New beat length for writing periodic structures using Bessel beams

Optics Communications 283 (2010) 447–450

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New beat length for writing periodic structures using Bessel beams Andrew W. Norfolk *, Edward J. Grace Photonics Group, Department of Physics, Imperial College London, London SW7 2BW, United Kingdom

a r t i c l e

i n f o

Article history: Received 20 August 2009 Received in revised form 5 October 2009 Accepted 5 October 2009

PACS: 42.82.Cr 42.60.Jf 42.65.k 42.65.Jx

a b s t r a c t With the ultimate aim of exploiting the self-focusing behaviour to create periodic structures, we have investigated the behaviour of Bessel–Gauss beams in Kerr-like nonlinear media and have identified that a previously proposed nonlinear beat length is inaccurate with increasing power. By studying the behaviour of the beam we suggest a correction; providing a much better description of the beat length. This correction is tested against results from numerical simulations confirming the improved accuracy. Within the, scalar, nonparaxial limit we show that this modified beat length is valid for beam powers surpassing the paraxial self-focusing threshold. From this modified beat length equation, the appropriate experimental variables may be chosen to create accurate periodic structures by direct laser writing in a single exposure. Ó 2009 Elsevier B.V. All rights reserved.

Keywords: Bessel beam Kerr nonlinearity Periodic nanostructures

1. Introduction The integration of optical designs by direct laser writing within materials such as heavy metal oxide (HMO) glass is often plagued by nonlinear beam reshaping due to Kerr-like self focusing [1]. This is particularly striking in the case of Gaussian beams where one can observe a significant variation in effective focal depth with beam power. By numerically studying the propagation of Bessel– Gauss beams we report on their potential application for writing periodic structures in a single shot, exploiting the natural selffocusing effect to our advantage. As previously reported [2] a Bessel–Gauss beam propagating in a Kerr-like nonlinear medium exhibits periodic modulation of the axial field intensity along the optical axis. As the power is increased the modulation depth grows, permitting the central lobe intensity to periodically exceed the threshold for material modification without adverse effects on the beam as a whole. This could enable periodic structures to be built in a single exposure, using optics with a far lower numerical aperture than required to build the structure point by point. Combining this with raster scanning; volume Bragg-like structures, similar to those described in [3], could be formed. As yet, one problem remains unanswered. As the beam power is increased not only does the modulation deepen, its associated beat * Corresponding author. Tel.: +44 (0) 207 594 7738; fax: +44 (0) 207 594 7714. E-mail address: [email protected] (A.W. Norfolk). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.10.014

length increasingly begins to deviate from the low power limit shown in [2] coupling together the power of the beam and the period of any generated structure. Although the variation of beat length with power is clearly observable in previous numerical results [2] it does not appear to have received significant attention. Accounting for this variation with increasing power would allow accurate control of the size and separation of these structures. 2. Bessel beams The Bessel beam is a well known exact, diffraction free solution to the scalar, linear, isotropic and homogeneous Helmholtz wave equation. In a linear medium these propagate as

Aðr; zÞ ¼ J 0 ðkr rÞ expðikz zÞ;

ð1Þ

where kr and kz are the radial and longitudinal wavenumbers respectively and a time convention of expðixtÞ has been assumed. The construction of a Bessel beam may be considered as the summation of an infinite set of plane waves with their optical axes aligned on the surface of a cone. The inner cone angle, h, denotes the angle any one of these plane waves makes with the principle optical axis of the beam. One such plane wave is shown in Fig. 1. The longitudinal and transverse wavenumbers are related by

kz ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k  kr ¼ k cos h:

ð2Þ

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Applying the above scaling to the inner cone angle, h in Fig. 1 gives the relation:

pffiffiffiffiffiffiffi sin h ¼ k~r 2j;

Fig. 1. Representation in k-space.

In the paraxial limit this reduces to 2

kz ¼ k  kr =ð2kÞ:

ð3Þ

Such Bessel beams are unphysical as the power is technically unbounded. A commonly used physical approximation is the Bessel–Gauss beam. Here a Gaussian envelope is used to ensure the beam is of finite power. Using this Gaussian window comes at a cost; the Bessel–Gauss beam is no longer diffraction free. By considering the shadow cast by the windowing function we may write this diffraction-free length as

zBD ¼

w0 ; tan h

ð4Þ

ð6Þ

from which the usual small angle approximations can be taken. Clearly, including the first term in Eq. (5) places a constraint on the relationship between k and kr . Any solution obtained from the NNSE will be valid only for this value of j and the associated inner cone angle, h. If, however, this term is neglected; any solution obtained by the NSE will be valid for any combination of k and kr . Put plainly, solutions obtained from the NSE, for a particular scaled transverse spatial frequency k~r , will be valid for any inner cone angle h. This is provided, of course, that we do not inadvertently violate the SVEA. As the Bessel beam field evolves in a Kerr-like nonlinear medium; the nonlinear interaction, which can be thought of as degenerate four wave mixing [2] or alternatively self-diffraction [4], generates a field propagating predominantly along the optical axis. From trivial geometrical arguments the interference between this new field and the Bessel beam, shown as Dkz on Fig. 1, leads to a beat length:

z0b ¼

2p : k  kz

ð7Þ

Fig. 2a shows the field intensity distribution as this plane wave beats with a linear Bessel–Gauss beam. This we shall consider our linear approximation.

where w0 is the 1=e radius of the Gaussian window. 4. Numerical simulation 3. Numerical model The propagation of the, potentially large cone angle, scalar beam in a Kerr-like medium may be described by the nonparaxial, nonlinear, Schrödinger equation (NNSE):

o2 u

ou

j ~2 þ i ~ þ r2? u þ juj2 u ¼ 0; oz oz

ð5Þ

where r2? is the transverse Laplacian, the form of which depends on the coordinate system. A scaling has been introduced appropriate to 2 a Bessel–Gauss beam such that ~r ¼ rkr , ~z ¼ z=LD and LD ¼ 2k=kr is the Rayleigh length of the isolated central Bessel lobe [4]. The constant j ¼ ðkr =kÞ2 is a measure of the nonparaxiality pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the beam. Finally, the field is scaled as uð~r ; ~zÞ ¼ kn2 =n0 Ld Að~r ; ~zÞ where Að~r; ~zÞ is the unscaled field and the forward phase expðikzÞ has been factored out. To recover the nonlinear Schrödinger equation (NSE) from Eq. (5) we must neglect the first term. Commonly, this is achieved by assuming the slowly varying envelope approximation (SVEA). It is worth remembering that we do not need the explicit condition that j ! 0 for the NSE to be valid. We may model beams with modest nonparaxiality providing that we do not inadvertently violate the slowly varying envelope approximation. For weak nonlinearity we do not expect the numerical results derived with the NNSE to manifestly differ from those derived from the NSE. By construction, providing the forward phase has been factored out, the Bessel–Gauss beam varies slowly with respect to ~z, this holds true for small amounts of nonlinearity. As the nonlinear influence on the beam increases, with increasing power, the self focusing begins to dominate the behaviour of the beam. As this occurs the beam is clearly no longer varying slowly and consequently the SVEA is violated. Above a threshold power; catastrophic self focusing is predicted by the NSE [5]. It is know that the full nonlinear scalar Helmholtz or NNSE do not permit such catastrophic self focusing [6]. We test our assumptions for the beat length using both the NNSE and the NSE.

The assumptions made above were tested by modelling the Bessel–Gauss beams with the Hankel-based Adaptive Radial Propagator (HARP). The Hankel transforms were implemented using the quasi discrete Hankel transform [7]. A symmetrised split step operator was used for the paraxial results and a finite difference operator, based on the NNSE, was used for the nonparaxial results [8]. The nonparaxial results were obtained for an inner cone angle of h ¼ 30 , corresponding to a central lobe FWHM of 0:36k. The numerical results in Fig. 2b shows the intensity distribution for slices across the beam path. The modulation of the Bessel– Gauss beam is clearly present. By comparing these numerical results with the linear approximation, Fig. 2a several deficiencies in the linear approximation are apparent. Energy is concentrated near the axis in the nonlinear numerical case to a far greater extent than the trivial linear case of Fig. 2a. In addition the modulation length varies, reducing for lobes closer to the central point. For the central lobe this is pronounced; eight periods in Fig. 2b as opposed to six in Fig. 2a. If this intensity modulation were to be used for writing periodic structure such unpredictable control over the spacing the points would be unacceptable; a correction to this beat length is required. 5. Power dependent beat length Two features are not taken into account in the linear approximation. The first deficiency arises because we have incorrectly assumed that the generated field has the form of a plane wave, whereas it is Gaussian-like [2]. This accounts for the concentration of power towards the central lobes. Additionally we have neglected to take into account the Kerr-induced phase retardation on the generated field. This increases the optical path for the generated field resulting in a reduction in the modulation length which scales as average intensity. We propose a new nonlinear beat length based on a correction to Eq. (7) [2]:

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Fig. 2. Bessel–Gauss inner cone angle, 30°. Central lobe intensity FWHM 0:36k.

zb ¼

2p : kð1 þ Ie n2 =n0 Þ  kz

ð8Þ

Here, Ie is an effective intensity which scales linearly with power with n0 and n2 the usual linear and nonlinear refractive indices. Since the beat length should be valid over a length for which the real field intensity jEj2 varies; Ie may be thought of as a modified average intensity; Ie ’ hIi. We assume that the intensity is averaged over the central lobe of the Bessel–Gauss beam; hIi ¼ 0:27I0 where I0 is the axial intensity of the central lobe. In the low power limit the initial beat length, z0b , is recovered. Fig. 3a shows the numerically determined beat length for both the paraxial and nonparaxial model. Both observed beat lengths appears to converge on the low power limit, z0b , as expected. By introducing a free parameter, , relating the effective intensity Ie to the average intensity hIi such that Ie ¼ ð1 þ ÞhIi, we can estimate how well assumptions match the expected observations in both cases. For the assumption  ¼ 0 the observed fits between the observations and modified beat lengths are R2 ¼ 0:97 and R2 ¼ 0:95 for the paraxial and nonparaxial cases respectively. By allowing the effective intensity to differ with respect to the average intensity the error, , was calculated using a best fit to Eq. (8). For the results displayed in Fig. 3a;  ¼ 0:05 ðR2 ¼ 0:96Þ for the nonparaxial results and  ¼ 0:04 ðR2 ¼ 0:98Þ for the paraxial results. Such small differences between the observation and Eq. (8) are likely to be too small to experimentally observe, lending support

(a)

to this modified beat length being an experimentally useful expression. As we approach the onset of beam blow-up at P c the modulation can become extremely large due to the self focusing of the beam, at this point our assumption that the initial intensity I0 is a good approximation for the average intensity over one beat length is less reasonable and a deviation from Eq. (8) is observed. Fig. 3b compares the central intensity behaviour for linear, high and relatively low power nonparaxial Bessel–Gauss beams; their positions are marked on Fig. 3a. From this it can be seen that the depth of modulation may greatly increase with beam power in this case exceeding 100% relative to the initial intensity. Although the maximum intensity varies slightly across each period; the threshold-like response of the optical breakdown within the material [1] should mean that the quality of each structure remains the same. In the paraxial approximation increasing beam power eventually leads to catastrophic self-focusing above a threshold power. The total Bessel–Gauss beam power may exceed the collapse power for a Gaussian beam as the power is distributed throughout the multiple side lobes. It is predominantly the power of the central lobe which characterises the threshold for collapse of the Bessel–Gauss beam. This may be determined through numerical modeling. The threshold power, Pc , is marked as the solid line on Fig. 3a. We have observed that when the Bessel–Gauss beam is considered nonparaxial; not only may the power exceed this limit, but the intensity behaviour remains largely similar to the sub-critical beam.

(b)

Fig. 3. (a) Beat length relative to geometric limit for the paraxial (green ‘+’) and nonparaxial (red ‘’). Large black crosses correspond to powers used in Fig. 3b. Blue line shows modified beat length (Eq. (8)). (b) Central intensity, Iðr ¼ 0; zÞ, clearly showing the interference behaviour for: a linear beam (dotted); a moderate power beam, P < P c , (dashed); and a nonparaxial, high power beam (solid). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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6. Conclusions The intensity modulation observed in Bessel–Gauss beams propagating in Kerr-like nonlinear materials may allow the writing of a periodic structure in a single exposure. The high power required to obtain this deep modulation causes a deviation from the low power beat length z0b . To account for this we have, for the first time, introduced a nonlinear beat length for Bessel–Gauss beams. This power dependence results in an improvement over the previously proposed beat length and allows a more accurate control over the size and separation of the periodic structures. This would allow the selection of the correct power and Bessel cone angle to accurately drive the local intensity above the threshold for material modification. A full treatment of the nonlinear behaviour has not been given and the precise dynamics of the phenomenon are subtle. Our numerical simulations show that the small correction leads to a significant improvement over the uncorrected beat length and would be suitable for improving the accuracy of any experimentally generated structures. We have also demonstrated that the nonparaxial Bessel–Gauss beam is robust for powers up to and surpassing the paraxial critical power. This robust and predictable behaviour suggests a strong po-

tential for the single shot generation of micro structures and the formation of Bragg-like scattering objects in highly nonlinear glasses. In future we hope these findings will be tested experimentally. Acknowledgements A. Norfolk is generously supported by an EPSRC doctoral training award. E. Grace is generously supported by a fellowship from the RAE/EPSRC. References [1] J. Siegel, J.M. Fernández-Navarro, A. García-Navarro, V. Diez-Blanco, O. Sanz, J. Solis, Appl. Phys. Lett. 86 (2005) 121109. [2] R. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevicius, A. Stabinis, Opt. Commun. 196 (2001) 309. [3] F. He, H. Sun, M. Huang, J. Xu, Y. Liao, Z. Zhou, Y. Cheng, Z. Xu, K. Sugioka, K. Midorikawa, Appl. Phys. A (2009), doi:10.1007/s00339-009-5338-4. [4] V. Pyragaite, K. Regelskis, V. Smilgevicius, A. Stabinis, Opt. Commun. 257 (2006) 139. [5] P.L. Kelley, Phys. Rev. Lett. 15 (1965) 1005. [6] G. Fibich, Phys. Rev. Lett. 76 (1996) 4356. [7] M.M. Guizar-Sicairos, J.C.J.C. Gutièrrez-Vega, J. Opt. Soc. Am. A 21 (2004) 53. [8] P. Chamorro-Posada, G.S. McDonald, G.H.C. New, Opt. Commun. 192 (2001) 1.