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Abstract
The combined effect of chromatic dispersion and conical diffraction (i.e., off-plane diffraction) in femtosecond laser inscription of fiber Bragg gratings using the phase mask technique is characterized by measuring the light intensity distribution after the phase mask. As the distance from the mask and the observation point grows, chromatic dispersion and conical diffraction introduced by the mask gradually decrease the peak intensity inside the line-shaped focal volume of the cylindrical lens that is used to focus the femtosecond pulses inside the fiber. We also show that at a certain distance from the mask spherical aberration introduced by the plane-parallel mask substrate is cancelled out by conical diffraction and, at a different distance, chromatic aberration of the cylindrical lens is cancelled out by chromatic dispersion of the mask. These two independent cancellation effects lead to sharpening of the line-shaped focus and the consequent growth of peak light intensity inside it. The above phenomena become especially pronounced for tightly focused femtosecond laser beams and small-pitch phase masks, which, in turn, allows one to choose experimental conditions to inscribe Bragg gratings in polymer-coated non-sensitized 50 µm fibers.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Femtosecond (fs) laser micromachining/microprocessing of both transparent and absorptive materials can be sub-divided into two main categories: sequential and parallel. In sequential fs-laser processing, structures are produced one after another, whereas in parallel processing a relatively large area is exposed to patterned fs-radiation and many structures are produced simultaneously. Even though parallel processing is faster than the sequential one, it is generally more complex as it utilizes micro-lens arrays [1–3] or various diffractive optical elements [4–10] to synthesize the required light pattern at the workpiece. In the case of diffractive optics, angular chromatic dispersion unavoidably stretches the focal intensity distribution of the broadband fs-pulses and a number of different approaches, including the use of additional diffractive optics and holograms, have been proposed to correct for this undesirable effect [11–17]. The above solutions, however, work well only when the diffraction angles and the associated chromatic dispersion are relatively small.
On the other hand, fs-laser inscription of fiber/volume Bragg gratings (FBG) using phase masks (i.e., transmission phase diffraction gratings) [18–20] relies on large diffraction angles, which is the necessary condition for the creation of small-period (<1 µm) interference patterns to be imprinted into the material. Surprisingly, despite the fact that FBGs fabricated by means of fs-radiation focused through a phase mask have been used for many scientific and industrial applications for more than a decade, the obvious relevance of chromatic dispersion and potential limitations imposed by it on this important laser writing technique have never been discussed in the literature.
Moreover, there is another neglected set of phenomena that is inherent to the phase mask technique regardless of whether the laser radiation used for FBG inscription is broadband or narrowband. By definition, the plane-parallel phase mask substrate introduces negative spherical aberration to the laser beam even when the focusing optics (i.e., cylindrical lens) is aberration free [21–23]. When the cylindrical lens focuses the laser beam tightly, which is required for the inscription of FBGs through the protective fiber coating in the case of fs-beams [24–28], the magnitude of the negative spherical aberration can be quite significant even for thin mask substrates. Additionally, focusing of a laser beam through a phase mask implies that the associated diffraction should be considered within the framework of conical diffraction or off-plane diffraction [29–32]. Again, the magnitude of the off-plane diffraction phenomenon increases dramatically for tight focusing geometries.
In this study we demonstrate the importance of chromatic dispersion and conical diffraction for the phase mask technique in its classic implementation for FBG inscription. Specifically, we demonstrate how the interplay of the chromatic and aberration phenomena introduced by the phase mask transforms the spatial intensity distribution in the line-shaped fs-laser focus as the distance between the observation point and the mask is changed. We also show, both analytically and experimentally, that there exists an optimum position from the mask along the beam propagation direction where the confocal parameter of the line-shaped laser focus is smallest and the peak intensity in the focus is highest. This effect becomes dramatic for tightly focused infrared fs-beams and for small-pitch (∼1 µm) large-diffraction-angle (∼45°) phase masks.
2. Background
A phase mask that splits a focused laser beam into different diffraction orders represents a robust and very simple interferometric setup for the inscription of Bragg gratings, both in optical fibers and bulk materials (Fig. 1). There are several unique features pertinent to this technique, which will be briefly mentioned below to introduce the necessary terminology and formalism.
Fig. 1. Interference of ultrashort pulses after a phase mask that produces four diffraction orders (m = 0,..3). M denotes the phase mask, CL is the cylindrical lens, ΔT is the transverse walk-off, ΔL is the longitudinal walk-off, L is the distance from M to the observation point (O), l is the distance from M to the pulse front of the 0th diffraction order. The pulse phase fronts are normal to the propagation direction of the respective diffraction orders.
2.1. Effects associated with the broadband spectrum of ultrashort pulses
Consider a spatially unchirped and untilted fs-pulse falling normally onto a phase mask. The pulse front is assumed to be parallel to the mask. After the interaction of the pulse with the phase mask, a set of diffracted pulses is formed. The pulse fronts of the pulses in different diffraction orders (i.e., 0th, 1st,.., mth order) remain parallel to the phase mask but, except for the 0th diffraction order, they are no longer parallel to their phase fronts, i.e., the diffracted pulses have become tilted and spatially chirped [33–35].
The diffraction orders are defined by the respective diffraction angles, whose magnitudes are given by ${\theta _m} = \arcsin (m{\lambda _0}/d)$, where λ0 is the central wavelength of the pulse, d is the mask pitch and m is an integer satisfying the condition $|{m{\lambda_0}/d} |\le 1$ in order to ensure the existence of propagating (rather than evanescent) diffraction orders [29,30].
If a phase mask generates several diffraction orders, a Talbot interference pattern [36–39], which can be easily observed using standard time-integrating detectors (e.g., CCD or CMOS cameras), is generated in the vicinity of the mask. At a certain distance L from the mask, a pure two-beam interference pattern (with a period Λ = d/2) produced by the pulses in the + m and –m diffraction orders will begin to emerge. This happens when the following condition is satisfied:
In this sub-section we would also like to mention that the lateral overlap (x-axis in Fig. 1) between the fs-pulses diffracted into the ± m orders decreases as they propagate away from the mask and the region where the interference fringes are formed shrinks as a result. This geometric effect—if we neglect spatial chirp introduced by it—is called transverse walk-off (ΔT in Fig. 1) [28]. The transverse walk-off for the mth diffraction order can be calculated using the following expression ${\Delta _\textrm{T}} = 2L\tan ({\theta _m})$ [28].
There is, however, another effect that is inherent to the phase mask technique. The chromatic effect originates from the broadband nature of fs-pulse and depends on the chromatic dispersion of both the phase mask and the focusing cylindrical lens.
A) The angular spread $\Delta {\theta _m}$ of the spectrum in the mth order corresponding to Δλ can be obtained by differentiating the grating equation. Hence, for normal incidence, one can write $\Delta {\theta _m} = \frac{m}{{d\cos ({\theta _m})}}\Delta \lambda $ [29,30]. As can be seen from Fig. 2(a), the long-wavelength spectral components (i.e., ‘red’) of the pulse are diffracted at a larger angle than the short-wavelength components (i.e., ‘blue’) and the cylindrical lens will therefore focus them closer to the mask. The difference in the positions of the ‘blue’ and ‘red’ foci along the z-axis is given by:
We also note that the plane-parallel glass substrate on the surface of which the phase mask is engraved introduces an additional chromatic focal elongation given in the paraxial approximation by:
B) The cylindrical lens also introduces a chromatic focal elongation, but in this case the ‘blue’ spectral components are focused closer to the mask than their ‘red’ counterparts (Fig. 2(b)). This elongation is nothing but chromatic aberration, which can be written as [43]:
Fig. 2. Focal elongation caused by chromatic dispersion of the mask (a) and chromatic aberration of the cylindrical lens (b). In (a), $\Delta {\theta _1}$ is the angular spread of the spectrum of the 1st diffraction order corresponding to a pulse bandwidth Δλ. For clarity, the mask produces only the 0th and 1st diffraction orders. Note that ‘red’ light is focused closer to M in (a) and farther from M in (b).
The $\cos ({\theta _m})$-factor is introduced because the observation point is moved along the z-axis, whereas the diffracted beams propagate at an angle ${\theta _m}$ with respect to the z-axis. Some salient features of the scenario when the phase mask technique is used with broadband fs-pulses can be summarized as follows:
- i) The focal elongation $\Delta z_{\textrm{mask}}^{\textrm{chrom}.}$ caused by angular chromatic dispersion of the mask linearly grows with the distance L from the mask and the bandwidth Δλ of the fs-pulses. However, its dependence on the diffraction angle ${\theta _m}$ is essentially nonlinear, which becomes clear if Eq. (2) is rewritten (using the identity ${\cos ^2}[{\arcsin ({{\theta_m}} )} ]\equiv 1 - {({{{m{\lambda_0}} \mathord{\left/ {\vphantom {{m{\lambda_0}} d}} \right.} d}} )^2}$) in terms of m, d and λ0 as:$$\Delta z_{\textrm{mask}}^{\textrm{chrom}.} = \frac{{{m^2}{\lambda _0}L\Delta \lambda }}{{{d^2} - {m^2}\lambda _0^2}}.$$For instance, for a 3.21 µm-pitch mask, λ0 = 800 nm and Δλ = 10 nm, $\Delta z_{\textrm{mask}}^{\textrm{chrom}.}$ increases by ∼0.1 µm for m = 1, ∼0.4 µm for m = 2, and ∼1.6 µm for m = 3 for each 100 µm the observation point is moved away from the mask. To put it in perspective, Eq. (3) gives $\Delta z_{\textrm{substr}.}^{\textrm{chrom}.}$ ∼0.2 µm for the same Δλ and a fused silica (SiO2) mask substrate of thickness t = 3 mm (n1 = 1.453 at λ0 = 800 nm, dn1/dλ = −0.0173 µm−1 at λ0 = 800 nm).
- ii) The focal elongation caused by chromatic aberration of the focusing cylindrical lens (see Eq. (4)) linearly depends on Δλ for a given lens shape and lens material. Based on Eq. (4), chromatic aberration of the cylindrical lens used in our experiments (plano-convex lens with f = 12 mm made of OHARA S-LAH64 glass; the curved surface is designed to correct spherical aberration in one dimension; nL = 1.776 at λ0 = 800 nm, dnL/dλ = −0.0371 µm−1 at λ0 = 800 nm) is ∼5.7 µm for Δλ = 10 nm. According to Eq. (4a), focusing through the 3.21 µm-pitch mask will reduce this focal elongation by a factor of $\cos ({\theta _m})$, making it ∼5.6 µm for m = 1, ∼5.0 µm for m = 2, and ∼3.8 µm for m = 3.
- iii) If chromatic dispersion were the only cause of changing the focal intensity distribution of the fs-beam (which is not the case; see Section 2.2), there would exist a distance L from the mask where $|{\Delta z_{\textrm{mask}}^{\textrm{chrom}.}} |= |{\Delta z_{\textrm{lens}}^{\textrm{chrom}.}} |$ and the confocal parameter of the line-shaped fs-laser focus would attain its minimum value. For given m and λ0, this distance is solely determined by the mask-lens combination and does not depend on Δλ.
The cancellation of the two counteracting chromatic dispersions at a certain L would lead to a more pronounced sharpening of the fs-beam focus for tight-focusing geometries. This can be shown based on the following considerations. The peak intensity I0 in the focus of a Gaussian beam is given by ${I_0} = {{4P} \mathord{\left/ {\vphantom {{4P} {({\lambda_0}{Z_0})}}} \right.} {({\lambda _0}{Z_0})}}$, where P0 is the total power in the beam and Z0 is the confocal parameter of the beam. The presence of chromatic aberrations (and other aberrations, see Section 2.2) elongates the confocal parameter by ΔZ and, as a result, decreases the peak intensity, which is now given by ${I^{\prime}_0} \approx {{4P} \mathord{\left/ {\vphantom {{4P} {[{{\lambda_0}({Z_0} + \Delta Z)} ]}}} \right.} {[{{\lambda_0}({Z_0} + \Delta Z)} ]}}$. For a given ΔZ, the ratio ${{I^{\prime}_0} \mathord{\left/ {\vphantom {{{{I^{\prime}}}_0} {{I_0}}}} \right.} {{I_0}}}$ can then be written as ${{I^{\prime}_0} \mathord{\left/ {\vphantom {{I^{\prime}_0} {{I_0}}}} \right.} {{I_0}}} \propto {({1 + {{\Delta Z} \mathord{\left/ {\vphantom {{\Delta Z} {{Z_0}}}} \right.} {{Z_0}}}} )^{ - 1}}$. Taking into account that ${Z_0} \approx {{2{\lambda _0}} \mathord{\left/ {\vphantom {{2{\lambda_0}} {[{{\pi }{{\sin }^2}(\varphi )} ]}}} \right.} {[{{\pi }{{\sin }^2}(\varphi )} ]}}$ (or ${Z_0} \approx {{2{\lambda _0}\cos ({\theta _m})} \mathord{\left/ {\vphantom {{2{\lambda_0}\cos ({\theta_m})} {[{{\pi }{{\sin }^2}(\varphi )} ]}}} \right.} {[{{\pi }{{\sin }^2}(\varphi )} ]}}$ when focusing through a mask), where φ is the maximal half-angle of the cone of light (at the 1/e2-intensity level) that exits the focusing optics, a stronger sharpening for tightly focused beams becomes evident.
2.2. Effects associated with spherical aberration and conical diffraction
C) A plane parallel plate introduces aberrations to a converging/diverging electromagnetic wave produced by focusing/defocusing a light beam with a spherical or cylindrical lens [21–23]. When the beam axis is normal to the plate, only spherical aberration needs to be considered. This type of spherical aberration originates from the fact that rays that have a larger angle of incidence with respect to the normal to the plate (i.e., marginal rays) are displaced more along the beam propagation direction than rays that have a smaller angle of incidence (i.e., paraxial rays). The magnitude of such a longitudinal separation $\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}$ between the marginal and paraxial foci is given by [21]:
Fig. 3. Focal elongation caused by the plane parallel mask substrate. Note that marginal rays are focused farther from the mask than paraxial rays (compare with Fig. 4(b)).
The phase mask technique is inherently based on focusing a laser beam with a cylindrical lens through a plane parallel glass plate whose one surface is covered with periodic linear grooves and the effect of the above-mentioned spherical aberration should therefore be taken into account. Similarly to the case of lens-induced chromatic aberration represented by Eq. (4a), Eq. (6) needs to be modified as follows:
D) The generic grating equation $m{\lambda _0} = d\sin ({\theta _m})$, which we used in Section 2.1, is valid when the incident and diffracted rays lie in a plane that is perpendicular to the grooves (i.e., in-plane diffraction). However, certain FBG inscription scenarios require that the laser beam be tightly focused, which implies that rays of the incident light are no longer perpendicular to the grooves. This type of diffraction is called off-plane diffraction or conical diffraction. The term ‘conical diffraction’ emphasizes the fact that in the case of off-plane incidence the diffracted light corresponding to different diffraction orders lies on a conical surface [29,30]. To visualize the behavior of diffraction orders produced by an off-plane incident beam, Harvey et al. [31] introduced direction cosines of the actual spatial coordinates to describe both the incident and diffracted rays (Fig. 4(a)). According to this formalism, the absolute values of the direction cosines of the incident ray are given by [46]:
Fig. 4. Focal elongation caused by conical (off-plane) diffraction. (a) Visualization of the direction cosine space for conical diffraction by a phase mask (i.e., transmission diffraction grating). (b) Ray propagation in the yz-plane plane (βγ-plane in (a)). In (b), F0,0 denotes the paraxial focus of the 0th diffraction order, Fm,0 and Fm,φ respectively denote the paraxial and marginal foci of the mth diffraction order, L and l respectively denote the distance from M and Fm,0 (observation point O coincides with Fm,0) and the distance from M to F0,0. The mask produces only 0th and 1st diffraction orders. Note that the marginal focus (i.e., Fm,φ) lies closer to the mask than the paraxial focus (i.e., Fm,0).(Compare with Fig. 3.)
In our case, the azimuthal angle is χ = 0, while the polar angle φ represents the angle at which the marginal rays (Figs. 3 and 4) impinge the mask substrate and the grooves on the mask. The spatial spectrum of propagating (i.e., nonevanescent) diffraction orders is defined by the condition $\alpha _m^2 + \beta _m^2 \le 1$.
For χ = 0 (Fig. 4(b)), γm and $\textrm{tan}({\varPsi _{m,\varphi }})$ can be written as:
The z-coordinate ${z_{m,\varphi }}$ of the focal line produced by diffracted marginal rays (i.e., point Fm,φ in Fig. 4(b)) is given by ${z_{m,\varphi }} = l\tan (\varphi ){\tan ^{ - 1}}({\varPsi _{m,\varphi }})$, where l is the distance from the back surface of the mask to the 0th-order paraxial focus F0,0. On the other hand, the z-coordinate ${z_{m,0}}$ of the focal line produced by diffracted paraxial rays (i.e., point Fm,0 in Fig. 4(b)) is given by ${z_{m,0}} = l\cos ({\theta _m})$. Finally, based on Fig. 4(b), the distance $\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.} = {z_{m,0}} - {z_{m,\varphi }}$ along the z-axis between the marginal focus Fm,φ and the paraxial focus Fm,0 corresponding to the mth diffraction order can be expressed as:
As in Section 2.1, we would like to summarize some key features of spherical aberration induced by the mask substrate and an aberration originating from off-plane diffraction by the mask.
- i) The focal elongations $\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}$ and $\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}$ do not depend on Δλ and therefore should be taken into account during FBG inscription irrespective of the laser source. However, $\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}$ and $\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}$ can be neglected for small focusing angles φ (see below), which is generally the case for FBG inscription using ultraviolet excimer lasers.
- ii) For a given material (and diffraction order m), $\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}$ depends nonlinearly on the focusing angle φ (i.e., ${\propto} {\varphi ^2}$) and depends linearly on the substrate thickness t.
- iii) $\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}$ is proportional to φ2 and linearly depends on the distance L from the mask. Similar to $\Delta z_{\textrm{mask}}^{\textrm{chrom}.}$ (compare Eq. (5) and Eq. (9a)), the dependence of $\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}$ on the diffraction angle θm is nonlinear. Based on the above ray optics analysis (Eq. (9)), the focal elongation caused by conical diffraction is expected to be quite significant. For instance, for a 3.21 µm-pitch mask, λ0 = 800 nm and φ = 15°, $\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}$ increases by ∼0.2 µm for m = 1, ∼1.2 µm for m = 2, and ∼4.7 µm for m = 3 for every 100 µm the observation is moved away from the mask.
- iv) If we ignore the chromatic effects considered in Section 2.1, there exists a distance L from the mask where $|{\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}} |= |{\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}} |$ and the confocal parameter of the line-shaped fs-laser focus attains its minimum value. For given m and λ0, this distance is solely determined by the mask parameters and does not depend on φ. However, the cancelation of the two counteracting aberrations at a certain L would lead to a more pronounced sharpening of the fs-beam focus for larger φ’s, that is for tight-focusing geometries.
- v) There exists a distance L from the mask where the conditions $|{\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}} |= |{\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}} |$ and $|{\Delta z_{\textrm{mask}}^{\textrm{chrom}.}} |= |{\Delta z_{\textrm{lens}}^{\textrm{chrom}.}} |$ can be fulfilled simultaneously, which is expected to result in a stronger maximum in the focal peak intensity ${I^{\prime}_0}$ than in the case when these conditions are separately met at different L’s. This can be achieved, for example, by first fulfilling the condition $|{\Delta z_{\textrm{mask}}^{\textrm{chrom}.}} |= |{\Delta z_{\textrm{lens}}^{\textrm{chrom}.}} |$ and then choosing the mask thickness t that would fulfill $|{\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}} |= |{\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}} |$.
3. Experimental results and discussion
Two Ti-sapphire regeneratively amplified laser systems (RA-1 and RA-2 in the following text), both operating at a central wavelength of λ0 = 800 nm, were used in the experiments. The bandwidth (FWHM) and output beam diameter (at the 1/e2-intensity level) of the fs-systems were Δλ ∼ 6 nm and 2w0 ∼ 7 mm for RA-1 and Δλ ∼ 33 nm and 2w0 ∼ 8 mm for RA-2. We note that an interference filter was placed in the RA-1 beam to reduce the original 14 nm bandwidth to the 6 nm bandwidth. To check the effect of substrate-induced spherical aberration, two different substrate thicknesses were used: 2.1 mm and 3.4 mm. The 2.1-mm mask was a holographic mask, whereas the 3.4-mm mask was manufactured using the e-beam engraving process. In each case, the mask pitch was 1.07 µm. To reconstruct 3D time-averaged intensity distributions after the mask, the technique described in [39] was employed. Briefly, the respective xy-intensity distributions with a 1 µm separation along the z-axis were projected onto a CMOS matrix by means of a high numerical aperture (i.e., NA = 0.9) objective lens, recorded and combined into 3D stacks. The yz-intensity distributions shown below in the text were obtained by averaging the values of points with fixed (yi, zi)-coordinates along the x-axis and projecting the respective mean values onto the yz-plane.
1) To estimate the quality of the focusing optic and the output beams of RA-1 and RA-2, the phase mask was initially removed from the beam path. Figure 5 shows the focal intensity distributions of the output beams of RA-1 and RA-2 in the yz-plane when the beams are focused with the 12 mm-focal-length acylindrical lens described in Section 2.1. The effective numerical aperture of the acylindrical lens (i.e., sin(φ)) is estimated at 0.26 for RA-1 and 0.30 for RA-2, due to the slightly different beam diameters. Under such conditions, aberration-free focusing of quasi-monochromatic light at λ0 = 800 nm would translate into a 7.6 µm confocal parameter (i.e., Z0 ∼ 7.6 µm) for sin(φ) = 0.26 and Z0 ∼ 5.7 µm for sin(φ) = 0.30. As it was mentioned earlier, the curved surface of the acylindrical lens is designed to correct spherical aberration in one dimension. However, in both cases, the focal shapes indicate that the lens still introduces a certain amount of negative spherical aberration, i.e., the marginal rays are focused farther from the lens than the paraxial ones. While the negative spherical aberration is comparable for both beams, chromatic aberration is noticeably stronger for the RA-2 beam as its bandwidth is approximately six times larger than that of the RA-1 beam (Eq. (4)). The effect of chromatic aberration can be deduced from Fig. 5 by measuring the respective confocal parameters Z0’s, which gives Z0 ∼ 27 µm for RA-2 versus Z0 ∼ 14 µm for RA-1. For reference, Eq. (4) gives $\Delta z_{\textrm{lens}}^{\textrm{chrom}.}$ ∼ 3.4 µm and ∼18.9 µm for RA-1 and RA-2, respectively.
Fig. 5. Focal intensity distributions in the yz-plane of the RA-1 beam (a) and the RA-2 beam (b) when the phase mask is removed from the beam path. The focusing is performed using the 12 mm-focal-length acylindrical lens whose effective numerical aperture is sin(φ) = 0.26 for (a) and sin(φ) = 0.30 for (b). In (a) and (b), the beam propagation is from left to right.
2) The effect of substrate-induced spherical aberration for the two substrate thicknesses under consideration, i.e., 2.1 mm and 3.4 mm, is demonstrated in Fig. 6. In this part of the experiments, the fs-beams were focused through the mask substrate without intercepting the phase mask grooves. Analysis of the respective yz-intensity distributions in Fig. 6 confirms that the substrate-induced negative spherical aberration significantly elongates the focal volume and increases with substrate thickness, as predicted. The measured Z0’s are presented in the respective panels of Fig. 6. For reference, longitudinal substrate-induced spherical aberration $\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}$ given by Eq. (6) is: i) ∼26 µm for t = 2.1 mm and ∼42 µm for t = 3.4 mm when sin(φ) = 0.26 and ii) ∼36 µm for t = 2.1 mm and ∼58 µm for t = 3.4 mm when sin(φ) = 0.30. One should also keep in mind that the above values for $\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}$ provide only a rough estimate as they define nothing else but two shadow boundaries on the z-axis between which the light rays cross the z-axis.
Fig. 6. Focal intensity distributions in the yz-plane of the RA-1 beam (a) and (c) and the RA-2 beam (b) and (d) when the beam is focused through the phase mask substrate without intercepting the mask grooves. (a) and (b) correspond to t = 2.1 mm, while (c) and (d) correspond to t = 3.4 mm. The focusing is performed using the 12 mm-focal-length acylindrical lens. sin(φ) = 0.26 and 0.30 for RA-1 and RA-2, respectively. In all the panels, the beam propagation is from left to right.
3) The interplay of the chromatic effects, which are characterized by $|{\Delta z_{\textrm{mask}}^{\textrm{chrom}.}} |$ and $|{\Delta z_{\textrm{lens}}^{\textrm{chrom}.}} |$, and the spherical aberration effects, which are characterized by $|{\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}} |$ and $|{\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}} |$, was first studied using the 1.07 µm-pitch 2.1 mm-thick mask. In this case, the fs-beams from RA-1 and RA-2 were focused through the mask grooves. The results pertinent to this experiment are presented in Figs. 7 and 8. The plot in Fig. 7 shows the focal peak intensity ${I^{\prime}_0}$ as a function of distance L from the mask, with a strong maximum being located at L ∼ 350 µm. The procedure for obtaining the data points on which the graphs in Figs. 7, 9, 11 and 12 are based was the following: i) the imaging high numerical aperture (NA = 0.9) objective lens was moved in 20 µm increments (50 µm increments were used for Fig. 12) along the beam propagation direction z; to follow the incremental movements of the objective lens, the focusing acylindrical lens was translated along z in increments given by 20 µm (or 50 µm)/cos(θ1) (θ1 is the diffraction angle), ii) the exact position of the acylindrical lens along z for each new increment was fine-tuned to maximize the intensity of the interference fringes in the respective xy-intensity distribution, and iii) a data point (i.e. focal peak intensity for given L) was obtained by averaging the peak intensities of the fringes. Hence, the above mentioned graphs contain 100 (Figs. 7, 9), 50 (Fig. 11), and 75 (Fig. 12) data points. Error bars for each data point are estimated at ± 2.5 arbitrary units (a.u.) regardless of the distance L from the mask.
Fig. 7. Focal peak intensity as a function of distance L from the 1.07 µm-pitch 2.1 mm-thick phase mask.
Fig. 8. Focal intensity distributions in the xy- and yz-planes of the RA-1 beam (a, c, e and g) and RA-2 beam (b, d, f and h) when the beams are focused through the 1.07 µm-pitch 2.1 mm-thick phase mask. The focusing is performed using the 12 mm-focal-length acylindrical lens. sin(φ) = 0.26 and 0.30 for RA-1 and RA-2, respectively. In all the panels corresponding to the yz-plane the beam propagation is from left to right.
Fig. 9. Focal intensity distributions of the RA-1 and RA-2 beams when the beams are focused through the 1.07 µm-pitch 3.4 mm-thick phase mask. (a) Focal peak intensity as a function of distance L from the mask. (b) and (c), focal intensity distributions in the xy- and yz-planes of RA-1 and RA-2 recorded respectively at L = 570 µm and L = 400 µm. The focusing is performed using the 12 mm-focal-length acylindrical lens. sin(φ) = 0.26 and 0.30 for RA-1 and RA-2, respectively. In all the panels corresponding to the yz-plane the beam propagation is from left to right.
The observed variations of ${I^{\prime}_0}$ with L can also be correlated with the transformations occurring with the focal intensity distributions in the yz-plane (Fig. 8). Substrate-induced negative spherical aberration remains quite strong at L = 100 µm, is almost neutralized by conical diffraction at L = 350 µm, and is completely reversed by conical diffraction at L = 900 µm. This trend holds for both RA-1 and RA-2. In the case of RA-2, however, the yz-intensity distributions are blurred out to a larger extent by chromatic aberration of the cylindrical lens and chromatic dispersion of the mask than in the case of RA-1.
Estimates based on Eqs. (4) and (5) show that chromatic aberration of the cylindrical lens $|{\Delta z_{\textrm{lens}}^{\textrm{chrom}.}} |$ and chromatic dispersion of the mask $|{\Delta z_{\textrm{mask}}^{\textrm{chrom}.}} |$ should compensate each other at L ∼ 240 µm (for both fs-systems). On the other hand, Eqs. (6b) and (9) predict that the condition $|{\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}} |= |{\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}} |$ should be fulfilled at L ∼ 380 µm (for both fs-systems). These data already provide evidence that the maximum in Fig. 7 is mainly caused by the compensation of substrate-induced spherical aberration by conical diffraction. After the maximum, i.e. at larger distances from the mask, the evolution of the focal intensity distribution is governed by the combined action of chromatic dispersion and conical diffraction originating from the mask. In the case of RA-2 (Fig. 8(h)), the chromatic focal spreading is very pronounced indeed.
4) The interplay of the chromatic effects and spherical aberration effects was then studied using a significantly thicker phase mask, i.e., the 1.07 µm-pitch 3.4 mm-thick mask (Fig. 9). In this case, the focal peak intensity plot for RA-1 (Fig. 9(a)) shows a sharp maximum at L ∼ 570 µm, whereas the plot obtained with RA-2 exhibits a much broader maximum centered at L ∼ 400 µm. Theoretically, the distance L at which the condition $|{\Delta z_{\textrm{mask}}^{\textrm{chrom}.}} |= |{\Delta z_{\textrm{lens}}^{\textrm{chrom}.}} |$ is fulfilled does not depend on the mask thickness t and should therefore be ∼240 µm (for both fs-systems). On the other hand, L, at which the condition $|{\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}} |= |{\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}} |$ is fulfilled, linearly depends on t and is thus expected to be ∼610 µm (for both fs-systems). This indicates that the distinctive maximum at L ∼ 570 µm for RA-1, whose bandwidth is relatively narrow (i.e., Δλ ∼ 6 nm), originates from the cancellation of substrate-induced spherical aberration by conical diffraction. For RA-2 with a Δλ ∼ 33 nm, the combined action of the two oppositely signed chromatic effects appears to be strong enough to produce a separate maximum much closer to the surface – nominally at ∼240 µm – when $|{\Delta z_{\textrm{mask}}^{\textrm{chrom}.}} |= |{\Delta z_{\textrm{lens}}^{\textrm{chrom}.}} |$. The existence of two distinct maxima determined by $|{\Delta z_{\textrm{mask}}^{\textrm{chrom}.}} |= |{\Delta z_{\textrm{lens}}^{\textrm{chrom}.}} |$ and $|{\Delta z_{\textrm{substr}.}^{\textrm{sph}.\textrm{aberr}.}} |= |{\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}} |$ is consistent with the width and position of the resultant maximum in the case of RA-2.
We also note that the observed enhancement of the focal peak intensity in the vicinity of the mask, i.e. at L < 150 µm (Fig. 9(a)), is related to a rather complex intensity distribution within the multi-beam interference pattern produced by the 0th, −1st and + 1st diffraction orders. Several examples of intensity distribution near a 1.07 µm-pitch phase mask can be found in [47]. The power in the 0th diffraction order of the 3.4 mm-thick mask is ∼20% of the total power in all the diffraction orders, which is sufficient to form a Talbot-like interference pattern containing regions where the peak intensity exceeds that of a classical two-beam interference pattern that would be generated solely by −1st and + 1st diffraction orders, each carrying 50% of the diffracted power. According to Eq. (1), the transformation of the Talbot-like interference pattern into the two-beam interference pattern would occur at L > 210 µm for RA-1 and at L > 40 µm for RA-2. The above enhancement is not observed with the 2.1 mm-thick mask because its design limits the power in the 0th diffraction order to less than 3% of the total diffracted power.
5) To further investigate the contribution of the chromatic effects to the focal intensity evolution, substrate-induced spherical aberration was compensated for by placing a plano-convex cylindrical lens having a focal length of 100 mm in front of the acylindrical lens, as shown in Fig. 10. This was done to introduce fixed positive spherical aberration into the fs-beam that would cancel out negative substrate-induced spherical aberration [48,49]. In the general case, this approach requires adjustment of the NA of the acylindrical lens, as the spherical aberration induced by passing through the substrate depends on this in a nonlinear fashion (see Eqs. (6) and (6b)). The adjustment procedure consisted of i) focusing the RA-1 and RA-2 beams through the substrate of the 3.4 mm-thick phase mask without intercepting the grooves and ii) optimizing the separation between the cylindrical lens and the acylindrical lens along the z-axis around 200 mm until the respective focal intensity distributions in the yz-plane showed that negative substrate-induced spherical aberration had been cancelled out. The results of this experiment are presented in Fig. 11.
Fig. 10. A procedure to compensate for substrate-induced negative spherical aberration using a plano-convex cylindrical lens (CL1) placed in front of the acylindrical lens.
Fig. 11. Focal intensity distributions of the RA-1 and RA-2 beams when the beams are focused through the 1.07 µm-pitch 3.4 mm-thick phase mask and substrate-induced negative spherical aberration has been corrected using the technique depicted in Figs. 10(a) and (b), focal intensity distributions in the yz-planes of RA-1 and RA-2, respectively, when the beams are focused through the phase mask substrate without intercepting the mask grooves. (c) Focal peak intensity as a function of distance L from the mask. The focusing is performed using the 12 mm-focal-length acylindrical lens. For RA-1 and RA-2, the effective numerical aperture is estimated at 0.25 < sin(φ) < 0.28. In (a) and (b), the beam propagation is from left to right.
Figures 11(a) and 11(b) demonstrate that using the above technique substrate-induced negative spherical aberration can be compensated with a high degree of accuracy – at least when the beams are focused through the phase mask substrate without intercepting the mask grooves. Focusing through the mask grooves, however, is not expected to introduce any complications as both the positive and negative spherical aberrations should be equally affected by the cos(θm)-factor (as in Eq. (6b)).
Similar to the previous results, the plot in Fig. 11(c) clearly shows an absence of the ‘chromatic’ maximum defined by the condition $|{\Delta z_{\textrm{mask}}^{\textrm{chrom}.}} |= |{\Delta z_{\textrm{lens}}^{\textrm{chrom}.}} |$ in the case of the narrowband RA-1. What is more important is that compensating substrate-induced negative spherical aberration allows one to estimate the contribution of the chromatic maximum to the focal peak intensity as a function of distance L for the broadband RA-2. The center of this relatively weak and broad chromatic maximum is estimated to be around L ∼ 300 µm, which is in reasonable agreement with the predicted L ∼ 240 µm.
6) Finally, the focal peak intensity as a function of distance L was measured for the 1st diffraction order (i.e., m = 1) of a 2.14 µm-pitch 2.4 mm-thick phase mask manufactured using the e-beam engraving process. This experiment was done with the 12 mm-focal-length acylindrical lens and RA-1. The above analysis for RA-1 suggests that a focal peak intensity maximum will be caused by the spherical aberration/conical diffraction effects and be located at L ∼ 4800 µm, based on Eqs. (6b) and (9). Also, the dependence of $|{\Delta z_{\textrm{mask}}^{\textrm{con}.\textrm{diffr}.}} |$ on L is only 0.6 µm per 100 µm and, as a consequence, the maximum is predicted to be weak and very broad.
To record intensity distributions millimeters away from the mask, the fs-beam was expanded approximately 7 times along the x-axis in order to minimize the effect of transverse walk-off [28]. Figure 12 summarizes the results of this experiment. A weak focal peak intensity maximum can indeed be seen on the graph, but accurately finding its center is difficult because of the large peak width. Nevertheless, with a high degree of probability the peak center lies somewhere between L = 3800 µm and L = 4200 µm, which is ∼20% less than the expected value. The confocal parameter Z0 of the fs-beam measured at L = 4000 µm is ∼16 µm, which is 60–70% larger than the value obtained for the two 1.07 µm-pitch masks. This variation can be explained by the difference in the cos(θm)-factors of the 1.07 µm-pitch masks and the 2.14 µm-pitch mask.
Fig. 12. Focal intensity distributions of the RA-1 beam when the beam is focused through the 2.14 µm-pitch 2.4 mm-thick phase mask. (a) Focal peak intensity as a function of distance L from the mask. (b) Focal intensity distributions the 1st diffraction order in the xy- and yz-planes. In the panel corresponding to the yz-plane, the beam propagation is from left to right. The focusing is performed using the 12 mm-focal-length acylindrical lens at sin(φ) = 0.26.
7) In the last set of experiments we studied whether the focal peak intensity plots can be used as a guide for through-the-coating FBG inscription in terms of maximizing the fs-light intensity at the fiber core and minimizing it at the fiber surface, i.e. at the coating.
i) Uncoated SMF-28 fiber samples were placed at different distances L from the 1.07 µm-pitch 2.1 mm-thick mask and exposed to the radiation of RA-1. The RA-1 beam was focused with the 12 mm-focal-length acylindrical lens with an effective numerical aperture of sin(φ) = 0.26, as before. During the inscription the acylindrical lens was scanned with a piezo actuator perpendicular (i.e., along the y-axis) to the fs-beam in order to maximize the overlap of the fs-laser-induced modification with the fiber core. For each sample, the laser fluence was kept at the same level. The peak intensity inside the fiber core was kept below the threshold for Type-II material modification [51], which was confirmed by monitoring the light-induced changes occurring during the laser-writing process using the dark-field microscopy technique described in [52]. The broadband insertion loss of the resultant Type-I FBGs [51] was less than 0.01 dB when measured at a wavelength of 1560 nm using the tunable laser mentioned earlier in the text.
Figure 13(a) shows that the FBG strength does follow the focal peak intensity plot shown in Fig. 7 for RA-1. This trend can be seen even more clearly in Fig. 13(b), where the induced change Δn in the refractive index of the fiber core is plotted as a function of L. In order to deduce Δn from the FBG strength, the well-known expression for the peak reflectivity R0 of a uniform FBG with a constant sinusoidal modulation was used [50]:
where $W^{\prime}$ is the FBG length, λB is the Bragg wavelength given by ${\lambda _\textrm{B}} = nd$ (n is the effective refractive index of the fiber core of the fundamental mode) [50], and $\eta (V) \approx 1 - 1/{V^2},\;V \approx 2.4$ is a function of the fiber parameter V that represents the fraction of the integrated fundamental-mode power in the core [50]. In this experiment, the FBGs were written with the expanded quasi-flat-top beam, as described earlier in the text. For each distance from the mask L, the effect of transverse walk-off was taken into account in the calculations by adjusting the grating width $W^{\prime}$ according to $W^{\prime} \approx {W_0} - {\Delta _\textrm{T}} = {W_0} - 2L\tan ({\theta _m})$[28], where W0 is the FBG length when the fiber touches the mask (W0 ∼ 13 mm, as defined by the clear aperture of the acylindrical lens along x).Fig. 13. (a) FBGs written in uncoated SMF-28 fiber. (b) Inferred refractive index change Δn for the corresponding FBGs.
ii) To conclude our experiments, approximately 5 mm-long Type-I gratings [51] were written in polyimide-coated 50 µm fibers (51.2 µm exactly). Importantly, the fibers were not hydrogen/deuterium-loaded. According to the manufacturer (Fibercore), the germanosilicate core has a diameter of 4.1 µm, and the polyimide coating is approximately 10 µm-thick. The 50 µm-fiber was placed 350 µm away from the 1.07 µm-pitch 2.1 mm-thick mask and exposed to the radiation of RA-1. The RA-1 beam was focused with the 12 mm-focal-length acylindrical lens with an effective numerical aperture of sin(φ) = 0.26. During the exposure, the acylindrical lens was scanned perpendicular to the fs-beam. The pertinent results are demonstrated in Fig. 14. We would also like to remark on the fact that writing in polyimide-coated 50 µm-fibers was possible only within a narrow range (250 µm < L < 450 µm) of fiber-to-mask distances around the optimum L = 350 µm. Outside this range, to induce refractive index change in the fiber core of the strength comparable to that shown in Fig. 14(a), the writing laser power has to be increased to a level where damage to the fiber coating would become inevitable. The broadband loss of the FBG in Fig. 14(a) was measured to be less than 0.01 dB, which, based on Eq. (10), gives Δn ∼ 1.5×10- 4.
Fig. 14. (a) Reflection spectrum (∼6 dB in transmission, Δn ∼ 1.5×10−4) of an FBG written in a 50 µm fiber through the polyimide coating. (b) An optical microscopy image of the 50 µm fiber containing the FBG whose spectrum is shown in (a). To visualize the FBG, red light at 637 nm was coupled into the fiber core [52].
All in all, the experimental results in Section 3 agree well with the predictions and estimates based on the semi-quantitative analysis in Section 2, where the chromatic effects were presented in terms of just two different wavelength (i.e., ‘blue’ and ‘red’) and monochromatic aberrations were introduced as ray optics phenomena.
Specifically,
- 1) The simple analytic expressions in Section 2 give the distance from the mask where i) spherical aberration introduced by the plane-parallel mask substrate is cancelled out by conical diffraction and ii) chromatic aberration of the acylindrical lens is cancelled out by chromatic dispersion of the mask. When these two distances are approximately equal, a sharpening of the laser focus and the accompanying growth of focal light intensity take place. For a 1.07 µm-pitch mask, the agreement between theory and experiment is within a few tens of micrometers. Most importantly, the strength of FBGs (Figs. 13(a) and (b)) recorded at a fixed fluence but different distances from the mask follows the focal peak intensity curve measured in free space (Fig. 7).
- 2) For the same focusing conditions, the above effects are easily observable with the 1.07 µm-pitch mask but become much less pronounced for the 2.14 µm-pitch mask, as predicted.
The limitations of the formalism presented in Section 2 are obvious. It provides only qualitative information about i) the intensity distribution in the line-shaped focal volume and ii) temporal pulse distortions caused by the rather complex optical setup, i.e., a combination of an acylindrical lens, a plane-parallel plate and a transmission phase diffraction grating. To calculate the temporal and spatial distribution of the electric field in the focal volume, diffraction needs to be taken into account, as it has been shown in the literature on this subject matter [53–56]. However, we note that a fully rigorous treatment of the problem should also include the electromagnetic diffraction of light focused through the highly curved cylindrical surface of the fiber. Even if the diffraction integrals describing the whole system could be derived in a practically usable form, a rigorous treatment would also need to include the residual aberrations of the acylindrical lens and the beam quality factor (i.e., M2) of the regeneratively amplified fs-system. In view of the above, our semi-quantitative formalism reinforced with intensity distribution measurements after the mask provides an important shortcut to identify optimum FBG laser writing conditions when the phase mask technique is used.
4. Conclusions
We have considered two independent sets of effects that are inherent to the phase mask technique, namely i) chromatic dispersion of the mask, which is counteracted by chromatic aberration of the cylindrical lens, and ii) conical diffraction by the mask, which is counteracted by spherical aberration introduced by the plane-parallel mask substrate. The interplay of these effects in the case of large diffraction angles (∼45°; 1.07 µm-pitch mask) and tight focusing leads to a distinctive maximum in the distribution of focal peak intensity as a function of distance from the mask. For a given laser central wavelength and bandwidth, the position of this maximum from the mask generally depends on the mask substrate (thickness, refractive index), focusing cylindrical lens (focal distance, refractive index), and diffraction angle of the mask (mask period). Under our experimental conditions, which are typical of fs-laser inscription of fiber Bragg gratings, the position of the maximum is essentially determined by the cancellation of spherical aberration by conical diffraction, even for the broadband RA-2. In this respect, one can tune the distance from the mask to the maximum of focal peak intensity by simply choosing a different substrate thicknesses, with the other parameters being kept fixed. This is especially true for relatively narrowband laser sources, such as RA-1.
After the maximum has been passed, the combined action of chromatic dispersion and conical diffraction introduced by the phase mask gradually decrease the peak intensity inside the focal volume of the cylindrical lens by stretching the focal volume along the beam propagation direction. Focal elongation caused by chromatic dispersion 1.5–2 mm away from the mask is so strong that through-the-coating inscription becomes impossible because of fs-radiation damage to the coating. Conversely, through-the-coating inscription inside very thin fibers (50 µm diameter) when they are placed at the optimum position from the mask becomes a readily achievable task even if the fibers are not hydrogen/deuterium-loaded to increase their photosensitivity.
The above chromatic and conical diffraction effects scale down nonlinearly as the diffraction angle is decreased and thus become barely noticeable when the diffraction angle is ∼22° (2.14 µm-pitch mask). Taking into account that chromatic aberration of the focusing acylindrical lens is generally small and negative spherical aberration introduced by the mask substrate can be relatively easily compensated, working with small diffraction angles may seem to provide a convenient laser-writing recipe in terms of its weak dependence on the distance from the mask. However, one should remember that the use of small diffraction angles implies that the resultant Bragg grating utilizes a higher-order resonance, which dramatically reduces the grating strength for a fixed grating length [57,58].
We would also like to mention that even though the geometric optics approach used in this work provides guidance regarding the distance from the mask where the maximum in the distribution of focal peak intensity should be located, the use of complimentary diagnostic techniques to characterize the intensity distribution after the mask is indispensable for obtaining accurate results for a given laser-writing setup.
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