1. Introduction

For many years lithium niobate (LiNbO₃) has been among the most widely used artificial materials in photonics [1]. The continuing popularity of LiNbO₃ in this area originates from its highly desirable properties, such as large electro-optic, acousto-optic, piezoelectric, and nonlinear optical coefficients, and also an important fact that this ferroelectric can be rather easily produced in a single-domain state [2, 3].

The significant progress in micro-structuring of LiNbO₃ crystals by means of electric field poling [4], metal diffusion [5], proton exchange [6], low- and high-energy ion implantation [7], focused ion beam milling [8], reactive ion etching [9], wet etching [10], optical grade dicing [11], and light-mediated domain engineering [12] has enabled the fabrication of various photonic devices. However, none of these techniques offers genuine capability to microstructure LiNbO₃ crystals in 3-D. Indeed, electric field poling allows only z-cut crystals to be efficiently poled [13], while the remaining techniques (except dicing), are inherently restricted to shallow (i.e., less than 10 μm) near-surface layers.

The technique of direct femtosecond laser writing can potentially overcome all these issues. The 3-D writing capability is derived from nonresonant multiphoton absorption of laser light inside transparent media. For instance, linear absorption in LiNbO₃, both pure and doped, is extremely weak in the near-infrared [14] and hence two or more photons must be absorbed simultaneously to bridge the relatively wide energy bandgap (~4 eV) of this material. The high intensity required for this nonlinear process is achieved by focusing the femtosecond pulses tightly. The material surrounding the focal volume then remains unaffected by the light passing through it, which allows one to inscribe structures at any depth in 3-D fashion [15]. As far as LiNbO₃ is concerned, this fundamental concept has been widely exploited in laser writing of Y-splitters [16], Bragg gratings [17, 18], optical waveguides [19–26], voxels for 3-D data storage [27, 28], photonic crystals [29, 30], structures for quasi phase matching [31], and ferroelectric domain engineering [32].

Despite the extensive literature on femtosecond laser writing in bulk LiNbO₃, high-resolution studies of the laser-induced modification morphology inside this material are still scant [33–35] and the very fact that the writing actually takes place in a birefringent medium is mostly ignored with some rare exceptions [36].

In an attempt to fill in the mentioned gaps, we firstly employ second-harmonic generation (SHG) microscopy to investigate how the morphology of laser-induced changes in crystalline LiNbO₃ samples is affected by the orientation of the pulse propagation direction with respect to the crystal optical axis. The theoretical model presented in the text allows us to accurately predict the shape and size of the laser-modified regions in LiNbO₃ when the focused light intensity of the laser pulses is near the threshold for modification and their peak power does not exceed the self-focusing threshold. Secondly, by using scanning electron microscopy (SEM) we demonstrate that laser writing performed in multi-shot irradiation mode results in the formation of polarization-sensitive periodic planar nanostructures inside both pure and MgO-doped LiNbO₃.

2. Theory

The LiNbO₃ crystal is a member of the trigonal crystal system wherein two different unit cells – hexagonal or rhombohedral – can be chosen. To describe the optical properties of LiNbO₃ it is more natural to use the hexagonal unit cell characterized by the c-axis about which the crystal exhibits three-fold rotation symmetry [2]. The c-axis coincides with the optical axis of this negative uniaxial crystal.

Propagation of plane electromagnetic waves in anisotropic media, including uniaxial crystals, is a standard topic in textbooks on optics. Without exception, the pertinent analysis is restricted to the propagation of a single wave and therefore generally unsuitable to describe the behavior of laser beams, which represent a superposition of an infinite number of plane waves whose wave vectors are oriented along different directions.

Journal articles offer several approaches to address this problem. Paraxial wave equations for beam propagation along an arbitrary direction inside uniform uniaxially anisotropic media were derived as early as 1983 [37]. It was shown that an ordinary-wave beam obeys a standard paraxial equation, whereas an extraordinary-wave beam is described by a paraxial equation that introduces both a transverse shift in the beam position and a rescaling of the transverse coordinate in the principal plane. Much later, Ciattoni et al obtained a plane-wave angular-spectrum representation of a vector beam inside uniaxially anisotropic media [38]. They also deduced paraxial expressions for its ordinary and extraordinary field components, and presented two decoupled parabolic equations that they obey [38]. In principle, the angular-spectrum representation allows one to obtain the distribution of a light field in the half-space z > 0 if the field is known at the reference plane z = 0. In reality, this and other relevant papers of Ciattoni et al [39–42] exclusively concentrate on the evolution of diverging beams inside uniaxial media as the z = 0 plane always coincides with the respective beam waists. Focusing of electromagnetic waves through a plane interface into a uniaxial medium with an arbitrary orientation of the optical axis was considered by Stamnes et al [43], where the authors derived explicit, albeit analytically complex and difficult to manipulate, expressions for the dyadic Green’s functions associated with the transmitted fields and obtained integral representations suitable for asymptotic analysis and numerical simulations.

Notwithstanding the obvious fact that some important laser applications (e.g., parametric light conversion, laser writing, nonlinear microscopy) deal with beam focusing inside optically anisotropic media, it was not until the late 2000’s that sequences of 2-D intra-focal intensity distributions inside different uniaxial crystals were clearly shown for two basic situations when a laser beam propagates along or perpendicular to their optical axes [44, 45].

Here, we consider these two most common geometries of beam propagation in the context of direct laser writing inside uniaxial media. We specifically concentrate on finding analytical expressions, which would allow one to straightforwardly estimate the light intensity distribution in the focal volume and predict how it may affect the capabilities of this technique. We also note that the previous analysis of the orientation-dependent femtosecond laser writing in uniaxial media (i.e., differently oriented LiNbO₃ crystals) [35], is generally inadequate to describe the situation when the pulses propagate along the optical axis and in contradiction to theory when the pulses propagate perpendicular to the optical axis [42–45].

In this work, the model equations governing the propagation of femtosecond laser pulses in LiNbO₃ crystals do not include any nonlinear terms. Many aspects of nonlinear pulse propagation in different isotropic media, such as gases and transparent solids, can be found in a review article by Couairon and Mysyrowicz [46]. However, applying the more rigorous models discussed in Reference [46] to an optically anisotropic medium would involve very complex numerical simulations, which could obscure the essence of the easily observable effects (see the experimental section) that directly follow from our linear model.

Our analysis of laser beam propagation in uniaxial media is performed in Cartesian coordinates (x, y, z) for the negative LiNbO₃ crystal, for which the ordinary refractive index n_o is larger than the extraordinary refractive index n_e (n_o < n_e for positive uniaxial crystals, e.g., MgF₂, YVO₄, GdVO₄, YLiF₄). The z-axis is always oriented parallel to the optical axis and the front surface of the crystal is cut perpendicular to the beam propagation direction.

A linearly polarized Gaussian beam is focused through a planar interface separating an ambient optically isotropic medium (vacuum) and a LiNbO₃ crystal, giving rise to a transmitted beam propagating inside the crystal. The propagation characteristics of this beam are analyzed in the paraxial approximation for two cases: the incident Gaussian beam impinges on the crystal 1) parallel and 2) perpendicular to its optical axis. The first case is somewhat simpler as it is insensitive to how the plane of polarization of the incident Gaussian beam is oriented – the transmitted beam remains completely unaffected by rotations of the crystal about the optical axis (i.e., z-axis). This is not true of the second case.

1) The transverse electric field vector E_⊥ of a beam propagating inside a uniaxial crystal along its optical axis, as shown in Fig. 1, can be written as $E_{\perp }=\tilde{E}{e}^{i(k{n}_{o}z-\omega t)}$, where the complex amplitude $\tilde{E}$ is a solution of the paraxial wave equation for a linear medium [47]:

 

Fig. 1 Beam splitting inside a negative uniaxial crystal when a focused Gaussian beam impinges on the crystal parallel to its optical axis. The incident beam always splits into two beams denoted by red and blue (Gaussian beams G₁ and G₂ in the text, respectively). The effective refractive index of the crystal for the red (blue) beam is n_o (n_e²/n_o < n_o). As a consequence, the red beam is focused farther from the front face of the crystal than the blue one. (E) denotes the electric field vector.

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It appears that after entering the crystal the incident Gaussian beam splits into two separate vector (i.e., inhomogeneously polarized) beams, as shown in the Fig. 2. The distances from the front face of the crystal to their waists are different because G₁ and G₂ have different Rayleigh ranges, which are respectively given by $\pi w_0^2{\lambda }^{-1}{n}_o^{}$and $\pi w_0^2{\lambda }^{-1}{n}_e^2{n}_o^{-1}$. For this geometry we define the focusing depth d as the distance measured along the z-axis from the front face of the crystal to the point lying halfway between the waists of these vector beams. This mid-point defines the reference point z = 0 and thus allows us to write $c_1=-c_2\equiv a$, where$a=d(n_o^2-n_e^2)/(n_o^2+n_e^2)$. For the sake of brevity, all the beams in this text are identified by their Gaussian envelopes; hence the beams G₁ and G₂.

 

Fig. 2 Beam splitting inside a negative uniaxial crystal when a focused Gaussian beam impinges on the crystal perpendicular to its optical axis. Generally, the incident beam splits into three beams denoted by red, blue and green (beams G, g₁, and g₂ in the text, respectively). The effective refractive index of the crystal is n_o for the red beam, n_e for the blue beam, and n_o²/n_e for the green beam. The effective refractive indices determine the corresponding beam waist positions inside the crystal.

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2) The Cartesian components of the transverse electric field vector of a beam propagating inside a uniaxial crystal perpendicular to its optical axis (see Fig. 2) can be written as $E_x={\tilde{E}}_x{e}^{i(k{n}_{o}y-\omega t)}$ and $E_z={\tilde{E}}_z{e}^{i(k{n}_{e}y-\omega t)}{e}^{i\Delta \varphi }$, where the complex amplitudes ${\tilde{E}}_x$ and ${\tilde{E}}_z$ satisfy the following two paraxial wave equations [42]:

The form of Eq. (3)b) suggests that we can write ${\tilde{E}}_z$ as ${\tilde{E}}_z(x,y,z)=g_1(x,y)g_2(z,y)$, where the functions g₁ and g₂ obey the one-dimensional equations $({\partial }^2/\partial x^2+2ik{n}_e\partial y)g_1(x,y)$and $({\partial }^2/\partial z^2+2ik{n}_e^{-1}{n}_o^2\partial y)g_2(z,y)$, respectively. As above, the Gaussian nature of the incident beam implies that the solutions of these equations are Gaussian envelopes $g_1={(E_0/{\xi }_x)}^{1/2}\exp \{-x^2/(w_0^2{\xi }_x)\}$ and $g_2={(E_0/{\xi }_z)}^{1/2}\exp \{-z^2/(w_0^2{\xi }_z)\},$ where ${\xi }_x=1+i\lambda (y+a_2)/(\pi n_e{w}_0^2)$, ${\xi }_z=1+i\lambda n_e(y-a_1)/(\pi n_o^2{w}_0^2)$, and the constants a₁ and a₂ determine the positions of the waists of g₁ and g₂. Since the reference point y = 0 coincides with the waist of G, $a_1=d(n_o-n_e)/n_e$ and $a_2=d(n_o-n_e)/n_o$.

When the incident beam is linearly polarized and the plane of polarization makes an angle γ with the optical axis (i.e., the z-axis) the Cartesian components E_x and E_z of the field inside the crystal can be written as:

One can see that the incident beam after entering the crystal splits into a superposition of two linearly polarized (i.e., homogeneously polarized) beams. The superposition now consists of a regular Gaussian beam G (expression (4a)) and an astigmatic beam (expression (4b)) represented as a product of two “one-dimensional” Gaussian beams g₁ and g₂. The Rayleigh ranges of G, g₁, and g₂ are $\pi w_0^2{\lambda }^{-1}{n}_o^{}$, $\pi w_0^2{\lambda }^{-1}{n}_e^{}$, and $\pi w_0^2{\lambda }^{-1}{n}_o^2{n}_e^{-1}$, respectively.

The expressions (2) and (4) provide a remarkably simple tool to analyze light distributions in uniaxial media produced by converging linearly polarized Gaussian beams. Moreover, these expressions can also be easily extended to the case when the incident Gaussian beam has an arbitrary homogeneous polarization state.

Some of the important characteristics of the transformations occurring with a linearly polarized Gaussian beam when it enters a crystal of pure LiNbO₃ along and perpendicular to its optical axis are presented in Fig. 3. To match the numerical simulations in Fig. 3 with our experimental conditions (see below), the wavelength and the waist radius of the incident beam are λ = 0.8 μm and w₀ = 0.4 μm, respectively. This implies that the numerical aperture (NA) of our focusing optic is NA = 0.65. The ordinary (extraordinary) refractive index of pure LiNbO₃ at this wavelength is n_o = 2.255 (n_e = 2.176) [51].

 

Fig. 3 Simulations of focus splitting inside a LiNbO₃ crystal when a linearly polarized Gaussian beam enters the crystal (a) along and (b) perpendicular to the optical axis (i.e., z-axis). In (a) and (b) the peak light intensity is normalized to unity at each focusing depth. The incident polarization is along the x-axis in (a) and at an angle γ = π/14 with respect to the optical axis in (b). The bottom panels of (a) and (b) show cross sectional intensity distributions of the respective vector beams at d = 300 μm. Intensity profiles after an analyzing polarizer, as indicated by the orientation of (E), are also provided. k denotes the beam propagation direction.

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Strictly speaking, a beam whose minimum spot size is comparable with the wavelength of light becomes nonparaxial, with the degree of nonparaxiality being determined by a parameter $f={(k{w}_0)}^{-1}$ [52, 53].

Nevertheless, it was shown that the electric field of a weakly nonparaxial beam (f < 0.5) propagating in a uniaxial crystal can be presented as a sum of the dominant component corresponding to the fully paraxial case and a small correction caused by the nonparaxiality [42], which we neglect. We also note that the expressions (2) and (4) provide physically accurate results only when the focal volume of the transmitted beam is well inside a birefringent medium. To avoid a scenario when one half of the focal volume is confined inside a LiNbO₃ crystal, whereas the other half is in vacuum, the minimum focusing depth in Fig. 3 is 10 μm.

The most salient feature of the intensity distributions shown in Fig. 3 is focus splitting along the beam propagation direction. For clarity, we again consider the propagation 1) along and 2) perpendicular to the optical axis separately.

1) If a converging linearly polarized Gaussian beam enters a LiNbO₃ crystal along its optical axis, two foci identified with the waists of G₁ and G₂ are always produced no matter how the electric field vector of the incident Gaussian beam is oriented in the xy-plane (Fig. 3(a)). On the other hand, distinct focus splitting takes place only when the separation between the two foci (i.e., Eq. (2)a)), which linearly grows with the focusing depth d, exceeds the sum of the Rayleigh ranges of G₁ and G₂.

The intensity profiles in the xy-plane show that the front focus, which lies closest to the front face of the crystal, is elongated perpendicular to the polarization of the incident beam, whereas the back focus is elongated along it (bottom panel of Fig. 3(a)). The two orthogonally oriented ellipsoidal foci can be rotated about the optical axis as a single object by rotating the plane of polarization of the incident beam.

The spatial confinement of both the foci perpendicular to the beam propagation direction does not appreciably depend on d and is comparable to that of a Gaussian focus inside an isotropic medium with the refractive index n_e < n <n_o. The extent of the front and back focus along the beam propagation direction does not depend on d and is determined by the Rayleigh range of the beam G₂ and G₁, respectively.

The maximum intensity in each focus is observed on the beam axis where the polarization is purely linear and aligned parallel to the incident polarization, i.e., x-polarization in this case. Our simulations also show that the maximum intensity in the y-polarized four-petal patterns is approximately ten times weaker than on the beam axis (bottom panel of Fig. 3(a)).

2) If a converging linearly polarized Gaussian beam impinges on a LiNbO₃ crystal perpendicular to its optical axis, three foci identified with the beam waists of G, g₁, and g₂ are generally produced (Fig. 3(b)). This interesting phenomenon will always take place unless the polarization of the incident beam is oriented either perpendicular or parallel to the optical axis. In the first situation a single focus is produced by the beam G, whereas in the other situation two foci are generated by the beams g₁ and g₂.

The intensity distribution in the focal volume critically depends on the orientation of the input polarization with respect to the optical axis and the focusing depth d. The latter determines the longitudinal separation between the front, central and back focus via a₁ and a₂, respectively.

When the foci are well-separated the maximum intensity in the front focus is always ~10% higher than that in the back focus. The top panel of Fig. 3(b) presents a sequence of 3-D focal intensity distributions when the incident polarization makes an angle γ = π/14 with the optical axis. This angle is chosen to make the maximum intensity in the front, central and back focus approximately equal at d = 300 μm and clearly visualize the production of three distinct foci.

The front focus is always elongated along the z-axis, while the back focus is elongated perpendicular to it, independent of the orientation of the incident polarization (bottom panels of Fig. 3(b)). The transverse elongation grows with focusing depth and can become very pronounced. On the other hand, the extent of both the front and back focus along the beam propagation direction does not change with d and is determined by the Rayleigh range of the beam g₁ and g₂, respectively. The dimensions of the central focus do not depend on d at all because this focus is produced by nothing else but a converging standard Gaussian beam G whose propagation is not affected at all by the optical anisotropy of LiNbO₃.

When the three foci are well separated the polarization in the front and back focus is nearly linear and aligned along the z-axis, while the central focus is predominantly x-polarized. The scale bars in the bottom panel of Fig. 3(b) present the relative light intensities of the x- and z-polarized beam components.

Figure 4 shows how the intra-focal intensity distribution inside a LiNbO₃ crystal varies with the focusing depth d for a fixed waist radius w₀ of the incident beam (Figs. 4(a)-4(c)) and also how the intensity depends on w₀ if d is kept fixed (Figs. 4(d)-4(f)).

 

Fig. 4 Parametric analysis of focus splitting inside a LiNbO₃ crystal. A linearly polarized Gaussian beam enters a LiNbO₃ crystal along the optical axis in (a) and (d), and perpendicular to the optical axis in (b), (c), (e), and (f). On-axis intensity distributions along the beam propagation direction are shown as a function of the focusing depth d at a fixed beam waist radius w₀ = 0.4 μm in (a)–(c) and as a function of w₀ at a fixed d = 300 μm in (d)–(f). The incident linear polarization is arbitrary-angle in (a) and (d), parallel to the optical axis in (b) and (e), and at an angle γ = π/14 with respect to the optical axis in (c) and (f). In each panel the peak intensity is normalized to the total power in the beam P, which is kept constant in each case. Under this condition, the electric field amplitude is given by $E_0=\sqrt{2P/(w_0^2\pi c{\epsilon }_0)}$, where ε₀ is the vacuum permittivity.

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When a converging linearly polarized Gaussian beam enters a LiNbO₃ crystal along its optical axis the on-axis peak intensity quickly drops as d changes from 0 to 50 μm and then asymptotically tends to ~1/4 of the peak intensity at d = 0 (Fig. 4(a)). The existence of this well-defined limit is a direct consequence of the constant spatial confinement of both the front and back focus, which is approximately equal to that of a Gaussian focus inside an isotropic medium, as discussed earlier in the text.

When the propagation takes place perpendicular to the optical axis and the incident polarization is aligned parallel to the optical axis the peak intensity in the front and back focus steadily decreases with d as the foci become more and more elongated perpendicular to the light propagation direction (Fig. 3(b)). On the other hand, if the incident polarization is set at an angle γ with respect to the optical axis, the central focus (beam G), whose peak intensity is proportional to ${\sin }^2\gamma$, will appear. The ratio of the peak intensities in the front/back and central focus varies with the focusing depth and at sufficiently large d’s the highest intensity will be observed in the central focus (Fig. 4(c)) because its peak intensity does not change with d.

Another parameter that strongly affects the intra-focal intensity pattern inside LiNbO₃, as well as any uniaxial media, is w₀. In Figs. 4(d)-4(f), this dependence is presented for d = 300 μm. At this focusing depth, distinct focus splitting in LiNbO₃ occurs even for a relatively loosely focused incident beam if it propagates along the optical axis. Conversely, to observe this effect for light propagation perpendicular to the optical axis the incident beam needs to be focused relatively tightly. This becomes especially true if there is even a small angle γ between the incident polarization and the optical axis (Fig. 4(f)).

The large difference between n_o and n_e in LiNbO₃ allows focus splitting to take place at moderate NA’s of the focusing optics and small focusing depths, representing the parameter space within which femtosecond laser writing in this material is usually carried out. For negative optically uniaxial materials with a stronger birefringence, such as CaCO₃, focus splitting will be easy to observe at even smaller focusing angles and shallower depths. On the other hand, sapphire – an important negative crystal – has an order of magnitude lower birefringence than LiNbO₃ and to observe focus splitting and other effects associated therewith in this case the beam has to be focused several hundreds of micrometers below the surface.

We note that although we do not specifically discuss any arbitrary states of polarization of the incident Gaussian beam, except linear state of polarization, it is assumed that the effect of arbitrary elliptical polarization can be seen as a superposition of two linearly polarized states of the same beam. The effect of each of these states separately is described by the expressions (2) and (4). We also note that the expressions (2) and (4) can be used to analyze intra-focal light distributions inside positive uniaxial media.

Finally, we would like to mention that the vacuum-LiNbO₃ interface introduces spherical aberration, which distorts the wave front of the incident beam and can cause significant spreading of the intensity distribution along the beam propagation direction near the focus. This interface-induced spherical aberration, which takes place irrespective of whether a beam is focused into an isotropic or anisotropic medium, has been extensively studied both theoretically and experimentally [54–57]. In the above simulations we assume that a predetermined wave front distortion has been imposed onto the incident wave front in order to compensate for the interface-induced aberration. In laser writing it is done by using either specialty objectives that allow adjustable compensation for spherical aberration or spatial light modulators [30, 58].

3. Results and discussion

To check how accurately the above theoretical model can predict the shape and size of laser-induced changes in bulk LiNbO₃ we used both z-cut samples (i.e., the z-axis is perpendicular to the front face) and y-cut samples (i.e., the y-axis is perpendicular to the front face). The laser writing was performed in ambient air using two different Ti:sapphire regenerative amplifiers. One operated at a central wavelength of 800 nm, pulse duration τ_p = 250 fs (FWHM) and pulse repetition rate of 100 kHz, whereas the other operated at a central wavelength of 775 nm, τ_p = 200 fs and pulse repetition rate of 10 Hz. In each case, the linearly polarized output beam was spatially filtered and then focused inside different LiNbO₃ samples with a NA = 0.65 microscope objective (O1) allowing depth-dependent compensation of spherical aberration induced by the air-LiNbO₃ planar interface (Fig. 5(a)). This objective (i.e.,LCPLN50XIR) is designed for inspecting internal structures in silicon (Si) wafers and has a correction collar that adjusts for varying thicknesses of silicon (and glass) substrates. The correction collar calibration for LiNbO₃ was obtained by maximizing the peak intensity of the reflected signal from the front and back surface of a LiNbO₃ sample of known thickness.

 

Fig. 5 (a) Schematic of the laser-writing setup: G – Glan polarizer, λ/2 – achromatic half-wave plate, O1 – NA = 0.65 microscope objective allowing depth-dependent compensation of spherical aberration. (b) Schematic of the scanning SHG microscopy setup: O2 – NA = 0.85 dry microscope objective to focus the 830 nm excitation beam, O3 – NA = 0.9 objective to collect the 415 nm SHG signal, F – band-pass optical filter to remove the excitation light, PMT – photomultiplier tube to detect the SHG signal. In (a) and (b) CR denotes a MgO-doped LiNbO₃ crystal.

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To vary the pulse energy E_p and the orientation of the plane of polarization of the incident pulses with respect to the crystal axes we used a Glan polarizer (G) sandwiched between two achromatic half-wave plates (λ/2). The low pulse repetition rate amplifier was used to conveniently induce single-spot modification with a low number of pulses (i.e., 1-10 pulses per spot), whereas the high pulse repetition rate amplifier allowed us to write either continuously by moving the sample with respect to the focused laser beam or induce single-spot modification with a large number of pulses (i.e., ~10⁵ pulses per spot).

A) To visualize laser-modified regions we employed scanning SHG microscopy, which was previously shown to be able to map the naturally occurring distribution of second-order nonlinearity and microstructure inside LiB₃O₅ crystals, such as regions of amorphized material and crystal fracture defects [59]. SHG microscopy was also successfully used in the characterization of He⁺-implanted KTiOPO₄ waveguides [60] and, which is directly relevant to our studies, ultrafast laser-written waveguiding structures in LiTaO₃ [61] and in a Nd:YVO₄ + KTiOPO₄ hybrid system [62].

Our commercial scanning SHG microscopy setup is schematically presented in Fig. 5(b). Linearly polarized pulses at a repetition rate of 80 MHz generated with a Ti:sapphire oscillator were focused onto the sample using a NA = 0.85 dry microscope objective (O2). The central wavelength of the pulses was 830 nm. The pulse duration at the sample was minimized to the nominal 80 fs FWHM by means of the oscillator’s dispersion compensation scheme designed to cancel the chromatic dispersion of the optical elements – mainly the focusing microscope objective – in the beam path. The forward-propagating light, which included the 830 nm excitation beam and the SHG signal at 415 nm, was collected with a NA = 0.9 objective (O3) and after a band-pass optical filter (F) directed to a photomultiplier tube (PMT) for detection. The detected light originated from an illuminated volume element within the sample. 2-D SHG images from selected depths were acquired by scanning the excitation laser beam. To produce a 3-D SHG image of the sample from the 2-D SHG images captured at different depths, the sample was translated along the excitation beam propagation direction by predetermined increments. The lateral and axial resolutions of the setup are estimated at ~0.5 μm and ~3 μm, respectively.

Obviously, the propagation of a focused excitation beam in LiNbO₃ should be influenced by the birefringence of this material in the context of our previous theoretical analysis. In the case of z-cut LiNbO₃, any homogeneously polarized excitation beam always produces two foci, which detrimentally affects the point spread function (PSF) of the SHG microscopy setup. Only when the excitation beam is radially or azimuthally polarized can a single focus be produced inside a uniaxial medium at any focusing depth d [41]. To minimize the deterioration of the PSF caused by d-dependent focus splitting, the excitation beam in our SHG microscopy studies of z-cut LiNbO₃ was always focused through the surface lying closest to the laser-induced modification. Specifically, i) in order to image modification produced near the front surface, the excitation beam was focused through the front surface (i. e., small d for both the laser-writing and excitation beam), ii) to image modification produced near the back surface the excitation beam was focused through the back surface (i. e., large d for the writing beam and small d for the excitation beam).

On the other hand, we could completely avoid these issues in our studies of y-cut LiNbO₃ by simply aligning the polarization of the excitation beam perpendicular to the optical axis of the samples. By definition, this polarization ensures that the excitation beam has a single focus irrespective of d.

Figure 6 shows 3-D SHG microscopy images of laser-modified regions in 5mol% MgO-doped z-cut LiNbO₃ crystals. The addition of MgO to LiNbO₃ significantly increases its resistance to the photorefractive damage and thus makes it suitable for high-power practical applications [1–3]. The structures were produced by irradiating single spots inside the material with five 200 fs FWHM pulses generated by the low pulse repetition rate amplifier. The images clearly show the presence of a single modified region near the front surface (i.e., d = 10 μm) and two distinct laser-modified regions aligned along the writing beam propagation direction at a large focusing depth (i.e., d = 340 μm), as predicted. The modification near the front surface was written at a pulse energy of 20 nJ, whereas the two-component structure near the back surface was produced with 80 nJ pulses. This scaling factor in the pulse energy E_p was introduced because the predicted peak intensity in the laser focus near the front surface is ~4 times higher than that in each of the two foci (beams G₁ and G₂) formed deep inside the sample. Under our experimental conditions, permanent structural changes that cannot be removed by annealing the sample for one hour at 250°C occurred at E_p ~10 nJ near the front surface and at E_p ~40 nJ at d > 50 μm. Additionally, we found that the separation between the two components of the laser-induced modification could indeed be predicted using the expression $2a=2d(n_o^2-n_e^2)/(n_o^2+n_e^2)$ (see Fig. 1), with the ordinary and extraordinary refractive indices for 5mol% MgO-doped LiNbO₃ being n_o = 2.252 and n_e = 2.167, respectively [50]. We note that the peak power P ~E_p/τ_p even in 80 nJ pulses did not significantly exceed the critical power for self-focusing P_cr for LiNbO₃, which was measured to be ~0.3 MW at λ = 800 nm [24, 63]. This implies that the wave Eq. (1), which was in fact derived to describe light propagation in linear media, can be used to correctly explain the main features of femtosecond laser writing inside z-cut LiNbO₃ under weakly nonlinear conditions.

 

Fig. 6 Comparison between SHG microscopy images of modification produced in z-cut MgO-doped LiNbO₃ after irradiation of one spot with five pulses and simulations based on the expressions (2). (a) modification near the front surface: E_p ~20 nJ. (b) modification deep inside the material: E_p ~80 nJ. In (a) and (b) τ_p = 200 fs.

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Writing in y-cut LiNbO₃ with pulses whose polarization is aligned perpendicular to the optical axis (i.e., x-polarized pulses) results in the formation a single focus (beam G) at any d (Figs. 7(a) and 7(b)). As a consequence, this allows one to easily suppress nonlinear effects by using low pulse energies. Indeed, the modification shown in Figs. 7(a)-7(c) was produced at E_p ~20 nJ, i.e., at P ~1/3 P_cr. The five-pulse energy threshold for inscribing permanent changes in this writing geometry was ~10 nJ irrespective of d.

 

Fig. 7 Comparison between SHG microscopy images of modification produced in z-cut MgO-doped LiNbO₃ after irradiation of one spot with five pulses and simulations based on the expressions (4). The incident polarization is along the x-axis in (a) and (b), along the z-axis in (c) and (d), and at an angle γ = π/14 with respect to the z-axis in (e) and (f). E_p ~20 nJ in (a), (b), (c), and (e); E_p ~1 μJ in (d) and (f). In (a)-(f) τ_p = 200 fs.

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On the other hand, it is difficult to avoid nonlinear effects when the incident polarization is aligned parallel to the optical axis (i.e., z-polarized pulses). For this geometry, the peak intensity in the front and back focus decreases with d quite rapidly and in order to produce any material modification in the respective focal region the pulse energy needs to be scaled up accordingly (Fig. 4(b)), easily making the peak power P in the pulse greater than P_cr. Moreover, the light propagation at P > P_cr can also be affected by the highly elliptical cross-sectional profile of the beam [64–66].

For instance, we noticed that modification at d = 440 μm solely occurred in the front focus, whereas according to the simulations presented in Fig. 4(b) the peak light intensity in the front focus is only slightly higher than in the back focus. In fact, to initiate optical breakdown in the vicinity of the back focus E_p had to be increased to ~1 μJ (Fig. 7(d)), whereas the energy threshold to induce permanent modification near the front focus was ~100 nJ, which, in fact, translates into P ~1.5P_cr. Interestingly, the shape and size of the produced modification could be still accurately described by the linear model. For instance, the axial separation between the modified regions produced by the front and back focus was in close agreement with the predicted value given by $a_1+a_2=d(n_o^2-n_e^2)/(n_o^{}{n}_e^{})$(see Fig. 2).

When the incident polarization was aligned at γ = π/14 with respect to the optical axis three distinct modified regions could be created by the front, central and back focus under certain conditions (Fig. 7(f)). As in the case of z-polarized pulses, only the front and central foci were able to modify the material at the threshold E_p, which was slightly above 100 nJ, and E_p ~1 μJ was required to inscribe the expected three-component structures.

To check whether this apparent disagreement between theory and experiment for y-cut LiNbO₃ was indeed caused by the nonlinear pulse propagation we repeated the experiment with z-polarized pulses at progressively decreasing peak powers, while keeping the corresponding focused intensities above the material modification threshold. Three pulse durations – 250 fs, 1 ps, and 2 ps – were used in that experiment. A pulse energy of 300 nJ was chosen to ensure that the peak power was well above, comparable to, and significantly lower than P_cr. As before, the pulses were focused at d = 440 μm. This time, however, we used the high pulse repetition rate amplifier and each spot was exposed to ~10⁵ pulses. The results are presented in Fig. 8. The irradiation with 250 fs pulses resulted in a single modified region whose orientation implies that it was produced by the front laser focus. With 1 ps pulses we were able to cause permanent changes in LiNbO₃ in the vicinity of both the front and back laser foci. However, the material modification threshold for the front focus was ~1.5 times lower than for the back one. On the other hand, at τ_p = 2 ps both the front and back focus produced changes in the material at almost the same threshold value of E_p.

 

Fig. 8 SHG microscopy images of modification produced in y-cut MgO-doped LiNbO₃ after irradiation of one spot with 10⁵ 300 nJ pulses: the role of peak power P in the beam. The incident polarization is along the z-axis.

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These observations indicate that under our experimental conditions the linear model summarized in the expressions (2) and (4) can be used to accurately describe the propagation of ultrashort laser pulses in both z-cut and y-cut LiNbO₃ if their peak power is kept below P_cr. At P < 0.5P_cr the agreement between theory and experiment is spectacular.

The SHG microscopy images in Fig. 9 show how the focus splitting described in the previous paragraphs can affect continuous laser writing in MgO-doped LiNbO₃. In LiNbO₃ and other materials this mode is used for the inscription of various waveguiding structures and therefore important for many applications in photonics [16, 19–26, 31].

 

Fig. 9 SHG microscopy images of modification produced in MgO-doped LiNbO₃ in continuous writing mode. The writing was performed at a velocity |v| = 100 μm/s, with ~10³ pulses being deposited into the sample per micrometer of its travel with respect to the laser focus. (a) z-cut LiNbO₃, τ_p = 250 fs, E_p ~100 nJ, d = 340 μm. (b)-(d) y-cut LiNbO₃. τ_p = 2 ps, E_p ~300 nJ in (b) and (c). τ_p = 250 fs, E_p ~50 nJ in (d). The incident polarization is along the z-axis in (b), at an angle γ = π/14 with respect to the z-axis in (c), and along the x-axis in (d).

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For z-cut LiNbO₃, the peak intensity inside both the foci remains essentially unchanged for d > 50 μm and is ~1/4 of the peak intensity inside the unsplit focus near the front surface, as we discussed earlier in the text. If the laser beam is scanned in the xy-plane and the peak intensity inside the foci is sufficiently high to induce optical breakdown, but the peak power in the beam is low enough to avoid nonlinear effects, two distinct lines of modified material separated along the z-axis are easily produced (Fig. 9(a)). For laser writing near the material modification threshold the parameter space for this phenomenon to occur or disappear can be identified based on the simulations presented in Figs. 4(a) and 4(d).

Continuous writing in y-cut LiNbO₃ results in the production of single-, double-, and triple-line structures depending on the orientation of the incident polarization with respect to the optical axis, as we show in Figs. 9(b) and 9(c). A unique feature of this laser writing geometry – especially from a practical point of view – is that by using x-polarized pulses it allows one to inscribe single lines of modified material at any d, as if LiNbO₃ were an optically isotropic medium (Fig. 9(d)). The laser writing parameters for each situation are provided in the respective figure captions.

To conclude this section we would like to point out that there can be a significant degree of similarity between light propagation in uniaxial and biaxial crystals. The latter case is substantially more complex, however, as the index ellipsoid is now described by three unequal principle refractive indices n_x < n_y < n_z that correspond to the principal crystal axis x, y and z. Here, we will only briefly summarize the results pertaining to the situation when a converging x-polarized beam impinges on a biaxial crystal along its z-axis. For this geometry, the beam inside the biaxial crystal accumulates astigmatism on propagation and when the incident beam is focused sufficiently tightly this astigmatic beam propagation gives rise to two elongated criss-crossed foci, akin to those shown in Fig. 7(d). The elongation of the foci perpendicular to the z-axis and the separation between them along the z-axis become larger as the focusing depth increases [67, 68]. As a result, the peak intensity inside the material and the bulk damage threshold also become depth-dependent [69].

B) So far, our studies of the laser-induced modification morphology in LiNbO₃ have been performed with a micrometre resolution, which can be achieved using optical microscopy. In the last section of the article we will investigate how multi-shot irradiation of LiNbO₃ modifies its structure at the nanosclale.

Figure 10(a) shows a schematic diagram of the laser writing configuration that we used to produce lines of modified material inside MgO-doped LiNbO₃. The continuous laser writing presented in Fig. 10 was performed inside z-cut samples at d = 340 μm using three different pulse durations: 250 fs, 1 ps, and 2 ps. In each case the pulse energy was E_p ~300 nJ and the polarization was perpendicular to the writing direction.

 

Fig. 10 SEM microscopy images of nanostructural changes produced in z-cut MgO-doped LiNbO₃ in continuous writing mode. Cross section is taken parallel to the yz plane. The writing was performed at a velocity |v| = 100 μm/s, with ~10³ pulses being deposited into the sample per micrometer of its travel with respect to the laser focus. In each case E_p ~300 nJ.

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After irradiation the samples were cut perpendicular to the lines of modified material (i.e., yz-plane in Fig. 10(a)), polished and then etched in concentrated hydrofluoric acid (49% solution in H₂O) for 10 s at 20°C [70]. At room temperature, single crystal LiNbO₃ is highly resistant to HF-etching: the ± x-faces and the + z-face do not etch at all, the ± y-faces etch extremely weakly, and the etch rate of the –z-face is only ~1 μm h⁻¹ [71, 72]. On the other hand, if the physical and chemical properties of LiNbO₃ crystal are locally changed using, for 9instance, ion beam irradiation or laser beam irradiation, the etch rate of the modified regions can increase dramatically [24, 73] and thus allow their visualization by combining wet chemical etching with subsequent microscopy.

For τ_p = 250 fs and 1 ps, SEM images of HF-etched samples (Fig. 10(b)) revealed the presence of self-organized periodically-assembled nanostructures aligned perpendicular to the laser polarization [74–76]. Under those conditions the nanogratings were produced in both the front and back focus. It is also interesting to note that at τ_p = 250 fs (P ~3 P_cr) several well-defined regions containing nanostructures were formed in the vicinity of the front focus, whereas for τ_p = 1 ps (P ~P_cr) the effect vanished.

To investigate whether the formation of nanogratings was affected by the orientation of the crystal axes with respect to the writing direction we produced planes of modified material inside z-cut and y-cut LiNbO₃ crystals with 250 fs pulses, as shown in Fig. 11. The planes were engraved by overlapping several layers containing lines of modified material. The lateral separation between the lines in each layer was Δl = 1 μm, and the writing direction was kept the same for each line. The separation between the layers along the beam propagation direction was Δd = 5 μm. To study the nanomorphology of laser-induced changes using SEM all the samples were cut (depicted in dotted lines), polished and then HF-etched, as discussed earlier.

 

Fig. 11 SEM microscopy images of nanostructural changes produced in LiNbO₃ in continuous writing mode. Cross sections are taken parallel to the xy (a)-(c) and zx (d)-(f) planes. The writing was performed at a velocity |v| = 100 μm/s, with ~10³ pulses being deposited into the sample per micrometer of its travel with respect to the laser focus. (a)-(c) z-cut pure LiNbO₃, τ_p = 250 fs, E_p ~100 nJ, 300 μm < d < 350 μm. (d)-(f) y-cut MgO-doped LiNbO₃, τ_p = 250 fs, 400 μm < d < 450 μm. E_p ~100 nJ in (d) and (e), and E_p ~300 nJ in (f). The cross-sections of the front (solid line) and back foci (dashed line) are schematically presented for each writing geometry.

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Figure 11 shows SEM images of nanogratings formed hundreds of micrometers inside z-cut pure LiNbO₃ (Figs. 11(a)-11(c)) and y-cut MgO-doped LiNbO₃ (Figs. 11(d)-11(f)). In the z-cut configuration nanogratings could be easily oriented at any angle with respect to the writing direction by rotating the plane of polarization of the incident pulses (Figs. 11(a)-11(c)).

In the y-cut crystal we limited laser writing only to two orientations of the incident light polarization – parallel and perpendicular to the optical axis – because only for these two situations the polarization in the foci is homogeneous, i.e., spatially invariant. When the polarization was perpendicular to the optical axis the nanogratings looked very similar to those in the z-cut crystal (Figs. 11(d) and 11(e)). In order to write nanogratings with the polarization oriented along the optical axis we had to use much higher pulse energies to compensate for the dramatic drop in the peak intensity caused by the depth-dependent astigmatism of the laser focus (Fig. 11(f)). Interestingly, the resultant nanogratings were less regular compared with those produced in other writing configurations shown in Fig. 11, which we attribute to the anisotropic crystalline structure of LiNbO₃.

The results clearly demonstrate that subwavelength periodic planar nanostructures, aligned perpendicular to the laser polarization, were produced throughout the laser-modified regions both in z-cut and y-cut LiNbO₃, regardless of whether it is MgO-doped or pure. Having analyzed a large number of SEM images we have also found that the nanogratings were formed no matter whether the laser beam propagation direction was parallel or antiparallel to the respective crystal axis (i.e., ± z or ± y). Taking into account that these structures are self-organized rather than directly written the reproducibility is quite remarkable.

The very fact that bulk nanogratings can be consistently produced in LiNbO₃ with ultrashort laser pulses is important in itself as this type of structured changes has so far been observed in only a scant assortment of materials [74, 75, 77–79]. We also note that bulk nanogratings should be distinguished from laser-induced periodic surface structures or surface ripples [80]. The surface phenomenon was first reported in the 60’s and since then has been observed in a variety of materials, including dielectrics, semiconductors, and metals under widely different illumination conditions. Continuous and pulsed (including femtosecond) laser radiation at wavelengths ranging from the middle infrared to the near ultraviolet was used in these experiments [80]. Only much later, in 2003, did Shimotsuma et al [74] discover ultrashort pulse-induced bulk nanogratings, which still remain the smallest embedded structures ever produced by light.

4. Conclusion

We have shown how the birefringence of LiNbO₃ affects the ability of ultrashort laser pulses to induce permanent changes inside this important material. The simple analytical expressions presented in the article clearly demonstrate that a tightly focused ultrafast Gaussian beam – the current mainstay of direct laser writing – after entering a uniaxial crystal generally has two or three foci inside the medium depending on the orientation of the pulse propagation direction with respect to the crystal optical axis. Only in one special case, namely when the writing beam is linearly polarized and its polarization is aligned perpendicular to the optical axis of a x- or y-cut crystal, will a single focus be formed at any focusing depth.

We have confirmed these predictions by engraving linear and spot structures in z-cut and y-cut LiNbO₃ samples using 250 fs – 2 ps pulses and then analyzing the induced changes using SHG microscopy and SEM. Specifically, we show that if the power of the light pulses is below or near the self-focusing threshold, but their focused intensity is above the multiphoton ionization threshold, our model very accurately predicts shape and size of the laser-induced changes in bulk LiNbO₃. We also show that with repeated irradiation nanograting structures emerge written throughout the focal region. These self-assembled planar nanostructures are oriented perpendicular to the electric vector of the light and order themselves to produce spacings of approximately half of the laser wavelength in the medium.

The presented results have important implications for the whole field of direct laser writing in crystalline dielectric materials as they show that the ultimate precision of the technique in this case can be drastically decreased by multifocusing effects and other distortions of the focal intensity distribution.