Grating Dispersion [Equation of the Week]

High-resolution astronomical spectroscopy has invariably been carried out with gratings. Even with the advent of the new calorimeter detectors, which can measure the energy of incoming photons to an accuracy of as low as 1 eV, gratings are still the preferred setups for hi-res work below energies of 1 keV or so. But how do they work? Where are the sources of uncertainty, statistical or systematic?

The basis of dispersion is Bragg’s Law, which relates the constructive interference pattern of diffraction from a regular structure to the wavelength of the incident photons. If you have photons of wavelength λ incident at an angle θ0, going through a transmission grating (such as the LETGS or the HETGS on Chandra) which has rulings spaced at distances d, the resulting diffracted photons will constructively interfere at dispersion angles θ such that

mλ = d (sinθ – sinθ0) ≈ dδθ

where m=0,±1,±2,…, and for small deviations, we can approximate sinθ≈θ.

Now here’s the neat trick. If the grating is curved, then the photons incident on different parts of it will be diffracted by different angles, and it turns out (long story, lots of geometry, perhaps another time) that (for small changes in the incident angles) these photons are brought to a focus on the opposite side of a circle whose diameter is equal to the radius of curvature D of the grating. This circle is called the Rowland Circle. If a detector is placed on the Rowland circle directly across from the grating, photons of different wavelengths will be dispersed to a focus at a lateral distance Y along the detector, following along the circumference of the Rowland circle:

Y = D (θ – θ0) = D δθ = mλ D/d

Note that for m=0, all the photons come to a focus at the same point Y=0 irrespective of λ. For m≠0, the photons are dispersed to different distances depending on the dispersion order m and the wavelength. Therefore, for any non-zero order, the wavelength of a photon can be estimated simply by measuring the dispersion distance Y:

λ = (1/mD) Y d

Some detectors that have rudimentary intrinsic spectral resolution (like ACIS) can separate out the photons based on orders. Others (like HRC), will have all the orders overlapping and unseparable. In such cases, modeling must be done carefully, using response matrices that include multiple orders.

The nominal statistical uncertainty on the estimated wavelength can be written as a combination of the measurement uncertainties in the grating periodicity d and the uncertainty in the photon’s position wrt the 0th-order:

δλ/λ = √‾ ( (δd/d)2+(δY/Y)2+(δD/D)2)

Generally, the uncertainties in the grating curvature δD and in the periodicity δd are quite negligible. The major contributor to the wavelength uncertainty is then due to the location, which is composed of two factors — the uncertainty in registering the location on the detector that the photon is deposited (at least ~1 pixel), and the size of the astrophysical source that is being observed, which for point sources is the same as the size of the PSF. The reciprocal of the fractional uncertainty in wavelength (or energy), λ/δλ, is called the Resolving Power. Good spectrometers in the X-ray regime nowadays reach resolving powers of thousands.

As an example, consider the Chandra LETGS, which has a Rowland diameter D=8637 mm, and a grating period d=0.99125±0.000086 μm. At say 100Å in the 1st order (i.e., m=±1) δθ ≈ ±0.0100883c ~ ±37.4 arcmin. Then Y=±87.1324 mm, for a plate scale of 1.148Å/mm. The HRMA on-axis PSF is approximately 5 HRC pixels (0.032 mm) wide, so the resolving power is ≈2700. The simple back-of-the-envelope calculation gives results quite close to the actual values.

PS: This was originally a slide in the Gratings talk at the 2nd X-ray Astronomy School, 2002, Berkeley Springs, WV.

  1. hlee:

    Irrelevant question but an extension from [Q]systematic error:
    From “The nominal statistical uncertainty on the estimated wavelength can be written as a combination of the measurement uncertainties in the grating periodicity d and the uncertainty in the photon’s position wrt the 0th-order …”
    since statistical uncertainty accompanies probability distribution (given or subject to be estimated, parametric or nonparametric), I wonder if this statement is implying that wavelength has a (parametric) distribution whose parameters are associated with the measurement uncertainties in the grating periodicity and the uncertainty in the photon’s position? Do people estimate this distribution in terms of these parameters? Is the nominal statistical uncertainty a joint distribution of these two parameters? or plainly assume Gaussian and additive errors? I’m still seeking answers to systematic errors, measurement errors, and how to model these uncertainties. Thanks~

    p.s. I think I’ll understand this post better if I scrutinize later. A first time reading hardly provided me anything.

    p.p.s. I came back to read again but another thought came to me. I do not recall that I saw Kalman filter or HMM occasionally from astro-ph but I imagine that the things I learned from nonlinear control and system could help modeling some of astronomical systems with systematic errors or measurement errors in addition to known statistical errors. (Here, known means not quantification but qualification; for example, not estimated σ but Poisson noise). As astrometry is a subject in astronomy, I hope handling errors to be regularized. I’ll appreciate any criticism and comments to my question.

    07-07-2008, 5:53 pm
  2. vlk:

    As written, the expression for the fractional error in wavelength is a direct consequence of propagating the statistical error via the method of moments delta method and does assume a Gaussian error model.

    Given the location of the photon on the detector, it is clearly not possible to assign a wavelength to it to an accuracy better than the PSF. The uncertainty in the grating period is usually determined by the tolerances in the construction, which are usually quite stringent and far better than the PSF size, and so are quite negligible. The line spread function (LSF) therefore is almost entirely determined by the PSF. This is not the systematic error. Any systematic errors that are present are in addition to these errors.

    07-07-2008, 11:51 pm
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