1. Introduction

Femtosecond pulsed lasers enable the fabrication of optical devices in glass such as waveguides [1], diffraction gratings or Fresnel lenses [2]. Laser direct writing also allows high degree of integration of different light guiding elements [3]. Furthermore, it enables fabrication methods based in a one-step process and free of chemicals, in contrast with photolithography, which is a commonly used method for surface micropatterning. In this respect, femtosecond lasers offer an alternative method with submicron resolutions for surface and thin film patterning, as well as the possibility of 3D modification inside transparent materials [4–6].

Volume phase gratings (VPG) are diffraction gratings embedded in transparent materials that are mainly used in astrophysics, spectroscopy or telecommunications [7]. Usually, VPGs are fabricated in dichromated gelatin and work in the Bragg regime with close to 100% diffraction efficiency over narrow spectral bandwidths. In addition to their dispersive properties, VPGs embedded in glass present added advantages: greater durability, cleanability and possibility of applying coatings. There are some previous works on direct laser writing of diffraction gratings, most of them aimed at fabrication of Bragg gratings in glass, based on different methods to increase grating thickness in order to achieve higher efficiency, but they are not focused on near-field diffraction [8–10]. Systems based on near-field diffraction where Moiré fringes, Talbot patterns or Talbot-Lau patterns are formed behind gratings, have received continuous attention due to the physics involved in that phenomena and their wide range of applications such as: metrology, Moiré interferometry, laser array illumination and spectroscopy [11]. Furthermore, near-field of VPGs is a little studied field [12,13].

In this paper we explore the potential of laser direct writing for the fabrication of phase diffraction gratings in borosilicate glass with periods much longer than the wavelength and high visibility in the near-field of diffraction. We studied how laser parameters and fabrication processes affect the phase modulation of the gratings, their diffraction efficiency and the visibility of self-images, besides the characterization of near-field and far-field diffraction.

2. Theoretical considerations

2.1 Far-field diffraction

Binary transmission phase gratings can work in three different regimes of diffraction depending on the grating thickness L, period p and amplitude of the refractive index modulation Δn = |n-n’|, see Fig. 1(a): Raman-Nath Regime (thin gratings), Intermediate Regime and Bragg Regime (thick gratings) [14]. The diffractive behavior of a grating is commonly evaluated by its thickness parameter, Q = 2πλL/n_avp², and modulation parameter, γ = πLΔn/2λ; being n_av the average refractive index of the grating. Usually, gratings are considered thin if Q<1 or thick if Q>10. But, when values of Δn are low, as is the case for the gratings presented in this publication, boundaries between different regimes are not so clear and the applicability of the approximated models must be studied.

 

Fig. 1 (a) Sketch of a VPG diffracting a green laser beam. n is the refractive index of the glass and n’ is the refractive index of the modified region. (b) Calculated first order diffraction efficiency as a function of thickness by RCWA simulations (solid lines) and Eq. (1). (dashed lines). The illumination wavelength is 532 nm and s-polarized. Incidence is normal and the gratings, with no tilt, were approximated to a rectangular index profile with 10 µm of period.

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Gratings working in well-defined regimes, Bragg or Raman-Nath, can be described by approximated models that allow the solution of the inverse diffraction problem, i.e, the calculation of their parameters from diffraction efficiency. In the case of thin binary gratings, scalar diffraction theory [15] predicts that the efficiency of the mth diffraction order, η_m, depends on the phase difference Δφ = 2πΔnL/λ; which is defined by the amplitude of the binary refractive index modulation, Δn, of a region of thickness L and the illumination wavelength, see Eq. (1). However, when fabricating phase objects embedded in glass by direct laser writing, the induced refractive index change in borosilicate glass, Δn, is in the range 10⁻⁴-10⁻². Hence, for such low refractive index contrast, a grating thickness of tens of micrometers is needed to maximize the efficiency for visible or infrared illumination.

Results obtained from RCWA (Rigorous Coupled Wave Analysis) simulations and Eq. (1) are presented in Fig. 1(b) in order to show the range in which Eq. (1) is still applicable for gratings with such high thickness and low index contrast. The fabrication of gratings with thin behavior is a desired result as they have lower dependence with the angle of illumination and wavelength of the source, and hence greater tolerance to misalignments and bandwidth of the light source. Furthermore, Δφ could be estimated from diffraction efficiency for this type of gratings.

2.2 Near-field diffraction

If a periodic object, in this case a diffraction grating, is illuminated by a coherent light beam, replicated images of the object are formed at equal spaced distances along the propagation direction called Talbot distance, Z_T [16]. In the paraxial approximation, Talbot distance can be expressed as Z_T = 2p²/λ. For planar phase gratings, self-images are located at distances z = (M + 1/2)p²/λ after the grating (being M a positive integer) and their visibility has a theoretical behavior given by V = sin(Δφ) [12]. Hence, Δφ = π/2 must be achieved to reach the maximum of visibility. There is no comprehensive study about near-field diffraction of VPGs, but taking the previous case as reference, the fabrication of gratings with phase difference of π/2 is taken as the objective in this work.

2.3 Direct writing inside glass

Among the different regimes of modification that can be achieved by direct laser writing inside glass, permanent changes in the refractive index of the glass take place in the vicinity of the focal volume at moderate energies. In order to achieve gratings thick enough, an objective of 0.4 NA has been selected. Low numerical apertures yield to higher confocal parameters and, therefore, longer modified regions. Furthermore, for low NA pulse energy must be increased to achieve the modification threshold of the material and that phenomenon may lead to peak powers above the critical one for self-focusing. Consequently, under the condition for self-focusing, written lines with longer axial sections are expected [17]. Additionally, spherical aberrations also play an important role when focusing inside glass. Due to the air-glass interface, the focal volume becomes elongated and displaced from the position of the geometrical focus, F, expected in the absence of the glass substrate [18–20]. For instance, longitudinal spherical aberration defined as the distance from the paraxial focus to the marginal focus, Δ = F_PF_M, depends on the NA of the lens and the focusing depth Z; defined as the distance from glass surface, z = 0, to the geometrical focus, z = F; see Fig. 2(a).

 

Fig. 2 (a) Sketch of the main phenomena involved in the formation of structures with different refractive index inside glass. I, sketch of a self-focused beam, Z_SF is the distance for self-focusing. II, longitudinal spherical aberration Δ at a focusing depth Z. III expected modified region in the absence of self-focusing and aberrations (C.P. is the confocal parameter). IV, modified region in a general case. (b) Sketch of the fabrication process of a binary VPG in glass.

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In this work, both self-focusing and spherical aberrations were exploited in order to achieve gratings with enough thickness in just one layer. This way, issues related to axial stitching of several layers were avoided and process time was notably reduced.

3. Experimental setup

3.1 Femtosecond irradiation setup

Pulses of 145 fs at a central wavelength of 800 nm were generated with a Ti:Sapphire laser system consisting of a mode-locked oscillator and a regenerative amplifier with 1kHz of repetition rate and up to 3mJ per pulse. Pulses were focused inside the sample by an objective with 0.4 NA and 8 mm of entrance aperture fully illuminated by 11 mm beam diameter. The selected material was a 0.9 mm-thick alkali-free borosilicate glass, Schott AF45.

3.2 Fabrication

A set of 20 gratings of 2x2 mm² and 10 μm of period is presented. Each line of the gratings is formed by five overlapped single lines written in the same direction with 1 μm of overlap distance, as it is sketched in Fig. 2(b). All the single lines were written in the same direction in order to avoid asymmetric orientational writing effects [21] and reach good overlapping. The overlap of several lines is expected to increase material modification and better define the duty cycle. Pulses with energies, E_P, of 1, 1.5, 2.6, 3.6 and 4.6 μJ were applied at a scanning speed, V_S, of 200 µm/s. For each selected energy, gratings were fabricated at four different depths, Z, of 200, 300, 400 and 550 µm. Pulse energies were measured at the entrance of the objective.

4. Results and discussion

4.1 Geometry of the modified region in glass

Visual inspection of the samples with optical microscopy revealed several square millimeters of grating homogenously written, see Fig. 3. Figure 3(b) shows that for gratings processed at Z = 200 μm, where aberrations can be neglected for 0.4 NA [19], the length of the modified region increases with pulse energy due to self-focusing effect. An elongation of the modified region due to the contribution of aberrations is observed at higher depths, Fig. 3(b) (bottom). The thickness, L, of the resulting gratings for the range of conditions tested is shown in Fig. 4(a), where L is presented as a function of the focusing depth, Z, and pulse energy.

 

Fig. 3 (a) Top view of a set of 2x2 mm² gratings made in AF45. (b) Cross-sectional images of groups of five overlapped lines made with phase contrast microscope, written with 1, 1.5, 2.6, 3.6, 4.6 μJ from right to left, and focusing depth Z = 200 μm (top) and Z = 550 μm (bottom).(c) Top view of some grating lines written in glass by the method described in section 3.3. Pulse energy of 3.6 μJ, scanning speed of 200 μm/s and 10 μm of period.

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Fig. 4 Plot of the measured thickness of the structures written in glass from microscope images, left. First order diffraction efficiency vs. focusing depth Z; wavelength of 532 nm and normal incidence, center. Plot of phase difference and optical path difference vs. focusing depth, values were obtained by the application of Eq. (1), right.

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4.2 Far-field diffraction

As mentioned in section 2.1, the diffractive behavior of gratings depends on their thickness and index modulation. In the previous section it has been shown how the geometry of the structures in glass is affected by writing parameters, Fig. 4(a). In this section, the effect of the writing parameters on diffraction efficiency, phase difference and refractive index modulation is studied. In Fig. 4(b), first order diffraction efficiency measured with a Nd:YAG laser source (532 nm) s-polarized is shown. It can be observed that if pulse energy and focusing depth are increased, gratings become more efficient. This means that higher E_P and Z produce greater contribution to the optical path difference, L·Δn.

The obtained thickness of the gratings, ranging from 31 to 61 μm, and the corresponding resulting efficiencies, enable an estimation of the effective refractive index variations from 0.0020 to 0.0029 by the application of Eq. (1). In this range of parameters and efficiencies the relative differences between results of RCWA and Eq. (1) are less than 5%, Fig. 1(b). Hence, Eq. (1) is considered suitable in order to obtain estimated values of Δφ and Δn; Δφ values from 0.27π up to 0.51π and optical path from 71 nm to 137 nm were achieved from measurements of efficiency. These results demonstrate that it is possible to fabricate gratings with phase differences of π/2 for visible illumination that theoretically yields to self-images and Lau fringes with maximum visibility, V = 1 [16]. These gratings can be considered as pure phase gratings, as absorption can be neglected due to their high transmittance. For instance, for a grating made at Z = 550 μm and 3.6 μJ the efficiencies of only the orders ± 3, ± 2, ± 1 and 0 sum 94.6%. It means that more than 95% of the transmitted power by the substrate (AF45 transmittance is 0.92 at 532 nm) is contained in the diffraction pattern, so absorption and scattering are almost unappreciated.

4.3 Near-field diffraction

In this section, the dependence of fringes visibility on the estimated phase difference is studied. For near-field characterization, the beam was collimated and filtered before illuminating the grating. Images of diffraction patterns were recorded at different distances from the grating by an imaging system formed by a 40X objective and CCD camera mounted in a translating stage. Talbot planes separated by 190 ± 3 µm were observed, value very close to the theoretical one, Z_T/2 = 188 μm. In order to quantify the visibility of self-images, grayscale profiles of the first self-image found outside the glass sample were fitted to a sinusoidal function, a + b·sin(x) (where b is the amplitude and a is the mean value), see Fig. 5(a). Visibility was then calculated as V = b/a and plotted in Fig. 5(b), in terms of the calculated Δφ. The maximum visibility, 0.96 ± 0.02, is obtained for Ep = 3.6 μJ, Z = 550 μm, and Vs = 200 μm/s.

 

Fig. 5 (a) Self-image of the grating fabricated with pulse energy of 3.6 μJ and Z = 550 μm (top) and its grayscale profile fitted to a sine function (bottom). (b) Plot of the measured visibility of the complete set of gratings vs. the estimated Δφ/π shown in Fig. 4c. Red line represents V = sin(Δφ).

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Values of visibility increase with Δφ, but are somewhat lower than the theoretical behavior of planar phase gratings given by V = sin(Δφ) (red line in the figure), specially for the highest energies at the lowest depths. Near field of VPGs is still not well understood and, furthermore, some factors could be introducing deviations from the binary model: small deviations of the refractive index profile from the binary case for different writing parameters or small variations of duty cycle.

5. Conclusions

Volume phase diffraction gratings in borosilicate glass with near-field patterns of visibility of 0.96 have been demonstrated. The gratings were fabricated by focusing infrared laser pulses by a 0.4 NA objective. Structures from 30 to 61 μm long were achieved by taking advantage of spherical aberrations and self-focusing and the enhancement of gratings efficiency with pulse energy and focusing depth was demonstrated. The thin grating model was applied in order to get approximate values of phase modulation and optical path modulation: Δφ from 0.27π to 0.51π and Δn from 71 to 137 nm (λ = 532 nm). Additionally, it has been demonstrated that a phase modulation of π/2 is achievable by direct laser writing, value that theoretically yields to maximum visibility of Talbot and Talbot-Lau patterns of planar gratings.

Finally, visibility of self-images has been studied as a function of phase modulation and results have been compared with the expected behavior of planar gratings. The volume grating fabricated with E_P = 3.6 μJ, Vs = 200 μm/s and Z = 550 μm presents the higher visibility. It has been proven that visibility increases with pulse energy and focusing depth. The gratings written at Z = 550 μm present a closer behavior to the planar phase gratings while at high energies and low depths they considerably differ from that model.