear model equations have been obtained by using a fluid model of plasma. We have solved the model equations using the paraxial ray approximation. The results are discussed later in this Letter. The multiphoton ionization of water gives rise to a changing plasma frequency ωp [27], ∂ω2p  Pω2pm − ω2p ; ∂t where P

βK I K Kℏωnpm

ω2p ⃗ 1 ∂2 D⃗ E;  c2 c2 ∂t2

⃗ Ex; y; z; t  E⃗ 0 x; y; z; teiωt−kz :

2

(6)

So, the wave equation of pulse propagation during the creation of plasma is given by 2ik

∂E 0 2iωε ∂E 0 − ∇2T E 0  2 ∂z ∂t c 2 2 ωp r ω2  22 E⃗ 0 2  2 ε00 g2c r 2 E 0  0; c r0 c

(7)

where k2  ω2 ε00 ∕c2 1 − ω2p0 ∕ω2 ε00   ω2 ε0p ∕c2 . For the slowly converging or diverging behavior of the laser beam, the scalar amplitude of the electric field can be written as E 0  Ax; y; z; te−ikSx;y;z;t ;

(1)

is the rate of multiphoton ionization by the laser of intensity I and βK is the nonlinear coefficient for K-photon absorption. The order of multiphoton ionization is evaluated from K  modE g ∕ℏω  1, which is defined as the minimum number of photons of energy ℏω required to overcome E g (i.e., the ionization energy needed to knock out an electron from water molecule), ω2pm  4πnpm e2 ∕m, npm is the initial density of neutral atoms, and h  2πℏ is the plank’s constant [17,20]. Equation (3) governs the dynamics of pulse during plasma generation in the crystalline lens, ∇2 E⃗ 

the y or x axis, and c1 ( − 0.02) [28,29]. The solution of Eq. (3) may be written as

(8)

where S is the eikonal of the laser beam. In Eq. (7) we have taken the slowly varying envelop approximation (SVEA). Putting Eq. (8) in Eq. (7) and separating the real and imaginary parts and on transforming the variable z; t to z0 ; t0  t − z∕vg , we get    2  2 ∂S ∂S ∂S 1 ∂2 A ∂2 A  2 0− 2  ∂x ∂y ∂z k A ∂x2 ∂y2 

ω2p2 r 2 ε00 2 2  gc r  0 c2 k2 r 20 ε0p

(9)

and  2    2    2  ∂A2 ∂2 S ∂S ∂A ∂S ∂A 2 ∂ S   0:  A   ∂z0 ∂x ∂y ∂x ∂y ∂x2 ∂y2

(3)

(10)

⃗  εE, ⃗ where ε  n2 denotes the dielectric conand D stant of the medium and refractive index profile of the crystalline lens is of the following form [28]:

To account for a significant radial profile change, we have expanded ω2p and P up to second order in r 2 . P in Eq. (2) can be written as P  P 0  P 2 r 2 ∕r 20 . Now we assume the solution of Eqs. (9) and (10) for a Gaussian profile laser beam as

  g2 z 2 2 x  y  ; nx; y; z  n0 z 1 − 2

(4)

where n0 z and gz give the variation of refractive index along the direction of propagation and the gradient parameter that specifies variation of refractive index, respectively. For the paraxial region, we retain terms only up to second order in x and y and will ignore the higher-order terms. We can evaluate the gradient parameter at the center (of z axis) by g2c

2c  − 12 ; nc b

  E 200 z0 ; t0  r2 A  exp − 2 2 0 0 r 0 f z ; t  f2 2

(5)

where nc (1.406) is central refractive index, b (4.5 mm) is the common semi-axis of ellipse along

(11)

and S  βz0 ; t0 

r2 ; 2

β

1 ∂f : f ∂z0

(12)

Here β and f denote the inverse of the radius of curvature and the dimensionless beam width parameter, respectively. The symbol r 0 is the initial beam width. We obtained the following equation for plasma frequency ωp0 , ωp2 and beam width parameter f in successive powers of r 2 as

December 15, 2014 / Vol. 39, No. 24 / OPTICS LETTERS

and

∂ω2p0  P 0 ω2pm − ω2p0 ; ∂t0

(13)

∂ω2p2  P 2 ω2pm − ω2p0  − P 0 ω2p2 ; ∂t0

(14)

ω2p2 ∂2 f 1 ε  − − 00 g2c f ; 02 2 3 2 2 2 ∂z Rd f k c r 0 ε0p

(15)

2K 2K2 where P 0 βK I K , P 2 βK I K , 0 ∕Kℏωnpm f 0 ∕ℏωnpm f Rayleigh range Rd  kr 20 . We can numerically solve Eqs. (13)–(15) simultaneously for λ  580 nm, applying a boundary condition, which is, for t0  0, ωp0  ωp2  0, for all z0 ; z0  0, f  1, ∂f ∕∂z0  0. The RHS of Eq. (15) contains three terms: the first one represents the diffraction, the second term stands for defocusing of the beam because of plasma generation, and the third term is the result of the GRIN structure. We have solved the differential equations and have obtained the curves for dimensionless beam width parameter f . As the plasma generation starts, the number density will increase thereby leading to an increase in plasma frequency. The refractive index of the medium depends on the plasma frequency ωp . Therefore, with an increase in ωp , the refractive index of the medium starts to decrease and the laser pulse starts to defocus. In Fig. 1, we have plotted the variation of beam width parameter f with normalized distance traversed by the laser pulse in plasma. When the pulse duration is small enough so the plasma density is negligible, the laser pulse focuses due to the GRIN structure of the crystalline lens. As the pulse duration increases, plasma density also will increase, which causes a decrease in the refractive index of medium due to an increase in plasma frequency and, therefore, the pulse starts to defocus. In Fig. 2, the effect of the variation of the initial beam width r 0 on the focusing mechanism for constant pulse duration of 200 fs has been plotted. We observed that with a smaller initial beam width, the laser pulse starts to defocus before undergoing any focusing phenomenon. When the value of r 0 increases, the focusing phenomenon starts to appear. This behavior results because the

Fig. 1. Variation of beam width parameter with respect to normalized distance for various pulse durations of laser pulse (P in  5 × 107 W).

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Fig. 2. Variation of beam width parameter with respect to normalized distance for various initial beam widths of laser pulse (P in  5 × 107 W).

strength of diffraction is inversely proportional to initial beam width. Figure 3 shows the effect of the input peak power on the focusing property. As input peak power and hence intensity increases, the laser pulse defocuses because the time rate of change of plasma frequency ωp is proportional to the intensity by Eqs. (1) and (2). Hence, an increase of pulse intensity changes the plasma frequency, thereby leading to a decrease in the refractive index of medium and causing the pulse to defocus. We have numerically investigated the propagation dynamics of a laser pulse of femtosecond duration inside a crystalline lens. The focusing properties of the laser pulse depend on the pulse duration, initial beam width, and laser power. When a laser pulse is incident on a crystalline lens, by virtue of the focusing phenomenon of a crystalline lens, the laser gets focused and, at the focusing point, water molecules are ionized. When these electrons interact with electric field E of the laser, they gain energy from field because of their high mobility. They dissipate this energy in inelastic collision with heavy particle (water molecule) and ions. The average fractional energy loss per collision δ  δ0 2m∕M depends on the mass ratio of electron mass m to mass of heavy particle M. The constant δ0 is a temperature-dependent quantity and the value of

Fig. 3. Variation of beam width parameter with respect to normalized distance for various peak power levels of laser pulses, pulse duration τp  200 fs, and initial beam width r 0  50 μm.

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constant δ0 is one for elastic collision while it varies from 10 to several hundred for inelastic collision, depending on the temperature of the heavy particle [30]. If δ1 and δ2 are the average energy transfer per collision for elastic and inelastic collisions, respectively, then by the principle of conservation of energy, we can write (in quasisteady state) an expression for the temperature of the heavy particle T eff  αE · E  T 0 1 − δ1 ∕δ2 , where T 0 is the initial temperature, α  e2 ∕6kβ T 0 mδ1 ω2 , kβ is Boltzmann’s constant, ω is the frequency of laser light, and e and m are charge and mass of electron, respectively. Using this expression, we can make a rough estimate that the temperature is 7000 K using laser and formed plasma parameters. It should be mentioned that Vogel [24] has reported experimental values for the temperature of 10,000 and 2200 K for laser pulse durations of 6 ns and 30 ps, respectively. Based on these preliminary results, we will develop a rigorous model for laserproduced plasma formation and heating in the future. Our work may have a potential application for corneal refractive surgery as well as for material processing (waveguide writing). A lot of work, however, still remains to be done, but we expect to address remaining issues in future work. This work was partially supported by UGC, India, and DST, India. References 1. B. G. Wang, I. Riemann, H. Schubert, K. J. Halbhuber, and K. Koenig, Cell Tissue Res. 328, 515 (2007). 2. A. Vogel, J. Noack, G. H. Uttman, and G. Paltauf, Appl. Phys. 81, 1015 (2005). 3. R. R. Krueger, J. Kuszak, H. Lubatschowski, R. I. Myers, T. Ripken, and A. Heisterkamp, J. Cataract Refract. Surg. 31, 2384 (2005). 4. B. Zysset, J. G. Fujimoto, and T. F. Deutsch, Appl. Phys. 48, 139 (1989). 5. A. Glasser and M. C. W. Campbell, Vis. Res. 39, 1991 (1999). 6. R. R. Krueger, X. K. Sun, J. Stroh, and R. Myers, Ophthalmology 108, 2122 (2001). 7. T. Ripken, U. Oberheide, M. Fromm, S. Schumacher, G. Gerten, and H. Lubatschowski, Graefes Archive Clin. Exp. Ophthalmol. 246, 897 (2008).

8. V. Venugopalan, A. Guerra, K. Nahen, and A. Vogel, Phys. Rev. Lett. 88, 078103 (2002). 9. F. Docchio, L. Dossi, and C. A. Sacchi, Lasers Life Sci. 1, 87 (1986). 10. A. Vogel, S. Busch, K. Jungnickel, and R. Birngruber, Lasers in Surg. Med. 15, 32 (1994). 11. F. Docchio, C. A. Sachhi, and J. Marshall, Lasers Ophthalmol. 1, 83 (1986). 12. L. V. Keldysh, J. Exp. Theor. Phys. 47, 1945 (1964). 13. V. S. Popov, Phys. Uspekhi 47, 855 (2004). 14. C. A. Sacchi, J. Opt. Soc. Am. 8, 337 (1991). 15. P. K. Kennedy, IEEE J. Quantum Electron. 31, 2241 (1995). 16. A. Vogel, M. R. C. Capon, M. N. A. Vogel, and R. Birngruber, Invest. Ophthalmol. Vis. Sci. 35, 3032 (1994). 17. 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. Thompson, IEEE J. Quantum Electron. 33, 127 (1997). 18. J. Novak and A. Vogel, IEEE J. Quantum Electron. 35, 1156 (1999). 19. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984), p. 528. 20. P. K. Kennedy, D. X. Hammer, and B. A. Rockwell, Prog. Quantum Electron. 21, 155 (1997). 21. P. K. Kennedy, S. A. Boppart, D. X. Hammer, B. A. Rockwell, G. D. Noojin, and W. P. Roach, IEEE J. Quantum Electron. 31, 2250 (1995). 22. A. Vogel, S. Busch, and U. Parlitz, J. Acoust. Soc. Amer. 100, 148 (1996). 23. M. J. C. van Gemert and A. J. Welch, IEEE. Eng. Med. Biol. 8, 10 (1989). 24. A. Vogel, Optical Breakdown in Water and Ocular Media, and Its Use for Intraocular Photodisruption (Shaker Verlag, 2001). 25. R. P. Sharma, D. Strickland, and M. C. W. Campbell, Opt. Commun. 274, 139 (2007). 26. R. P. Sharma, D. Strickland, and M. C. W. Campbell, Opt. Commun. 304, 23 (2013). 27. A. Couairon and A. Mysyrowicz, Phys. Rep. 441, 47 (2007). 28. M. A. Rama, M. V. Perez, C. Bao, M. T. F. Arias, and C. G. Reino, Opt. Commun. 249, 595 (2005). 29. M. C. W. Campbell and A. Hughes, Vis. Res. 21, 1129 (1981). 30. R. J. Rosa, Magnetohydrodynamic Energy Conversion (McGraw-Hill, 1968), p. 94.

Effect of multiphoton ionization on performance of crystalline lens.

This Letter presents a model for propagation of a laser pulse in a human crystalline lens. The model contains a transverse beam diffraction effect, la...
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