Open Access
Add to BibSonomy
Abstract
Using the Kostenbauder matrix formalism, we analyze the complex spatio-temporal behaviour of focused femtosecond pulses propagating in three tunable phase mask interferometers for writing fiber Bragg gratings. In interferometers that use a second half-period mask to reverse the phase front, we show that at zero detuning, and with a proper amount of pre-dispersion, the pulses can be perfectly focused on the fiber, whereas with a Talbot interferometer, the pulses suffer from large, unavoidable spatial and temporal broadening. We quantify the enhancement of grating writing efficiency with focusing as a function of beam diameter and interferometer detuning, for a photoinduced index change that is due to a two-photon or higher order process. Based on previously published experimental results, we estimate the potential improvement in grating writing speed.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, there have been many reports of the use of femtosecond pulses to write fiber Bragg gratings (FBGs), at wavelengths from the infrared to the ultraviolet [1–6]. The main motivation is that such pulses have very high peak power, which means that two- or multi-photon effects are involved in creating a permanent refractive index change in the fiber. The use of infrared or visible wavelengths also allows the FBG to be written through the fiber cladding [7], which simplifies fabrication and helps preserve the fiber mechanical integrity. It is also useful for fiber materials that have too much absorption in the cladding at shorter wavelengths. Along with these demonstrations has come the development of Point-by-Point (PbP) writing techniques [1], where the FBG is written line by line by a tightly focused femtosecond laser beam, sometimes with a single pulse per line. One advantage of the PbP technique, the Bragg wavelength can be changed at will by tuning either the translation stage speed, or the pulse repetition rate.
The PbP technique is not a universal solution, however. The highly nonlinear nature of single pulse writing, and quasi threshold behaviour of the index change, make it difficult to get precise control of the grating modulation depth, and may result in strong losses at short wavelengths that are detrimental to fiber laser operation [5]. Furthermore, for good and repeatable Bragg wavelength accuracy, the precision requirements on the translation stage velocity are quite high. For those reasons, there have been many reports on the use of phase masks to write FBGs with femtosecond lasers [2–6,8,9].
In a typical phase mask setup, the fiber is positioned very close to the mask, and the beam is focused, at least in the transverse direction to the fibre axis, in order to maximize the power density. When using femtosecond pulses, this can lead to problems. For example, the high peak intensity can create multi-photon absorption in the phase mask glass itself [2]. Also, to get rid of the zero order beam, the fiber is often moved away from the mask, at the expense of fringe contrast since the two diffracted beams no longer have perfect overlap. Therefore, in most reports, the beam is not focused in the direction parallel to the fiber axis, even when a scanning beam technique is used [10]. This means that pulses with significant energy are required in order to reach a high enough peak power.
Most of these problems can be alleviated by using phase mask interferometers [11–13]. With such interferometers, the fiber is positioned away from the mask, so that tight focusing can be performed in both directions. The fiber can also be positioned at the perfect overlap of the two interfering beams, without any zero-, or higher-order diffracted beams. Thus one can take full advantage of the high peak power of the pulses, and improve the grating writing efficiency by a large factor, even with much smaller pulse energies. Phase mask interferometers also allow tuning of the Bragg wavelength, making them a very flexible tool that can compete with PbP writing for moderate tuning ranges, for example across the gain bandwidth of a given fiber amplifier or fiber laser.
We discuss here the performance of three such interferometers, shown in Fig. 1, with regards to their ability to focus the pulses and enhance the peak power for more efficient grating writing. The Talbot design is well known and has been used with different types of lasers [2,14]. The other two, which we call the achromatic and the ring interferometers, use one, or a pair of second masks with half the first mask period to rediffract the beams The second mask reverses the spectral dispersion of the first one, allowing perfect imaging of the pulses on the fiber at zero detuning, no matter what the temporal coherence of the source is. The achromatic design has been known in photolithography for a long time [15], and was recently used with femtosecond pulses, albeit in a zero-detuning configuration [13]. The ring interferometer is a variant of the Sagnac interferometer used in Ref [11]. Though not published before, it was developed and used by one of the authors (F.O.) for industrial production using a relatively low coherence Q-switched solid-state UV laser. In the achromatic interferometer, tuning the Bragg wavelength is achieved by rotation of the pair of second masks in opposite directions [16], whereas it is done by rotation of the mirrors in the ring. As with a standard phase mask setup, the phase of the grating is only sensitive to the relative lateral movements of the masks and fiber, which makes them more robust compared with the Talbot, which is very sensitive to back and forth vibrations of the mirrors.
In this paper, we will analyze the advantages and drawbacks of these interferometers when using femtosecond pulses, specifically when it comes to focusing the pulses. We will first describe their basic characteristics, such as the tuning range, and path delay. We will then show how the writing efficiency can be enhanced by using focused pulses when the index change is due to a nonlinear process. Using the Kostenbauder matrix formalism, we will calculate the evolution of the pulse as it travels toward the fiber, derive expressions for the width and duration of the resulting interference pattern, and define enhancement parameters to quantify the improvement in grating writing speed with respect to unfocused pulses. Finally, we will use data from experimental reports to estimate the attainable writing speed in cases of practical interest.
2. Tuning and path delay
In both the Talbot and the ring, wavelength tuning is achieved by tilting both mirrors symmetrically. The detuning from the nominal Bragg wavelength (at zero tilt) is given by the angle of the beams intersecting on the fiber, which is the diffraction angle from the mask plus or minus two times the tilt angle of the mirrors (for the Talbot), or four times for the ring, since the beam is reflected twice. For example, tilting by one degree in the ring is sufficient to tune the Bragg wavelength by more than 200 nm. In the achromatic interferometer, the tuning occurs because the difference between the incident and diffracted angles is not constant with incident angle. This is a second order effect, and therefore a much larger tilt angle is required. Figure 2 shows the Bragg wavelength as a function of tilt angle for a nominal Bragg wavelength of 2000 nm, where the tilt angle is defined as positive for clockwise rotation of the right side mask. Since this is a second order effect, tuning can only go towards shorter wavelengths. It is also slightly larger for positive angles. The same 2000 nm wavelength will be used in all our examples, because of its interest for mid-infrared lasers.
Fig. 2. Bragg wavelength as a function of the phase mask tilt angle $\theta _{M}$ for the achromatic interferometer in Fig. 1 (b).
The first characteristic of the interferometers affecting their performance with femtosecond pulses is the increasing path delay between the two branches at positions away from the central position. At zero detuning, the paths in the two branches are always the same. A relative path delay occurs when the interferometers are tuned to other wavelengths, and it increases linearly with distance from the central position, and with detuning from the nominal Bragg wavelength.
The path delay can be found by geometrically tracing the rays for both arms and calculating their length. Since it increases linearly with position, we can quantify it with the parameter $D$, expressed in femtoseconds per millimeter of grating. Figure 3 shows the value of $D$ as a function of detuning for the three interferometers. To first order, it is independent of the writing wavelength. The Talbot and the ring have nearly indistinguishable values of $D$, and thus only one curve is shown. The achromatic has much smaller path delay, and is thus a better choice for writing longer gratings. For example, with 200 fs pulses, the grating length for 200 nm detuning would be limited to a little less than 2 mm for the Talbot and the ring, but can be as long as 7 mm for the achromatic. Therefore, the path delay puts a first limitation to the tuning range when using femtosecond pulses. On the other hand, this effect can also be used to provide a natural apodization of the grating [11], especially since many commercial fs lasers have adjustable pulse widths.
3. Enhancement in writing efficiency with focusing
When the index change is a nonlinear function of intensity, focusing the beam can greatly increase the writing efficiency for a given pulse energy. The other advantage of a focused scanning beam is the better control on the grating profile due to the higher spatial resolution, using various methods for apodization and phase shifting. We use a simple model for the index change where it is linearly cumulative with each pulse, but has a nonlinear dependence on the peak intensity of the pulse, as observed in many reports [2,4,5,8]. At a given location, after exposing to $N_{p}$ pulses of peak intensity $I_{pk}$ during a time $\tau _{g}$, which may or may not be equal to the pulse duration $\tau _{p}$, the total index change is:
where $k$ is a constant, which can be estimated from experimental results using the peak intensity and the total fluence required for a given $\delta n_{tot}$: The peak intensity is linked to the pulse energy $K_{p}$, the tranverse and longitudinal diameters $w_{t}$ and $w_{p}$, and the pulse duration $\tau _{p}$: The time $T_{loc}$ required to reach $\delta n_{tot}$ at a given location is then the number of pulses divided by the laser repetition rate $R$, and can be expressed as:4. Kostenbauder matrix calculation
The pulse propagation can in principle be computed with the Kirchoff-Fresnel integral. However, when dealing with Gaussian beams, and temporally Gaussian pulses, the matrix formalism introduced by Kostenbauder has proven to be very useful and more convenient to use [17,18], although other simplified techniques have been used [19,20]. The Kostenbauder matrices are 4x4 matrices, as opposed to the 2×2 matrices commonly used to calculate Gaussian beam propagation, where the additional elements take into account the time dimension. All optical elements can be described by a matrix, and the final beam parameters is computed by multiplying the various matrices of the optical system.
We consider a setup where a cylindrical lens is placed at a distance $d_{f }$ upstream of the first phase mask, to focus the beam with a diameter $w_{p0}$ in a direction parallel to the fiber axis. That lens can be translated along with the beam, to write a longer grating [21], ultimately limited by the increasing path delay between the pulses [11]. Therefore it is necessarily placed upstream of the interferometer. Another fixed cylindrical lens can be used to focus the beam in the direction perpendicular to the fiber, down to a diameter $w_{t}$. Such a lens can be positioned either outside or inside the interferometer, and can stay fixed if it is long enough to accomodate the scanning range of the beam.
When the Bragg wavelength is tuned by tilting either the mirrors or the second masks, the $z$ distance (normal to the first mask) between that mask and the intersection point (where the fiber is placed) will vary, so for a fixed focal length, the distance $d_{f }$ of the lens useed for focusing in the parallel direction should be adjusted accordingly. In our examples, we will use a fixed focal length of $f=200$ mm, with $d_{f}$ being optimized for each Bragg wavelength. Focusing in the perpendicular direction is assumed to remain the same whatever the detuning, which will be the case if the lens used is placed inside the interferometer, and at a fixed distance from the fiber. In all our examples, we consider an achromatic interferometer with second masks placed at $z=50$ mm away from the first one, for a total propagation distance of approximately 108 mm (varying with detuning). To have the same propagation distance with the ring interferometer, the rotation center of the side mirrors must be 35.5 mm away from the mask in the $z$-direction, and ${\pm} 14.5$ mm away in the $x$ direction. There is no need to calculate the position of the Talbot mirrors since they do not enter into the matrix calculations. Those are quite compact designs, but compactness is desirable, since the effects described here increase with travel distance. In all our examples, we will assume a nominal Bragg wavelength of 2 $\mu$m at zero detuning.The writing wavelength is taken as 515 nm, typical of commercially available frequency-doubled Yb-based femtosecond lasers, and the original pulse duration is taken as 200 fs. These parameters represent a worst case scenario. The deleterious effects found here are smaller for shorter writing wavelengths, such as the fourth harmonic, and for longer pulses. We will consider only negative detunings (toward shorter wavelengths), since this is the only direction achievable with the achromatic interferometer, and is also more practical to implement with the ring.
To perform the K-matrix calculation, we need up to six matrices. The Talbot interferometer requires four: (1) $k_{lens}$ for the lens , (2) $k_{df}$ for the travel from the lens to the mask, (3) $k_{m1}$ for the phase mask itself , and (4) $k_{l1}$ for the travel from the mask to the fiber, which is also the matrix for propagation from the first mask to the second mask in the other two interferometers. The mirrors have no effect on the spatio-temporal properties of the pulses, other than redirecting them, and do not require a matrix. The achromatic interferometer and the ring interferometer have additional matrices (5) $k_{m2,}$ for the second mask and (6) $k_{l2}$ for the distance from the second mask to the fiber. As with the Talbot, the mirrors of the ring interferometer have no effect on the pulses. Later, we will add another matrix $k_{D}$, which is simply a certain amount of pre-dispersion added to the pulse before the interferometer, that can compensate for the temporal chirp acquired by the pulse after it is dispersed by the mask.
In all our calculations, we assume that the interferometer is symmetrical. Therefore we only need to calculate the propagation in one arm, and the pulse from the other arm is simply a mirror image of the first one. In the formulas below, we show the matrices for the right-side arm.
The first step is to calculate the various distances and angles. This was done using the same ray-tracing calculation already performed to calculate the path delays. The matrix for a free space propagation over a distance $d$ only has $k(1,2)=d$ as a non-zero non-diagonal element (diagonal elements are equal to 1). For the lens, the only non-diagonal element is $k(2,1)={-}1/f$. For dispersion, we will use a matrix with $k(3,4)=D$, where $D$ is the dispersion in ps$^{2}$. The matrices for mask 1 and 2 need to be carefully written down, using the right value and sign for the incident and diffracted angles. There is no clear convention in the literature about those. Considering the right arm of the interferometer, and taking the angle with respect to the mask plane, with a positive value in the range 0 to $\pi /2$, we get the following matrices:
The pulse parameters, such as their spatial width and temporal duration, can be readily obtained from the $Q_{out}^{i}$ matrix. Expressing the electric fields on the right and left sides $E_{R}$ and $E_{L}$ in terms of the new coordinates, and with some rearrangement, we find:
where:The interference pattern, on the other hand, is due to the sum of both fields, and is proportional to the product $E_{R}E_{L}^{*}$. Because of symmetry, it does not have any tilt, but has a phase term proportional to $xt$. We find:
5. Interferometers at zero detuning
Two effects have to be considered to understand the behavior of the femtosecond pulses in the interferometers. The first one is the combined effect of focusing and diffracting the pulses, shown in Fig. 4, and the second one is the effect of spectral dispersion by the masks, shown in Fig. 5. In order to understand the former, we start with the simpler configuration of the Talbot interferometer in Fig. 4(a), which will also reveal how unsuited it is for use with femtosecond pulses.
Fig. 4. Pulse front behaviour when focusing the beam at zero detuning in the direction parallel to the fiber axis.
Fig. 5. (a) spectral spread in the Talbot interferometer (b) spectral spread in the achromatic interferometer at zero detuning, and (c) for non-zero detuning
Focusing a femtosecond pulse with a lens can have a strong effect on the pulse, due to the delay between the phase front and the pulse front, as well as chromatic dispersion in the lens material. Pulses can suffer large temporal broadening at the focal point [22]. Achromats avoid this effect but introduce a uniform GVD across the lens, that can however be precompensated. In our case, we assume a perfect lens and so the K-matrix remains very simple. In any case, we are dealing with relatively long pulses (200 fs), and are not looking for extreme focusing, and using small numerical apertures (${\approx}$0.01), so both conditions make these effects less severe. On the other hand, if the beam is diffracted by a phase mask right after the lens at an angle $\theta _{d}$ with respect to the mask plane, as illustrated in Fig. 4(a), a delay between the rays on both sides of the beam is introduced, that increases from right to left (for the beam diffracted to the right). Thus the pulse front is gradually tilted, as indicated by the dashed lines. The maximum delay corresponds to the path difference between the left-side ray and the right-side ray, given by the formula:
To this effect is added the effect of spectral spread, illustrated in Fig. 5(a). In that case, rays that originate from the same central point diverge according to their wavelength, so that the focused pulse is actually an image of the pulse spectrum. This also contributes to a spatially broader pulse on the fiber plane. The spectral dispersion also causes a temporal broadening because of the wavelength-dependent travel time to the fiber, further stretching the pulse temporally.
The full behaviour of the pulse is therefore a combination of these effects, and is shown by the animation in Fig. 6(a), which is taken in the reference frame of the pulse, with an observing plane parallel to the mask (the preview only shows the final shape and dimensions of the pulse). The animation starts once the beam has been diffracted by the first mask, which is positioned at $z=0$. For this example, the original beam waist is 2 mm, the 200 mm focal length lens is placed at $d_{f}=71.3$ mm from the mask, and the focal point is at a distance $z= 100$ mm from the mask, for a total beam propagation distance from the mask to the fiber of 108 mm. As expected, the pulse front is seen to tilt with distance, until the tilt angle reaches 90 degrees at the focal point. All the while the pulse duration is broadened. Figure 7 shows the evolution of the beam width $w_{p}$ and duration $\tau _{p}$ as a function of the $z$-distance from the mask as it propagates toward the fiber, as well as the values of $e_{p}$ and $e_{g}$. The pulse duration starts to evolve after the lens, at $z={-}d_{f}$, while the parameters $e_{p}$ and $e_{g}$ are only defined after the mask ($z=0$). The FWHM pulse duration on the fiber is found to be 1620 ps, and the FWHM beam waist is 184 $\mu$m. The pulse enhancement factor is found to be $e_{p}=0.67<1$, showing no enhancement at all for the pulse itself, due to the large temporal broadening. Furthermore, the interference pattern width $w_{g}$ is found to be only 23 $\mu$m, so that the final enhancement factor $e_{g}$ is only 0.083, which is a much worse situation than for no focusing at all! As can be seen, $e_{g}$ just keeps decreasing after the mask, making any beam focusing useless. As a matter of fact, we have found that the highest possible $e_{g}$ was only 1.6, and occurs for an unfocused pulse with an initial width of 250 $\mu$m on the mask. We thus find the surprising conclusion that no enhancement of the writing efficiency with focusing is possible with a Talbot interforemeter.
Fig. 6. Pulse evolution animation previews : (a) Talbot interferometer (see Visualization 1); (b) Achromatic and Ring at zero detuning (Visualization 2); (c) Achromatic for 200 nm detuning (Visualization 3). On these previews, only the final pulse dimensions are shown.
Fig. 7. (a) Evolution of the pulse width and duration and (b) enhancement parameters for a Talbot interferometer.
Now consider the achromatic interferometer set up at zero detuning, with all masks parallel, as illustrated in Fig. 4(b). In such a configuration, it is formally equivalent to the ring interferometer, so the same reasoning applies to both. From the first mask to the second mask, the pulse behaves as in the Talbot interferometer: the pulse front is gradually tilted, and stretched, since the rays on the right side are faster than on the left side. However, past the second mask, the phase front is reversed, so that now the faster rays become the slower ones. Since the diffracted angle from the second mask is the same as that of the first mask (for a period P/2), everything is symmetrical. Thus at the focal point, all the rays have traveled an equal distance, and there is no more spatial pulse broadening due to focusing. By the same token, the spectral dispersion is also reversed by the second mask, as shown in Fig. 5(b), and at zero detuning, all the wavelengths converge on the same point, avoiding the spectral spread that affects the Talbot interferometer. As a result, the pulse can be much more tightly focused.
There remains, however, a temporal broadening due to the fact that although all wavelengths converge to the same point, they still do not travel the same distance, and hence do not arrive on the fiber at the same time. This means that the pulse has acquired a temporal chirp, and is correspondingly broadened. This temporal broadening due to spectral dispersion is the same as in the Talbot interferometer, as illustrated in Fig. 5(b). This is not, however, an unavoidable situation: if the input pulse is given a same amount of chirp, but of opposite sign, then all the wavelengths will effectively arrive at the same time.
The animation in Fig. 6(b) illustrates the evolution of a 200 fs pulse of width $w_{p0}=2$mm, starting again once the pulse is diffracted by the first mask. As in the Talbot, the pulse is gradually tilted. However, after the second mask, the tilting is stopped and reversed, and the pulse gets shorter and narrower, until it reaches the focal point. Figure 8 (a) and (b) shows the pulse width $w_{p}$ and pulse duration $\tau _{p}$ as it travels, and the enhancement parameters $e_{p}$ and $e_{g}$. At the focal point on the fiber the two enhancement factors have the same value $e_{p}=e_{g}\approx 29$. The interference pattern width $w_{g} =23 \mu$m, and the duration $\tau _{g}=303$ fs. Furthermore, if the right amount of pre-dispersion is added to the pulse, then even that broadening can be avoided. Figure 8(c) and (d) shows the pulse evolution in that case. A dispersion of $D=0.1$ ps$^{2}$ is imparted, leading to an input broadened pulse duration of 303 fs (which is the pulse duration on the fiber without pre-dispersion). The pulse duration on the fiber is now reverted back to its original duration of 200 fs. The enhancement ratio $e_{g}$ now reaches 42. Such a large enhancement can make a huge difference in writing efficiency, especially for higher order effects ($n>2$). Pre-dispersion can readily be imparted to pulses from amplified femtosecond lasers, since these lasers already have a stretcher and adjustable compressor for chirped pulse amplification. The compressor just needs to be adjusted to give the right amount of pre-chirp.
Fig. 8. Pulse evolution and enhancement parameters at zero detuning without (a) and (b) and with pre-dispersion (c) and (d) for the ring and achromatic interferometers
6. Effect of detuning
One of the further advantages of both the achromatic and the ring interferometer is the ability to tune the wavelength over a broad range, by simply tilting the masks or the mirrors. Hundreds of nanometers of wavelength range can potentially be reached that way. The Talbot interferometer has the same wavelength tuning capability, but as we have shown, it cannot be used with focused pulses, on top of being overly sensitive to mechanical vibrations. We will therefore not consider it any further.
For the achromatic interferometer, when the second masks have an angle, the rays on the right and the left do not travel the same distance to the mask, and the propagation angle changes after being rediffracted, as illustrated in Fig. 9(a). In effect, the tilt rate after the second mask is accelerated. As a result, the two rays will have an equal propagation distance at a $z$ position that is now in front of the focal position where the fiber is. It is therefore no more possible to have both spatial and temporal refocusing of the pulse at the intersection point on the fiber. Also, because of the change in propagation angle after the second mask, the pulse appears to be de-magnified when viewed in the fiber plane, which also affects the distance to focus. In fact, we found that the distance $d_{f}$ has to be changed from 73.8 mm at zero detuning to about 35 mm for 200 nm detuning (mask angle of 17.89 degrees). What happens in the ring interferometer can be understood by unfolding it, as shown in Fig. 9(b), which shows its similarity with the achromatic design, but also its difference. The beam is slightly deflected by the first mirror (represented by the ${\approx} 45$ degree line that crosses the beam), changing the incidence angle on the mask, and deflected again by the second mirror. For 200 nm detuning, the mirror angle is only 0.65 degrees, and each deflection is only 1.3 degrees. There will also be a magnification due to the deflection, but it is partly cancelled by a demagnification from the second mirror. As a result, the focal distance only varies from 73 mm to 78 mm at 200 nm detuning, and the position of equal propagation distance is much closer to the fiber than with the detuned achromatic interferometer.
If we now look at the rays from a spectral perspective, we find another set of phenomena affecting the pulse. At non-zero detuning, the point of intersection of different wavelengths is moved away from the fiber axis, in both the $z$ and the $x$ directions. On the fiber axis, the pulses will appear to have transverse chirps which are mirrors of each others, due to this residual spectral spread, as illustrated in Fig. 5(c). Combined with the temporal spread, this results in a slight pulse tilt in opposite directions for the right and left side pulses. As a result, the width $w_{g}$ of the interference pattern will be reduced.
The effects of spectral and temporal spread due to focusing combine into a rather complex behaviour when approaching the focal point, where the rotation changes sign. It can be visualized in the animation of Fig. 6 (c) for 200 nm detuning of the achromatic interferometer. Figure 10 (a) and (b) shows the spatial and temporal widths, as well as the enhancement factors. The temporal focal point (shortest $\tau _{p}$) is now in front of the fiber, which is postioned at the spatial focal point (tilt angle of 90 degrees). Most of the temporal broadening is not due to a linear temporal chirp, and cannot be reversed by pre-dispersing the pulse. This is of course accompanied by a larger than ideal beam waist at focus. The temporal broadening due to focusing gets worse when the initial beam waist is larger, or the focal length is shorter. Therefore, it is no use trying to get a smaller beam waist (and larger enhancement) this way. Because of the pulse and phase front tilt (nonzero $\beta$), $e_{g}$ is now much smaller than $e_{p}$, and peaks at 8.7. We found that pre-dispersing the pulse does not even improve $e_{g}$ in this case.
Fig. 10. Pulse width, duration and enhancement parameters for 200 nm detuning with the Achromatic (a) and (b) and Ring interferometer (c) and (d).
In the case of the ring interferometer, these effects are found to be much less severe, and a much larger enhancement factor is achievable even for large detunings. Figure 10 (c) and (d) shows the behavior of the pulse and enhancement factors for a pre-dispersed pulse at 200 nm detuning (0.65 degree angle of the mirrors). The performance is only stlightly decreased compared with zero detuning, and $e_{g}$ still reaches 27.8. Furthermore, pre-dispersing the pulse with $D=0.1$ ps$^{2}$ will still augment $e_{g}$ to 39.7.
Figure 11 summarizes the enhancement ratio $e_{g}$ for the achromatic and the ring interferometer, for different initial beam waists (0.5, 1, and 2 mm), with and without pre-dispersion, and detunings from 0 to 200 nm. The performance of the achromatic interferometer degrades rapidly with detuning, whereas the ring interferometer can have $e_{g} >30$ and 40 for $w_{0}=2$ mm over the entire 200 nm tuning range with and without pre-chirping, respectively.
Fig. 11. Enhancement factor $e_{g}$ as a function of detuning for 3 beam diameters: (a) Achromatic ; (b) Achromatic with pre-dispersion; (c) Ring; (d) Ring with pre-dispersion
7. Discussion and conclusion
Based on our results, and using experimental results from the published litterature, we can now estimate how efficiently gratings can be written, by tightly focusing the femtosecond pulses. We will compare gratings written in non-hydrogenated fibers, with an index modulation of about $10^{{-}3 }$, as this represents the most interesting case for a simplified manufacturing process. This is also a worst case scenario, as using hydrogen dramatically reduces the required fluence. As will be seen the pulse energy required when focusing is in the low microjoule range. Even though most reports use high-energy, low repetition-rate Ti:sapphire pulses, Ytterbium-based commercial lasers are now readily available, with microjoule-level energies at 200 kHz or larger repetition rates. Thus for the sake of consistency, we will consider visible (515 nm) and UV (257 nm) wavelengths, 200 fs pulses, typical of such lasers. The pulses are assumed to be focused down to 25 microns (parallel to the fiber axis) by 10 microns (perpendicular to the fiber axis). For comparison with the published results, we consider the same average and peak power. Ultimately, the peak power incident on the fiber is limited by higher order effects, such as filamentation. The threshold for these will depend on the wavelength and the fiber used. The gratings considered here are all Type I gratings. It should be noted that in most reports, there was little indication of saturation with fluence, so it can fairly be assumed that index changes larger than $10^{{-}3 }$ should be obtained by extending the writing time.
Slattery et al. [2] used 300 $\mu$J, 200 fs pulses at 264 nm, and showed a clear nonlinear dependence of the index change. The peak power was about $2\times 10^{11}$ W/cm$^{2}$, and the total fluence required for $10^{{-}3 }$ index modulation in H$_{2}$-free Fibercore photosensitive fiber was about 3 kJ/cm$^{2}$. Similar results were obtained by Dragomir et al. [8] with the same laser. In their experiments, the peak power was limited by the two-photon absorption in the mask, which would not be a problem with the phase mask interferometers discussed here. For a fair comparison, we will assume the same (small) average power of only 8 mW, but a larger repetition rate of 24 kHz (as compared with their rate of 27 Hz), and a much smaller pulse energy of 0.33 $\mu$J, with a total transmission of only 0.30 through the system, to account for the low diffraction efficiency of the second mask and other losses. Thus the peak power is the same. Their writing time was 40 minutes for a 3 mm long grating. For a two-photon effect ($n=2$), using Eq. (5), we calculate a writing rate of 37.5 s/mm (113 s total). On the other hand, if we consider a higher average power of 100 mW (well within reach of commercial Yb-based lasers), with a repetition rate of 150 kHz, for 0.68 $\mu$J, pulses, and a doubled peak power of $4\times 10^{11}$ J/cm$^{2}$, probably still below the damage threshold, the writing time would be as small as 1.5 s/mm.
Zagorulko et al. [4] also used 267 nm, 150 fs pulses with 130 $\mu$J at 1 kHz, and could get $10^{{-}3}$ index modulation with $2\times 10^{11}$W/cm$^{2}$, and 58 kJ/cm$^{2}$ fluence, in non-hydrogenated SMF-28 fiber. With focusing, this could be achieved with the same average power (130 mW) and peak power, and 0.27 $\mu$J, pulses at a rate of 490 kHz, for a writing speed of 45 s/mm of grating, and a total writing time of 135 s, as compared with their writing time of more than 30 minutes.
Bernier et al.[5] used 400 nm pulses of only 40 fs duration, with 0.9 mJ energy, and wrote gratings in double-clad, germanium-free, Yb-doped fiber, an important industrial application. A 20 s exposure led to $10^{{-}3}$ index modulation for a 6.9 mm grating. Assuming $n=4$ in Eq. (5, for similar average and peak power, we find that pulses with only 36 $\mu$J, and 200 fs duration (25 kHz) would achieve the same result, at a writing speed of 1.7 s/mm, for a total writing time of 12 s, as compared with 20 s under their conditions (1kHz repetition rate). The difference in writing time is not as important because we assume much longer pulses, which are in practice much easier to handle. The low energy required also gives much more margin to increase the pulse energy, and greatly reduce the writing time, due to the higher-order nonlinearity. In effect, doubling the peak intensity could reduce the writing time by a factor of eight.
It is therefore clear that using either the ring or the achromatic interferometer, short grating writing times and large index modulations can be achieved even in H$_{2}$-free fiber using either 257 nm or 515 nm as the writing wavelength, at energies not exceeding a few $\mu$J per pulse. Furthermore, both interferometers are tunable over ranges similar to the gain bandwidths of most fiber lasers or amplifiers. Shorter periods, such as those required for 1 $\mu$m fiber laser reflectors, are readily reachable with commercial phase masks. Our calculations were made for 515 nm writing wavelength, and are a worst case scenario for the effects of focusing and spectral spreading discussed here. All of these effects (except the path delay) are actually less severe at shorter writing wavelengths.
In conclusion, we have shown that the achromatic and ring phase mask interferometer analyzed here are very promising for writing gratings with focused femtosecond pulses, as opposed to the Talbot design. The achromatic interferometer has much less path delay than the ring, but the combined effects of spatial and spectral dispersion limit the ability to tightly focus the pulses at non-zero detuning, whereas this is much less of a problem with the ring interferometer. We have also shown that pre-chirping the pulses can further enhance the peak power on the fiber. Compared with a typical phase mask setup, where the fiber is often placed a few mm behind the mask, these interferometers allow a perfect overlap of the beams on the fiber, and essentially perfect contrast of the interference pattern, further enhancing the writing efficiency. Based on previous results by other groups, we estimate that with a commercial femtosecond laser, such as an Yb-based laser, and by focusing the beam to about $10\times 25\mu$ m$^{2}$, gratings with more than $10^{{-}3 }$ index modulation could be written at a rate of a few seconds per millimeter in most fibers, with pulse energies in the low $\mu$J range or smaller. With their combination of large tuning range, insensitivity to vibrations, and ability to tightly focus the pulses, these interferometers are an attractive alternative to the standard phase mask setups.
Funding
National Natural Science Foundation of China (NSFC) (61421002, 61435003, 61722503).
References
1. A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, “Direct writing of fiber bragg gratings by femtosecond laser,” Electron. Lett. 40(19), 1170–1172 (2004). [CrossRef]
2. S. A. Slattery, D. N. Nikogosyan, and G. Brambilla, “Fiber bragg grating inscription by high-intensity femtosecond uv laser light: comparison with other existing methods of fabrication,” J. Opt. Soc. Am. B 22(2), 354–361 (2005). [CrossRef]
3. S. J. Mihailov, D. Grobnic, C. W. Smelser, P. Lu, R. B. Walker, and H. Ding, “Bragg grating inscription in various optical fibers with femtosecond infrared lasers and a phase mask,” Opt. Mater. Express 1(4), 754–765 (2011). [CrossRef]
4. A. Zagorulko, P. Kryukov, Y. Larionov, A. Rybaltovsky, and E. Dianov, “Fabrication of fiber bragg gratings with 267 nm femtosecond radiation,” Opt. Express 12(24), 5996–6001 (2004). [CrossRef]
5. M. Bernier, R. Vallée, B. Morasse, C. Desrosiers, A. Saliminia, and Y. Sheng, “Ytterbium fiber laser based on first-order fiber bragg gratings written with 400nm femtosecond pulses and a phase-mask,” Opt. Express 17(21), 18887–18893 (2009). [CrossRef]
6. M. Becker, L. Fernandes, M. Rothhardt, S. Brückner, K. Schuster, J. Kobelke, O. Frazão, H. Bartelt, and P. V. S. Marques, “Inscription of fiber bragg grating arrays in pure silica suspended core fibers,” IEEE Photon. Technol. Lett. 21(19), 1453–1455 (2009). [CrossRef]
7. A. Martinez, I. Y. Krushchev, and I. Bennion, “Direct inscription of bragg gratings in coated fibers by an infrared femtosecond laser,” Opt. Lett. 31(11), 1603–1605 (2006). [CrossRef]
8. A. Dragomir and D. N. Nikogosyan, “Inscription of fiber bragg gratings by ultraviolet femtosecond radiation,” Opt. Lett. 28(22), 2171–2173 (2003). [CrossRef]
9. M. Livitziis and S. Pissadakis, “Bragg grating recording in low-defect optical fibers using ultraviolet femtosecond radiation and a double-phase mask interferometer,” Opt. Lett. 33(13), 1449–1451 (2008). [CrossRef]
10. J. Thomas, E. Wikszak, T. Clausnitzer, U. Fuchs, U. Zeitner, S. Nolte, and A. Tünnermann, “Inscription of fiber bragg gratings with femtosecond pulses using a phase mask scanning technique,” Appl. Phys. A 86(2), 153–157 (2007). [CrossRef]
11. P. Cortes, F. Ouellette, and S. LaRochelle, “Intrinsic apodization of bragg gratings written using uv-pulse interferometry,” Electron. Lett. 34(4), 396–397 (1998). [CrossRef]
12. R. Kashyap, J. Armitage, R. Campbell, D. L. Williams, G. Maxwell, and B. Ainslie, “Light sensitive optical fibres and planar waveguides,” BT Technol. J. 11, 150–158 (1993).
13. M. Sozzi, A. Rahman, and S. Pissadakis, “Non-monotonous refractive index changes recorded in a phosphate glass optical fibre using 248nm, 500fs laser radiation,” Opt. Mater. Express 1(1), 121–127 (2011). [CrossRef]
14. R. Kashyap, Fiber Bragg Gratings, 2nd ed (Academic Press, 2010).
15. F. J. Weinberg and N. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36(5), 227–230 (1959). [CrossRef]
16. P. St-Hilaire and A. Grunnet-Jepsen, US Patent 6,904,201 (2002).
17. A. Kostenbauder, “Ray-pulse matrices: A rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26(6), 1148–1157 (1990). [CrossRef]
18. S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of gaussian pulses and beams,” Opt. Express 13(21), 8642–8661 (2005). [CrossRef]
19. D. Li, X. Lv, S. Zeng, and Q. Luo, “A generalized analysis of femtosecond laser pulse broadening after angular dispersion,” Opt. Express 16(1), 237–247 (2008). [CrossRef]
20. C. G. Durfee, M. Greco, E. Block, D. Vitek, and J. A. Squier, “Intuitive analysis of space-time focusing with double-abcd calculation,” Opt. Express 20(13), 14244–14259 (2012). [CrossRef]
21. J. Martin and F. Ouellette, “Novel writing technique of long and highly reflective in-fibre gratings,” Electron. Lett. 30(10), 811–812 (1994). [CrossRef]
22. Z. Bor, “Distortion of femtosecond laser pulses in lenses,” Opt. Lett. 14(2), 119–121 (1989). [CrossRef]
