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6. Spectral Techniques

Spectroscopy is the study of

spectra the quantitative

measurement of spectral lines:

their width, depth and shape.

We measure a line profile - a

graph of the specific intensity,

or flux density, of radiationreceived from a source as a

function of frequency.

Often the shape of the line profile is simply characterised by a measure of

its width (e.g. equivalent width see A2 Theoretical Astrophysics).

Spectroscopy is of fundamental importance to astrophysics because it

allows us to deduce many physical characteristics of planets, stars and

galaxies even though we observe them remotely, from enormous distances.

Intensity

frequency

Continuum

0Schematic diagram of an absorption line


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From analysis of spectral lines we can learn about:-

Characteristic from

1. Chemical elements frequency

2. Chemical abundances intensity

3. Bulk velocity (i.e. velocity frequency

of atmosphere as a whole)

4. Temperature, pressure, line width

gravity

5. Spread of velocities line width

6. Magnetic and electric field fine structure in lines

(e.g. Zeeman splitting)

0

0


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We can collect light in a narrow frequency range using a filter

Effectiveness measured by spectral resolving power,R

Examples: Dye filter

Interference filter

But we need or higher to be useful (i.e. to be

sensitive to Doppler shifts of a few km/s)

=

=

00R (6.1)

10010~ R

410~R

510~R


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Much higher spectral resolving powers can be achieved using aPRISM or a DIFFRACTION GRATING

White light Red light

Blue light

White light

Red light

Blue light

Blue light

Red light

Note that a prism disperses blue

light more strongly than red light,while for a diffraction grating red

light is dispersed more strongly

than blue light.


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a

For an infinite diffraction

grating with light incident atright angles (normal incidence),

the dispersed light has an

intensity maximum when the

path difference betweenadjacent light rays satisfies

na =sin (6.2)

sinaPath difference =

Constructive interference

a = spacing between lines of grating

n = order of the maximum


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If the light is incident at an

oblique angle, , maximaoccur at path differences

which satisfy:-

( ) na =+ sinsin

a

)sin(sin +aTotal path difference =

(6.3)

To determine the angular dispersion

(i.e. the spread in angular deflection

corresponding to a given spread in

wavelength) we differentiate (6.2) or

(6.3) with respect to wavelength.


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nd

da =

cosDifferentiating

Hence

cosa

n

d

d=

(6.4)

(requires to be in radians)

Angular dispersion

We can achieve a highangular dispersion via:

o high order

o small

o large

a

Suppose the dispersed light is focussed on a detector (e.g. a CCD) using a

lens or mirror of focal length f

Then

d

d

dfdfdx ==

Linear separation of lines on detectorActual separation of lines in spectrum

(6.5)


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nd

da =

cosDifferentiating

Hence

cosa

n

d

d= (6.4)

(requires to be in radians)

Angular dispersion

We can achieve a highangular dispersion via:

o high order

o small

o large

a

Linear Dispersion

Often also defined is the Reciprocal Linear Dispersion (RLD)

d

df

d

dx= (6.6)

dx

d=


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Intensity

sin

a a2 a3aa2a30

Intensity profile from a monochromatic light source, of wavelength ,

diffracted by an infinite diffraction grating, is a series of infinitely thin

peaks equally spaced in

In practice, of course, any grating is finite (i.e. only a finite number of

lines is illuminated)

sin


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Finite diffraction grating

Consider grating with rules.

For the aperture,

Phase difference

By principle of superposition, total wave amplitude diffracted throughan angle is (for an incident wave of unit amplitude)

(modelling the wave as complex simplifies the algebra)

1

2

3

4

5

N - 1

NN

thm

mamm == sin

2(6.7)

)1(2tot 1

++++=Niii eee K (6.8)


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iNiii eeee +++= K2tot

Multiplying through by

Subtracting (6.7) from (6.6)

i.e.

ie

(6.9)

iNi ee = 11 tot (6.10)

i

iN

e

e

= 1

1tot

(6.11)


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We can rewrite the RHS as

i.e.

using the result

And the Intensity in direction is then given by

( )

( )2/2/2/

2/2/2/

tot

iii

iNiNiN

eee

eee

= (6.12)

( )( )2/sin

2/sin)1(tot

Ne

Ni

= (6.13)

ixix

i eex

= 21sin

*


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Thus

Primary Maxima occur at (same as for infinite grating)

But maxima are not infinitely narrow.

Width of maxima

Also N 2 secondary maxima in between

( ) [ ][ ]2/sinsin2/sinsin 2

2

0

kaNkaII =

Incident intensity

2=k = wave number

na =sin

aN

W

~

(6.14)


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Thus

e.g. For N = 5


2

0

kaNkaII =

Incident intensity

2=k = wave number

sin

Intensity

a

a2

a

a2 0

(6.14)


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Thus

e.g. For N = 6


2

0

kaNkaII =

Incident intensity2

=k = wave number

sin

Intensity

a

a2

a

a2 0

(6.14)


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The finite width of the primary maxima limits the resolving power of

a diffraction grating.

Consider two spectral lines,

of wavelength and

Lines are observed at order

light diffracted

through angles satisfyingsin

Intensity

1 22

1

n

a

n 11sin

=

a

n 22sin

= (6.15)


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We can only resolve the two

lines provided they are separatedby (at least) their width.

In other words, the resolution

limit is

Or

sin

Intensity

21

(6.17)

aNa

n =

nNR =

=

Can use an echelle grating to

achieve a high resolving power;

lines are blazed cut in a

special pattern to concentrate

light in high orders


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We can only resolve the two

lines provided they are separatedby (at least) their width.

In other words, the resolution

limit is

Or

sin

Intensity

21

(6.17)

aNa

n =

nNR =

=

Spectral resolving power dependsnot on ruling separation, but onthe total number of lines on the

grating and the order of the

maximum observed.


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Design of a Slit Spectrometer

Key features of the design are summarised in the following diagram

Detector

Focussinglens

Grating

CollimatinglensSlitTelescope

aperture

The slit cuts out unwanted light. Its angular size at the collimating lens

defines the range of angles entering the grating

Diameter of collimating lens width of grating, so that little light is lost


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Detector

Focussinglens

Grating


aperture

Grating response width = width of primary maxima =

Focussing lens also produces a diffraction pattern (see next section)

which smears out light, with width

Choose so that - i.e.

gratingDNa =

focusD

gratingfocus DD


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Detector

Focussinglens

Grating


aperture

Choose focal length of focussing lens so that width of diffraction peak

at the detector width of pixel on detector.

i.e. the diffraction maxima cover several pixels

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