Self-focusing and defocusing of twisted light in non-linear media

Anita Thakur; Jamal Berakdar

doi:10.1364/OE.18.027691

1. Introduction

The self-focusing of light beams is a basic phenomena in nonlinear optics [1] with a variety of important applications [2] that rely on the manipulation and control of the photon beam [3]. Generally, the theory of self focusing is well established with the propagation characteristics found to be closely related to the properties of the medium [1–4] and to the pulse width of laser beams [5]. Self-focusing and de-focusing of electromagnetic beams in nonlinear media was reviewed by Akhmanov et. al [6]. Recently, several investigations were conducted to study the propagation properties of Cosh Gaussian and Hermite Gaussian beams in different media [7, 8]. Here, we investigate the self focusing of Laguerre Gaussian beams [9–12] in a nonlinear medium using the the Wentzel-Kramers-Brillouin (WKB) and the paraxial approximations [6, 13]. Laguerre Gaussian (LG) beams with a central hole singularity have been shown to play an important role in several areas of optics [14, 15]. In particular, light carrying orbital angular momentum l, also called twisted light, is described by LG modes with a term describing the on-axis phase singularity of strength l; hence the name optical vortices [16–19] for this type of intensity distribution. In addition to the winding number l, LG modes are characterized by their radial index p and their waist size w₀. Here we use LG modes with p = 0; for l ≠ 0. In this case, the intensity cross-sections perpendicular to the propagation direction consists of one bright ring with no on-axis intensity. This feature makes them ideal for applications in optical trapping and optical tweezers. Furthermore, as LG beams can transfer orbital momentum to the trapped particle, it can also act with a torque on the trapped particle [16–19]. LG tweezers can also trap metallic particles with a refractive index higher than that of the surrounding medium [20, 21].

All of these applications rely on the light scattering and hence they are related to the strength and the distribution of the intensity. The focusing and defocusing are thus important, e.g. in the above context they allow the manipulation of the trapping spot size and the strength of the tweezers.

The paper is structured as follows: starting from the amplitude distribution of LG beams propagating in a nonlinear dielectric medium in section 2, we derive a general differential equation for the beam width parameter. Utilizing the WKB and paraxial approximations in section 3 we derive analytical expression for the intensity distribution as a function of the beam’s parameters. In section 4 results are presented graphically and discussed. A brief conclusion and future prospectives are given in section 5.

2. Theoretical background

2.1. Laguerre Gaussian beams

The amplitude distribution of the LG beam $u_{l p}^{L G} (r, ϕ, z)$ in a cylindrical coordinate with z axis being along the beam propagation direction, is [24]

\begin{array}{l} u_{l p}^{L G} (r, ϕ, z) & = & \frac{C_{p}^{l}}{w (z)} {[\frac{\sqrt{2 r}}{w (z)}]}^{l} exp [\frac{- r^{2}}{w^{2} (z)}] L_{p}^{l} (\frac{2 r^{2}}{w^{2} (z)}) \\ \times exp [- i \frac{k r^{2} z}{2 (z^{2} + z_{R}^{2})} - i l ϕ + i (2 p + l + 1) arctan (\frac{z}{z_{R}})], \end{array}

where r is the radial coordinate and ϕ is the azimuthal angle.

w (z) = w_{0} \sqrt{1 + (x^{2} / z_{R}^{2})}

is the radius of the beam at z, and z_R is the Rayleigh range. w₀ is the beam waist at z = 0.

L_{p}^{l} (x)

is the associated Laguerre Polynomial,

C_{p}^{l}

is the normalization constant, and

(2 p + l + 1) arctan (\frac{z}{z_{R}})

is the Gouy phase. At the beam waist, z = 0, the amplitude of a Laguerre-Gaussian beam simplifies to

u_{l p}^{L G} (r, ϕ, z = 0) = C_{p}^{l} {[\frac{\sqrt{2 r}}{w_{0}}]}^{l} exp [\frac{- r^{2}}{w_{0}^{2}}] L_{p}^{l} (\frac{2 r^{2}}{w_{0}^{2}}) exp (- i l ϕ) .

2.2. Self-focusing and defocusing in a nonlinear medium

We consider a nonlinear medium characterized by the dielectric function ɛ = ɛ₀ + F(EE*), i.e. ɛ(r,z) depends upon the beam irradiance; the functional dependence of F is determined by the physical situation/mechanism under study. In turn |E|² depends on z in a manner yet to be determined. In the spirit of the paraxial approximation, we expand F in a Taylor series in powers of r² and retain terms up to r². This leads to

ɛ (r, z) \approx ɛ_{0} (z) - r^{2} ɛ_{2} (z) .

In the wave equation governing the propagation of the laser beam

\nabla^{2} E + \frac{ω^{2}}{c^{2}} ɛ E + \nabla (\frac{E \nabla ɛ}{ɛ}) = 0

the third term can be neglected if k^–2∇²(lnɛ) ≪ 1, where k is the wave vector. This inequality is satisfied in almost all cases of practical interest. For a cylindrically symmetric beam a solution for

\nabla^{2} E + \frac{ω^{2}}{c^{2}} ɛ E = 0

we obtain using WKB and the paraxial approximation Refs. [6, 13] as

E (r, ϕ, z) = A (r, ϕ, z) exp [i (ω t - k z)]

where

k = \frac{ω}{c} \sqrt{ɛ_{0}}

and A(r, ϕ, z) is the complex amplitude of the electric field. Substituting for E(r, ϕ, z) and neglecting

\frac{\partial^{2} A}{\partial z^{2}}

on the basis of WKB approximation which implies that the characteristic distance of intensity variation is much greater than wavelength. We obtain

2 i k \frac{\partial A}{\partial z} = \frac{\partial^{2} A}{\partial r^{2}} + \frac{1}{r} \frac{\partial A}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} A}{\partial ϕ^{2}} - \frac{ω^{2}}{c^{2}} ɛ_{2} r^{2} A .

To solve Eq. (7) we express A(r, ϕ, z) as

A (r, ϕ, z) = A_{0} (r, z) exp {i [- k S (r, z) - l ϕ]}

where A₀ and S are real functions of r, ϕ and z and the Eikonal S is

S = \frac{r^{2}}{2} β (z) + Θ (z) .

Θ(z) is an additive function where

β (z) = \frac{1}{f} \frac{d f}{d z} .

The parameter β(z) is the curvature of the wavefront. Substituting for A(r, ϕ,z) and S from Eq. (9) and Eq. (10) in Eq. (8) one obtains

2 \frac{\partial S}{\partial z} + {(\frac{\partial S}{\partial r})}^{2} = \frac{1}{k^{2} A_{0}} [\frac{\partial^{2} A_{0}}{\partial r^{2}} + \frac{1}{r} \frac{\partial A_{0}}{\partial r} - \frac{l^{2}}{r^{2}} A_{0}] - \frac{ɛ_{2}}{ɛ_{0}} r^{2}

\frac{\partial A_{0}^{2}}{\partial z} + \frac{\partial A_{0}^{2}}{\partial r} \frac{\partial S}{\partial r} + A_{0}^{2} (\frac{\partial^{2} S}{\partial r^{2}} + \frac{1}{r} \frac{\partial S}{\partial r}) = 0

The solution of Eq. (11) for LG beam can be written as

A_{0} (r, z) = \frac{E_{0}}{f} {(\frac{\sqrt{2 r}}{w_{0} f})}^{l} exp (\frac{- r^{2}}{w_{0}^{2} f^{2}}) L_{P}^{l} (\frac{2 r^{2}}{w_{0}^{2}}) .

For l = 1 and p = 0, substituting for S and A₀ from Eqs. (9), (13), (11) yields

\frac{1}{f} \frac{d^{2} f}{d z^{2}} = \frac{4 c^{2}}{ω^{2} ɛ_{0} (z) w_{0}^{4} f^{4}} - \frac{ɛ_{2} (z)}{ɛ_{0} (z)} .

Equation (14) can always be solved by considering the conditions

f = 1 and \frac{d f}{d z} = 0 at z = 0 .

It is, however, convenient to reduce Eq. (14) to a dimensionless form by transforming the coordinate z to the dimensionless distance of propagation

ξ = \frac{z c}{w_{0}^{2} ω}

and the beam width w₀ to the dimensionless beam width

ρ = \frac{w_{0} ω}{c}

Substituting Eq. (16) and (17) in Eq. (14) yields

\frac{ɛ_{0} (z)}{f} \frac{d^{2} f}{d ξ^{2}} = \frac{4}{f^{4}} - ρ^{2} w_{0}^{2} ω_{2} (z) .

In case of a parabolic nonlinearity, that is when the nonlinear term is proportional to E² we have the r dependent term

ɛ_{2} (f) r^{2} = \frac{α E_{0}^{2}}{f^{2}} \frac{r^{2}}{w_{0}^{2} f^{2}}

where α is a constant. Substitution of ɛ₂(f) in Eq. (18) yields

\frac{1}{f} \frac{d^{2} f}{d ξ^{2}} = \frac{1}{ɛ_{0} f^{4}} (4 - ρ^{2} α E_{0}^{2}) .

The analytical solution of Eq. (20) under the conditions (15) is

f = \sqrt{\frac{ɛ_{0} + 4 ξ^{2} - E_{0}^{2} α ξ^{2} ρ^{2}}{ɛ_{0}} .}

For further analysis it is useful to write Eq. (20) in the form

\frac{d^{2} f}{d z^{2}} = \frac{c^{2}}{ɛ_{0} w_{0}^{4} ω^{2}} (4 - \frac{α w_{0}^{2} ω^{2}}{c^{2}} E_{0}^{2}) f^{- 3} .

3. Results and discussions

Equation (20), or (22) is the fundamental second order differential equation governing self-focusing/defocusing of LG beams in a parabolic medium. Essentially, the effect of the non linearity is dictated by the second term on the right of Eq. (20), or (22). In the absence of this term $\frac{d^{2} f}{d ξ^{2}}$ remains positive causing the beam width parameter (f) to increase continuously leading to a steady divergence. This effect is the natural diffraction divergence. The second term containing the nonlinear effect is negative and acts in the opposite direction tending to converge the beam. The convergence (focusing) or divergence (defocusing) of the beam depends on which of the two terms predominates. Equation (22) makes also clear that for the focusing the product (w₀ωE₀)² is relevant, i.e. for a focused beam the frequency has to be increased when the intensity (or the waist) is lowered to maintain focusing.

Figures 1(a) and 1(b) illustrate the focusing effects for typical experimentally feasible parameters.

Fig. 1 (color online) (a) Intensity in CGS units of LG beam verses the radial distance from the propagation direction (in cm) for the l = 1, and p = 0. The angular frequency is ω = 2 × 10¹⁴ rad/sec, w₀ = 1 [cm], ɛ₀ = 1, α = 1, E₀ = 0.3 [StatV /cm]. ρ = 0.66 × 10⁴. (b) Initial intensity profile (dotted curve) compared to the propagated intensity at ξ = 4 × 10^–4.

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As clear from this figure, focusing and intensity increase occur until at certain value of the normalized distance of propagation (ξ) = 0.00062, after that de-focusing sets in. This is due to the fact that at higher intensity nonlinear refractive term dominates over the diffractive term for some initial distance of propagation after that the diffractive term strongly overcomes the nonlinear refractive term and therefore the beam defocuses.

From Fig. 1 it is also obvious that the focusing effect can be utilized, e.g. for creating tighter and stronger three-dimensional optical traps by crossing two LG beams at the focused distance. The predicted focusing effect can also be used for the realization of more versatile optical tweezers. From Eq. (22) we infer that to achieve results similar to those in Fig. 1 for a smaller staring waist one has either to increase the intensity (or the frequency) by the roughly the same amount.

4. Conclusions

We studied the self focusing of a twisted light beam in a nonlinear dielectric medium by using the paraxial approximation. The differential equation for the beam waist is solved analytically. The occurrence of the focussing is pointed out and its dependence on the beam’s parameters is worked out analytically and illustrated by numerical calculations. Some practical applications of the predicted effect are pointed out.

References and links

1. Y. R. Shen, The Principles of Nonlinear Optics, (Wiley, 1984).

2. W. S. Williams, J. T. Hunt, and W. E. Warren, “Light propagation through large laser system,” IEEE J. Quantum Electron. 17, 1727 (1981). [CrossRef]

3. A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102, 233902 (2009). [CrossRef] [PubMed]

4. G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, “Self-focusing threshold in normally dispersive media,” Opt. Lett. 19, 862 (1994). [CrossRef] [PubMed]

5. E. G. Hanson, Y. R. Shen, and G. K. L. Wong, “Experimental study of self-focusing in a liquid crystalline medium,” Appl. Phys. 14, 65 (1977). [CrossRef]

6. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10, 609 (1968). [CrossRef]

7. S. D. Patil, M. V. Takale, and M. B. Dongare, “Propagation of Hermite-cosh-Gaussian laser beams in n-InSb,” Opt. Commun. 281, 4776 (2008). [CrossRef]

8. S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, “Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption,” Opt. Laser Eng. 47, 604 (2009). [CrossRef]

9. M. W. Beijersbergen, L. Allen, H. van Der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123 (1993). [CrossRef]

10. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305 (2007). [CrossRef]

11. M. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57, 35 (2004). [CrossRef]

12. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. XXIX, 291 (1999). [CrossRef]

13. M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Self Focusing of lasers in plasmas and semiconductors, (New Delhi: Tata McGraw-Hill; 1974).

14. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992). [CrossRef] [PubMed]

15. S. Furhapter, A. Esacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13, 689 (2005). [CrossRef] [PubMed]

16. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002). [CrossRef]

17. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “The mechanical equivalence of the spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52 (1997). [CrossRef] [PubMed]

18. K. T. Gahagan and G. A. Schwartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827 (1996). [CrossRef] [PubMed]

19. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348 (1998). [CrossRef]

20. A. T. O’Neil and M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the de-coupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185, 139 (2000). [CrossRef]

21. M. P. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, and P. E. Bryant, “Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap,” Opt. Lett. 26, 863 (2001). [CrossRef]

22. A. E. Siegmann, Lasers, (University Science Book, Mill- Valley, CA, 1986), Sec. 17.5.

23. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56, 4193 (1997). [CrossRef]

24. L. C. Dvila-Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B. Quantum Semiclass. Opt. 4, S66 (2002). [CrossRef]

Self-focusing and defocusing of twisted light in non-linear media

Abstract

1. Introduction

2. Theoretical background

2.1. Laguerre Gaussian beams

2.2. Self-focusing and defocusing in a nonlinear medium

3. Results and discussions

4. Conclusions

References and links

Cited By

Figures (1)

Equations (22)

Optics Express