We propose a comprehensive analysis of the effects that spherical aberration may have on the process of ultrafast laser photowriting in bulk transparent materials and discuss the consequences for the generated refractive index changes. Practical aspects for a longitudinal photowriting configuration are emphasized. Laser-induced index variation in BK7 optical glass and fused silica (a-SiO2) affected by spherical aberration are characterized experimentally using phase-contrast optical microscopy. Experimental data are matched by analytical equations describing light propagation through dielectric interfaces. Corrective solutions are proposed with a particular focus on the spatial resolution achievable and on the conditions to obtain homogeneously photo-induced waveguides in a longitudinal writing configuration.
© 2007 Optical Society of America
The interest for using ultrashort pulsed lasers in the field of advanced material processing is well established and this technology is now commonly used in industrial applications requiring a high level of accuracy. Exploiting the particular material dynamics at ultra-short time scales, a reduced thermal affected zone is obtained leading to a high quality machining [1, 2]. Moreover, a range of possibilities are still explored for 3D processing of transparent materials using the laser-induced nonlinearities characteristics to the ultrafast scales. Indeed, when focused inside the bulk of a transparent material, the high peak-intensity achievable in the focal volume enables to define a well located interaction region. The limits of this region are determined by non-linear absorption mechanisms including multiphoton ionization, field-induced electron tunneling, or collisional ionization. An electronic plasma with spatially variable density and temperature is formed within the excited region which subsequently deposits the energy to the glass matrix. Consequently, the material is structurally modified, and, depending on the input fluence, index variation or catastrophic optical damage may be induced [3–7]. This has stimulated technological development in the field of photonics or analytics including the fabrication of buried 3D optical devices such as guided optics , optical memories , microfluidics , diffraction gratings  or Bragg gratings .
Especially in the case of low-repetition rate photowriting, the performances of these embedded 3D photonic components depend on the spatial resolution of the photomodification process, which is itself affected by the presence of a dielectric interface between materials of different refractive indices. In the case under study here, the interface is between air and glass. When such an interface is present in the path of a focusing beam, spherical aberration is induced, causing in particular a variation of the focus position as a function of the incidence angle. As a result, both radial and longitudinal spreadings of the light distribution in the focal volume are induced.
In the frame of femtosecond material processing, the importance of spherical aberration for depth-dependent processes has only recently been underlined. Indeed, experimental investigations have been carried out in silica glass as a function of depth [13–16]: a significant increase in damage and index variation thresholds with depth has been observed [13, 14]. Moreover, the length of the photoinduced plasma channel increases with depth . A correction of spherical aberration using a specially designed adjustable dry focusing objective was previously demonstrated and used to avoid the spatial spreading of light for increasing focusing depth .
Spherical aberration due to an index mismatch at a dielectric interface has already been investigated to improve the performance of confocal or multi-photon microscopy. Both vectorial [17, 18] or scalar [19, 120] theories of diffraction have been employed to describe the light distribution in the focal volume. Experimental dynamic compensation of spherical aberration involving a feedback-loop and adaptive optics has also been performed to optimize the writing of optical memories [21, 22]. These previous studies were carried out with high numerical aperture (NA) objectives.
In this paper, we propose a detailed investigation of the effects of spherical aberration on the formation of embedded photomodified regions, notably regions exhibiting a change in the refractive index under ultrafast light exposure. These effects occur for different photowriting geometries which are usually employed for creating buried structures. We will focus here on the longitudinal configuration, where the sample is scanned parallel to the laser beam . Indeed, in this case, the photo-written waveguide maintains the radial symmetry of the writing beam. This geometry is therefore preferable to the transversal writing (perpendicular scan) to obtain circularly symmetric guides in a simpler and more straightforward manner. The challenges are thus to maintain the symmetry and the energy distribution over long distances. To accomplish this task, this geometry usually employs long enough working distance objectives (usually associated with low NA) to manufacture sufficiently-long waveguides. On the other hand, a high enough NA is necessary to reduce non-linear effects that appear at high input powers, such as self-channelling and filamentation . These artefacts perform a less-controllable axial energy redistribution function. Consequently, a numerical aperture of about 0.45 appears like a good compromise to perform longitudinal femtosecond laser photowriting with lower non-linear effects and cm-long achievable waveguides. In these conditions, the limiting factor is mainly the wave front distortion. We show below that simple analytical expressions may be used to describe the major effects of spherical aberration.
The paper is organized as follows. In section 2, we derive a theoretical expression for the aberrated optical path difference using Nijboer formulas and introduce a standard decomposition on Zernike polynomials. In section 3, we describe the most significant but detrimental consequences of spherical aberration, which increase with depth: radial and longitudinal spreading of the focal volume and distance from paraxial to best focus. In section 4, theoretical predictions will be compared to experimental characteristics of femtosecond-induced refractive index variation measured by optical phase contrast microscopy in two widely used types of glasses: fused silica and BK7 borosilicate crown optical glass. In section 5, the effects of a correction of spherical aberration will be theoretically studied. In particular, a quick idea of which resolution is easily reached at a given depth for a given correction will be provided, together with experimental required conditions to induce a homogeneous longitudinally photowritten waveguide.
Such a detailed analysis represents a significant issue prior to any programmable correction of aberrations. Indeed, the higher-order Zernike polynomial used to characterize the aberration defines the required spatial frequency of the phase modulator. Moreover, the number of required Zernike polynomials is essential to optimize the convergence speed of the feedback-loop. And finally, an easy predictive access to the spatial resolution of the set-up makes the design of the desired photonic device much more accurate.
2. Aberrated optical path difference
The analysis that is conducted in this work is based on geometrical considerations and on the scalar theory of diffraction. To be fairly rigorous, this approximation stays valid for an aperture number N<1.5 as usually used by commercially available optical design softwares, corresponding to numerical apertures NA<0.33. Nevertheless, it has already been used for high NA objectives [19,120] and it has proven to be sufficient for NA<0.6. Above this value, scalar theory still gives a good guideline.
Let us consider the focusing geometry described in Fig. 1. The focus in a matched medium is A. Due to refraction at the dielectric interface, the real focal point is shifted and its position depends on the opening angle α’: for low values of α ’, rays converge to the paraxial focus A’P. For high values of α’, rays converge to the marginal focus A’M. The longitudinal spherical aberration (LSA) l, i.e. the algebric distance between the paraxial and the marginal focus, is then easily expressed as:
α’ is the opening angle in the image space, is the algebric distance between the interface S and the focal point A in a matched medium, n1 and n2 are the refractive indices of the media before and after the interface respectively.
The aberrated Optical Path Difference (OPD) in the output pupil plane of the objective Δaberr may be evaluated using the wave aberration in the object and image space respectively Δ1 and Δ2. Indeed, according to Nijboer formulas in the case of a rotationally invariant problem , the wave aberration Δ2, as defined as the algebric distance along the optical ray in the image space between a reference sphere centered in the paraxial focus A’P and the actual aberrated wavefront, is given by:
Using Gouy theorem, specifying that the product ni.Διt keeps constant along an optical ray , and getting rid of constant terms that do not affect the point spread function (PSF), one can deduce Δaberr as follows:
One can check that Δaberr is negative for n1<n2, as it should be the case in an overcorrected optical system. Equation (3) cannot be directly compared to expressions obtained in [19, 20]. Indeed, in these references, the calculation of Δaberr is conducted in the object space, and the reference sphere is centered in A. By applying Gouy theorem from object to image spaces, and operating a change in the center of the reference sphere from A to A’P, the two expressions become identical, except the sign.
Δ2 deduced from Eq. (2) may be compared to the Seidel expansion of spherical aberration, widely used in optical system design. Indeed, the lowest-order spherical aberration term in Seidel expansion is expressed as Δ2=-a α′ 4/4, where a is the Seidel spherical aberration coefficient. A development of Eq. (2) to fourth order in α’ leads to:
with n being an even integer and ρ the normalized radial coordinate. For simplicity, R0n(ρ) may be expressed in terms of tabulated Legendre polynomials of the first kind . The first Zernike polynomials are given in Table 1. One should note that, following this definition, and with Z0n(ρ) and A’n0 respectively the Zernike polynomial and coefficient defined in [19,20].
with Bn(γ) given by Eq.(13) of Ref. .
In the following sections, we describe how the aberrated OPD enables to characterize the focal volume in the presence of spherical aberration. Theoretical results will be compared to experimental measurements. Moreover, thanks to the decomposition of the OPD on Zernike polynomials detailed above, we will predict the effect of a total or partial correction.
3. Characterization of the focal volume
The most visible consequence of spherical aberration is the longitudinal spreading of the focal volume. The distance from paraxial to marginal focii is the longitudinal spherical aberration l (LSA).
Equation (1) gives a simple geometrical value of l. By developing Eq. (1) at lowest order in NA, l is simplified into lSeidel=a NA2/n22, with a given by Eq. (4), in agreement with Seidel theory valid for low NA. Another expression for the LSA may be deduced from scalar diffraction theory limited to the on-axis intensity, following the procedure detailed in . To summerize, the two normalized on-axis positions u1 and u2 characterizing the longitudinal extension of spherical aberration are defined such that:
with I the on-axis intensity, 0<ε<1 and
u1 and u2 may be determined analytically by relating the normalized radius at which the ray passes through the pupil plane to the normalized on-axis position u at which it crosses the focal area.
The predictions of these analytical models are compared in Fig. 2 as a function of NA for a fixed depth of x=3 mm in fused silica. As expected, Seidel approximation underestimates the LSA for NA>0.2. The two other approaches, deduced from geometrical considerations or from the on-axis PSF are in good agreement. Nevertheless, these analytical descriptions based on geometrical considerations and involving mainly ray tracing become incorrect in the case of low NA. Indeed, for low NA, the longitudinal spreading of light is mainly due to diffraction effects that may be described by the confocal parameter of the beam lconfocal, given by :
The evolution of lconfocal with NA is added in Fig. 2. For high enough NA (depending on depth), spherical aberration governs.
To check the validity of these analytical models, we compared them to a prediction of vectorial diffraction theory [17, 18] as a function of NA. For doing this, we implemented a numerical solution of Eq. (7) involving the on-axis intensity, as calculated in [17, 18]. This approach naturally accounts for diffraction effects that dominate for low NA. As depicted in Fig. 2, analytical predictions are close to the numerical ones involving vectorial diffraction theory.
Another relevant parameter to characterize the light distribution in the focal volume is the position on the optical axis where intensity is maximum, that is to say the best focus denoted A ”. A noticeable consequence of spherical aberration is a depth-dependent shift in the best focus position, which is located somewhere between AP ’ and A’M (Fig. 1). This effect is highly detrimental for accurate material processing but it may be estimated thanks to the decomposition of the aberrated OPD on Zernike polynomials. Indeed, Zernike polynomials describe aberrations whose reference is located at the best focus A ”. This change from the paraxial focus to the best focus is expressed through the parabolic component of R02(ρ). By identifying the parabolic term of the OPD related to R02(ρ) deduced from Eq. (5) to a purely defocusing OPD, dz is expressed as:
with A2,0 given by Eq. (6). By developing Eq. (10) at lowest order in NA, dz is simply expressed as the half of the LSA, in agreement with Seidel theory. Evolution of dz with depth for different NA is plotted in Fig. 3 for fused silica. As expected, for NA>0.4, the difference between half Seidel LSA and Eq. (10) overpasses 11%.
The last consequence of spherical aberration to be considered in this paper is the radial spreading of the focal volume, leading to an additional decrease in the maximum intensity. This effect may be described by the Strehl ratio defined as the ratio between the maximum intensity in the plane of the best focus to the absolute (i.e. without aberration) maximum intensity . In our case, the maximum intensity is located on the optical axis. If aberrations are weak enough so that the variance of the OPD σΔ satisfies σ Δ≪λ/(2π), the Strehl ratio Rs may be expressed analytically as :
with A2j,0 the Zernike coefficient defined by Eq. (6). In Eq. (11), the involved OPD is Δaberr where the focus term has been corrected. Indeed, this enables to have a straightforward access to the OPD in the plane of the best focus.
In the next section, we propose to link relevant parameters characterizing the focal volume in the presence of spherical aberration to experimental measurements. In particular, the LSA will be compared to the longitudinal extension of the photo-modified area and the theoretical position of the best focus will be compared to the first position where the peak intensity overpasses the threshold of photo-modification in the material.
The Strehl ratio will be used in section 5 to deduce the spatial resolution of a focusing optical system for a given correction.
The experimental probing of the consequences of aberrations in femtosecond photo-induced refractive index variations in glasses requires the observation of the spatial characteristics of the imprinted structures. For creating the structures we use a NA=0.45 objective, considered as a trade-off between a long enough working distance and a low enough input energy in order to decrease nonlinear optical propagation effects in the material . Consequently, we believe that the size of the modified zone of the material is mainly governed by aberrations since they establish dominantly the exposure region. Nevertheless, nonlinear effects should carry a certain contribution in redistributing energy within the exposed area due to competing focusing and defocusing self-induced phenomena. We will concentrate in the following on estimations coming from linear propagation behavior. The consequences of an aberrated wavefront on the nonlinear propagation are not quantified here. Investigations on the nonlinear contribution to the size and aspect of the modifications are in progress and will be treated elsewhere.
Two types of glasses are investigated. Fused silica is characterized by a refractive index of 1.453 at an input wavelength of λ=800 nm. The forbidden energy gap of approximately 9 eV can be bridged during laser excitation by a k=6 multiple photon absorption process at 800 nm incident radiation. This glass exhibits dominantly a positive index variation under femtosecond irradiation due to a post-irradiation material compaction . BK7, with a refractive index of 1.511 at λ=800 nm, and a k=3 photon absorption process, has a different behavior and exhibits dominantly a negative index variation in standard ultrafast irradiation conditions. Though the mechanisms responsible for this behavior are not the focus of this work, it can nevertheless be related to a very efficient thermal expansion. Observations are performed using a Zernike type positive optical phase contrast microscope, similar to the set-up detailed in [6, 26]. The choice of phase contrast microscopy is natural for observations of phase objects and has a higher sensitivity than standard optical microscopy techniques.
Shortly, polished fused silica and BK7 parallelepiped samples are irradiated at atmospheric pressure. 170 fs 800 nm laser pulses generated by a Ti:sapphire amplified laser system are employed at a repetition rate of 100Hz. Single pulses or controllable sequences of pulses are selected with the help of an electro-mechanical shutter. Care is taken that the laser illuminates uniformly the aperture of the focusing microscope objective. The objective is not specially designed to correct for optical distortions induced by refraction at air-dielectric interfaces. In particular, chromatic aberrations induced by the 8 nm FWHM spectrum are negligible compared to geometrical ones. The observations are made perpendicular to the propagation axis of the laser pulse at different depths with respect to the air-dielectric interface. Phase contrast microscopy delivers a two dimensional map of the phase object where the amplitude of the index variation relative to the background value corresponds to a grey level in the color map.
Typical images of index variations induced in a-SiO2 and BK7 by single and multiple pulses of 170 fs at different depths are given in Fig. 4 for a certain range of input energies. Illustrative for the elongation effect of aberrations, the affected regions extend on several tens of µm depending on the focusing depth, and several gray level regions are visible. A black region corresponds to a positive refractive index change, while the white regions are equally attributed to a decrease of the refractive index or to the appearance of a light scattering center. It was shown before that white regions can be associated with a maximum rate of energy deposition and highest temperatures achieved in the material  due a nonlinear energy redistribution. For both types of glasses, the structures produced deeper in the bulk are accompanied by an increased modulation in the structure appearance.
The length of the modified area as well as the distance from paraxial to best focus are experimentally characterized. The length is defined as being the maximal dimension of a continuous modification. As depicted in Fig. 4, phase contrast images are highly different for both glasses and we point out the difficulty to characterize accurately the irradiated area because of multiple index variations along the beam path. Moreover, the size of the modified zone is energy and pulse-number dependent. Nevertheless, these data illustrate how significant the influence of spherical aberration may be in femtosecond photo-writing processes. Indeed, experimental characterization is conducted as follows: for a given depth, several measurements are performed in the same conditions (same pulse number Num and same energy) and typical measured dimensions are averaged. In Figs. 5 and 6, an input energy of 2.7 µJ is considered. This value corresponds to the energy necessary to keep above photo-writing threshold up to a depth as high as 2 mm for which the light distribution is highly elongated by spherical aberration. Data for pulse number Num=1, 10 and 1000 are plotted. In addition, one should mention that we have also measured the relative increase in size for several depths at various energy levels and the ratio appears not to strongly depend on energy within the measurements errors. Results obtained for both glasses are very close. Consequently, for clarity, only data obtained for SiO2 are plotted in Figs. 5 and 6.
The length of the index variation area is estimated within an error bar of 10% characterizing the difficulty to define a sharp border of the modified zone (Fig. 5). The measurements are normalized to the value obtained at the deepest position. Indeed, deep in the bulk and for NA=0.45, the length of the modified region is mainly governed by aberrations, and not by diffraction effect. Consequently, the measurements performed at the deepest position are a reliable reference for aberration characterization. It is to be noted that a straightforward estimation of the distance from linear to non linear focii  is only about 30% of the measured length of the modified area, which confirms that nonlinear effects mainly govern a redistribution of energy within the exposed area and that geometrical aberrations have a significant contribution to its size. Theoretical predictions of the LSA given in sections 2–3 are added to the graph, together with a bottom limit defined by the confocal parameter of the beam (Eq. (9)), and not described by geometrical optics considerations. The longitudinal extension of the processed area only weakly deviates from the theoretical predictions of sections 2–3, despite the fact that the model does not account for multi-photon excitation. Indeed, the multiphoton excitation order would normaly reduce the length by restricting excitation only to the high intensity regions. Nevertheless remaining non linear optical effects and particularly channeling seem to balance this shortening.
The distance from paraxial to best focus as a function of depth is reported in Fig. 6. The best focus is the position that first overpasses the index variation threshold in a linear approximation. It is thus determined at low input energy as being located within the central region of the lowest energy structures still observable under phase contrast microscopy. Anyway, even at low energy, the modified area is not punctual, which explains an error bar of 20%. We mention here that the nonlinear energy redistribution within the illuminated area would make it impossible to associate best focus position with the area of maximum modification, leading us to the above-mentioned approximation. The experimental paraxial focus is supposed to coincide with the beginning of the modified area. Consequently, at low depth, its distance to the best focus is mainly given by half of the confocal parameter of the incident beam. The general slope of the distance to the best focus as a function of depth is only weakly over-estimated by Eq. (10). Non-linear effects appear not to affect significantly this data, which is of particular interest when an optimization of the maximum induced index variation is involved.
The main consequences of spherical aberration in the case of longitudinal femtosecond photowriting, which are a lengthening of the photo-induced area with depth together with a shift in the position of the best focus are well described by analytical equations detailed in sections 2–3. In the following section, these equations will be used to predict the benefit of any correction of spherical aberration for femtosecond longitudinal photowriting in bulk glasses, in particular concerning spatial resolution.
5. Correction of spherical aberration
In this section, we are interested in the maximum NA which still ensures a nearly diffraction limited beam, i.e. a Strehl ratio Rs above 0.8 as defined by the Rayleigh criterion , for various corrections of spherical aberration, and for various depths. In Eq. (5), the aberrated OPD Δaberr is expressed as a function of Zernike polynomials. Supposing that the first N Zernike polynomials are corrected, the remaining OPD after correction is Δremain:
with An0 given by Eq. (6).
For a given depth, by introducing Eq. (12) in Eq. (11) and solving numerically as An0 is also NA-dependent, one obtains the maximum NA that still satisfies Rs>0.8. We checked that this result is in good agreement with the direct resolution of Rayleigh criterion, specifying that the maximum peak-valley OPD affected by spherical aberration that ensures Rs>0.8 is λ/4 .
This problem may be expressed in another way to find the minimum spot diameter achievable at a given depth for a given correction, which also provides the spatial optical resolution of the system. Indeed, the minimum spot diameter corresponds to the Airy disk diameter ϕ=1.22λ/(2NA) FWHM for the maximum tolerable NA. The spatial optical resolution is plotted in Fig. (7) for various values of N. As refractive indices of BK7 and fused silica are close, the two curves nearly overlap. Moreover, it appears that N=6, i.e. only 3 Zernike polynomials, is largely enough to correct nearly all spherical aberration for depth up to several millimeters. It is to be noted that Fig. (7) does not describe the minimum size of modified area in the material but the minimum size of the optical spot. Physical phenomena inducing the index variation are not described here.
Additionally, one can determine the maximum NA tolerable for a given correction to obtain a constant peak index variation whatever the depth may be, up to a given value. This result is of prime interest to ensure a good homogeneity in longitudinally photowritten waveguides. For doing this, Rs is simply replaced by R′S=RkS with k being the number of near-infrared photons necessary to bridge the bang gap of the material. The results, plotted in Fig. (8), confirm that, for both glasses, homogeneous ~cm-long waveguides may be longitudinally photo-induced with a NA=0.45 objective as long as spherical aberration is corrected up to N=6.
An experimental investigation of the ways to compensate for these aberrations using a spatial beam shaping set-up that has already proven its efficiency in the frame of femtosecond micromachining [28–30] is in progress.
We presented a detailed analysis of the effect of spherical aberration in ultrafast laser-induced refractive index variation in glass, in particular when longitudinal photowriting processes are involved. Analytical predictions that were derived for moderate NA values are in good agreement with experimental data measured in fused silica and BK7, two widely used glasses that exhibit respectively positive and negative index variations under femtosecond near-infrared irradiation in usual conditions. We studied theoretically the effect of a correction of spherical aberration focusing on the spatial resolution achievable at a given depth and the condition on the numerical aperture to obtain a homogeneous longitudinally-photowritten waveguide. The application of this analysis to programmable correction of aberrations for femtosecond processing in transparent media is in progress.
We would like to thank T. Lépine for illuminating discussions. The financial support of GIP-ANR and of PICS 2005 is gratefully acknowledged.
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