A VIPA Spectrograph with Ultra-high Resolution and Wavelength Calibration for Astronomical Applications

The bottom panel of Figure 1 shows a partial image of a flat-field white light with simultaneous comb references recorded by the CCD. The present parameters lead to the comb points being well Nyquist-sampled, and the average FWHM is around 5.4 pixels along the dispersion directions of both the VIPA and the échelle. The separation of signals from two channels is 18.88 pixels and distinguishable to some extent. The measured resolution of our spectrograph is ${ \mathcal R }\approx 1.12\times {10}^{6}$. Notable differences of the resolution between the theoretical and the experimental results originate primarily from the finite size of the VIPA. In contrast to the infinite dimension assumption in Equation (1), we take into account the interference from a finite number of waves in our simulations such that a resolution of ${ \mathcal R }\approx 1.19\times {10}^{6}$ is consistent with the experimental value results. On the other hand, the spectral range ${\rm{\Delta }}\lambda$ is approximately 4.9 nm wide, which is restrained by the detector size and can be improved significantly if necessary.

The reciprocal dispersion relationship of the VIPA can be derived as

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}\lambda }{{\lambda }_{0}}=\displaystyle \frac{\sin 2{\theta }_{i}}{2({n}_{r}^{2}-{\sin }^{2}{\theta }_{i})}\cdot \displaystyle \frac{{\rm{\Delta }}x}{{F}_{\mathrm{cam}}}+\displaystyle \frac{{\rm{\Delta }}{x}^{2}}{2{F}_{\mathrm{cam}}^{2}},\end{eqnarray} \tag{ 2 }$$

which depends on the VIPA material (n_r) and the angle of incidence on the VIPA (${\theta }_{i}$).

To be more concrete, we envisage a spectrograph using a customized VIPA with a resolution of ${ \mathcal R }=30{\rm{,0}}000$. Based on numerical calculations, such resolution requires a VIPA volume of 20 × 20 × 0.3 mm³ and an angle of incidence of 2°. According to Equation (2), the focus length of the camera ${F}_{\mathrm{cam}}\approx 250$ mm ensures a five-pixel sampling of the resolution element if the pitch size of the CCD detector is 10 μm. If the cross-dispersion element is chosen properly and the order spacing is 10 × FWHM = 50 pixels, the wavelength range of 106 nm (wavelength center at 760 nm) can be covered with a detector of 1400 × 8000 pixels. Accordingly, a wavelength coverage of 458 nm (594–1052 nm) at ${ \mathcal R }=30{\rm{,0}}000$ is achievable with four VIPAs in parallel and a large-format detector (8000 × 8000 pixels). Fortunately, the four VIPAs are installed independently and bring no additional technical difficulties during the commissioning process.

The measured transmission of the VIPA is around 60%, which is comparable to a classical échelle grating and can be further optimized because the theoretical value of the transmission equals 78.76% for our VIPA according to the formula (Weiner 2012)

$$\begin{eqnarray}&&T=\displaystyle \frac{1-| {r}_{2}{| }^{2}}{1-| {r}_{1}{| }^{2}| {r}_{2}{| }^{2}}.\end{eqnarray} \tag{ 3 }$$

It should be noted that the cross-disperser is an échelle in the operational spectrograph. For practical utilization in future, especially for night astronomical observations, a ruled grating (e.g., 750 nm blaze, 1800 grooves per mm, 30 × 30 mm² in Littrow configuration for the operational spectrograph) or a volume-phase holographic grating (Barden et al. 1998) is expected to perform the same dispersion capability to guarantee the higher efficiency. Moreover, the spectral scattered light accounts for 1.26% after averaging over the whole wavelength range.

Wavelength calibration is another key component of the spectrograph. Extremely high precision in wavelength calibration is indispensable in searching for exoplanets with the RV method. The tide is turning toward the LFC as the calibration light source for astronomical spectrographs. Our spectrograph is also equipped with a homemade Yb:fiber ring LFC as the wavelength calibration reference source. It has a repetition frequency (f_rep) of 810 MHz, which is sufficiently high for a spectrograph with spectral resolution ${ \mathcal R }\gt {10}^{6}$. Both the carrier-envelope-offset frequency (${f}_{\mathrm{ceo}}$) and f_rep are stabilized and referenced to a rubidium clock with a relative accuracy of about 10⁻¹¹ in 1 s. More detail on the LFC can be found in Xu et al. (2017), Jiang et al. (2014). An external cavity laser diode (LD) with a superior thermal stability ($\leqslant 100\,\mathrm{MHz}$/°C) and a narrow line width (<100 kHz) is used to calibrate the LFC modes. The LD working at the exact wavelength of 760.106870 nm is coupled into an SMF binding with the LFC light via the fiber coupler.

It is noteworthy that the VIPA interferometer demonstrates an Airy–Lorentzian line shape with respect to the output spatial intensity along the dispersion direction in the focal plane (Xiao et al. 2004). It is therefore inevitable to consider the influence of the line-shape characteristics of the VIPA on the wavelength calibration. In the present and next sections, the wavelength calibration of the VIPA spectrograph is investigated with the LFC spectral data using the reduction method following Probst (2015).

The absolute frequencies of the observed comb lines can be evaluated in terms of ${f}_{n}={f}_{\mathrm{ceo}}+n\cdot {f}_{\mathrm{rep}}$, where ${f}_{\mathrm{ceo}}=30.000000$ MHz and ${f}_{\mathrm{rep}}=807.905949\,\mathrm{MHz}$ have been defined precisely in our system. Then, the mode number n is determined by inspecting the LD and comb spectra as shown in the top panel of Figure 2. It is trivial to derive the mode number to be n = 486,923 in the present configuration as the spectrum of the LD is built up and superimposes on the comb mode corresponding to the wavelength 760.106941 nm.

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Another key role of the narrow-line-width LD is to resolve the FSR of the VIPA. As shown in the bottom panel of Figure. 2, two noticeable bright points highlighted by red circles correspond to the LD in two consecutive orders of the VIPA. Two dashed yellow lines can be defined in such a way that they go through the two circles and are simultaneously vertical to the line connecting the two circles (dashed white line). Then, the modes located between the two dashed yellow lines are assigned to one FSR in each separate order of VIPA. The solid white lines with arrows indicate how the continuous modes are indexed and counted. It reveals that there are a total of 3145 observed comb lines conforming to the mode numbers ranging from 484,815 to 487,959 in a single measurement, corresponding to the wavelength range of 758.1 to 763.0 nm.

The line centers μ of different LFC modes are determined by fitting the comb line shapes using the Levenberg–Marquardt algorithm. The standard error of each line center σ_μ can be obtained from error propagation of the photon noise, which is assumed to be $w=\sqrt{{\left(\sqrt{N}\right)}^{2}+{R}^{2}}$, with N being the number of detected photoelectrons excluding the dark and bias currents of the CCD and R the CCD readout noise. The photon noise limit (PNL) is calculated using

$$\begin{eqnarray}&&{\sigma }_{\mu }=\sqrt{\mathrm{diag}({[{J}^{T}{WJ}]}^{-1})},\end{eqnarray} \tag{ 4 }$$

where ${\boldsymbol{J}}$ is the Jacobian matrix and the weighting matrix ${\boldsymbol{W}}$ is diagonal with ${W}_{{ii}}=1/{w}_{i}^{2}$ (Gill & Murray 1978).

Here we fit the data with three different functions: Gaussian, Lorentzian, and hyperbolic secant with an additional linear term,

$$\begin{eqnarray}\begin{array}{rcl}{f}_{g}(x) & = & {a}_{1}+{a}_{2}\cdot x+{a}_{3}\cdot \exp \left(-\displaystyle \frac{{\left(x-\mu \right)}^{2}}{2{\sigma }^{2}}\right),\\ {f}_{l}(x) & = & {a}_{1}+{a}_{2}\cdot x+\displaystyle \frac{{a}_{3}}{{\left(x-\mu \right)}^{2}+{\sigma }^{2}},\\ {f}_{h}(x) & = & {a}_{1}+{a}_{2}\cdot x\\ & & +\,\displaystyle \frac{{a}_{3}}{\exp \left({a}_{4}\cdot (x-\mu )\right)+\exp \left(-{a}_{4}\cdot (x-\mu )\right)},\end{array}\end{eqnarray} \tag{ 5 }$$

where x is the position on the detector within one of the VIPA orders and μ the expected position of the line center. The FWHM of the comb lines can be written as FWHM = 2.355σ, 2σ, and $2.634/\sigma$ for the Gaussian, Lorentzian, and hyperbolic secant functions, respectively.

The PNL values 0.65, 0.54, and 0.70 m s⁻¹ are obtained for the three functions according to Equation (4), respectively. Data reductions of the astrocomb in échelle spectrographs (e.g., Wilken et al. 2012; Probst et al. 2015, 2016; Hao et al. 2018) have shown that the Gaussian function is a good approximation of the instrument's PSF. However, in our VIPA spectrograph with diffraction-limited input, the Lorentzian function is much better than the Gaussian or other fitting functions. This result may be attributable to the Lorentzian distribution of the PSFs in the VIPA spectrograph.

Absolute calibration is performed as a pixel-to-frequency relation by polynomial-fitting the line centers on the CCD as well as the known frequencies of the corresponding comb modes. The results are displayed in the top panel of Figure 3. A span of 53.18 m s⁻¹ over an average CCD pixel is obtained from the fitted parameters. The rms deviation of the residuals fitted with a fifth-order polynomial after removing 3σ outliers is 1.49 m s⁻¹ (middle panel of Figure 3), which cannot be significantly further improved by fitting with higher-order polynomials. The deviations of the comb line centers from the frequency solution in our case can be attributed dominantly to the dispersion characteristics of the VIPA itself. Both our simulation results following Xiao et al. (2004; see the bottom panel of Figure 3) and the experimental images show that the FWHMs of comb spectral lines vary gradually within one dispersive order of the VIPA. The spectral lines with larger FWHM will introduce more significant deviations in the data analysis. The quantile distribution of the residuals in the middle panel of Figure 3 clearly shows the residuals from the left-half side of the comb lines (blue) are larger than those from the right-half side (red). This indicates that the larger FWHMs of the comb lines will induce larger residuals. Furthermore, the overlap of the comb spectral lines and lower intensities of the comb teeth may also augment the errors. Because the FWHMs and the signal-noise ratios of the comb lines in one dispersive order of the VIPA have separate contributions to residuals, it is expected that the errors accumulated over whole spectral lines have non-Gaussian distributions.

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Irregular spectral drifts are inevitable during the measurements due to the limited stability of the system, and so it is an essential process to examine the calibration repeatability of the system. The repeatability test of our spectrograph was carried out using two methods. The first is a two-channel simultaneous calibration (Probst et al. 2016), i.e., the light of the calibration source together with the LD signal is simultaneously sent to both channels A and B of the spectrograph. A number of subsequent acquisitions are recorded at equal intervals, and the first acquisition is assigned to be the reference acquisition (n = 0). The spectrograph drifts can be traced by the calibration shifts relative to the reference acquisition in both A and B, while the calibration precision of the spectrograph characterized by the calibration repeatability is derived from the relative shifts A−B. The averaged line shifts of acquisition n with respect to the reference acquisition 0 in units of RV m s⁻¹ are calculated by considering the photon noise weighting as

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}_{n}^{A} & = & \displaystyle \frac{\displaystyle \sum _{i}{{\rm{w}}}_{{ni}}^{A}\cdot ({\mu }_{{ni}}^{A}-{\mu }_{0i}^{A})\cdot {p}_{i}^{A}{\lambda }_{i}}{\displaystyle \sum _{i}{{\rm{w}}}_{{ni}}^{A}},\\ {{\rm{\Delta }}}_{n}^{B} & = & \displaystyle \frac{\displaystyle \sum _{i}{{\rm{w}}}_{{ni}}^{B}\cdot ({\mu }_{{ni}}^{B}-{\mu }_{0i}^{B})\cdot {p}_{i}^{B}{\lambda }_{i}}{\displaystyle \sum _{i}{{\rm{w}}}_{{ni}}^{B}},\end{array}\end{eqnarray} \tag{ 6 }$$

where ${p}_{i}=\tfrac{\partial f}{\partial x}{| }_{x={\mu }_{i}}$ and λ_i are obtained from the absolute calibration, and the weighting of (μ_ni − μ_0i) can be determined by

$$\begin{eqnarray}&&{{\rm{w}}}_{{ni}}=\displaystyle \frac{1}{{\sigma }_{\mu ,{ni}}^{2}+{\sigma }_{\mu ,0i}^{2}},\end{eqnarray} \tag{ 7 }$$

with ${\sigma }_{\mu ,{ni}}$ and ${\sigma }_{\mu ,0i}$ being the uncertainties of μ_ni and μ_0i which have been prepared in the line-position fittings.

In the process of subsequent acquisitions, the standard derivation (SD) of $({{\rm{\Delta }}}_{n}^{A}-{{\rm{\Delta }}}_{n}^{B})$ denoted by σ^A–B quantifies the calibration repeatability of the spectrograph. In principle, it is expected that σ^A–B should approach the PNL ${\sigma }_{p}^{A-B}$ when all noises are well suppressed except those of photons. From Equation (6), ${\sigma }_{p}^{A-B}$ can be calculated as

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{p,n}^{A-B} & = & \sqrt{{\left({\sigma }_{p,n}^{A}\right)}^{2}+{\left({\sigma }_{p,n}^{B}\right)}^{2}}\\ & = & \sqrt{\displaystyle \frac{{\displaystyle \sum }_{i}{{\rm{w}}}_{{ni}}^{A}\cdot {\left({p}_{i}^{A}{\lambda }_{i}\right)}^{2}}{{\left({\displaystyle \sum }_{i}{{\rm{w}}}_{{ni}}^{A}\right)}^{2}}+\displaystyle \frac{{\displaystyle \sum }_{i}{{\rm{w}}}_{{ni}}^{B}\cdot {\left({p}_{i}^{B}{\lambda }_{i}\right)}^{2}}{{\left({\displaystyle \sum }_{i}{{\rm{w}}}_{{ni}}^{B}\right)}^{2}}}.\end{array}\end{eqnarray} \tag{ 8 }$$

In the first test method, more than 4800 acquisitions were recorded every 5 s, and the recording time spans over 6.5 hr. The individual and relative shifts of the two channels (A and B) are shown in the top panel of Figure 4. The SDs of the two channels are both greater than 98 m s⁻¹ (PNL: 0.0246 and 0.0308 m s⁻¹), while the SD of the relative shifts amounts to 12.889 m s⁻¹ (PNL: 0.0399 m s⁻¹). Considerable drifts are expected in view of the fact that our spectrograph is sitting in an ambient condition with temperature fluctuations, air turbulence, and barometric pressure. In addition, the drifts between the two fiber inputs are inevitable even with the rigid mechanical connection. However, simultaneous calibration precision is not our major concern in this work. Here we focus on the data reduction method and the feasibility of simultaneous calibration on the VIPA spectrograph.

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We also notice that the channel drifts fluctuate periodically in phase with the ambient temperature, whereas the humidity and barometric pressure play minor roles. The relative shifts A−B are enhanced significantly at the individual drift valleys corresponding to temperature peaks, indicating that the temperature is a critical factor and influences the two channels independently. In spite of this, the above results confirm that simultaneous calibration is implemented successfully in our VIPA spectrograph. The long-term stability of the instrument will be improved prospectively, benefiting from such fiber-to-fiber tracking after the necessary environmental controls, in particular the temperature of the instrument.

We also examined the comb calibration repeatability with the scheme proposed by Probst et al. (2015) in which the comb modes in one channel were divided into two groups labeled by odd and even numbers. The SD of the relative shifts between the two groups evaluates the calibration repeatability, which incorporates random ingredients, such as photon noise, random spectral instability from comb teeth, part of the systemic errors, etc., and can measure the stability of the comb as the calibration source. The results of the odd and even modes and their difference are shown in the bottom panel of Figure 4. The SDs of the odd and even mode shifts are quantitatively consistent with those in the first method. Meanwhile, the SD of the relative shift is 0.4084 m s⁻¹ (PNL 0.0491 m s⁻¹). Similar to the case of the first method, the relative shifts have strong fluctuations at the temperature peaks. The SDs of the relative shifts in the short term (5 minutes) measured in the two methods are recorded as, respectively, 1.7612 and 0.1728 m s⁻¹ in one of the most stable campaigns.

The repeatability of a two-channel simultaneous calibration quantifies the short-term drifts between subsequent acquisitions of the spectrograph. Deviations from the PNL of the relative shifts stem mostly from the instrumental drifts. Various mechanisms including the mechanical connection between the two-channel fiber inputs, environmental perturbations in the measurements, and random fluctuations in the comb-teeth intensity can result in uncertainties in the repeatability. On the other hand, the repeatability measured in the odd–even mode method is capable of eliminating the uncertainties from spectrograph drifts and characterizing the performance of the Yb:fiber ring LFC as the calibrator for UHRSs. If all of the error sources except photon noise are well suppressed, the SD of the relative shifts should be at the PNL level. However, our UHRS is neither evacuated nor temperature- and pressure-stabilized, the deviations in this case originate mainly from the beam perturbation caused by environment disturbance and the random variations of the comb-teeth intensity. For our instrument, it is apparent that the temperature is a key factor for the long-term stability of both the VIPA spectrograph and the LFC. Nevertheless, our experimental results verify that the VIPA spectrograph can make use of the LFC as an ideal calibration source.

The VIPA spectrograph has shown its excellent performance in spectral resolution and wavelength calibration with the LFC under diffraction-limited conditions. It has potential applications in solar spectral research by virtue of the brightness of the Sun, which may be able to tolerate the relatively low efficiency of SMF couplings. Also, the highly precise Doppler velocities derived from the calibration precision may facilitate research at the solar convective blueshift and limb effect (e.g., Beckers & Nelson 1978; Balthasar 1984; de la Cruz Rodríguez et al. 2011), meridional motion (e.g., Dikpati & Gilman 2007; Doerr et al. 2012), and Sun-as-a-star RV measurements (e.g., Lagrange et al. 2010; Makarov 2010), etc.

All of the results presented in the previous sections are based on a spectrograph in diffraction-limited conditions. A challenging and appealing task is to apply the VIPA spectrograph to night-sky observation under seeing-limited conditions, such as exoplanet searching with the RV method and O2 detection in the atmosphere of an Earth-like planet. At present, most spectrographs are fed the starlight from telescopes via multimode fibers (MMFs) combined with the calibration source.

Now, we hypothesize a VIPA spectrograph with the MMF input of 100 μm core diameter, which corresponds to the seeing of 1'' on a 2 m $f/10$ telescope. With other parameters intact, the resolution of ${ \mathcal R }=30{\rm{,0}}000$ requires the volume of the VIPA to be $20\times 20\times 0.45$ mm³ if the f numbers of the collimator and the camera are 10 and 10/3, respectively. The thicker the VIPA is, the smaller FSR will result. A wavelength range 70 nm can be covered by a single VIPA on a CCD detector with 1200 × 8000 pixels. Consequently, six VIPAs in parallel can lead to a wavelength coverage of 460 nm on a detector with 8000 × 8000 pixels.

As mentioned before, the resolving power is predominantly dependent on the thickness of a VIPA instead of its area, leaving the beam diameter unrestrained in the VIPA spectrograph. The small beam diameter can substantially reduce the volume of the components, for example, the collimator, making the spectrograph more compact and low cost. The total volume of such a VIPA spectrograph is estimated to be 60 × 60 × 30 cm³, facilitating the controls of temperature and pressure.

The existing astronomical spectrographs generally use échelle grating as the disperser, and the dependence of the spectral resolution on the field accepted by the fiber/slit ϕ and the telescope aperture D is given by (Pepe et al. 2000)

$$\begin{eqnarray}&&{{ \mathcal R }}_{\max }=\displaystyle \frac{2h\cdot \tan \beta }{\phi D},\end{eqnarray} \tag{ 9 }$$

where h is the height of the échelle grating, and β is the grating's blaze angle. It is clear that to achieve an equivalent spectral resolution ${ \mathcal R }$, a larger échelle grating is required for a telescope with a larger aperture D. HARPS, a cross-dispersed échelle spectrograph (${ \mathcal R }\approx 1.15\times {10}^{5}$), has the échelle size of $840\times 214\times 125$ mm³. Evidently, the manufacturing costs and technical difficulties of using the échelle will increase dramatically as the telescope diameter and spectral resolution increase, thereby restricting their applications. Following our previous discussion, the volume of a VIPA spectrograph with comparable performance is around 0.1 m³, much smaller than that of conventional échelle spectrographs. The compactness gives advantages in controlling the instrument temperature and pressure, and the long-term stability of the astronomical spectroscopic systems.

For current UHRSs, for example, the double-pass coudé spectrograph (${ \mathcal R }=(5\mbox{--}6)\times {10}^{5}$) at the McDonald Observatory 2.7 m telescope and the Ultra-High-Resolution Facility (UHRF; ${ \mathcal R }=9.9\times {10}^{5}$) at the Anglo-Australian Telescope (AAT), the total efficiencies, including sky and telescope transmission, slit loss, and spectrograph and CCD quantum efficiency, are smaller than 0.5%, and the wavelength coverage is merely a few angstroms (Ge et al. 2002). Such low throughputs primarily owe to the unique optics used to achieve the high resolving power in these instruments, namely a very narrow slit and double-path optics in the McDonald coudé 2.7 m telescope operated by the spectrograph and a confocal image slicer in the UHRF at the AAT. As a result, the current UHRSs at seeing-limited telescopes are generally restricted to the Sun and bright stars. For a spectrograph with a VIPA disperser of $40\times 40\times 3.37$ mm³, the simulations have shown that the resolving power attains ${ \mathcal R }={10}^{6}$ even with a 50 μm core-diameter MMF input, thereby enhancing the efficiency greatly compared with conventional spectrographs.

The comb light couples into MMFs, inducing laser speckles and modal interference due to its high degree of coherence, so that uncertainties in the measurements may result. To diminish multipath interferences and the sensitivity to the light injection, Wilken et al. (2010) employed a dynamic mode scrambler to average over a large number of fiber modes. They achieved a short-term repeatability of Doppler shift 2.5 cm s⁻¹ (Wilken et al. 2012). Moreover, Ye et al. (2016) coupled the comb light into an octagonal fiber by applying a dynamic agitation driven by a vibrator on the fiber to suppress the fiber noise and further tested it on the Chinese 2.16 m telescope at Xinglong Observatory, where a short-term repeatability of 0.1 m s⁻¹ is realized (Hao et al. 2018). It is therefore quite viable to feed the VIPA spectrograph with MMFs by utilizing a scrambler mechanism to preserve the desired spectral resolution and wavelength calibration.