2. Diffraction Grating Basic principles: Diffraction Basic principles: Fraunhofer and Fresnel diffraction Single- & Multi-slit The grating equation again Resolving power Blazed gratings

Consider the electromagnetic wave E and H. We define them as or for a plane wave where (because ) Thus In general, the above equation can be written as

Applying Fourier inversion transform for the above equation, At z=0 Applying Fourier inversion transform for the above equation, However, at z=0, f=0 outside the hole R. Then we have x k R z y

Fraunhofer vs. Fresnel diffraction In Fraunhofer diffraction, both incident and diffracted waves are considered to be plane (i.e. both S and P are at large distance) If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction S P

EX. Fraunhofer diffraction by very long slit First, let's assume the rectangle slit (left figure). The length and the width of the slit are assumed to be 2b and 2a, respectively. The incident plane wave is described as , f=A when z=0. Therefore, from z y x 2a 2b x k  z 2a y

Assuming k1=k and k2=k, where =sin and =cos, x x k  z z 2b 2a 2a y y

2a 2b k  2a The intensity ratio of A(k1,k2) is therefore, When b>>1, 0. Accordingly, Or supposing =ka sin()=2a/sin() z y x 2a 2b x k  z 2a y

Single Slit Fraunhofer diffraction: Effect of slit width Minima for sin [k a sin] = 0 k a sin = p First minima at sin  = /2a  2a k a sin

Single Slit Fraunhofer diffraction: Effect of slit width Width of central max = /a This relation is characteristic of all Fraunhofer diffraction If a is very large  0 and a point source is imaged as a point If a is very small (~) /2 and light spreads out across screen

Similary, you can consider Fraunhofer diffraction from a circular aperture

Fraunhofer vs. Fresnel diffraction (cont.)  ’  r’ r h’ h d’ S d P

Now calculate variation in (r+r’) in going from one side of aperture to the other. Call it   ’  h h’ r’ r

sin’≈ h’/d’ and sin ≈ h/d sin’ + sin =  ( h’/d + h/d ) (cont.) By the way… sin’≈ h’/d’ and sin ≈ h/d sin’ + sin =  ( h’/d + h/d ) First term = path difference Thus the second term is measure of curvature of wavefront. Fraunhofer Diffraction is defined when the curvature of the wavefront is small enough, that is,

If aperture is a square, the area is 2. Then we have the Fraunhofer diffraction if, Fraunhofer or far field limit

Diffraction from two slits: Now, suppose the collimated beam falls on two slits of width b, separated by a distance a: Then we end up with the pattern at right: a series of bright interference fringes modulated by the envelope of the single slit. Note that, so far, we assumed the slit width = 2a

Diffraction from two slits: Mathematically, one can show that the bright fringes are regularly spaced, and modulated by the single slit envelope: Same as for single slit (the envelope) Interference term

Diffraction from an array of N slits: … separated by a distance a and of width b y=(N-1)a + b  y=(N-1)a y=3a+b P  y=3a y=2a+b  y=2a y=a+b  y=a y=b  y=0

Diffraction from an array of N slits It can be shown that, where, Same as for single slit (the envelope) Interference term and

Diffraction and interference for N slits The diffraction term Minima for sin  = 0  = ±p = k(b/2)sin  or sin = ±p(/b) because k=2/ The interference term Amplitude due to N coherent sources It can see this by adding N phasors that are 2 out of phase (see next page).

Diffraction and interference for N slits: Interference term Maxima occur at  = ±m (m = 0,1, 2, 3,..) Thus maxima occur at sin  = ±m/a This is the same result we have derived for Young’s double slit Intensity of principal maxima, I = N2Io

Diffraction and interference for N slits: Interference term (cont.) Minima occur for  = /N, 2/N, … (N-1)/N Thus principal maxima have a width determined by zeros on each side Since  = (/)a sin  = ±/N , the angular width is determined by sin  = /(Na) Thus peaks are N times narrower than in a single slit pattern (also a > b).

Diffraction gratings As we learned, it is composed of systems with many slits per unit length – usually about 1000/mm Also it is usually used in reflection Thus principal maxima are vary sharp The width of peaks Δ = 2/N As N gets large the peak gets very narrow

Diffraction gratings Diffraction gratings are quite useful for achieving moderately high resolving powers in astronomy. In particular, at far-infrared wavelengths, reflective gratings are commonly used. For non-normal incidence, the relationship between the incoming and outgoing rays is given by the grating equation: sin  + sin  = n/d

Diffraction gratings (Blazed Grating) A problem with simple gratings is that the diffraction envelope is broadest at = 0, where the chromatic dispersion is zero. That is most of the power is dumped into the zero order :') This problem is solved by 'Blazing', shifting the diffraction envelope relative to the interference pattern to dump the power into a more useful angle, such as 1st (or higher) order of the interference pattern.

Diffraction gratings (Blazed Grating) For a reflection grating, the facets are tilted by the blaze angle, , and the maximum then occurs for: +=2 when we have specular reflection. The manufacturer normally defines the blaze angle when the grating is used in Littrow mode, i.e. =: nl/d = sin a + sin b = 2sin

Diffraction limit and Airy disks The diffraction pattern resulting from a uniformly-illuminated circular aperture has a bright region in the center (Airy disk) together with the series of concentric bright rings (Airy pattern). The angle at which the first minimum occurs, measured from the direction of incoming light, is given by the approximate formula: sin = 1.22/D   where θ is in radians and λ is the wavelength of the light and D is the diameter of the aperture. f D  

Fraunhofer diffraction and spatial resolution Suppose two point sources or objects are far away (e.g. two stars) Imaged with some optical system Two Airy patterns If S1, S2 are too close together the Airy patterns will overlap and become indistinguishable S1  S2

Fraunhofer diffraction and spatial resolution Assume S1, S2 can just be resolved when maximum of one pattern just falls on minimum (first) of the other Then the angular separation at lens, e.g. telescope D = 1 m and  = 500  10-9 m e.g. human eye D ~ 7mm  Comparable to the apparent size of Saturn

Limitations for High Dispersion Problem: detector size, shape generally square or 1x2 format a conventional grating spectrograph produces a very LONG high dispersion spectrum that does not fit on a CCD Solution: the echelle grating works in high orders (n=100) a second dispersing element spreads the light in a perpendicular direction

Echelle Gratings To increase spectral resolution, increase the order at which a grating is used For high orders, we must increase a and b in the grating equation (to ~50-75o) The spectral range for each order is small so the orders overlap Separate the orders with a second disperser (cross disperser) acting in a perpendicular direction.

Multi-object Spectroscopy Observing one star at a time is inefficient When many stars are available in a field (e.g. a star cluster) use multi-object spectroscopy Put an optical fiber at locations of objects to take spectra. Feed the optical fibers into a spectrograph.

Grism A grism is a combination of a prism and grating arranged so that light at a chosen central wavelength passes straight through. The advantage of a grism is that it can be placed in a filter wheel and treated like another filter. The basic relationships required to design a grism are where  is the central wavelength; T is the number of lines per millimeter of the grating; n is the refractive index of the prism material;  is the prism apex angle.

The Fourier Transform Spectrometer (FTS) The FTS is a scanning interferometer with collimated light as a ninput. A typical scheme is shown below. For a collimated monochromatic beam, the intensity at the detector is determined by the phase difference 2(xa+xb). The primary disadvantage for astronomical work is the fact that the measurements require a time sequence to determine the spectrum.

Fabry-Perot Interferometer The Fabry-Perot interferometer is an imaging spectrometer which is formed by placing a device called an "etalon" in the collimated beam. The etalon consists of two plane- parallel plates with thin, highly reflective coatings on their inner faces. The plates are in near contact but separated by a distance d. Assuming that the refractive index of the medium in the gap is n (usually n=1) and  is the angle of incidence of a ray on the etalon (usually very small), then multiple reflections and interference within the gap occurs and the wavelengths transmitted with maximum intensity obey the relation: m=2ndcos