PHYS 362 - Advanced Observational Astronomy

Optical Spectroscopy Notes

Diffraction Grating

In this derivation:

$i$ = angle of incidence onto the grating.

$\theta$ = angle of diffraction from the grating.

$A$ = Amplitude of diffracted wave.

$a(\theta)$ = diffraction pattern from a single slit.

$N$ = number of slits.

$d$ = spacing of slits.

$w$ = width on individual slits.

$\delta$ = phase change between successive slits.

$p$ = path difference between successive slits.

$\lambda$ = wavelength.

$\phi$ is the phase of the summed wave.

$m$ = order of diffraction.

$$p = d \sin{i} + d \sin{\theta}$$

$$\delta = {{2\pi p} \over \lambda}$$

Summing the complex amplitude contributed by each slit:

$$Ae^{i\phi} = {a(1 + e^{i\delta} + e^{i2\delta} + e^{i3\delta} + .... + e^{i(N-1)\delta})}$$

Summing the series:

$$Ae^{i\phi} = {{a (1 - e^{iN\delta})} \over {1 - e^{i\delta}}}$$

This is an amplitude, to find the intensity multiply by the complex conjugate:

$$A^2 = {{a^2 (1 - e^{iN\delta})(1 - e^{-iN\delta})} \over {(1 - e^{i\delta})(1 - e^{-i\delta})}}$$

$$A^2 = {{a^2 (1 - \cos{N\delta})} \over (1 - \cos{\delta})}$$

$$A^2 = {{a^2 \sin^2{N\delta \over 2}} \over \sin^2{\delta \over 2}}$$

Maxima of this function are when: ${\delta \over 2} = m \pi$, where m is an integer.

$$d (\sin i + \sin \theta) = m \lambda$$

$a^2$ is the intensity diffracted by a single slit:

$$a^2 \propto {\sin^2{\beta} \over \beta^2}$$

where:

$$\beta = {{\pi w (\sin\theta - \sin{i})} \over \lambda}$$

This provides the envelope of efficiency.

$$I \propto A^2 = {{A_0^2} {\sin^2{\beta} \over \beta^2} {\sin^2{N\delta \over 2} \over \sin^2{\delta \over 2}}}$$

As $N \rightarrow \infty$ the principal maxima become sharper.

Blaze is a tilt on the grroves of a reflection grating to concentrate the light at a higher angle, and therefore in orders M $>$ 0. Thus light is concentrated in the dispersed orders.

For a blazed grating we get an efficiency curve of the form $sinc^2\gamma$, where:

$$\gamma = {{{\pi a \cos{\theta_B}} \over \lambda} {(\sin{(-d - \theta_B)} + \sin{(i - \theta_B)})}}$$

where $\theta_B = {{(i+d)} \over 2}$

Now call the angle of diffraction d instead of $\theta$. Angle of incidence is still i

Blaze angle $\theta_B = {(i + d) \over 2}$.

Spectrograph Design

A reflection spectrograph consists of:

-

A slit in the focal plane of the telescope to limit the entrance aperture. This slit defines the resolution of the spectrograph, for a monochromatic light source an image of this slit is formed on the detector.

-

A collimator to take the expanding beam from the telescope which has passed through the slit and make it parallel. A collimator can be a mirror, or an achromatic combination of lenses.

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A blazed reflection grating which reflects/diffracts light in the parallel beam. The beam can be reflected back along its incomimg path, this is known as the Littrow configuration, but more often it is not, so there can be a spatial separation between collimator/slit and camera/detector. The grating can be rotated to direct light of different wavelengths into the camera.

-

A camera which takes the parallel beam and focusses it on the detector. A camera like a collimator consists of either a mirror, a combination of lenses, or both.

-

A detector which is either a CCD or a photon counting detector.An image of the slit is formed on this detector.

In this derivation:

$i$ = angle of incidence onto the grating.

$d$ = angle of diffraction from the grating.

$a$ = groove spacing on grating.

$\lambda$ = wavelength.

$A_{tel}$ = Aperture of Telescope (metres).

$A_{coll}$ = Aperture of Collimator (metres).

$A_{cam}$ = Aperture of Camera (metres).

$f_{tel}$ = Focal length of Telescope (metres).

$f_{coll}$ = Focal length of Collimator (metres).

$f_{cam}$ = Focal Length of Camera (metres).

$m$ = order of diffraction.

$\alpha$ = entrance slit width on the sky (radians).

$p$ = projected slit width on detector (metres).

$x$ = detector pixel size (metres).

$L$ = detector linear size (metres).

$W$ = illuminated width of grating (metres).

$D$ = linear dispersion (strictly dimensionless but usually quoted in nm of spectral coverage per mm of detector

$M$ = Magnification

$R$ = Resolving power expressed as $\Delta\lambda \over \lambda$.

$\theta_B$ = Blaze angle (radians).

$\theta_C$ = angle between camera and collimator optical axes, which is fixed. (dimensionless)

The Disperser Equation for a grating is expressed as $g(i,d,\lambda) = 0$.

In this case:

$$g(i,d,\lambda) = {\sin i + \sin d - {m\lambda \over a}} = 0$$

Magnification is given by:

$$M = {p \over {\alpha f_{tel}}} = {- f_{cam} \over f_{coll}} {({\partial d \over \partial i})}$$

The first term is just the ratio of focal lengths as normal, the second term is an anamorphic magnification, caused by the fact that the beam coming off the grating has a different diameter to the beam incident upon it.

$$M = {f_{cam} \over f_{coll}} \times {({{\partial g} \over {\partial i}}) \over ({{\partial g} \over {\partial d}})}$$

Dispersion is given by the linear rate of change of wavelength with distance on the detector, which is the rate of change of wavelength with diffracted angle, times the focal length of the camera:

$$D = {f_{cam}} \times {({{\partial d} \over {\partial \lambda}... ...\partial \lambda}}) \over ({{\partial g} \over {\partial d}})}}$$

Resolving power $R = {\lambda \over \Delta\lambda}$, and $\Delta\lambda = {p \over D}$

$$R = {{\lambda D} \over p} = {{-\lambda f_{cam}} \over p} {({... ...{\partial \lambda}}) \over ({{\partial g} \over {\partial d}})}$$

And substituting in the Magnification:

$$R = {{-\lambda f_{coll}} \over {\alpha f_{tel}}} {({{\partia... ...{\partial \lambda}}) \over ({{\partial g} \over {\partial i}})}$$

But differentiating g with respect to each variable:

$${\partial g \over \partial \lambda} = - {m \over a};\,\, {\pa... ... \partial i} = cos i;\,\, {\partial g \over \partial d} = cos d$$

.

Also, from simple geometry:

$${A_{coll} \over A_{tel}} = {f_{coll} \over f_{tel}}$$

and:

$$W = {A_{coll} \over \cos{i}} = {{A_{cam} \over \cos{d}}}$$

The properties of the spectrograph are then:

$$M = {{f_{cam} \over f_{coll}}{\cos{i} \over \cos{d}}} = {{f_... ... \over A_{cam}} {A_{tel} \over f_{tel}}{\cos{i} \over \cos{d}}}$$

$$D = {{f_{cam} m} \over {a \cos{d}}}$$

$$R = {{\lambda f_{coll} m} \over {\alpha f_{tel} a \cos{d}}} = {{\lambda A_{coll} m} \over {\alpha A_{tel} a \cos{d}}}$$

Note that the magnification given is appropriate for the dispersion direction only, the anamorphic factor does not work along the slit (which is why it is an anamorphic factor). The slit direction magnification is:

$$M_{slit} = {{f_{cam} \over f_{coll}}} = {{f_{cam} \over A_{cam}} {A_{tel} \over f_{tel}}}$$

So the resolving power is increased if $A_{coll}$ is large and all of $\alpha$, $a$ and $A_{tel}$ are small. It is harder to design a high resolution spectrograph the larger the telescope aperture.

Blaze angle $\theta_B = {(i+d) / 2}$, and the camera-collimator angle $\theta_C = (i-d)$. From the disperser equation $\sin{i} + \sin{d} = m \lambda / a$.

$$\sin{i} + \sin{(i-\theta_C)} = {{m \lambda} \over a}$$

.

$$\sin{i} + \sin{i} \times \cos{\theta_C} - \cos{i} \times \sin{\theta_C} = {{m \lambda} \over a}$$

Using $\cos{i} = \sqrt{1 - \sin^2{i}}$:

$${\sin{i} {(1 + \cos{\theta_C})} - {{m \lambda} \over a}} = {\sin{\theta_C} {\sqrt{1 - \sin^2{i}}}}$$

If $m$, $\lambda$, $a$, and $\theta_C$ are all known, this is a quadratic equation in $\sin{i}$ which can be solved. As a quadratic equation it has two solutions. these correspond to two different configurations of the spectrograph, of which is called blaze-to-collimator which is illustrated at the top of this section, and one which is called blaze-to-camera which is illustrated below.

In the blaze-to-collimator configuation the beam expands as it comes off the grating, $\cos{d} > \cos{i}$, and there is anamorphic demagnification of the slit image, so the slit can be wider for the same resolving power. The penalty is that the beam is larger, and so the camera optics need to be larger and consequently more expensive. In the blaze-to-camera configuration $\cos{d} < \cos{i}$, and there is anamorphic magnification.

Going back to the formula for resolving power:

$$R = {{\lambda A_{coll} m} \over {\alpha A_{tel} a \cos{d}}}$$

$${{\lambda m} \over a} = {\sin{i} + \sin{d}} = {2 \sin({{i + d} \over 2}) \cos({{i - d} \over 2})}$$

But:

$${W = {A_{coll} \over \cos{d}}};\,\,\,{\theta_B = ({{i + d} \over 2})};\,\,\, {\theta_C = ({i - d})}$$

Therefore:

$$R = {{2 W \sin{\theta_B} \cos({\theta_C \over 2})} \over {\alpha A_{tel}}}$$

So to get good resolving power we need large gratings, large blaze angle, small camera angle (in practice this is usually of order 30$^{\circ}$), small entrance aperture size, and its easier with smaller telescopes. Of course at high resolving power you need the large telescopes in order to get enough signal for the science goals, so we need to build the big spectrographs for the large telescopes.

Grating Fabrication

There are three principal ways of making reflection gratings:

1. Diamond ruling on a low expansion glass substrate.
2. Epoxy resin cast of a diamond ruled master.
3. Imaging the interference pattern of a laser fed interferometer onto photosensitive emulsion, then etching the unexposed regions with acid. Gratings made this way are called holographic gratings, not to be confused with Volume Phase Holographic (VPH) gratings which work quite differently. Holographic gratings can be made with much smaller groove spacings than ruled gratings, but on the other hand it is not possible to make an efficient blazed grating using holographic techniques.

Blazed Transmission Gratings and Grisms

Transpission gratings are used only for low resolution applications. It is hard to make effectively blazed transmission gratings. More common is a grism, where a transmission grating is replicated on the back of a prism, the prism is usually specified so that the deviation for the central wavelength of interest is equal and opposite to that of the grating. This means that the instrument can be straight. Grism spectrographs still operate at comparatively low dispersion.

Prismatic Dispersion

Prisms disperse light because the refractive index is different for different wavelengths. The refractive index as a function of wavelength is given by an empirical formula due to Hartmann:

$${\mu(\lambda)} = {A + {B \over (\lambda - C)}}$$

Where $A$, $B$ and $C$ are properties of the refracting material. The between the direction of the final ray and the initial ray is given by:

$$\theta = {i_1 - \alpha + \sin^{-1}\{{({A + {B \over (\lambda ... ...n^{-1}({\sin{i_1} \over {(A + {B \over (\lambda - C)})}})]}}\}$$

Dispersion is a much stronger function of wavelength that with a grating, so the relationship between wavelength and position on the detector will be more non-linear. Deviation is highest at shorter wavelengths, as opposed to a grating where deviation is highest at longer wavelengths. Prisms are made from glasses at optical wavelengths, and out of more exotic materials like Selenium Fluoride for infra-red use. Prisms are rarely used in spectrographs by themselves, but are much more commonly used as cross dispersers for order separation in conjunction with echelle gratings.

Echelle Gratings

An echelle grating is an extreme example of a blazed grating. The blaze angle $\theta_B = {(i+d) / 2}$ is set very high (to of order 65$^{\circ}$), with this blaze angle the camera angle $\theta_C = (i-d)$ must be small to avoid groove shadowing.

Because echelle gratings cannot be made with small groove spacing $a$, they are used in high order $m$. $m$ can be inthe range 5-10 for a low dispersion echelle (echellette) spectrograph, but for the highest resolution stellar echelle spectrographs orders of a few hundred can be used. This leads to order overlap because the position on the detector depends only upon $m \lambda$; for instance light at wavelength 550nm in order 100 overlaps with light at wavelength 555.5nm in order 99.

In a classical echelle this order overlap problem is solved by cross dispersion, in which a low dispersion is provided in an orthogonal direction to the main dispersion provided by the echelle grating. This separates the orders which then appear side by side on the detector. This technique is suitable only for point sources (e.g. stars), otherwise the orders still overlap. The cross dispersing element must have a different functional form of the dispersion from the main disperser (the echelle), otherwise order coincidence still occurs. Usually a prism or series of prisms are used.

In the diagram above the layout of the echelle spectrograph built by University College London for the Anglo-Australian telescope is shown. Three prisms provide the cross dispersion, and the direction of the main dispersion provided by the echelle grating is out of the plane of the paper. This provides a series of orders, each covering a short wavelength range, separated on the detector. This is shown in the diagram below.

An alternative for the future to using cross dispersion is to use energy sensitive detectors, such as an array of Superconducting Tunnel Junction detectors or of Transition Edge Sensor Quantum Calorimeters, to separate the orders. If this can be done (currently these arrays have too few pixels to be useful) then the echelle spectrograph can also work on extended objects.

Bragg Diffraction

Diffraction of X-rays was used as a tool to analyse the structure of crystals in the early 20th century, but a crystal with a regular lattice spacing can be used as a diffraction element at X-ray wavelengths.

In this form of diffraction the phase difference is provided by the spacing between the crystal planes, reflection from successive planes produces a phase difference $\delta = {{4 \pi d_s \sin{\theta}} \over \lambda}$, where $d_s$ is the spacing between the crystal planes. We can sum the amplitudes from successive planes as with the analysis of the diffraction grating, and find that constructive intereference occurs at the Bragg angle $\theta$, given by:

$${m \lambda} = {2 d_s \sin{\theta}}$$

A crystal like this with the crystal planes parallel to the crystal surface acts as a monochromater, i.e. only one wavelength of radiation is reflected from it. At X-ray wavelengths crystal spectrometers are constructed by either scanning the angle of the crystal, or alternatively using a bent crystal, where different parts of the surface reflect X-rays of different wavelengths at different angles to a position sensitive detector (e.g. a CCD).

Volume Phase Holographic gratings

A Volume Phase Holographic (VPH) grating works by a mechanism similar to Bragg diffraction, but these are used at optical and near infra-red wavelengths. The planes in this case are modulations of the refractive index in the form of fringe planes, parallel and equally spaced, running through the grating material, terminating at the surfaces of the volume. They are manufactured using holographic techniques, using material whose refractive index can be changed by a process similar to the photographic process, and consist of a layer of gelatin in which the fringes are developed, sanwiched between glass plates which provide mechanical support and protection from dust, water etc.

If the fringes are parallel to the grating surface, then as in the case of the crystal at X-ray wavelengths, the grating acts as a reflecting monochromater which can be scanned in angle.

If the fringes are nearly but not quite parallel to the grating surface, the grating acts as a reflection grating.

If the fringes are perpendicular or nearly so to the grating surface, the grating acts as a transmission grating. Using this technology it is possible to make transmission gratings of much higher fringe frequency that the ruling frequency possible with conventional gratings.

Notation for this section:

$\alpha$ - angle of incidence.

$\beta$ - angle of diffraction.

$\gamma$ - angle of the fringes relative to the grating surface.

$\Lambda$ - fringe spacing in the medium.

$\nu$ - grating frequency ( $\nu = (\Lambda \sin{\gamma})^{-1}$.

$n_0$ - refractive index of air.

$n_1$ - refractive index of substrate.

$n_2$ - refractive index in the medium.

$\lambda$ - wavelength.

$m$ - order of diffraction.

$d$ - thickness of grating medium.

$\eta$ - efficiency of the grating.

$\alpha_{2B}$ - Bragg angle in the medium.

Refractive index is modulated according to:

$$n_2(x,z) = {n_2(0) + {{\Delta}n_2 \cos[({2 \pi \over \Lambda}) {(x \sin{\gamma} + z \cos{\gamma})}]}}$$

We will consider the case of fringes perpendicular to the grating surface, $\gamma$ = 90$^{\circ}$.

$$n_2(x,z) = {n_2(0) + {{\Delta}n_2 \cos[{2 \pi x \over \Lambda}]}}$$

Bragg condition is:

$${m \lambda} = {\Lambda 2 n_2(0) \sin{\alpha_{2B}}}$$

Where $\alpha_{2B}$ is the Bragg angle in the grating medium (gelatin). The Bragg condition is met when $m$ is integer and $\lambda$ and $\Lambda$ are such that the angles of incidence and diffraction are symmetric within the grating medium about the fringe planes.

$$\sin{\alpha_{2B}} = {{n_0 \over n_2} \sin{\alpha}}$$

Near the Bragg condition there is a Bragg envelope, where light at different wavelengths is diffracted at different angles, but at less efficiency than the light diffracted at the Bragg angle. This is analagous to the blaze envelope of a conventional grating.

The disperser equation of the configuration in panel A of the plot above is:

$${{M \lambda} \over \Lambda} = {\sin{\alpha} - \sin{\beta}}$$

So:

$${\partial\lambda \over \partial\beta} = {{-\Lambda \cos{\beta}} \over m}$$

Unlike with a blazed grating, the angle can be rotated to an angle at which the Bragg condition is satisfied for a different wavelength $\lambda$ or different order $m$. With a VPH transmission grating or ant transmission grating the camera must be rotated as well.

The Diffraction efficiency of a VPH grating depends upon the thickness $d$, the intensity of the modulations in the refractive index ${\Delta}n_2$, the wavelength, and the Bragg angle. For $\gamma = 90^{\circ}$ in first order:

$$\eta = {\sin^2[{{\pi {\Delta}n_2 d} \over {\lambda \cos{\alpha_{2B}}}}]}$$

And the passband over which the grating is effective is given approximately by:

$${{\Delta}\lambda \over \lambda} \approx ({\Lambda \over d}) cot(\alpha_{2B})$$

${{\Delta}\lambda \over \lambda}$ here is the bandwidth, not the resolution.

In practice the amplitude of the refractive index modulations and the thickness must be tuned to produce a grating with the desired properties.

Fabry-Perot Etalons

A Fabry-Perot etalon works by constructive interference of light from multiple reflections between two exactly parallel surfaces, this is usually a gap between two precisely manufactured plates of some material (glass for an optical etalon, sometimes sapphire for an infra-red etalon) the inner surfaces of which are coated with a reflective surface. For the purposes of this discussion we call the material that the plates are made out of glass and the material in the gap air. Other materials are possible.

Notation for this section:

$\mu$ - refractive index of air.

$a$ - amplitude of incident ray.

$r$ - proportion of amplitude reflected at air - glass surface.

$r'$ - proportion of amplitude reflected at glass - air surface.

$t$ - proportion of amplitude transmitted at air - glass surface.

$t'$ - proportion of amplitude transmitted at glass - air surface.

$\theta$ - angle of incidence.

$d$ - gap between reflective surfaces.

$m$ - order of interference.

$\lambda$ - wavelength of radiation.

$\delta$ - phase difference between successively reflected rays.

$k$ - wavenumber (= $2 \pi c / \lambda$).

$I_T$ - transmitted intensity.

$R$ - proportion of Intensity reflected.

$N_R$ - reflective finesse of etalon.

$N_A$ - aperture finesse of etalon.

$N_D$ - defect finesse of etalon.

$N_E$ - effective or total finesse of etalon.

$\Sigma$ - free spectral range.

$D_{tel}$ - Diameter of telescope.

$D_{coll}$ - Diameter of Collimator, Etalon and Camera (in a Fabry-Perot spectrograph the collimated beam does not change diameter, there is no anamorphic beam expansion).

$f_{cam}$ - camera focal length.

$\epsilon$ - spatial resolution element.

$\delta\lambda_{\epsilon}$ - wavelength change across a spatial resolution element.

$\alpha$ - angular resolution (seeing).

$\theta_F$ - limiting field angle on the etalon.

$\theta_S$ - limiting field angle on the sky.

$R_P$ - resolving power.

We begin by considering Stokes' treatment of reflection at a surface. The principal of reversibility states that if you reverse the direction of all rays at a surface, the amplitudes will remain the same.

By the principal of reversibility:

$att' + arr = a$, $art + ar't = 0$.

solving these equations:

$r' = -r$; $tt' = (1-r^2)$.

Summing the complex amplitude from each reflected ray:

$${Ae^{i\phi}} = att' + att'r^2e^{i\delta} + att'r^4e^{2i\delta} + ......$$

$${Ae^{i\phi}} = a (1 - r^2) (1 + r^2e^{i\delta} + r^4e^{2i\delta} + ......$$

Summing the series:

$${Ae^{i\phi}} = {{a (1 - r^2)} \over {1 - r^2e^{i\delta}}}$$

To get the intensity multiply this by its complex conjugate:

$$I_T = {{a^2 (1 - r^2)^2} \over 1 - r^2(e^{i\delta} + e^{-i\delta}) + r^4}$$

$$I_T = {{I_0 (1 - r^2)^2} \over 1 - 2r^2\cos{\delta} + r^4}$$

Using $\cos{2\delta} = 1 - 2 \sin^2{\delta}$:

$$I_T = {{I_0} \over {[1 + {4r^2 \over (1 - r^2)^2}\sin^2{\delta \over 2}}]}$$

The phase difference is given by:

$$\delta = {{4 \pi d \mu \cos{\theta}} \over \lambda}$$

,

and the reflected intensity $R = r^2$.

$$I_T = {{I_0} \over {[1 + {4R \over (1 - R)^2}\sin^2({{2 \pi d \mu \cos{\theta}} \over \lambda}})]}$$

High reflectivity $R$ gives sharper fringes, and the peaks occur when $m \lambda = {2 \mu d \cos{\theta}}$.

The reflective finesse of an etalon $N_R = {{\pi \sqrt{R}} \over (1-R)}$ is used as a measure of its theoretical performance. It gives the ratio of the free spectral range to the spectral purity.

The dispersion of an etalon is given by ${{d\lambda} \over {d\theta}} = {{-2 d \mu \sin{\theta}} \over m}$, or at small $\theta$, ${{d\lambda} \over {d\theta}} = \lambda\theta$.

Resolving Power of an Etalon

We define the resolving power in a way analagous to the Rayleigh criterion for resolving images.

To get to the half intensity point we require the wavelength to change by $\Delta\lambda$ or the angle to change by $\Delta\theta$.

$$I_T = {{I_0} \over {[1 + {4R \over (1 - R)^2}\sin^2({{2 \pi d \mu \cos{\theta}} \over \lambda}})]}$$

At maximum, $2 d \mu \cos{\theta} = m \lambda$.

At the 50% point the denominator above must be 2, so:

$${\sin^2[{{2 \pi d \mu \cos{(\theta + \Delta\theta)}} \over \lambda}]} = {{(1 - R)^2} \over {4R}}$$

$${\sin[{{2 \pi d \mu \cos{(\theta + \Delta\theta)}} \over \lambda}]} = {{1 - R} \over {2\sqrt{R}}}$$

Using the trigonometric addition formulae:

$$\sin{\{{{2 \pi d \mu} \over \lambda} [\cos{\theta}\cos{\Delta... ...sin{\theta}\sin{\Delta\theta}]\}} = {{1 - R} \over {2\sqrt{R}}}$$

For small $\Delta\theta$; $\cos{\Delta\theta} \approx 1;\,\, \sin{\Delta\theta} \approx \Delta\theta = {{m \Delta\lambda} \over {2 d \mu \sin{\theta}}}$.

$$\sin{\{{{2 \pi d \mu \cos{\theta}} \over \lambda} - {{\pi m \Delta\lambda} \over \lambda}\}} = {{1 - R} \over {2\sqrt{R}}}$$

Using the trigonometric addition formulae again:

$$\sin{({{2 \pi d \mu \cos{\theta}} \over \lambda})}\cos{({{\pi... ...m \Delta\lambda}\over \lambda})} = {{1 - R} \over {2\sqrt{R}}}$$

${{m \Delta\lambda} \over \lambda}$ is small, so $cos{({{\pi m \Delta\lambda} \over \lambda})} \approx 1$; and $\sin{({{\pi m \Delta\lambda} \over \lambda})} \approx {{\pi m \Delta\lambda} \over \lambda}$. Also $2 d \mu \cos{\theta} = m \lambda$.

$$\sin{\pi m} - {{\pi m \Delta\lambda} \over \lambda} \cos{\pi m} ={{1 - R} \over {2\sqrt{R}}}$$

$\sin{\pi m} = 0$, and $\cos{\pi m} = \pm 1$, (we use -1 for convenience).

$${{\pi m \Delta\lambda} \over \lambda} = {{1 - R} \over {2\sqrt{R}}}$$

$${\lambda \over \Delta\lambda} = {{2 \pi m \sqrt{R}} \over {1 - R}}$$

But according to the Rayleigh criterion you have to go to twice this $\Delta\lambda$ to resolve the line.

$$R_P = {\lambda \over \Delta'\lambda} = {{\pi m \sqrt{R}} \over {1 - R}}$$

In terms of finesse:

$$R_P = m N_R$$

To increase the finesse, and hence the resolving power, of the etalon we require a high reflectivity, and a large gap d.

In practice the profile of an etalon does not achieve the theoretical sharpness, it is degraded by other contributions to the finesse:

-

The aperture finesse $N_A = {{2 \pi} \over {m \Omega}}$, where $\Omega$ is the solid angle set by the focal ratio of the incoming rays.

-

The defect finessed $N_D = {\lambda \over {2 \delta_L}}$, where $\delta_L$ is the amplitude of defects on the etalon surface, because the plates are not perfectly flat.

The effective finesse is given by:

$${1 \over N_E^2} = {{1 \over N_R^2}+ {1 \over N_A^2}+ {1 \over N_D^2}}$$

Fabry-Perots are used in high order m to give high resolving power, the free spectral range $\Sigma = {\lambda \over m}$ is therefore small. They are used in astrophysical applications where the light is concentrated in a small number of emission lines, such as HII regions and Planetary Nebulae, where the light is concentrated in H$\alpha$ and in [OIII] 500.7 nm respectively. Orders are sorted with narrow band filters, allowing through only the light in that particular line. The etalon is scanned either by varying the gap $d$, or by varying the refractive index $\mu$ by varying the pressure, and a three dimensional data array, in which two dimensions are spatial dimensions on the sky, and the third is wavelength, is built up. Because the etalon has a dispersion, an individual etalon setting does not give a unique wavelength, the wavelength map must be built up by scanning the etalon and then applying a phase correction as a function of the distance of the position on the sky from the field centre.

Imaging Fabry-Perot consists of:

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A collimator, usually a combination of lenses to provide a parallel beam through the etalon.

-

An interference filter to isolate the spectral line of interest.

-

The Etalon, whose faces are wedged in order that scattered light caused by internal reflections should be scattered out of the opical path.

-

A camera, again usually a combination of lenses, to image the light onto the detector.

-

The detector, either a CCD or a photon counting detector.

Field Size limitation of a Fabry-Perot

The dispersion of a Fabry-Perot is ${{d\lambda}\over{d\theta}} = {\lambda \sin{\theta}} \approx {\lambda\theta}$. This increases with the angle away from the optical axis, $\theta$. Eventually this becomes so large that the change of wavelength over one detector pixel, or over the spatial resolution element set by the seeing $\alpha$, is large enough to limit the wavelength resolution.

If the pixel size on the detector is $\epsilon$, then the change across a pixel is:

$$\delta\lambda_\epsilon = {{\epsilon \lambda \theta} \over f_{cam}}$$

If the spatial pixel size is matched to the seeing then:

$${f_{cam} \over D_{coll}} = {\epsilon \over {D_{tel} \alpha}}$$

Subject to ${f_{cam} \over D_{coll}} > 1.5$ which is a practical limit set by the difficulty of manufacturing faster focal ratio optics.

The incident field angle $\theta_F$ at which the wavelength change across one resolution element is equal to the intrinsic wavelength resolution is:

$$\theta_F = {f_{cam} \over {R_P \epsilon}} = {{D_{coll} \over {D_{tel} R_P \alpha}}}$$

However the angle incident upon the etalon is related to the field angle on the sky by:

$${\theta_F \over \theta_S} = {D_{tel} \over D_{coll}}$$

So:

$$\theta_S = {({D_{coll} \over D_{tel}})^2} \times {1 \over {R_P \alpha}}$$

The field diameter $\beta_S$ is twice this.

$$\beta_S = {{2 \over {R_P \alpha}} \times {({D_{coll} \over D_{tel}})^2}}$$

Michelson or Fourier Transform Spectrometer

The Michelson or Fourier transform spectrograph was popular for infra-red applications at a time when infra-red astronomy only had single pixel detectors. Imaging Michelson spectrometers are a possibility, but in general have been supplanted by imaging Fabry-Perot instruments which are easier to construct. The Michelson spectrograph relies on the same principle as the Michelson-Morley experiment. Light from the source is split into two beams by a half-silvered mirror, one is reflected off a fixed mirror and one off a moving mirror. The beams interfere, and by making measurements of the signal at many discrete positions of the moving mirror, the spectrum can be reconstructed. Michelson spectrographs are capable of very high spectral resolution observations of very bright sources.

For a monochromatic source of wavelength $\lambda$, and at a particular position of the movable mirror, there will be a path difference ${\Delta}p$ between the two interfering beams. The Intensity measured at the detector will be:

$$I(\lambda,\Delta{p}) = {{I(\lambda) \over K} {{[1 + \cos({{2 \pi \Delta{p}} \over \lambda})}]}}$$

K is a constant. For a source with a general spectral distribution:

$$I(\Delta{p}) = \int_0^{\infty}{{I(\lambda) \over K} {{[1 + \cos({{2 \pi \Delta{p}} \over \lambda})}]} d\lambda}$$

$$I(\Delta{p}) = {{{1 \over K}\int_0^{\infty}{I(\lambda) d\lamb... ...y}{I(\lambda)\cos({{2 \pi \Delta{p}} \over \lambda})} d\lambda}$$

The first term is a DC offset which we can neglect. This leaves:

$$I(\Delta{p}) = { {1 \over K}\int_0^{\infty}{I(\lambda)\cos({{2 \pi \Delta{p}} \over \lambda})} d\lambda}$$

Which is the cosine transform of the spectrum. We change variables now to frequency $\nu = {c \over \lambda}$:

$$I(\Delta{p}) = { {1 \over K'}\int_0^{\infty}{I(\nu)\cos({{2 \pi \Delta{p} \nu} \over c})} d\nu}$$

Defining $I(-\nu) = I(\nu)$:

$$I(\Delta{p}) = { {1 \over K'} \times Real{\{\int_{-\infty}^{\infty}{I(\nu)e^{-i({{2 \pi \Delta{p}} \over c})\nu}} d\nu\}}}$$

Let $x = {{2 \pi \Delta{p}} \over c}$:

$$I(x) = { {1 \over K''} \times Real{\{\int_{-\infty}^{\infty}{I(\nu)e^{-i x \nu}} d\nu\}}}$$

And the spectrum may be obtained from the data by the inverse transform:

$$I(\nu) = { {1 \over K'''} \times Real{\{\int_{-\infty}^{\infty}{I(x)e^{i x \nu}} dx\}}}$$

$$I(\nu) = { {1 \over K''''} \times \int_0^{\infty}{I(x) \cos{(x \nu)} dx}}$$

In practice the intensity is sampled at discrete values of x, and the discrete transform is used.