FIBER BRAGG GRATING SENSOR IN POLYMER-COATED ULTRA-THIN OPTICAL FIBERS AND METHOD FOR PRODUCING SAME

Abstract

A method and apparatus for inscribing a Bragg grating in an optical waveguide, comprising: providing electromagnetic radiation from an ultrashort pulse duration laser, wherein the electromagnetic radiation has a pulse duration of less than or equal to 5 picoseconds, and wherein the wavelength of the electromagnetic radiation has a characteristic wavelength in the wavelength range from 150 nanometers (nm) to 2.0 microns (.Math.m); providing cylindrical focusing optics corrected for spherical aberration; providing a diffractive optical element that when exposed to the focused ultrashort laser pulse, creates an interference pattern on the optical waveguide, wherein the irradiation step comprises irradiating a surface of the diffractive optical element with the focused electromagnetic radiation, the electromagnetic radiation incident on the optical waveguide, from the diffractive optical element, being sufficiently intense to cause the permanent change in the index of refraction in the core of the optical waveguide.

Claims

1. A method for inscribing a Bragg grating in an optical waveguide, comprising the steps of: providing electromagnetic radiation from an ultrashort pulse duration laser; providing a focusing optical element to focus the electromagnetic radiation from an ultrashort pulse duration laser; providing a diffractive optical element that when exposed to the focused electromagnetic radiation generates a beam on the optical waveguide having an interference pattern, said diffractive optical element having a substrate; positioning the optical fiber at a distance with respect to the diffractive optical element where the confocal parameter of a line-shaped laser focus created by the focusing optical element is smallest and the peak intensity in the focus is highest due to substantial or complete cancelation of i) chromatic aberration of the diffractive optical element $\left(\Delta z_{\text{mask}}^{\text{chrom}\text{.}}\right)$ and chromatic aberration of the focusing optical element $\left(\Delta z_{\text{lens}}^{\text{chrom}\text{.}}\right)$ and ii) negative spherical aberration caused by the substrate of the diffractive optical element $\left(\Delta z_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}\right)$ and conical diffraction caused by the diffractive optical element $\left(\Delta z_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}\right);$ and irradiating the optical waveguide with the electromagnetic radiation to form a Bragg grating, the electromagnetic radiation incident on the optical waveguide being sufficiently intense to cause Type I or Type II permanent change in the index of refraction within a core of the optical waveguide when exposed to a single pulse or a succession of laser pulses.

2. The method of claim 1, wherein the electromagnetic radiation has a pulse duration of less than or equal to 5 picoseconds.

3. The method of claim 1, wherein the wavelength of the electromagnetic radiation is in a range from 150 nm to 2.0 microns.

4. The method of claim 1, further comprising providing a focusing element corrected for spherical aberration for focusing the electromagnetic radiation on the diffractive optical element.

5. The method of claim 1, further comprising providing a cylindrical lens corrected for spherical aberration for focusing the electromagnetic radiation on the diffractive optical element.

6. The method of claim 4, further comprising scanning the focusing element corrected for spherical aberration for focusing the electromagnetic radiation using a piezo actuator.

7. The method of claim 5, further comprising scanning the cylindrical lens corrected for spherical aberration focusing the electromagnetic radiation using a piezo actuator.

8. Apparatus for inscribing a Bragg grating in an optical waveguide, comprising: an ultrashort pulse duration laser for providing electromagnetic radiation; a focusing optical element to focus the electromagnetic radiation from an ultrashort pulse duration laser; and a diffractive optical element that when exposed to the focused electromagnetic radiation from the focusing element produces an interference pattern in the optical waveguide: wherein positioning the optical waveguide at a distance with respect to the diffractive optical element along the propagation direction of the electromagnetic radiation where the confocal parameter of line-shaped laser focus is smallest and the peak intensity in the focus is highest results in the effects of i) negative spherical aberration and conical diffraction caused by the diffractive optical element and ii) chromatic aberration of the focusing element and chromatic dispersion of the diffractive optical element substantially or completely cancelling each other out; and wherein irradiating the optical waveguide with the electromagnetic radiation forms a Bragg grating, the electromagnetic radiation incident on the optical waveguide being sufficiently intense to cause Type I or Type II permanent change in the index of refraction within a core of the optical waveguide when exposed to a single pulse or a succession of laser pulses.

9. The apparatus of claim 8, wherein the optical waveguide is an optical fiber.

10. The apparatus of claim 8, wherein the optical waveguide is a non-photosensitized optical fiber.

11. The apparatus of claim 8, wherein the optical waveguide is a polymer-coated optical fiber.

12. The apparatus of claim 8, wherein the optical waveguide is a polymer-coated non-photosensitized optical fiber.

13. The apparatus of claim 8, wherein the optical waveguide is an optical fiber with a diameter less than or equal to 50 .Math.m.

14. The apparatus of claim 8, wherein the optical waveguide is a non-photosensitized (no hydrogen or deuterium loading) optical fiber with a diameter less than or equal to 50 .Math.m.

15. The apparatus of claim 8, wherein the optical waveguide is a polymer-coated optical fiber with a diameter less than or equal to 50 .Math.m.

16. The apparatus of claim 8, wherein the optical waveguide is a polymer-coated non-photosensitized optical fiber with a diameter less than or equal to 50 .Math.m.

17. The apparatus of claim 8, wherein the optical waveguide is a buried channel waveguide.

18. The apparatus of claim 8, wherein the optical waveguide is a ridge waveguide.

19. The apparatus of claim 8, wherein the optical waveguide is a tapered optical fiber.

20. The apparatus of claim 8, wherein the electromagnetic radiation has a pulse duration of less than or equal to 5 picoseconds.

21. The apparatus of claim 8, wherein the wavelength of the electromagnetic radiation is in a range from 150 nm to 2.0 microns.

22. The apparatus of claim 8, wherein the ultrashort pulse duration laser comprises a Ti-sapphire regeneratively amplified laser system operating at a central wavelength of 800 nm.

23. The apparatus of claim 8, wherein the diffractive optical element comprises a uniformly pitched phase mask.

24. The apparatus of claim 8, wherein the diffractive optical element comprises a chirped phase mask.

25. The apparatus of claim 8, wherein the diffractive optical element comprises a phase-shifted phase mask.

26. The apparatus of claim 8, further comprising providing a focusing element corrected for spherical aberration for focusing the electromagnetic radiation on the diffractive optical element.

27. The apparatus of claim 26, further comprising providing a piezo actuator for scanning the focusing element corrected for spherical aberration for focusing the electromagnetic radiation.

28. The apparatus of claim 8, further comprising providing a cylindrical lens corrected for spherical aberration for focusing the electromagnetic radiation on the diffractive optical element.

29. The apparatus of claim 28, further comprising a piezo actuator for scanning the cylindrical lens corrected for spherical aberration for focusing the electromagnetic radiation using a piezo actuator.

30. The method of claim 1, wherein the distance with respect to the diffractive optical element is a distance at which the condition $\left|\Delta z_{\text{mask}}^{\text{chrom}\text{.}}\right|=\left|\Delta z_{\text{lens}}^{\text{chrom}\text{.}}\right|$ is fulfilled simultaneously.

31. The method of claim 30, wherein after the condition $\left|\Delta z_{\text{mask}}^{\text{chrom}\text{.}}\right|=\left|\Delta z_{\text{lens}}^{\text{chrom}\text{.}}\right|$ is fulfilled, choosing a thickness of the diffractive optical element that fulfils the condition $\left|\Delta z_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}\right|=\left|\Delta z_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}\right|.$ .

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0025] FIG. 1 depicts an interferometric setup for the inscription of a fiber Bragg grating in an optical waveguide, according to an aspect of this specification.

[0026] FIG. 2a depicts the transverse and longitudinal order walk-off effect from a uniform phase mask of diffracted femtosecond pulses.

[0027] FIG. 2b depicts the longitudinal elongation of the focal spot caused by chromatic dispersion from the mask.

[0028] FIG. 3 depicts the chromatic aberration of the pulse due to a cylindrical lens.

[0029] FIG. 4 depicts the elongation caused by the plane parallel mask substrate, wherein marginal rays are focused farther from the mask than paraxial rays.

[0030] FIG. 5a depicts the visualization of the direction cosine space for conical diffraction by a phase mask (i.e., transmission diffraction grating).

[0031] FIG. 5b depicts the ray propagation in the yz-plane plane (βγ-plane in FIG. 5a) leading to the focal elongation.

[0032] FIG. 6 shows the focal intensity distribution in the yz-plane of an 80 fs, 800 nm wavelength beam with a beam diameter 2w.sub.0~ 7 mm that is focused in free space with a 12 mm focal-length cylindrical lens corrected for spherical aberration, where the beam propagation is from left to right.

[0033] FIG. 7 shows the focal intensity distribution in the yz-plane of the laser beam shown in FIG. 6 when the beam is focused (with the 12 mm focal-length cylindrical lens corrected for spherical aberration) through a phase mask substrate with a thickness of 2.1 mm.

[0034] FIG. 8 shows the focal intensity distribution at the optimal distance from the phase mask in the yz-plane and the xy-plane of the laser pulse shown in FIG. 6 and FIG. 7 when the beam is focused (with the 12 mm focal-length cylindrical lens corrected for spherical aberration) through a phase mask having a pitch of 1.071 .Math.m.

[0035] FIG. 9 shows the variation of the focal spot peak intensity as a function of distance from the mask when the beam is focused through the phase mask with the 12 mm focal length cylindrical lens corrected for spherical aberration.

[0036] FIG. 10a shows reflection spectra of Type I FBGs written (with the 12 mm focal-length cylindrical lens corrected for spherical aberration) in uncoated SMF-28 fiber at different separations between the phase mask and the fiber, and FIG. 10b shows the inferred refractive index change for the corresponding Type I FBGs.

[0037] FIG. 11a is a reflection spectrum of a Type I FBG written (with the 12 mm focal-length cylindrical lens corrected for spherical aberration) in a 50 .Math.m diameter optical fiber with a 10 .Math.m thick polyimide coating using a method according to an embodiment, where the total reflection of the grating is 75 % (-6 dB in transmission).

[0038] FIG. 11b is an optical microscopy image of a 50 .Math.m fiber with a 10 .Math.m thick polyimide coating containing the Type I FBG whose spectrum is shown in FIG. 11a.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0039] It is an object of an aspect of this specification to provide for fabrication of FBG sensors in small diameter optical fibers and optical fiber tapers. It is another object of an aspect of this specification to provide for fabrication of FBG sensors in small diameter optical fibers and optical fiber tapers when said optical fibers and optical fiber tapers are non-photosensitized (i.e. (no hydrogen or deuterium loading) and polymer-coated. Such sensors can be used for measurement of pressure, strain, temperature, or displacement or index of refraction. Advantageously, when fabricated in silica-based waveguide filaments, such sensors can also be made to have high thermal stability, for example up to 1000° C. They can also be easily integrated into composite materials to create ‘smart skins’.

[0040] FIG. 1 depicts an interferometric setup for the inscription of a fiber Bragg grating in an optical waveguide, according to an aspect of this specification. An incident femtosecond beam 10 from an ultrashort pulse duration laser is focused by a cylindrical lens corrected for spherical aberration 11 onto a uniformly pitched phase mask 12. Hereinafter, the cylindrical lens corrected for spherical aberration is referred to as a cylindrical lens. Interference of the ultrashort pulses by the phase mask 12 produces multiple diffraction orders (m= 0,..3), where Δ.sub.T is the transverse walk-off and Δ.sub.Lis the longitudinal walk-off, as discussed in greater detail below with reference to FIG. 2b, L is the distance from the mask 12 to an observation point (O) and / is the distance from the mask 12 to the pulse front of the 0.sup.th diffraction order. In FIG. 1, the pulse phase fronts are normal to the propagation direction of the respective diffraction orders.

[0041] As taught by Mihailov et al. in U.S. Pat. 7,031,571, femtosecond pulses interact differently with a transmission diffraction grating or phase mask when compared to continuous wave (CW) UV sources. In FIG. 2 a), the incident femtosecond beam 10 with a width 2w.sub.0, impinges on the phase mask 12 at normal incidence. The femtosecond beam 10 is composed of a number of femtosecond light pulses 13 which have a spatial dimension or pulse envelope of the electromagnetic radiation 14 which is dependent on the pulse duration. For example, a 100 femtosecond pulse has a 30 .Math.m width of its electromagnetic pulse envelope in the direction of the beam propagation. On passing through the phase mask, the femtosecond beam 10 is split into a number of beams that is dependent upon the periodicity Λ.sub.M of the corrugation structure of the mask and the wavelength λ.sub.o of the femtosecond beam such that:

[00002]$\theta ={\sin }^{-1}\left(\frac{m{\text{λ}}_o}{{\text{Λ}}_M}\right)$

where m is the order number of the diffracted beam and is an integer satisfying the condition |λ.sub.0 • m / Λ .sub.M | ≤ 1 in order to ensure the existence of propagating (rather than evanescent) diffraction orders. For example, a phase mask 12 that has a nominal periodicity of 1.071 .Math.m when irradiated with 800 nm infrared radiation at normal incidence will produce 3 femtosecond beams related to the diffracted orders of the mask, namely a 0.sup.th order that is in line with the incident beam and + 1 and -1 orders that diffract with angles θ ~ ± 48° with respect to the normal to the mask. By optimizing the depth of the troughs of the corrugation pattern of the mask with respect to the wavelength of the irradiating beam, coupling of the incident beam energy into the ± 1 orders can be maximized. At the phase mask surface, the generated pulses in each of the diffracted orders overlap spatially. The pulses propagate away from the surface at the speed of light along the beam path defined by the diffracted order. At a given distance L (15 in FIG. 2a) from the phase mask 12, the projection of the propagating ± 1 orders 16 on to the normal of the mask no longer overlap with the 0 order 17. Only the portions of the ± 1 orders that overlap will interfere resulting in a sinusoidal interference pattern 18. This is the longitudinal order walk-off effect taught by Mihailov in U.S. Pat. 7,031,571. For the example of the 1.071 .Math.m pitched phase mask referred to above, irradiated with 100 fs pulse duration 800 nm infrared radiation, the walk-off distance 15 is ~ 90 .Math.m from the mask surface.

[0042] Expressed mathematically, at a certain distance L (15), a pure two-beam interference pattern 18 (with a period Λ.sub.G = Λ.sub.M/2) produced by the pulses in the +m and -m diffraction orders will begin to emerge. This happens when the following condition is satisfied:

[00003]$L\ge \frac{{\lambda }_o^2}{\text{Δ}\lambda }\frac{\cos \left({\sin }^{-1}\left(\frac{m{\lambda }_o}{{\text{Λ}}_M}\right)\right)}{\cos \left({\sin }^{-1}\left(\frac{\left(m-1\right){\lambda }_o}{{\text{Λ}}_M}\right)\right)-\cos \left({\sin }^{-1}\left(\frac{m{\lambda }_o}{{\text{Λ}}_M}\right)\right)}$

where Δλ is the FWHM (full width at half maximum) bandwidth of the fs-pulses and m is assumed to be positive. Equation (3) simply states that when the separation of the pulse fronts of the m.sup.th and (m-1).sup.th order along the beam propagation direction, exceeds the coherence length of the fs-pulses (i.e., λ.sub.0.sup.2/Δλ),the multi-beam Talbot-like interference pattern produced by these two adjacent diffraction orders disappears. Importantly, when a time-integrating detector is used to observe the intensity distribution after the phase mask12, this effect is solely defined by the spectral bandwidth of the pulse (rather than its spectral phase) and, therefore, does not depend on whether the pulse is transform-limited or chirped. This also means that the minimum distance at which this effect can be observed with time-integrating detectors is identical for an ultrashort optical pulse and an incoherent broadband light with the same spectral density.

[0043] The lateral overlap of the ± m diffracted orders that defines the interference fringes 18, decreases as the diffracted orders propagate away from the mask. This geometric effect— neglecting spatial chirp introduced by it—is called transverse walk-off. The transverse walk-off Δ.sub.T for the m.sup.th diffraction order can be calculated using the following expression Δ.sub.T = 2 L tan (θ.sub.m), where θ.sub.m is the diffraction angle of the m.sup.th order.

[0044] There is, however, another effect (the chromatic effect) that is inherent to the phase mask technique inscription method set forth above. The chromatic effect originates from the broadband nature of the fs-pulse and depends on the chromatic dispersion of both the phase mask 12 and the focusing cylindrical lens 11. The angular spread Δθ.sub.m of the spectrum in the m.sup.th order corresponding to Δλ can be obtained by differentiating the grating equation. Hence, for normal incidence, this can be expressed as

[00004]$\text{Δ}{\theta }_m=\frac{m}{{\Lambda }_m\cos \left({\theta }_m\right)}\text{Δ}\lambda \mathrm{..}$

SAs can be seen from FIG. 2b, the long-wavelength spectral components of the pulse 21 (denoted by vertical hash lines) are diffracted at a larger angle than the short-wavelength components 22 (denoted by horizontal hash lines) and the cylindrical lens 11 will therefore focus them closer to the mask 12. The difference in the positions of the ‘blue’ and ‘red’ foci along the z-axis 23 is given by:

[00005]$\text{Δ}{Z}_{mask}^{chrom}=L\frac{\sin \left({\theta }_m\right)}{\cos \left({\theta }_m\right)}\text{Δ}{\theta }_m=\frac{mL\text{Δ}\lambda }{{\text{Λ}}_M}\frac{\sin \left({\sin }^{-1}\left(\frac{\left|m\right|{\lambda }_o}{{\text{Λ}}_M}\right)\right)}{{\cos }^2\left({\sin }^{-1}\left(\frac{\left|m\right|{\lambda }_o}{{\text{Λ}}_M}\right)\right)}$

[0045] The plane-parallel glass substrate on the surface of which the phase mask 12 is engraved introduces an additional chromatic focal elongation given in the paraxial approximation by:

[00006]$\text{Δ}{z}_{\text{substr}.}^{\text{chrom}\text{.}}=\frac{t}{n_1^2}\left(\text{d}{n}_1/\text{d}\lambda \right)\text{Δ}\lambda$

where t is the thickness of the mask substrate and n.sub.1 is the refractive index of the mask substrate. As before, the substrate moves the ‘red’ focus closer to and the ‘blue’ focus farther from the mask 12, however this fixed elongation appears to be very small compared to the elongation caused by the mask and can be neglected.

[0046] The cylindrical lens also introduces a chromatic focal elongation along the beam propagation 30 or z axis 36, as denoted in FIG. 3. An incident fs pulse of beam diameter 31 propagates through cylindrical lens 11. Unlike the case of the chromatic elongation due to the phase mask, ‘blue’ spectral components 33 (denoted by horizontal hash lines in FIG. 3) are focused closer to the mask than their ‘red’ counterparts 34 (denoted by vertical hash lines in FIG. 3). This elongation is chromatic aberration 35, which can be written as:

[00007]$\text{Δ}{z}_{\text{lens}}^{\text{chrom}\text{.}}=-\frac{f}{n_2-1}\left(\text{d}{n}_2/\text{d}\lambda \right)\text{Δ}\lambda$

where f is the focal length of the cylindrical lens and n.sub.2 is its refractive index. It can be shown that Eq. (6) remains valid even for thick lenses, provided that the lens shape is plano-convex. The latter condition is fulfilled, as set forth below. In the presence of an angularly dispersive element after the lens 11, i.e. phase mask 12, Eq. (6) needs to be modified as follows:

[00008]$\text{Δ}{z}_{\text{lens}}^{\text{chrom}\text{.}}=-\frac{f\cos \left({\theta }_m\right)}{n_2-1}\left(\text{d}{n}_2/\text{d}\lambda \right)\text{Δ}\lambda$

[0047] Some salient features of the scenario when the phase mask technique is used with broadband fs-pulses can be summarized as follows:

[0048] First, the focal elongation

[00009]$\text{Δ}{z}_{\text{mask}}^{\text{chrom}\text{.}}$

caused by angular chromatic dispersion of the mask 12 linearly grows with the distance L from the mask and the bandwidth Δλ of the fs-pulses. However, its dependence on the diffraction angle θ.sub.m is essentially nonlinear, which becomes clear if Eq. (4) is rewritten in terms of m, d and λ.sub.0 as:

[00010]$\Delta z_{\text{mask}}^{\text{chrom}\text{.}}=\frac{m^2{\lambda }_{0}L\text{Δ}\lambda }{{\text{Λ}}_M^2-m^2{\lambda }_0^2}$

[0049] For instance, for a 3.21 .Math.m-pitch mask 12 Λ.sub.M, λ.sub.0= 800 nm and Δλ = 10 nm,

[00011]$\text{Δ}{z}_{\text{mask}}^{\text{chrom}\text{.}}$

increases by ~0.1 .Math.m for m = 1, ~0.4 .Math.m for m = 2, and ~1.6 .Math.m for m = 3 when the observation point is moved every 100 .Math.m away from the mask. To put it in perspective, Eq. (5) gives

[00012]$\text{Δ}{z}_{\text{substr}\text{.}}^{\text{chrom}\text{.}}$

~0.2 .Math.m for the same Δλ and a fused silica (SiO.sub.2) mask substrate of thickness t = 3 mm (n.sub.1 = 1.453 at λ.sub.0= 800 nm, dn.sub.1/dλ, = -0.0173 at λ.sub.0= 800 nm). It should be noted that the 10 nm-bandwidth in the above example is a ‘round’ number and does not represent experimental conditions.

[0050] Second, the focal elongation caused by chromatic aberration of the beam-focusing cylindrical lens 11 (see Eq. (6)) linearly depends on Δλ for a given lens shape and lens material. Based on Eq. (6), chromatic aberration of the cylindrical lens used in the examples set forth herein (a plano-convex cylindrical lens with f = 12 mm made of OHARA S-LAH64 glass; the curved surface of the lens is designed to correct spherical aberration in one dimension; n.sub.2 = 1.776 at λ.sub.0= 800 nm, dn.sub.2/dλ = -0.0371 .Math.m.sup.-1 at λ.sub.0= 800 nm) is ~5.7 .Math.m for Δλ = 10 nm, since it does not concentrate incident light into one focal point, but instead along a focal line without the influence of spherical aberrations. According to Eq. (7), focusing through the 3.21 .Math.m-pitch mask 12 will reduce this focal elongation by a factor of cos(θ.sub.m) , making it ~5.6 .Math.m for m = 1, ~5.0 .Math.m for m = 2, and ~3.8 .Math.m for m = 3.

[0051] Third, if chromatic dispersion were the only cause of changing the focal intensity distribution of the fs-beam (which is not the case, as discussed below), there would exist a distance .sub.L from the mask 12 where

[00013]$\left|\text{Δ}{z}_{\text{mask}}^{\text{chrom}\text{.}}\right|=\left|\text{Δ}{z}_{\text{lens}}^{\text{chrom}\text{.}}\right|$

and the confocal parameter of the lineshaped fs-laser focus would attain its minimum value. For given m and λ.sub.0, this distance is solely determined by the mask-lens combination and does not depend on Δλ.

[0052] The cancelation 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 /.sub.0 in the focus of a Gaussian beam is given by I.sub.0 = 4P/(λ.sub.0Z.sub.0), where P.sub.0 is the total power in the beam and Z.sub.0 is the confocal parameter of the beam, which is twice the Rayleigh length. The presence of chromatic aberrations (and other aberrations, discussed below) elongates the confocal parameter by ΔZ and, as a result, decreases the peak intensity, which is given by I′.sub.0≈ 4P/[λ.sub.0(Z.sub.0 + ΔZ)]. For a given ΔZ, the ratio I′.sub.0/I.sub.0 can then be written as I′.sub.0/I.sub.0∝ (1 + ΔZ/Z.sub.0).sup.-1 . Taking into account that Z.sub.0 ≈ 2λ.sub.0/[πsin.sup.2 (φ)] (or Z.sub.0 ≈ 2λ.sub.0cos(θ.sub.m)/[πsin.sup.2(φ)] when focusing through mask 12), where φ is the maximal halfangle of the cone of light (at the 1/e.sup.2-intensity level) that exits the focusing optics, a stronger sharpening for tightly focused beams becomes evident.

[0053] In order to consider the effects associated with spherical aberration and conical diffraction, FIG. 4 denotes aberrations to a converging/diverging electromagnetic wave 40 produced by focusing/defocusing a light beam with a spherical or cylindrical lens 11 introduced by a plane parallel plate 41. 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 42 making angle φ with the normal of the substrate) are displaced more along the beam propagation direction than rays that have a smaller angle of incidence (i.e., paraxial rays 43). The magnitude of such a longitudinal separation

[00014]$\text{Δ}{z}_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}$

(44) between the marginal and paraxial foci is given by:

[00015]$\text{Δ}{z}_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}=\frac{t}{n_1}\left\{1-{\left[\frac{n_1^2\left(1-{\sin }^2\left(\phi \right)\right)}{n_1^2-{\sin }^2\left(\phi \right)}\right]}^{1/2}\right\}$

where t is the thickness 45 of the phase mask substrate. In the 3.sup.rd-order approximation Eq. (9) reduces to:

[00016]$\text{Δ}{z}_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}=\frac{\left(n_1^2-1\right)t{\phi }^2}{2n_1^3}$

[0054] Even though Eqs. (9) and (10) provide no information about the intensity distribution in the focal volume, they define two shadow boundaries on the z-axis between which the light rays cross the z-axis and in this respect give a rough estimate of the focal volume extent in the beam propagation direction. The related geometric optics ray tracing formalism is schematically presented in FIG. 4.

[0055] The phase mask technique is inherently based on focusing a laser beam with a cylindrical lens 11 through a plane parallel glass plate 41 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. (7), Eq. (9) needs to be modified as follows:

[00017]$\text{Δ}{z}_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}=\frac{t\cos \left({\theta }_m\right)}{n_1}\left\{1-{\left[\frac{n_1^2\left(1-{\sin }^2\left(\phi \right)\right)}{n_1^2-{\sin }^2\left(\phi \right)}\right]}^{1/2}\right\}$

[0056] In the 3.sup.rd-order approximation, Eq. (11) reads:

[00018]$\text{Δ}{z}_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}=\frac{\left(n_1^2-1\right)t\cos \left({\theta }_m\right){\phi }^2}{2n_1^3}$

[0057] As an example, for a typical substrate (t = 3 mm, n.sub.1 = 1.453 (SiO.sub.2) at λ.sub.0= 800 nm), φ = 15°, and a 3.21 .Math.m-pitch mask 12 Eq. (11) gives

[00019]$\text{Δ}{z}_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}$

~37 .Math.m for m = 1, ~33 .Math.m for m =2, and ~25 .Math.m for m =3. It is clear that

[00020]$\text{Δ}{z}_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}$

significantly exceed the theoretical confocal parameter Z.sub.0 ≈ 2λ.sub.0 cos(θ.sub.m)/[Πsin.sup.2(.sub.φ)], which is ~7.4 .Math.m for m = 1, ~6.6 .Math.m for m =2, and ~5.0 .Math.m for m =3.

[0058] The generic grating equation mλ.sub.0 = Λ.sub.M sin(θ.sub.m) , used above, 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. To visualize the behavior of diffraction orders produced by an off-plane incident beam, Harvey et alin Applied Optics, vol. 42 no. 7, pg. 1167-1174 (2003), introduced direction cosines of the actual spatial coordinates to describe both the incident and diffracted rays (FIG. 5a). According to this formalism, the absolute values of the direction cosines of the incident ray are given by:

[00021]$\begin{array}{l}{\alpha }_{\text{i}}=\sin \left(\chi \right)\cos \left(\phi \right);{\beta }_{\text{i}}=\sin \left(\phi \right);\\ {\gamma }_{\text{i}}=\cos \left(\chi \right)\cos \left(\phi \right)\text{with}{\alpha }_{\text{i}}^2+{\beta }_{\text{i}}^2+{\gamma }_{\text{i}}^2=1\end{array}$

where x is the in-plane angle of incidence. The absolute values of the direction cosines of the ray diffracted into the m.sup.th order can be found from:

[00022]$\begin{array}{l}{\alpha }_m-{\alpha }_i=m{\lambda }_0/{\Lambda }_M,\\ {\alpha }_m=\sin \left({\theta }_m\right)\cos \left(\phi \right);{\beta }_m={\beta }_i=\sin \left(\phi \right)\text{with}{\alpha }_m^2+{\beta }_m^2+{\gamma }_m^2=1\end{array}$

[0059] In one embodiment, the azimuthal angle is x = 0, while the polar angle φ represents the angle at which the marginal rays (FIGS. 4 and 5) impinge the mask substrate and the grooves on the mask 12. The spatial spectrum of propagating (i.e., nonevanescent) diffraction orders is defined by the condition

[00023]${\alpha }_m^2+{\beta }_m^2\le 1.$

[0060] For x = 0 (FIG. 5b), γ.sub.m and tan(.sub.Ψm,φ) can be written as:

[00024]$\begin{array}{l}{\gamma }_m={\left[1-m^2{\lambda }_0^2/{\text{Λ}}_M^2-{\sin }^2\left(\phi \right)\right]}^{1/2};\\ \tan \left({\psi }_{m,\phi }\right)=\sin \left(\phi \right){\left[1-m^2{\lambda }_0^2/{\text{Λ}}_M^2-{\sin }^2\left(\phi \right)\right]}^{-1/2}\end{array}$

[0061] The z-coordinate z.sub.m,φ of the focal line produced by diffracted marginal rays (i.e., point F.sub.m,.sub.φ in FIG. 5b) is given by

[00025]$z_{m,\phi }=l\tan \left(\phi \right){\tan }^{-1}\left({\Psi }_{m,\phi }\right)$

, where / is the distance from the back surface of mask 12 to the 0.sup.th-order paraxial focus Fo,o. On the other hand, the z-coordinate z.sub.m,0 of the focal line produced by diffracted paraxial rays (i.e., point F.sub.m,.sub.0 in FIG. 5b) is given by

[00026]$z_{m,0}=l\cos \left({\theta }_m\right).$

Finally, the distance

[00027]$\text{Δ}{z}_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}=z_{m,0}-z_{m,\phi }$

along the z-axis between the marginal focus F.sub.m,.sub.φ and the paraxial focus F.sub.m,.sub.0 corresponding to the m.sup.th diffraction order can be expressed as:

[00028]$\text{Δ}{z}_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}=L\left\{1-{\left[\frac{{\cos }^2\left(\phi \right)-m^2{\lambda }_0^2/{\text{Λ}}_M^2}{{\cos }^2\left(\phi \right)\left(1-m^2{\lambda }_0^2/{\text{Λ}}_M^2\right)}\right]}^{1/2}\right\}$

, where L = l/cos(θ.sub.m) is the distance from the back surface of mask 12 to the paraxial focus F.sub.m,.sub.0, as defined in FIG. 5b. When φ is small (i.e., in the 3.sup.rd-order approximation), Eq. (16) can be presented in a more compact form:

[00029]$\text{Δ}{z}_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}=\frac{m^2{\lambda }_0^{2}L\phi }{2\left({\text{Λ}}_M^2-m^2{\lambda }_0^2\right)}$

[0062] As above, the geometric optics formalism used to derive Eq. (16) yields no information about the intensity distribution in the focal volume and only defines two shadow boundaries on the z-axis between which the light rays cross the z-axis.

[0063] To summarize, some key features of spherical aberration induced by the mask substrate and an aberration originating from off-plane diffraction by the mask 12 are as follows:

[0064] First, the focal elongations

[00030]$\text{Δ}{z}_{\text{substr}\text{.}}^{\text{sub}\text{.aberr}\text{.}}$

and

[00031]$\text{Δ}{z}_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}$

do not depend on Δλ and therefore should be taken into account during FBG inscription irrespective of the laser source.

[0065] Second, for a given material (and diffraction order m),

[00032]$\Delta z_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}$

nonlinearly depends on the focusing angle φ (i.e., ∝ φ.sup.2) and linearly depends on the substrate thickness t.

[0066] Third,

[00033]$\Delta z_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}$

is proportional to φ.sup.2 and linearly depends on the distance L from the mask 12. Similar to

[00034]$\Delta z_{\text{mask}}^{\text{chrom}\text{.}}$

(compare Eq. (8) and Eq. (17)), the dependence of

[00035]$\Delta z_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}$

on the diffraction angle θ.sub.m is nonlinear. Based on the above ray optics analysis (Eq. (16)), the focal elongation caused by conical diffraction is expected to be quite significant. For instance, for a 3.21 .Math.m-pitch mask 12, λ.sub.0= 800 nm and φ= 15°,

[00036]$\Delta z_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}$

increases by ~0.2 .Math.m for m = 1, ~1.2 .Math.m for m = 2, and ~4.7 .Math.m for m = 3 when the observation point is moved every 100 .Math.m away from the mask 12.

[0067] Fourth, if the chromatic effects considered above are ignored, there exists a distance L from the mask 12 where

[00037]$\left|\Delta z_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}\right|=\left|\Delta z_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}\right|$

and the confocal parameter of the line-shaped fs-laser focus attains its minimum value. For given m and λ.sub.0, this distance is solely determined by the mask parameters and in the 3.sup.rd-order approximation 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.

[0068] Fifth, there exists a distance L from the mask 12 where the conditions

[00038]$\left|\Delta z_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}\right|=\left|\Delta z_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}\right|\text{and}\left|\Delta z_{\text{mask}}^{\text{chrom}\text{.}}\right|=\left|\Delta z_{\text{lens}}^{\text{chrom}\text{.}}\right|$

can be fulfilled simultaneously, which is expected to result in a stronger maximum in the focal peak intensity I′.sub.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

[00039]$\left|\Delta z_{\text{mask}}^{\text{chrom}\text{.}}\right|=\left|\Delta z_{\text{lens}}^{\text{chrom}\text{.}}\right|$

and then choosing the mask thickness t that would fulfill the condition

[00040]$\left|\Delta z_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}\right|=\left|\Delta z_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}\right|$

.

EXPERIMENTAL RESULTS

[0069] ATi-sapphire regeneratively amplified laser system operating at a central wavelength of λ.sub.0= 800 nm, was used in experiments implementing embodiments set forth herein. The bandwidth (FWHM) and output beam diameter (at the 1/e.sup.2-intensity level) of the fs-system were Δλ ~ 6 nm and 2wo ~ 7 mm, wherein an interference filter was placed in the regenerative amplifier’s beam to reduce the original 14 nm bandwidth to the 6 nm bandwidth. The phase mask used had a mask pitch of 1.07 .Math.m and was 2.1 mm thick. To reconstruct 3D time-averaged intensity distributions after the mask, the technique described in Hnatovsky et al. Optics Express vol. 25 no. 13 pg. 14247-14259 (2017) was employed. Briefly, the respective xy-intensity distributions with a 1 .Math.m 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 in FIGS. 6, 7 and 8 were obtained by averaging the values of points with fixed (y.sub.i,z.sub.i)-coordinates along the x-axis and projecting the respective mean values onto the yz-plane.

[0070] To estimate the quality of the focusing optic and the output beam from the regenerative amplifier, the phase mask was initially removed from the beam path. FIG. 6 shows the focal intensity distributions of the output beam in the yz-plane when the beam is focused with the 12 mm-focal-length cylindrical lens corrected for spherical aberration 11 (hereinafter cylindrical lens 11) as described previously. The effective numerical aperture of the cylindrical lens (i.e., sin(φ)) is estimated at 0.26. Under such conditions, aberration-free focusing of quasi-monochromatic light at λ.sub.0= 800 nm would translate into a 7.6 .Math.m confocal parameter (i.e., Z.sub.0~ 7.6 .Math.m) for sin(φ) = 0.26. As mentioned above, the curved surface of the cylindrical lens is designed to correct spherical aberration in one dimension. However, the focal shape indicates 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. The effect of chromatic aberration can be deduced from FIG. 6 by measuring the confocal parameter Zo, which is ~ 14 .Math.m. For reference, Eq. (6) gives

[00041]$\Delta z_{\text{lens}}^{\text{chrom}\text{.}}\sim 3.4\text{μ}\text{}\text{m}\text{.}$

[0071] The effect of substrate-induced spherical aberration for the 2.1 mm thick substrate is demonstrated in FIG. 7, where the fs-beam was focused through the mask substrate without intercepting the phase mask grooves. Analysis of the respective yz-intensity distributions in FIG. 7 confirms that the substrate-induced negative spherical aberration significantly elongates the focal volume. The measured Zo is presented in FIG. 7. For reference, longitudinal substrate-induced spherical aberration

[00042]$\Delta z_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}$

given by Eq. (9) is ~42 .Math.m for t = 2.1 mm when sin(φ) = 0.26. It should be noted that the above value for

[00043]$\Delta z_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}$

provides only a rough estimate as it defines only two shadow boundaries on the z-axis between which the light rays cross the z-axis.

[0072] The interplay of the chromatic effects, which are characterized by

[00044]$\left|\Delta z_{\text{mask}}^{\text{chrom}\text{.}}\right|$

and

[00045]$\left|\Delta z_{\text{lens}}^{\text{chrom}\text{.}}\right|,$

, and the effects associated with spherical aberration and conical diffraction, which are characterized by |

[00046]$\left|\Delta z_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}\right|$

and

[00047]$\left|\Delta z_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}\right|,$

,was studied using the 1.07 .Math.m-pitch 2.1 mm-thick mask 12. The fs-beam from the regenerative amplifier was focused through the mask grooves. The results pertinent to this experiment are presented in FIGS. 8 and 9. The plot in FIG. 9 shows the focal peak intensity I′.sub.0 as a function of distance L from the mask 12, with a strong sharp maximum being located at L ~ 350 .Math.m. The observed variations of I′.sub.0 with L can also be correlated with the transformations occurring with the focal intensity distributions in the yz-plane. Substrate-induced negative spherical aberration is quite strong at L = 100 .Math.m, almost neutralized by conical diffraction at L = 350 .Math.m, and completely reversed by conical diffraction at L = 900 .Math.m. Theoretically, the distance L at which the condition |

[00048]$\left|\Delta z_{\text{mask}}^{\text{chrom}\text{.}}\right|$

=

[00049]$\left|\Delta z_{\text{lens}}^{\text{chrom}\text{.}}\right|$

is fulfilled does not depend on the mask thickness t and should therefore be ~240 .Math.m. On the other hand, L, at which the condition

[00050]$\left|\Delta z_{\text{substr}\text{.}}^{\text{sph}\text{.aberr}\text{.}}\right|$

=

[00051]$\left|\Delta z_{\text{mask}}^{\text{con}\text{.diffr}\text{.}}\right|$

is fulfilled, linearly depends on t and is thus expected to be ~380 .Math.m. This data provides evidence that the sharp maximum in FIG. 9 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 12, the evolution of the focal intensity distribution is governed by the combined action of chromatic dispersion and conical diffraction originating from the mask 12.

[0073] A final set of experiments investigated 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.

[0074] First, uncoated SMF-28 fiber samples were placed at different distances L from a 1.07 .Math.m-pitch mask 12 that was 2.1 mm in thickness and exposed to radiation from the regenerative amplifier. The femtosecond beam was focused with the 12 mm-focal-leng.sub.th cylindrical lens 11 with an effective numerical aperture of sin(φ) = 0.26, as above. During the inscription the cylindrical 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.

[0075] FIG. 10 shows that the FBG strength does follow the focal peak intensity plot shown in FIG. 7. This trend can be seen even more clearly in FIG. 10b, 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 Ro of a uniform FBG with a constant sinusoidal modulation was used:

[00052]$R_0\approx {\text{tanh}}^2\left[\pi \Delta n{W}^{\prime }\eta \left(V\right)/{\lambda }_{\text{B}}\right]$

[0076] In Eq. (18), W is the FBG length, λ.sub.B is the Bragg wavelength given by A.sub.B = nΛ.sub.M (n is the refractive index of the fiber core), and .sub.η(V) ≈ 1-1/V.sup.2,V ≈ 2.4 is a function of the fiber parameter V that represents the fraction of the integrated fundamental-mode power in the core. 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 according to W′ ≈ W.sub.0 - Δ.sub.T = W.sub.0 - 2Ltan(θ.sub.m) , where W.sub.0 is the FBG length when the fiber touches the mask (W.sub.0~ 13 mm, as defined by the clear aperture of the cylindrical lens 11 along x).

[0077] Next, approximately 5 mm-long Type-I gratings were written in polyimide-coated 50 .Math.m fibers (51.2 .Math.m exactly). It is important to note that the fibers were not hydrogen/deuterium-loaded. According to the manufacturer (FIBERCORE), the germanosilicate core has a diameter of 4.1 .Math.m, and the polyimide coating is approximately 10 .Math.m-thick. The 50 .Math.m-fiberwas placed 350 .Math.m away from the 1.07 .Math.m-pitch 2.1 mm-thick mask 12 and exposed to the radiation from the regenerative amplifier. The beam was focused with the 12 mm-focal-length cylindrical lens 11 with an effective numerical aperture of sin(φ) = 0.26. During the exposure the cylindrical lens was scanned perpendicular to the fs-beam. The pertinent results are demonstrated in FIG. 11. It should be noted that writing in polyimide-coated 50 .Math.m-fibers was possible only within a narrow range (250 .Math.m < L < 450 .Math.m) of fiber-to-mask distances around the optimum L = 350 .Math.m.

[0078] The experimental results agree with the predictions and estimates based on the semi-quantitative analysis above, where the chromatic effects were presented in terms of only two different wavelength (i.e., ‘blue’ and ‘red’) and monochromatic aberrations were introduced as ray optics phenomena.

[0079] Specifically, the simple analytic expressions given in Eqs. (3) through (17) give the distance from the mask 12 where i) spherical aberration introduced by the plane-parallel mask substrate is cancelled out by conical diffraction and ii) chromatic aberration of the cylindrical lens 11 is cancelled out by chromatic dispersion of the mask. When these two distances are approximately equal, a dramatic sharpening of the laser focus and the accompanying growth of focal light intensity take place. For a 1.07 .Math.m-pitch mask 12, the agreement between theory and experiment is within a few tens of micrometers. Most importantly, the strength of FBGs (FIGS. 11a and b) recorded at a fixed fluence but different distances from the mask follows the focal peak intensity curve measured in free space (FIG. 9).

[0080] The formalism presented above provides 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 cylindrical lens 11, a plane-parallel plate 41 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 set forth in the prior art. However, it should be noted 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, the beam quality factor (i.e., M.sup.2) of the regeneratively amplified fs-system and residual aberrations of the cylindrical lens 11 would still remain unknown. In view of the above, the semi-quantitative formalism set forth above reinforced with intensity distribution measurements after the mask 12 provides an important shortcut to identify optimum FBG laser writing conditions when the phase mask technique is used.

[0081] Two independent sets of effects have been considered that are inherent to the phase mask technique, namely i) chromatic dispersion of the mask 12, which is counteracted by chromatic aberration of the cylindrical lens 11, 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 (~48°; 1.07 .Math.m-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 12. 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 11 (focal distance, refractive index), and diffraction angle of the mask (mask period). Under 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. In this respect, the distance from the mask 12 can be tuned 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.

[0082] After the maximum has been passed, the combined action of chromatic dispersion and conical diffraction introduced by the phase mask 12 gradually decrease the peak intensity inside the focal volume of the cylindrical lens 11 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 .Math.m cladding) when they are placed at the optimum position from the mask 12 becomes a readily achievable task even if the fibers are not hydrogen/deuterium-loaded to increase their photosensitivity.

[0083] The above chromatic and conical diffraction effects nonlinearly scale down as the diffraction angle is decreased and thus become barely noticeable when the diffraction angle is ~22° (2.14 .Math.m-pitch mask). Taking into account that chromatic aberration of the focusing cylindrical lens 11 is generally small and negative spherical aberration introduced by the mask substrate can be relatively easily compensated, working with small diffraction angles provides a convenient laser-writing technique in terms of its weak dependence on the distance from the mask 12. However, it should be remembered that the use of small diffraction angles implies that the resultant Bragg grating utilizes a higher-order resonance, which reduces the grating strength for a fixed grating length.

[0084] In addition, even though the geometric optics approach set forth above provides guidance regarding the distance from the mask 12 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 important for obtaining accurate results for a given laser-writing setup.

[0085] The many features and advantages of the invention are apparent from the detailed specification and, thus, it is intended by the appended claims to cover all such features and advantages of the invention that fall within the true spirit and scope of the invention. Further, since numerous modifications and changes will readily occur to those skilled in the art, it is not desired to limit the invention to the exact construction and operation illustrated and described, and accordingly all suitable modifications and equivalents may be resorted to, falling within the scope of the invention.