HTML
-
The ultrafast laser-induced refractive index change of transparent materials is a complex phenomenon that relies on different physical processes. First, a physical rearrangement of the structural network was observed1, 20. It is believed that the densification induced by various complex phenomena, such as a fast temperature change21, 22 and plasma shock waves23, 24, has a great impact on the fs-laser-induced refractive index change. Another process is related to the variation in the absorption spectrum through the Kramers–Kronig relations25. Increased absorption due to photoinduced defects such as color centers26, 27 produced by self-trapped excitons28 leads to a variation in the refractive index. Since most of the defects can be annealed while a partial refractive index change remains29, defects can only partially explain the laser-induced refractive index change.
To date, no one has explicitly studied the effect of the FLIBGS on the refractive index. A first way of approaching this problem is via the Kramers–Kronig relations, relating the refractive index to the absorption coefficient α integrated over frequency:30
$$ n\left( \omega \right) = 1 + \frac{\pi }{c}{{\wp}} \int\nolimits_{0}^{+ \infty} {\frac{{\alpha \left( {\omega^{\prime}} \right)}}{{\omega^{\prime2} - \omega^{2}}}} d\omega ^{\prime} $$ (1) where c is the speed of light, ω′ is the angular frequency variable running through the whole integration range, and ℘ denotes the Cauchy principal value. Clearly, from this relation, a change in the absorption α(ω′) curve will in turn affect n(ω). To illustrate this effect, Fig. 1 shows the transmission spectrum through a zinc selenide (ZnSe) crystal with a thickness of d = 1 mm (gray curve, associated with the right axis). When a band-gap shift occurs, the absorption edge (near the electronic resonance) of the transmission spectrum shifts horizontally. For illustrative purposes, a dashed gray curve has been added to represent the shifted spectrum. At a wavelength near the absorption edge (where the transmission slope is significant, denoted by the FLIBGS window in Fig. 1), the shift greatly affects the absorption (double gray arrows) and thus the refractive index.
Fig. 1
Left axis: refractive index of ZnSe as a function of the propagating wavelength using the Sellmeier coefficient from ref. 33 (black curve). Potential effects of the fs-laser-induced variation in the number of charged particles per volume unit (blue dashed curve) and the FLIBGS (green dotted curve), based on Eq. 4. Right axis: ZnSe transmission spectrum (gray curve). The double gray arrows show the potential absorption variation in the FLIBGS window if a shift in the absorption edge occurs (dashed gray curve)Equation 1 is not convenient to use experimentally since it requires measurements over a very wide spectral band. Alternatively, the Lorentz dispersion relation with the Clausius–Mossotti form allows one to express the refractive index in terms of the number of charged particles per volume unit Nk31:
$$ \frac{{3\left( {\bar n^2 - 1} \right)}}{{\left( {\bar n^2 + 2} \right)}} = \mathop {\sum}\limits_{\mathrm{k}} {\frac{{4\pi N_{\mathrm{k}}\varepsilon _{\mathrm{k}}^2/m_{\mathrm{k}}}}{{\omega _{0{\mathrm{k}}}^2 - \omega ^2 + i\gamma \omega }}} $$ (2) where $\bar n$ is the complex refractive index and mk is the mass of particle k with charge εk. The number of charged particles per volume unit Nk, the resonance frequency $\omega _{0k}$, and the damping coefficient $\gamma$ are the only terms that can potentially be modified using fs-laser irradiation. The real part of the refractive index n can be experimentally obtained using the well-known Sellmeier equation, an empirical equation related to Eq. 2, as a function of the wavelength λ:
$$ n^2 = A + \mathop {\sum}\limits_{\mathrm{k}} {\frac{{B_{\mathrm{k}}\lambda ^2}}{{\lambda ^2 - C_{\mathrm{k}}^2}}} $$ (3) where the first (A) and second (k = 1) terms of this series represent the contributions to the refractive index due to the higher- and lower-energy bandgaps of electronic absorption, respectively, whereas the remaining terms (k > 1) account for a refractive index modification due to lattice resonance32. Equation 2 suggests that Bk is closely linked to the number of charged particles per volume unit Nk and Ck to the resonance frequency $\omega _{0k}$ (or wavelength λ0k). Note that the damping is not considered in the Sellmeier equation (also neglected in this work) since it is only significant in the close vicinity of the resonances. In addition, note that the damping is related to the absorption coefficient α and the Cauchy principal value ℘ in Eq. 1. Since the bandgap, absorption edge, and resonance frequency of a material are directly connected, the three models (using Eq. 1, Eq. 2, or Eq. 3) are similar in terms of studying the FLIBGS.
To experimentally study the FLIBGS, the following modified Sellmeier empirical equation that includes the effect of fs-laser irradiation is suggested:
$$ n_{irr}^2 = A + \mathop {\sum}\limits_k {\frac{{\left( {B_{\mathrm{k}} + dN_{\mathrm{k}}} \right)\lambda ^2}}{{\lambda ^2 - \left( {C_{\mathrm{k}} + d\lambda _{\mathrm{k}}} \right)^2}}} \approx A + \frac{{\left( {B_1 + dN_1} \right)\lambda ^2}}{{\lambda ^2 - \left( {C_1 + d\lambda _1} \right)^2}} $$ (4) where dNk is proportional to the laser-induced variation in the number of charged particles per volume unit and dλk is the laser-induced resonance shift (linked to the FLIGBS). The remaining terms (k > 1) are assumed to be negligible for wavelengths relatively close to the λ1 (or C1) electronic resonance, which is the case in this work. The fs-laser-induced refractive index contrast is Δn = nirr − n, where nirr is the refractive index of the irradiated region.
For illustrative purposes, Fig. 1 shows the ZnSe refractive index curve with the Sellmeier coefficients A = 4, B1 = 1.90, and C1 = 336.15 nm from ref. 33 (black curve, associated with the left axis). The effects of the variation in the number of charged particles per volume unit (blue dashed curves with dN1 = ±0.1) and the FLIBGS (green dotted curves with dλ1 = ±30 nm), both exaggerated to clearly observe their effect over the full spectrum, are plotted. The variation in the number of charged particles per volume unit tends to vertically displace the curve (blue arrows), which affects the refractive index similarly at all wavelengths, whereas the resonance shift tends to horizontally displace the curve (green arrows), which increasingly varies the refractive index when approaching the electronic resonance at lower wavelengths.
To demonstrate the effect of the FLIBGS on the refractive index contrast of a waveguide, fs-laser inscription was performed using a Ti:sapphire laser system (Coherent RegA). The system was operated at a wavelength of 795 nm with a repetition rate of 250 kHz. The temporal FWHM of the pulses was measured to be ~65 fs at the laser output. To estimate the electronic resonance shift dλ1 induced by the fs laser in ZnSe, several lines were inscribed with a scan speed of 5 mm/s and a pulse energy of 100 nJ. The inset in Fig. 2 shows the transmission spectrum of a ZnSe sample with a thickness of d = 1 mm before (black curve) and after (blue curve) photoinscription, measured using an Agilent Cary 5000 UV–vis–NIR spectrophotometer. Unfortunately, uniform irradiation over a 1-mm3 volume would take weeks. Therefore, 3300 lines were inscribed with a lateral displacement of 3 μm to form a layer (1 cm2), and 7 layers were inscribed with a vertical displacement of 10 μm, from a depth of 40–100 μm. The beam was focused beneath the surface of the sample using a 100× (1.25 NA) oil immersion microscope objective. The immersion oil refractive index (1.5) was beneficial for reducing the high aberration generated by the ZnSe refractive index (approximately 2.5 at 795 nm). However, it was impossible to write deeper due to the aberration and closer to the surface due to bubble formation in the oil.
Fig. 2 Calculating the FLIGBS value using the Tauc plot.
Relationship between (αhω)2 and hω for a ZnSe crystal sample before (black curve) and after (blue curve) photoinscription (pulse energy of 100 nJ with a scan speed of 5mm/s). Their transmission spectra through the sample with a thickness of d = 1 mm (including Fresnel losses) are shown in the insetThe shift in the absorption edge in BaAlBO3F2 and borosilicate glasses has been observed by two other groups, but has not been investigated34, 35. It is very difficult to obtain a quantitative measurement of the electronic resonance shift dλ1 from the transmission spectrum (inset of Fig. 2). Nevertheless, the absorption spectrum provides an efficient means to assess the band structure and width of the energy bandgap of optical materials, from which the electronic resonance frequency can be inferred. The optical bandgap Eopt can be expressed according to the Tauc law36:
$$ \left( {\alpha \left( \omega \right)h\omega } \right) = B\left( {h\omega - E_{\mathrm{opt}}} \right)^m $$ (5) where B is a constant depending on the transition probability, α is the absorption coefficient, and is calculated using the expression α = −2.303log(T)/d (d is the thickness of the sample and T is the transmission), ω is the incident light angular frequency, Eopt is the width of the bandgap, and m = 1/2 is the refractive index characterizing the direct transition process.
From the experimental transmission spectrum, (α(ω)hω)2 can be plotted as a function of hω in eV, as shown in Fig. 2. The optical bandgap Eopt is obtained as the intersection of the extrapolated linear portion of the curve with the photon energy hω axis. The bandgap shifts from approximately 2.627–2.620 eV, which corresponds to an electronic resonance shift of dλ1 = 1.26 nm. As a comparison, in typical semiconductors (Eopt ~10 eV) deformed using the piezospectroscopic effect, the strain-induced shift of an electronic resonance may be approximately 100 meV (dλ1 = 1.23 nm)37. Although this demonstrates an FLIBGS, the result is a lower bound since the sample is not irradiated over its whole volume.
-
Except for a few demonstrations, such as in ZnSe38, LiNbO3, and Nd:YCa4O(BO3)3, the refractive index change is generally negative in crystals5. Therefore, direct writing of waveguides in crystals is impractical. This can be explained because a positive refractive index change typically requires an increase in the material density, which is difficult to achieve in crystalline materials due to the compact structural order of the lattice, in contrast to vitreous materials with structural disorder and the existence of free space within the network. Figure 3 shows the refractive index contrast Δn for waveguides inscribed in a ZnSe crystal using the same parameters mentioned previously, with pulse energies from 100 to 195 nJ, as a function of the propagating wavelength. The results demonstrate a sign inversion of the refractive index change between 550 and 650 nm, depending on the energy. To the best of our knowledge, this is the first observation of a sign inversion of refractive index contrast as a function of the propagating wavelength. Details on the refractive index contrast measurement are provided in the "Materials and methods" section.
Fig. 3 Sign inversion of the photoinduced refractive index change in ZnSe.
Experimental values of the fs-laser-induced refractive index change as a function of the propagating wavelength for different laser pulse energies and the theoretical (green dotted) curve from Eq. 4 with dλ1 = 1.26 nm and dN1 = −6.5 × 10−3 and using the Sellmeier coefficients from ref. 33The green dotted curve represents the refractive index change calculated using Eq. 4 with dλ1 = 1.26 nm and dN1 = −6.5 × 10−3 (chosen to fit the experimental value at 700 nm) and using the Sellmeier coefficients from ref. 33. Although the experimental points agree well with the theoretical green dotted curve, a significant discrepancy is observed at shorter wavelengths, which supports the hypothesis of an underestimation of the band-gap shift dλ1. The inaccuracy of the empirical Sellmeier coefficients from ref. 33 could also contribute to the error, which is supported by the large difference between the different values found in the literature39.
Figure 4 shows the near-field mode profiles of the two waveguides inscribed in crystalline ZnSe with pulse energies of 115 and 195 nJ. With the 115-nJ pulses, the light is weakly confined at 520 nm and not guided at 633 nm. With the 195-nJ pulses, the light is weakly confined at 633 nm and not guided at 1550 nm. At lower wavelengths, the light is strongly confined in the waveguide at both pulse energies. The trend follows the sign inversion of the refractive index contrast. These results are of great interest, since many applications, such as waveguide lasers9, electro-optic modulators10, and frequency converters11, currently require multi-scan-depressed cladding structures due to the decrease in the refractive index that arises in most crystals5 and a wide variety of glasses6-8.
-
A peculiar phenomenon can be observed in Fig. 3 when the sign of the refractive index change is inverted as a function of the wavelength. At a specific wavelength, the refractive index contrast becomes zero, which means that the laser inscription should be invisible at this wavelength. At Δn = 0, i.e., when n = nirr (cf. Equations 3 and 4), the propagated light is not affected by the structural modification, which appears to be invisible. Due to the highly nonlinear effect of dλ1 compared with the effect of dN1, invisibility occurs at different wavelengths depending on the laser inscription parameters. Therefore, the FLIGBS allows for the direct inscription of invisible structures, which does not require invisibility cloaking14-17 to be hidden. As a preliminary experimental proof of concept, the left side of Fig. 5 shows the top view of the waveguide inscribed in ZnSe using a pulse energy of 170 nJ. The five pictures were taken with a microscope using filters at 500, 550, 600, and 650 nm. The visibility of the waveguide follows the trend of the refractive index contrast profile shown on the right of Fig. 5 (also see Fig. 3). At 500 and 550 nm, the waveguide is clearly seen. At 700 nm, the waveguide is fairly visible. At 600 and 650 nm, the waveguide is completely invisible to the naked eye and barely visible under the microscope, especially at 600 nm, where it is necessary to fine-tune the microscope focus position to make the waveguide barely visible.
Fig. 5 A waveguide inscribed using a fs laser in a ZnSe crystal becomes nearly invisible near 600 nm.
Top views of the waveguide taken with a microscope using filters at 500, 550, 600, 650, and 700 nm (left) and their respective refractive index change profiles (right)However, the fs-laser-induced refractive index contrast is not perfectly uniform over the whole inscribed cross section, mostly due to the stress induced around the focal region. This prevents the refractive index contrast from being zero over the full cross-section area of the waveguide, as shown in the refractive index profile at 600 nm (see Fig. 5, right). The perfect step refractive index induced by the fs laser should theoretically enable perfect invisibility, a field that has gained much interest in the last decade14-17, including fs-laser-written devices in smartphone screens, such as temperature sensors40 and on-surface refractometric sensors for liquids41, that are effectively invisible to the naked eye. In these previous works40, 41, the waveguides are undetectable to the naked eye due to the low laser-induced refractive index change, which limits the waveguide bend radii and thus the applications. Therefore, enhancing the invisibility in the visible region while increasing the refractive index change at the operating wavelength due to the FLIBGS would be of great interest. These invisible waveguide-based devices also have great potential in any see-through protection screen, such as car windshields, industrial displays, army helmets, and plane dashboards. The use of the multiscan technique or low repetition rates to avoid the heating effect42, and methods to minimize aberration such as using a spatial light modulator43 or a dual-beam technique44 in order to sharpen the Gaussian intensity profile should help obtain step refractive index inscriptions. Invisibility at specific wavelengths could enable interesting applications in photonic circuitry and gratings.
Note that a sign inversion of the refractive index contrast and invisibility is not possible via a type III modification (damage tracks). The negative refractive index contrast produced by voids formed due to microexplosions remains negative at any optical wavelength. Thus, invisibility can only be obtained via a negative refractive index change with a type I modification, which has been achieved in many materials5, 45.
-
An exponential increase in the refractive index contrast is observed when approaching the electronic resonance at shorter wavelengths (see Fig. 3). This feature is very interesting for the fabrication of photonic devices, such as splitters, couplers, and ring resonators, with a submillimeter size. In fact, submillimeter devices are still nearly impossible to fabricate using fs-laser writing due to the minimum waveguide bend radius limited by the refractive index contrast3, 18. No one has used wavelengths near a material electronic resonance for photonics applications obviously because of the higher material absorption. A centimeter-long device would be too lossy to be useful. However, for very compact devices, the intrinsic material absorption becomes less problematic. In the following paragraph, we address the possible benefits of the FLIBGS for the miniaturization of fs-laser-written photonic circuits.
To isolate waveguide bend losses, irradiation experiments were performed on GeS4 glass, which has an electronic bandgap lying in the visible region, in which it is easy to photoinscribe type I waveguides46. Figure 6 shows the refractive index contrast Δn as a function of the propagating wavelength for waveguides inscribed in GeS4 glass using the same parameters mentioned previously, with pulse energies from 50 to 120 nJ focused 100 μm beneath the surface using a 50× objective (Edmund Optics LWD 0.55 NA). The exponential increase in the refractive index contrast is clearly observed at short wavelengths. Positive refractive index changes up to ~1.7 × 10−2 are obtained at 500 nm for a pulse energy of 90 nJ. Note that this value of 1.7 × 10−2 is, to the best of our knowledge, the highest fs-laser-induced smooth positive type I refractive index change observed in any chalcogenide glass waveguide. The gray curve shows the transmission spectrum of the GeS4 glass through a 1.22-mm-thick sample (including Fresnel losses). For a fixed pulse energy, it is interesting to see that a significant enhancement of the refractive index change is still obtained at wavelengths within the highly transparent region. This extends the range of applications of FLIBGS-based devices.
Fig. 6 Exponential increase in the refractive index change in GeS4 glass as a function of the propagating wavelength for different laser pulse energies.
The gray curve shows the transmission spectrum of GeS4 through a 1.22-mm-thick sample (including Fresnel losses). For a fixed pulse energy, a significant enhancement of the refractive index change is still observed at wavelengths within the highly transparent regionTo ensure a smooth inscription of the tightly curved waveguides, the scan speed was reduced to 1 mm/s. Then, 20-nJ pulses were focused 100 μm beneath the surface using a 100× oil immersion objective (1.25 NA). To isolate the curvature loss, several S-bend waveguides were written in a 6-mm-long GeS4 sample, as shown in Fig. 7. Six S-bend waveguides with a fixed lateral displacement of 200 μm with lengths L ranging from 0.5 mm (with a radius curvature R of 0.363 mm) to 6 mm (R = 45.05 mm) were written.
Fig. 7 Top-view microscope image of an S-bend waveguide (with R = 363 μm) photoinscribed in GeS4 glass and (inset) its near-field mode profile at 520 nm with a width of 9.5 μm.
See the dynamics of the guided light in the Supplementary MovieThe S-bend waveguides were characterized using 520-, 633-, and 1550-nm laser sources. The light injection was performed by butt-coupling with a single-mode fiber. Simply by measuring the additional loss relative to a straight waveguide written under the same conditions, the additional loss from each S bend can be isolated. The bend loss in dB/mm is obtained by dividing this additional loss over the S-bend waveguide length. The results are plotted in Fig. 8a. At 1550 nm, the results are in agreement with prior results from the literature3, 18. For a radius curvature of 5 mm, the loss is less than 0.5 dB/mm at 520 nm, while it is over four times higher at 1550 nm. For a radius curvature of 1.3 mm, the signal is completely lost at 1550 nm, while the loss is less than 6 dB/mm at 520 nm. For a radius of curvature of 363 μm, guiding occurs only with the 520-nm light, with a bend loss of 17 ± 2 dB/mm, which seems promising for sub-millimeter-size devices, considering that 520 nm is not the optimized wavelength. Note that we have not been able to guide 520-nm light through waveguides with submillimeter bend radii in a material with a bandgap far from this wavelength, such as standard glasses (e.g., soda lime, borosilicate, and fused silica). In addition to the high refractive index contrast obtained due to the FLIBGS, the smooth type I-positive refractive index change may have an important impact on the guiding property of waveguides with submillimeter bend radii. In fact, a high refractive index contrast can be achieved with mixes of positive and negative refractive index changes or with type III (microexplosion or damage tracks) waveguides. However, the high asymmetry or roughness typically obtained from these methods induces additional losses in waveguide bends.
Fig. 8 Optical losses as a function of the bend radius.
a Waveguide bend loss and b effective loss (bend loss plus material absorption) at 520, 633, and 1550nm as a function of the radius curvature of an S bend photoinscribed in GeS4 glass. The solid curves are the theoretical curves obtained from Eq. 6. Despite the high material absorption loss, it is advantageous to use smaller wavelengths for very small bend radii (e.g., 480-nm solid curve)The experimental values can be compared with the theoretical formula of the waveguide bend loss LB (dB/mm)47:
$$ \begin{array}{l}L_{\mathrm{B}} = \frac{{2.171\pi ^{1/2}}}{{\left( {\rho R} \right)^{1/2}}}\left( {\frac{{V^4}}{{\left( {V + 1} \right)^2\left( {V - 1} \right)^{1/2}}}} \right)\\ \times \exp \left[ {\frac{{\left( {V - 1} \right)^2}}{{V + 1}} - \frac{{4R\left( {V - 1} \right)^3}}{{3\rho V^2}}\left( {\frac{{n_{irr}^2 - n^2}}{{2n_{irr}^2}}} \right)} \right]\end{array} $$ (6) where ρ is the waveguide core radius and V is the waveguide parameter given by:
$$ V = \frac{{2\pi \rho }}{\lambda }\left( {n_{irr}^2 - n^2} \right)^{1/2} $$ (7) The theoretical bend loss curves for 520, 633, and 1550 nm are plotted in Fig. 8a (solid curves). The differences between the experimental values and the theoretical curves can be explained by the perturbation at the transition point (halfway point of the S bend) where the curve changes the direction of its rotation47, which is not taken into account in Eq. 6, and the fact that defects and waveguide roughness have more significant effects for curved segments. Moreover, Eq. 6 is an approximation for perfectly symmetrical single-mode waveguides, which is not exactly the case in our experiment. As shown in the inset of Fig. 7, the mode profile is slightly elongated, and few modes appear at smaller wavelengths. The refractive index values of the GeS4 glass were obtained using an interpolation from five measurements (n = 2.153, 2.109, 2.058, 2.044, and 2.039 at wavelengths λ = 532, 633, 972, 1303, and 1538 nm, respectively) using a Metricon 2010/M prism coupler.
However, the most important parameter is the total loss of such curved waveguide-based devices. The mode mismatch and Fresnel losses (at the input and output) can be easily reduced to less than 1 dB47 and remain the same for any S-bend size; therefore, they are not taken into account in the following loss estimation. At wavelengths far from the resonances, the propagation loss in straight waveguides can be as low as 0.01 dB/mm1. This waveguide propagation loss is negligible compared with the bend loss and material absorption at wavelengths near electronic resonance and even more negligible for compact devices, which is the subject of this study. Figure 8b shows the sum of the two main optical losses (bend loss and material absorption), which will be referred to as the "effective loss", for several wavelengths as a function of the waveguide bend radius. The absorption spectrum of the GeS4 glass was measured using an Agilent Cary 5000 UV–vis–NIR system. Despite the higher absorption near electronic resonance, the experimental values and the theoretical curves in Fig. 8b clearly show the advantage of using wavelengths near resonance for tightly curved waveguides. For example, from the experimental measurements, a 1-mm-long optical splitter with a lateral displacement of the outputs of 400 μm, which is made of two S bends, as shown in Fig. 7, with a waveguide bend radius of 1.3 mm, exhibits an effective loss of 6.1 dB at 520 nm, while the signal is completely lost at 1550 nm. For a 1.6-mm-long splitter with a lateral displacement of the outputs of 250 μm, with a waveguide bend radius of 5 mm, the experimental effective loss is 2.16 dB at 633 nm. These relatively low losses are due to the fact that at 520 and 633 nm, the material absorption is still low, while the refractive index is significantly increased (see Fig. 6) due to the FLIBGS.
Despite the differences between the experimental points and the theoretical curves in Fig. 8b, both clearly show the same trend. Therefore, the theoretical calculation can be used to provide an optimized wavelength for a specific bend radius required for a specific application. The curves in Fig. 9 show the theoretical effective loss as a function of the wavelength for different waveguide bend radii photoinscribed in GeS4 glass. Optimized wavelengths of 895, 620, 545, 525, 505, 480, and 467 nm are obtained for bend radii of 5, 2, 1.3, 1, 0.75, 0.5, and 0.375 mm, respectively. Moreover, as shown in Fig. 9a, low-loss compact devices made of waveguides with a bend radius of 5 mm should be achievable over a bandwidth of ~600 nm (from ~550 to ~1150 nm).
Fig. 9 Effective loss as a function of the wavelength.
a Theoretical effective loss (bend loss plus material absorption) as a function of the propagating wavelength for various waveguide bend radii photoinscribed in GeS4 glass. b Zoom-in of the microbend region. The experimental points (black circles) include the Fresnel, mode mismatch, and misalignment lossesFigure 9b shows the experimental effective loss measurement (black circles) using an optical spectrum analyzer (Yokogawa AQ6373B) from a white-light source (Koheras SuperK Power supercontinuum source) launched in an S-bend waveguide (as shown in Fig. 7) with a bend radius of 1.3 mm. While this method of analysis is not precise enough to obtain a reliable measurement of the losses (it also includes Fresnel, mode mismatch, and misalignment losses), it provides a relative value of losses as a function of the wavelength. Therefore, the experimental values show the real optimized wavelength (524 nm), which is 21 nm shorter than the theoretical wavelength. This can be explained by any waveguide fluctuation, roughness, or defects caused by laser inscription power fluctuations, scratches on the surface, motor vibrations, or material imperfection, which results in a lower effective bend radius. Note that the Fresnel and mode mismatch losses are wavelength-dependent but should not significantly affect the value of the obtained optimized wavelength.
As shown in Fig. 9b, for very tight bends, the wavelength is more critical. In the case where the application requires the tightest bend, the use of the Tauc law (see Eq. 5 and Fig. 2) seems to be a practical way to obtain an efficient and reliable wavelength (or a good material choice for a fixed wavelength of interest). For GeS4 glass, a bandgap of 464 nm (2.67 eV) is obtained. At this wavelength, the losses (1.13 dB/100 μm) are mostly due to material absorption down to a bend radius of 430 μm. For a bend radius of 375 μm, an effective loss of 1.2 dB/100 μm is calculated.
As shown in Fig. 10, to obtain a lower bound of the FLIBGS in GeS4 glass, the same procedure using the Tauc law was executed (see section "FLIBGS theory and experiment"). The sample was irradiated from a depth of 60–660 μm over the sample with a thickness of d = 1.22 mm. The bandgap shifts from approximately 2.67–2.655 eV, which corresponds to an electronic resonance shift of dλ1 = 2.62 nm. The lower refractive index of GeS4 (2.1089 at 633 nm) makes deeper writing feasible, which probably contributes to the larger calculated band-gap shift compared with the shift for ZnSe. Unfortunately, since no Sellmeier coefficients were found in the literature for GeS4 glass, the theoretical curve of the refractive index contrast as a function of the wavelength could not be plotted in Fig. 6. As a comparison, a band-gap shift of approximately 0.06 eV (dλ1 ~10 nm) was observed after illuminating a GeS2.33 film for 4 h using a 400-W high-pressure Hg lamp48. One may notice a surprising increase in the absorption in the full spectrum for the irradiated samples compared with that of the pristine samples (see the inset in Figs. 2 and 10). This is due to the light scattered from the non-uniformly inscribed sample, which is not detected by the Cary detector. To ensure that this scattered light did not affect the band-gap shift calculation, a few measurements were performed using a detector close to the sample to measure all of the scattered light, which provided the same results but with a higher experimental error. These measurements also ensured that the laser inscription did not induce significant absorption loss, which was also demonstrated by Tong et al.49.
Fig. 10 Calculating the FLIGBS value using the Tauc plot.
Relationship between (αhω)2 and hω for a GeS4 glass sample before (black squares) and after (blue triangles) photoinscription. Their transmission spectra through the sample with a thickness of d = 1.22 mm (including Fresnel losses) are shown in the inset