Transcript Lecture 4 Supplement - Department of Physics, HKU

Chap. 4 Components of Spectroscopic Instruments
In this chapter we discuss basic spectroscopic instruments
and techniques employed to measure wavelength and line
profiles or to realize the sensitive detection of radiation.
4.1 Spectrographs and Monochromators
Spectrographs are optical instruments which form images S2(l) of an
entrance slit S1 which are laterally separated for different wavelengths λ of the
incident radiation. This lateral dispersion is achieved either by spectral
dispersion in prisms or by diffraction on plane or concave reflection gratings.
Figure 4.1 shows the schematic arrangement of optical components in a
prism spectrograph. The light source L illuminates the entrance slit S1 which is
placed in the focal plane of the collimator lens L1. Behind L1 the parallel light
beam passes through the prism P where it is diffracted by an angle q(l)
depending on the wavelength λ. The camera lens L2 forms an images S2(l) of
the entrance slit S1. The position x(l) of this image in the focal plane of L2 is a
Laser Spectroscopy/SJX
1
Chap. 4 Components of Spectroscopic Instruments
function of the wavelength λ. The linear dispersion dx/dλ of the
spectrograph depends on the spectral dispersion dn/dλ of the prism
material and on the focal length of L2.
x
S2(l2)
S2(l1)
L
L0
S1
L1
Prism
L2
Fig. 4.1. Prism spectrograph.
Laser Spectroscopy/SJX
2
Chap. 4 Components of Spectroscopic Instruments
In spectrographs a photoplate or photographic film is placed in the focal
plane of L2. The whole spectral range ∆l=l1(x1)-l2(x2) covered by the
lateral extension Dx=x1-x2 of the photoplate can be simultaneously
recorded. If the exposure of the photoplate remains within the linear part
of the photographic density range, the density Da(x) of the developed
photoplate at the position x(l)
T
Da ( x)  C (l )  I (l )dt
(4.1)
is proportional to the spectral irradiance I(l) integrated over the exposure
time T. The sensitivity factor C(l) depends on the wavelength l and
further more on the developing procedure and the history of the
photoplate. The photoplate can accumulate the incident radiant power
over long periods (up to 50 hours). Photographic detection can be used
for both pulsed and continuous wave light sources. The spectral range is
limited by the spectral sensitivity of available photoplates and covers the
wavelength range between 200~1000nm.
0
Laser Spectroscopy/SJX
3
Chap. 4 Components of Spectroscopic Instruments
When a reflecting grating is used to separate the spectral line S2(l), the
two lenses L1 and L2 are commonly replaced by two spherical mirrors M1
and M2 which image the entrance slit onto the plane of observation, as
shown in Fig. 4.2. An exit slit S2, selecting an interval Dx2 in the focal
plane, lets only a limited range through to the photoelectric detector.
Turning the grating allows the different spectral regions to be turned
across the fixed exit slit S2. Note that different spectral regions are
detected not simultaneously, but successively. The signal received by the
detector is proportional to the product of the exit-slit area hDx2 with the
spectral intensity I (l )dl
, where the integration extends over the
spectral range dispersed within the width Dx2 of S2.

Just according to the kind of detection we distinguish between
spectrographs and monochromators. In general, the name spectrometer is
often used for both types of spectroscopic instruments in literature.
Laser Spectroscopy/SJX
4
Chap. 4 Components of Spectroscopic Instruments
S2
Photodetector
M2
G
M1
S1
Fig. 4.2. Grating monochromator.
Laser Spectroscopy/SJX
5
Chap. 4 Components of Spectroscopic Instruments
1.
2.
3.
4.
Some basic characteristics of spectrometers are listed as follows:
Light-gathering power. It is determined by the maximum
acceptance angle for the incident radiation, measured by the ratio
a/f of diameter a to focal length f of the collimating lens L1 or of the
mirror M1.
Spectral transmittance T(l). It is limited by the transparency of the
lenses and prism in the prism spectrograph or by the reflectivity R(l)
of the mirrors and grating in grating spectrograph.
Spectral resolving power l/D(l) which specifies the minimum
separation Dl of two spectral lines that can just be resolved.
Free spectral range of the instrument, i.e., the wavelength dl in
which the wavelength l can unambiguously determined from the
position x(l).
The light-gathering power of a spectrometer is defined as the
product of the area A of the entrance slit and the maximum
acceptance angle W. That is
U  AW.
Laser Spectroscopy/SJX
(4.2)
6
Chap. 4 Components of Spectroscopic Instruments
For a prism spectrometer the maximum solid angle of acceptance, W=F/f12, is
limited by the effective area F=ha of the prism with height h and width a, f1
is the focal length of collimator lens L1.
For a grating spectrometer the optimized imaging of a light source onto the
entrance slit is achieved when the solid angle W of the incoming light
matches the acceptance angle (a/d)2 of the spectrometer, as shown in Fig. 4.3.
The spectral resolving power of any dispersing instrument is defined by the
expression
l .
(4.3)
R
Dl
where Dll1-l2 stands for the minimum separation of the central
wavelengths l1 and l2 of two closely spaced lines which are considered to be
just resolved.
Rayleigh’s criterion for the resolution of two nearly overlapping lines is
shown in Fig.4.4. We define two lines with equal intensities to be just
resolved if the dip between the two maxima drops to (8/p2) 0.8 of Imax.
Laser Spectroscopy/SJX
7
Chap. 4 Components of Spectroscopic Instruments
The fundamental limit on the spectral resolving power
R
l
dq
a
,
Dl
dl
(4.4)
which clearly depends on the size a of the limiting aperture and on the
angular dispersion dq dl of the instrument. The limiting aperture is
determined by the size of prism or grating. So, the spectral resolving power
W=(a/d)2
W’=(g2/g1)W
W’
g1
W
a
g2
d
M1
Fig. 4.3. Optimized grating
spectrometer.
Laser Spectroscopy/SJX
8
Chap. 4 Components of Spectroscopic Instruments
1
8/p
2
l1
l2
Fig. 4.4 Rayleigh’s criterion.
of a spectrometer is basically determined by the prism or grating.
Laser Spectroscopy/SJX
9
Chap. 4 Components of Spectroscopic Instruments
Note that the entrance slit imposes a limitation on the transmitted
intensity at small slit widths. The useful width bmin of the entrance slit is
given by
bmin  2l f1 a.
(4.5)
It is demonstrated that the resolution of the instrument cannot be increased
much by decreasing the entrance slit width b below bmin.
A practically attainable resolving power of a spectrometer for an
entrance slit width b below bmin is
R
Laser Spectroscopy/SJX
l
 a  dq
 
.
Dl  3  d l
(4.6)
10
Chap. 4 Components of Spectroscopic Instruments
3.1.2 Grating Spectrometer
In a grating spectrometer the grating is a central component which
consists of straight grooves with great number (~105). The grooves of the
grating are parallel to the entrance slit. The grooves have been ruled onto
an optically smooth glass substrate or have been produced by holographic
techniques. The whole grating surface is coating with a highly reflecting
layer (metal or dielectric film).
The many grooves, which are illuminated coherently, can be regarded as
small radiation sources, each of them diffracting the light incident onto this
small groove with a width of about the wavelength l, into a large range of
angles b around the direction of geometrical reflection. The total reflected
light is a coherent super-position of these many partial contributions. Only
in those directions where all partial waves emitted from the different
grooves are in phase the constructive interference of these partial waves
results in a large total intensity, while in all directions the total destructive
interference occur.
Laser Spectroscopy/SJX
11
Chap. 4 Components of Spectroscopic Instruments
Fig. 4.5. Illustration of the grating equation (4.7). Note that
the path length difference between the reflected lights by the
two adjacent grooves is DS=d(sin±sinb).
Laser Spectroscopy/SJX
12
Chap. 4 Components of Spectroscopic Instruments
Figure 4.8 shows a parallel light beam incident onto two adjacent
grooves. At an angle of incidence α to the grating normal one obtains
constructive inference condition for those directions β of the reflected light
d (sin   sin b )  ml , m=0, ±1, ±2,
(4.7)

where λ is the wavelength of the incident monochromatic light. In (4.7) the
plus sign means that β and α are on the same side of the grating normal;
otherwise the minus sign is taken, which is the case shown in Fig. 4.5.
For a special case in which α=βwhich means the light is reflected back
into the direction of the incident light. Such an arrangement is called a
Littrow grating mount, the condition (4.7) for constructive interference
reduces to
2d sin   ml , m  0,1,2
Laser Spectroscopy/SJX
(4.8)
13
Chap. 4 Components of Spectroscopic Instruments
The Littrow grating acts as a wavelength-selective reflector because light is
only reflected if the incident light wavelength λ satisfies the condition (4.8).
We now examine the intensity distribution I(β) of the reflected light
when a monochromatic plane wave is incident onto a grating. For
simplicity, consider the case in which the plane wave is normally incident
onto the grating, that is α=0. The incident plane wave can be expressed as
E=Aexp(i(ωt-kz)). The path difference between partial waves reflected by
any two adjacent grooves is ∆S=dsinβ, and the corresponding phase
difference is given by
2p
d  DS 
 (2p d sin b ) l .
(4.9)
l
the total amplitude of the partial waves reflected from all N grooves in the
direction β is
N
Ar  R  Ag e -imd
m 0
Laser Spectroscopy/SJX
1 - e -iNd
 R Ag
,
1 - e -id
(4.10)
14
Chap. 4 Components of Spectroscopic Instruments
where R is reflectivity of the grating, which depends on the reflection angle
β, and Ag is the amplitude of the partial wave incident onto each groove.
Because the intensity of the reflected wave is related to its amplitude AR by
IR=ε0cARAR*, we find from(4.10)
sin 2 ( Nd )
2
I R  RI 0
sin 2 (d )
2
with
I 0   0cAg Ag* .
(4.11)
The intensity distribution IR is plotted in Fig4.6 for two different gratings
with different total groove number N. The principal maxima occur for
δ=2mπ, which is , according to (4.9) equivalent to
dsinb=ml.
(4.12)
This is the grating equation (4.7) for the special case =0 and means that
the path difference between the partial waves reflected by adjacent grooves
Laser Spectroscopy/SJX
15
Chap. 4 Components of Spectroscopic Instruments
is an integer multiple of the wavelength λ. The integer m is called the order
of the interference. The function (4.11) has N-1 minima with
Fig. 4.6. Intensity distribution I(b) for different
numbers of grooves.
Laser Spectroscopy/SJX
16
Chap. 4 Components of Spectroscopic Instruments
IR=0 between two successive principal maxima. The intensity of the N-2
small maxima, which are caused by incomplete destructive interference,
decreases proportional to 1/N with increasing groove number N. It is not
hard to image that for a real grating with great number of grooves, the
reflected intensity IR(l) at a given wavelength l will have very sharply
defined maxima only in those direction as defined by (4.12). The small side
maxima are completely negligible.
Note that the reflectivity R(b, θ) is not only dependent on the reflection
angle b but also on the slope θ of the grooves. In order to achieve the
optimum value of R(β, θ) the slope θ of the grooves must be carefully
designed. We define a particular angle (blaze angle) θ for obtaining the
optimum value of the reflectivity. From Fig. 4.7 one infers in the case of
specular reflection i=r, with i=-q and r=q+b, the condition for the blaze
angle q
q=(-b)/2 ,
The incident angle  is determined by the particular construction of the
spectrometer and the angle β for which the constructive interference occurs
Laser Spectroscopy/SJX
17
Chap. 4 Components of Spectroscopic Instruments
Fig. 4.7. Illustration of blaze angle.
depends on the wavelength l. Therefore the blaze angle has to be specified
for the derived spectral range and the spectrometer type. The corresponding
wavelength is called the blazed wavelength of the grating.
Usually the second order diffraction (m=2) is employed to increase the
spectral resolution by a factor 2 without losing much intensity in the
practical spectrometer, if the blaze angle θ is correctly chosen to satisfy
(4.19) and (4.18) with m=2.
Differentiating the grating equation (4.7) with respect to l we obtain
Laser Spectroscopy/SJX
18
Chap. 4 Components of Spectroscopic Instruments
the angular dispersion at a given angle of incident α
db
Substituting from (4.7),
db
dl
dl
 m /( d cos b ).
(4.13)
(m )  (sin   sin b ) / l , We find
d
 (sin   sin b ) /(l cos b )
(4.14)
This illustrates that the angular dispersion is determined solely by the
angles  and b and not by the number of grooves.
The resolving power can be derived as following
R
l
 mN
Dl
(4.15)
The spectral resolving power is the product of the diffraction order m with
the total number N of grooves.
Laser Spectroscopy/SJX
19
Chap. 4 Components of Spectroscopic Instruments
Summarizing the considerations above we find that the grating acts as a
wavelength-selective mirror, reflecting light of a given wavelength only
into definite directions bm, m is the mth diffraction order.
Laser Spectroscopy/SJX
20