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Abstract
The design of a complex phase mask (CPM) for inscribing multi-notch fiber Bragg grating filters in optical fibers for OH suppression in astronomy is presented. We demonstrate the steps involved in the design of a complex mask with discrete phase steps, following a detailed analysis of fabrication constraints. The phase and amplitude of the complex grating is derived through inverse modelling from the desired aperiodic filter spectrum, following which the phase alone is encoded into the surface relief of a CPM. Compared to a complicated “running-light” Talbot interferometer based inscription setup where the phase of the inscribing beam is controlled by electro- or acousto-optic modulators and synchronized to a moving fiber translation stage, CPM offers the well-known convenience and reproducibility of the standard phase mask inscription technique. We have fabricated a CPM that can suppress 37 sky emission lines between 1508 nm to 1593 nm, with a potential of increasing to 99 channels for suppressing near-infrared (NIR) OH-emission lines generated in the upper atmosphere and improving the performance of ground-based astronomical telescopes.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ground-based astronomical observations are adversely affected by various properties of the atmosphere, such as molecular absorption, extinction from dusts and aerosols, turbulence-induced wavefront aberrations, but also the overwhelming presence of sky background emission lines spanning 0.9 to 2.5 $\mu$m, produced by hydroxyl (OH) radicals at an altitude of 90 km [1]. Such sky emissions can be 1000x brighter than the science light from a distant object, e.g. a high redshift galaxy. While the interline continuum between the OH emission lines is intrinsically faint, the sensitivity of low to medium resolution NIR spectrographs between the lines is severely degraded by the scattered-light induced broad wings of OH-emission lines [2]. This is a very disturbing limitation for a new generation of extremely large telescopes, e.g. the ELT [3]. As astronomy is photon-starved, with suppression of $>$25 dB and interline transmission losses $<$0.5 dB, a high order filter with negligible side-lobes is required.
Different methods like high dispersion masking, Rugate filters and holographic filters have been employed for OH-suppression; however, these methods inherently suffer from scattering properties due to dispersive optics and as a result, these methods were discovered to be inadequate when it comes to a suppression of the OH-lines that would be sufficient for astronomical use [4]. An extraordinary approach to suppress OH-lines in the H-band centered at 1.65 $\mu$m, was first introduced by Bland-Hawthorn et al. [5] using aperiodic fiber Bragg gratings (FBGs). As opposed to conventional single-notch gratings, the design and fabrication of such filters with the order of 100 notches is complicated, which is why in what follows we are using the term “complex” (not to be confused with mathematical meaning of complex numbers). FBG based filters bring in immense promise for OH-suppression as they are capable of filtering the OH-lines before the light enters a fiber-fed spectrograph [6–8], hence prohibiting any contribution of scattered light in the spectrum.
Although a challenging task to realize, complex gratings can be fabricated using ultraviolet (UV) or femtosecond lasers by dephasing partial gratings [9] in optical fibers/waveguides [10,11]. Point-by-point [12,13], line-by-line [14] and acousto-optic modulators based UV Talbot interferometry [10] have been explored to fabricate complex gratings. In these methods precise control on the phase and intensity of the inscribing beam is required. In a "running-light" Talbot interferometer, an electro- or acousto- optic phase modulator is used to control the relative phase of the two inscribing beams. The rate of change of the voltage (or phase) applied to the modulators is synchronized to the velocity of translation of the fiber, thereby controlling the phase and amplitude of the grating. The phase and amplitude control has to be performed in real-time while the grating is being inscribed. Such a setup requires high precision velocity control of the fiber translation stage with minimal linear or rotational errors, and precise alignment of two-beam optics.
In what follows, we will advocate to adapt a popular method for inscribing gratings in optical fibers. Phase masks (PM) are known for their high reproducibility and can be made available off-the-shelf. However, inscribing several individual gratings in series, using uniform phase masks results in excessive losses for the photon-starved applications in astronomy [7]. Such solutions also result in long fiber lengths requiring strain and temperature management. Even a fractional drift in center wavelength can result in drastic losses in science light throughput in the interline continuum. Superimposing multiple gratings in the same location is not a feasible option [15]. A single complex grating that generates multiple non-uniformly located filter lines matched to the OH sky emission background is required.
In ground-based observatories, the light from the telescope, coupled into a multimode fiber, is scrambled into multiple single mode fibers containing identical complex filters using a photonic lantern [7]. Fabricating identical filters on multiple fibers is challenging in interferometric inscription methods, where linear translation (x, y, z, pitch, roll, yaw) accuracy over a large length is in the order of a few microns, even for air-bearing translation stages. By transferring the phase of the complex grating onto a PM through a one-time mask manufacturing process, a CPM can be fabricated and can be used off-the-shelf in any standard PM fabrication setup with amplitude modulation, without requiring high precision linear translation.
In this paper, we introduce for the first time, the complete design methodology of complex phase masks by selecting the optimum parameters for complex grating inscription that leads to the design of the discrete phase steps to be encoded within a given length of the mask. We design a complex phase mask for 37 emission lines and using numerical modelling, we introduce the fabrication constraints for fabricating complex gratings in single mode optical fibers using such complex phase masks. Finally, we conclude by emphasizing the potential of such special phase masks for astronomy; a CPM can circumvent the need for a complicated interferometric setup for fabrication of identical and precise filters that are critical for ground-based observatories.
2. Design methodology
The Fourier relationship between non-uniform grating index modulation or phase for generating multi-channel filters have been studied before [16–19]. However, a complex phase mask with sub-nanometer surface relief resolution that inscribes grating phases between $-\pi$ to $+\pi$ has never been demonstrated. This is attributed to the fact that the fabrication of complex phase masks with thousands of precisely defined phase structures requires sub-nanometer e-beam alignment and etching processes which were demonstrated only recently [20].
The multi-channel filter can be defined as [16]
The phase $\phi _g$ and the index modulation $\Delta n_g$ of the physical grating in the fiber can be derived from the desired reflection spectrum $|r(\lambda )|$ by using the Layer Peeling (LP) method [21,22].
Figure 1 (top) and (bottom) show simulated results of desired filter spectra consisting of $N$=99 filter lines in H-band, and phase $\phi _g$ of the complex grating, respectively.
Without channel dephasing, all the gratings overlap at the same location in the fiber, increasing $\Delta n_g$ to values that cannot be achieved, as shown in Fig. 2, where a large index peak of $10^{-2}$ can be seen between 2 cm and 3 cm. By introducing a linear group delay $g_i$, we can ‘dephase’ or re-position the individual physical gratings along the fiber, essentially ‘spreading’ the complex grating along the full length $L$ of the grating, as shown in Fig. 3. Simulated annealing can be used to optimize $g_i$ so as to minimize $[max(\Delta n_g) - min(\Delta n_g)]$.
Fig. 2. $\Delta n_g$ derived from LP for $N\!=$99 without dephasing shows that the gratings overlap at the same location (top) causing high modulation index at the center of a grating of length $L$=5 cm (bottom).
Fig. 3. $\Delta n_g$ derived from LP for $N\!=$99 with dephasing shows low overall modulation index. With dephasing, the gratings are spread over the length of the grating.
All dephased individual gratings have the same length, but have different index modulation strengths. By ‘dephasing’ the individual gratings, the maximum value of index modulation at any location of the complex grating is reduced. Typically, in photosensitive optical fibers, depending on the germanium doping and/or boron co-doping, the maximum index modulation achievable is limited to the order of $10^{-4}$ for a Type-I grating [23,24]. Continuous exposure to the inscribing UV laser after the grating index modulation has reached its maximum will result in a decrease in reflectivity as well as an increase in the full-width half maximum of the filter. Increase in UV intensity or prolonged exposure results in a Type-II [25] or Type-IIa [26] gratings, respectively, which have high losses and are mechanically fragile.
In the design of a physical CPM, two critical parameters are, a) the filter bandwidth, or number of emission lines, and b) the diameter of the core of the optical fiber in which the complex grating will be inscribed. Standard optical fibers have mode field diameters (MFD) of 9 to 10.4 $\mu m$, whereas reduced clad fibers have MFD of 4 to 6 $\mu m$. In the following subsection we discuss the criteria for optimal selection of these two parameters that finally lead to the design of a complex phase mask.
2.1 Grating bandwidth and MFD
The critical parameter of fabricating multi-channel gratings using UV based interferometric methods such as Talbot interferometer or, as described in this paper, a complex phase mask, is defined by the spot size of the inscribing beam, $\Delta z$, which also defines the total bandwidth $\beta _\omega$ of the filter. The size of the inscribing laser spot can be defined as [15]
To cover the required filter bandwidth $\beta _w$, a maximum value for Eq. (2) exists, which we define as $\Delta z_{max}$, and if $\Delta z_{f\!ab}$ is the spot size of the UV beam used for inscribing such a filter, for a requisite number of emission lines, we requireThe physical dimension of the spot size of the inscribing beam must be comparable to the MFD of the optical fiber in order to accommodate the positioning errors of linear translation stages during inscription, i.e.,
Therefore, the fabrication process is defined by the writing boundaries
For reconstruction of the physical grating from the filter spectrum using LP, we represent the grating as a concatenation of many layers, each with thickness $\Delta z$, given by [22]
where, $L$ is the grating length and $M$ is the number of layers.The quality of $\phi _g$ and $\Delta n_g$ of the complex grating reconstructed through LP, and ultimately the filter, is dependent on $M$. The quality of the filter increases with increasing $M$ (shown in Fig. 6, for $N$=37). However, increasing $M$, also increases the length of the complex phase mask. Through numerical modelling, we see that increasing $M$ above $M_{min}$, does not improve the quality of the filter significantly. From Eq. (2) and Eq. (6), for a given $L$, $M_{min}$ is defined by $\beta _w$.
Table. 1 lists examples of the numerically derived boundaries of grating inscription parameters, i.e., the minimum required inscribing beam spot size $\Delta z$ and the number of layers $M$, for a given bandwidth $\beta _w$ and length $L$ = 5 cm. The design steps are as follows:
Table 1. Limits of grating parameters for $L$= 5 cm
At the location of individual focused UV spot of size $\Delta z_{f\!ab}$, a uniform seed grating is inscribed with its own unique index of refraction $\Delta n_g^{(j)}$, and each grating layer is dephased from its neighbouring layers by the complex phase $\phi _g^{(j)}$ where $j=1,2,\ldots ,M$.
2.2 Phase structure
The optimized $\Delta n_g$, $\phi _g$, $M$, and $L$ can now be used to define the information encoded in the complex phase mask. $\Delta n_g$ can be inscribed by using an amplitude mask [27], acousto/electro-optic amplitude modulator, or by directly controlling the exposure time during the grating inscription. $\phi _g$ can be realized in two ways: (I) by sampling the grating using established sampling methods [17,18] or (II) as a non-linear chirp resulting from partially overlapping dephased individual gratings [9,19]. Since $\Delta n_g$ can be implemented relatively easily compared to $\phi _g$, here we encode $\phi _g$ into the phase mask using method I. However, it is worth mentioning that method II can also be used to fabricate complex gratings. In this paper we demonstrated method I for the reason of simplicity of the mask fabrication over method II. In method I, the mask pitch is constant with discrete and variable phase steps introduced at specific locations along the surface relief, whereas in method II, a continuous non-linear chirp in the pitch will be required. The later is comparatively challenging. A comparison of complex phase mask designs following method I and method II needs to be studied further. It is important to assess the feasibility of the fabrication of a complex phase mask designed using method II as this involves introducing non-linear chirp in the pitch. The derived phase $\phi _g$ can be recorded into the surface relief structure of a mask by choosing the proportional groove width. For a standard uniform phase mask, the groove depth is defined by the wavelength of the laser used for the fabrication and the refractive index of the material used for the phase mask, whereas, the groove width is equal to the wavelength of the uniform Bragg grating divided by twice the effective index of the optical fiber used. In the CPM, discrete steps or the groove widths $\delta _m$, as shown in Fig. 4 (top), are introduced at intervals of $\Delta z$. $\delta _m$ is defined as
where, $m=1,2\cdots ,M$ and $\Lambda _0$ is the seed pitch of the CPM, defined by the seed grating $\lambda _0$. For a uniform Bragg grating, the groove widths are constant ($\Lambda _0/2$) over the entire length of the phase mask. For $\phi _g\in [-\pi ,+\pi ]\implies \delta _m\in [\Lambda _0/4,3\Lambda _0/4]$. We see that if $\delta _m$=$\Lambda _0/4$ (or $3\Lambda _0/4$), we get $\phi _g$=$-\pi$ (or $+\pi$) for a $\pi$-phase shifted mask. $M$ discrete groove widths $\delta _m$ are recorded into the surface relief of the phase mask (Fig. 4) (top, middle) at intervals given by Eq. (2). It is also important to note that the diffraction of orders of the mask at the location of the step $\delta _m$ splits the grating phase into two half-phases [28]. Figure 4 (bottom) shows the projection of the mask phase to the grating plane; where, if $\phi _m$ is the phase shift in the mask, then in the core of the fiber two half-phase shifts $\phi _m/2$ are inscribed, separated by $2y\tan \theta$, where $y$ is the distance between the mask and the fiber core, and $\theta$ is the angle of diffraction of $\pm 1$ order. The discrete half-phases accumulate along the length of the fiber as light propagates through the grating, resulting in a desired phase $\phi _g$ of the filter.Fig. 4. (Top) Representative 3D model of a section of the CPM showing two $\Lambda _0/4$ shifted grooves corresponding to ($-\pi$, $+\pi$) phase, separated by $\Delta z$. $\rho =\lambda _{uv}/2(n_{uv}-1)$ is the groove depth of the phase mask, defined by the wavelength $\lambda _{uv}$ of the laser used for fabrication, and the refractive index $n_{uv}$ of the mask material. (Middle) Three different groove widths are shown as an example (not to scale). (Bottom) Schematic showing the propagation of phase mask phase to fiber grating phase.
3. Results and discussion
3.1 CPM design
We select $N$=37 emission lines in the H-band from 1508 nm to 1593 nm, $\lambda _0$=1550 nm and using Eq. (1) we obtain the filter spectrum shown in Fig. 5 (top). The phase $\phi _g$ and index modulation $\Delta n_g$ of the physical grating in the fiber derived from the desired reflection spectrum $|r(\lambda )|$ by using the Layer Peeling (LP) method is given in Fig. 5 (middle and bottom). Figure 5 (bottom) also shows a comparison of $\Delta n_g$ with and without (inset) dephasing. $g_i$ is optimized using a simulated annealing algorithm to further reduce the maximum index modulation in the fiber.
For the given $\beta _w$ and optimised $\Delta n_g$ and $\phi _g$, we find (Fig. 6) the optimum $M$=5000 which in turn gives $L\!\approx$4.8 cm. In order to cover $2\pi$ for $\beta _w$=100 nm, $\sim$6100 groove widths ($\delta _m$) in the CPM will vary between 266 nm to 798 nm, where $\lambda _0$=1550 nm and $\Lambda _0\!\approx$1064 nm. The value for $M$ is chosen to accommodate the e-beam processes. Each individual $\delta _m$ is fabricated with sub-nanometer precision to faithfully record $\phi _g$. Figure 7 shows a CPM fabricated by Fraunhofer IOF. The fabrication processes of the CPM to achieve sub-nanometer precision will be described in a forthcoming paper along with Fraunhofer IOF.
Fig. 5. (Top) Filter spectra consisting of 37 filter lines in H-band, and (middle) the phase $\phi _g$ and (bottom) index profile $\Delta n_g$ of the complex grating. Inset (bottom) shows $\Delta n_g$ without dephasing.
Fig. 6. Reproducibility of reconstructed reflection spectra (red) with respect to the target spectra (blue) when $M$=2650 (top), $M$=5000 (middle) and $M$=7500 (bottom); For M$\geq$ 5000 the improvement in filter spectra for $N$=37 is negligible.
3.2 Fabrication constraints
The influence of $\Delta n_g$ and $\phi _g$ on the filter properties is numerically modelled to understand the fabrication constraints of using CPM in the standard phase mask inscription setup. We preserve $\phi _g$, and vary $\Delta n_g$ by reducing the modulation index. We see that if the details in $\Delta n_g$ are not recorded, even if the phase $\phi _g$ is inscribed without errors, the resulting filter spectra will be noisy (Fig. 8), and the interline continuum, which contains the faint optical science light from the telescope is completely overwhelmed.
Fig. 8. Reducing the modulation index of $\Delta n_g$ results in a noisy filter spectrum even if the phase $\phi _g$ is completely fabricated into the complex filter. (Top) preserving $\Delta n_g$. (Bottom) Reduced modulation index. The continuum between 1545 nm and 1553 nm shows noise floor $>0.5$ dB.
The resolution of e-beam lithography processes, $\delta _e$, will result in quantization of $\delta _m$ over the range 266 nm to 798 nm. This in turn quantizes $\phi _g$. The state-of-the art techniques developed by Fraunhofer IOF [20], have achieved sub-nanometer accuracy in recording $\delta _m$, resulting in highly precise reproduction of the grating phase $\phi _g$, shown in Fig. 5 (middle). Typically, inscribing $\phi _g$ and $\Delta n_g$ of the complex grating into an optical fiber requires a highly complex ‘running light’ Talbot-interferometer setup with sensitive optical alignment, inscription intensity control, and moving fiber mounts, where the velocity of translation is synchronized to the relative phases. In comparison, with the CPM we designed, where $\phi _g$ is pre-encoded through highly precise processes, by controlling only the exposure time, $\Delta n_g$ can be easily inscribed into the fiber in a standard phase mask based fabrication setup.
Based on the numerical results as shown in Fig. 8, an in-depth study of the effects of amplitude modulation and grating evolution dynamics [29] for a given photosensitive fiber has to be conducted. In the FBG filter inscription, as the optical fiber is held taut during inscription, a blue shift is seen in the filter lines after dismounting the fiber. For the photon-starved astronomical application, any shift in the central wavelengths of the multi-notch filter can overwhelm the starlight in the interline continuum. Hence, we are in the process of developing a suitable fiber core tracking system that will be used while inscribing gratings in a loosely mounted fiber in order to achieve the desired filter lines based on which the CPM is designed and fabricated. The results will be communicated in the forthcoming papers.
4. Conclusion
Prior works on nonlinear or sampled phase mask design addressed applications in telecommunication, e.g. dispersion compensation, wavelength division multiplexing etc. However, the requirements for astronomical applications are stringent, stemming from the fact that it is photon-starved and there are 99 aperiodically spaced OH-lines spread over 250 nm. Uniform phase masks introduce excessive losses in the filter lines and interferometric methods involve additional optics as well as high precision translational stages, thus making the fabrication process tedious and cumbersome. The challenge in fabricating reproducible notch filters still remains. To our knowledge, this is the first demonstration of a design of a unique complex phase mask with thousands of discreetly positioned structural information that records the phase information, to solve a critical and long-standing issue in astronomy. In part I, we present the complete design of a complex phase mask. In the subsequent parts II and III future articles, we will communicate the novel fabrication processes of the CPM, and the effect of e-beam accuracy on the filter spectra. Amplitude modulation along with CPM not only simplifies the process of inscription but also holds a promise for reproducible complex grating fabrication which is essential for fiber-fed spectrographs for future extremely large ground-based telescopes.
Funding
Bundesministerium für Bildung und Forschung (03Z22A511).
Acknowledgments
The authors thank Prof. Dr. Uwe D. Zeitner, Fraunhofer IOF, and Dr. Martin Heusinger, Institute of Applied Optics, Friedrich-Schiller-Universität, Jena, for manufacturing the complex phase mask.
Disclosures
The authors declare no conflicts of interest.
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