Laser Spectroscopy/SJX Chap. 4 Components of Spectroscopic
Instruments 1 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 S pectrographs are optical
instruments which form images S 2 ( ) of an entrance slit S 1 which
are laterally separated for different wavelengths 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 S 1 which is placed in the focal
plane of the collimator lens L 1. Behind L 1 the parallel light
beam passes through the prism P where it is diffracted by an angle
depending on the wavelength . The camera lens L 2 forms an images S
2 ( ) of the entrance slit S 1. The position x( ) of this image in
the focal plane of L 2 is a Laser Spectroscopy/SJX Chap. 4
Components of Spectroscopic Instruments 2 Fig Prism spectrograph.
function of the wavelength . The linear dispersion dx/d of the
spectrograph depends on the spectral dispersion dn/d of the prism
material and on the focal length of L 2. LL0L0 L1L1 L2L2 S1S1 S 2 (
2 ) S 2 ( 1 ) Prism x Laser Spectroscopy/SJX Chap. 4 Components of
Spectroscopic Instruments 3 In spectrographs a photoplate or
photographic film is placed in the focal plane of L 2. The whole
spectral range = 1 (x 1 )- 2 (x 2 ) covered by the lateral
extension x=x 1 -x 2 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 D a (x)
of the developed photoplate at the position x( ) (4.1) is
proportional to the spectral irradiance I( ) integrated over the
exposure time T. The sensitivity factor C( ) depends on the
wavelength 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. Laser Spectroscopy/SJX Chap. 4 Components
of Spectroscopic Instruments 4 When a reflecting grating is used to
separate the spectral line S 2 ( ), the two lenses L 1 and L 2 are
commonly replaced by two spherical mirrors M 1 and M 2 which image
the entrance slit onto the plane of observation, as shown in Fig An
exit slit S 2, selecting an interval x 2 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 S 2. 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 h x 2 with the spectral intensity, where the
integration extends over the spectral range dispersed within the
width x 2 of S 2. 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 Chap. 4
Components of Spectroscopic Instruments 5 Fig Grating
monochromator. S1S1 G M1M1 M2M2 Photodetector S2S2 Laser
Spectroscopy/SJX Chap. 4 Components of Spectroscopic Instruments 6
Some basic characteristics of spectrometers are listed as follows:
1.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 L 1 or of the
mirror M 1. 2.Spectral transmittance T( ). It is limited by the
transparency of the lenses and prism in the prism spectrograph or
by the reflectivity R( ) of the mirrors and grating in grating
spectrograph. 3.Spectral resolving power / ( ) which specifies the
minimum separation of two spectral lines that can just be resolved.
4.Free spectral range of the instrument, i.e., the wavelength in
which the wavelength can unambiguously determined from the position
x( ). 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 . That is Laser Spectroscopy/SJX Chap. 4
Components of Spectroscopic Instruments 7 For a prism spectrometer
the maximum solid angle of acceptance, =F/f 1 2, is limited by the
effective area F=h a of the prism with height h and width a, f 1 is
the focal length of collimator lens L 1. For a grating spectrometer
the optimized imaging of a light source onto the entrance slit is
achieved when the solid angle of the incoming light matches the
acceptance angle (a/d) 2 of the spectrometer, as shown in Fig The
spectral resolving power of any dispersing instrument is defined by
the expression. (4.3) where stands for the minimum separation of
the central wavelengths 1 and 2 of two closely spaced lines which
are considered to be just resolved. Rayleighs 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/ 2 ) 0.8 of I max. Laser
Spectroscopy/SJX Chap. 4 Components of Spectroscopic Instruments 8
Fig Optimized grating spectrometer. a M1M1 =(a/d) 2 d g2g2 g1g1 =(g
2 /g 1 ) The fundamental limit on the spectral resolving power
which clearly depends on the size a of the limiting aperture and on
the angular dispersion of the instrument. The limiting aperture is
determined by the size of prism or grating. So, the spectral
resolving power Laser Spectroscopy/SJX Chap. 4 Components of
Spectroscopic Instruments 9 Fig. 4.4 Rayleighs criterion. of a
spectrometer is basically determined by the prism or grating. Laser
Spectroscopy/SJX Chap. 4 Components of Spectroscopic Instruments 10
Note that the entrance slit imposes a limitation on the transmitted
intensity at small slit widths. The useful width b min of the
entrance slit is given by It is demonstrated that the resolution of
the instrument cannot be increased much by decreasing the entrance
slit width b below b min. A practically attainable resolving power
of a spectrometer for an entrance slit width b below b min is Laser
Spectroscopy/SJX Chap. 4 Components of Spectroscopic Instruments
Grating Spectrometer In a grating spectrometer the grating is a
central component which consists of straight grooves with great
number (~10 5 ). 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, into a large range of angles 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 Chap. 4 Components of Spectroscopic Instruments 12
Fig Illustration of the grating equation (4.7). Note that the path
length difference between the reflected lights by the two adjacent
grooves is S=d(sin sin ). Laser Spectroscopy/SJX Chap. 4 Components
of Spectroscopic Instruments 13 Figure 4.8 shows a parallel light
beam incident onto two adjacent grooves. At an angle of incidence
to the grating normal one obtains constructive inference condition
for those directions of the reflected light m=0, 1, 2, (4.7) where
is the wavelength of the incident monochromatic light. In (4.7) the
plus sign means that and are on the same side of the grating
normal; otherwise the minus sign is taken, which is the case shown
in Fig For a special case in which =which 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 (4.8) Laser
Spectroscopy/SJX Chap. 4 Components of Spectroscopic Instruments 14
We now examine the intensity distribution I() 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 =0. The incident plane
wave can be expressed as E=Aexp(i(t-kz)). The path difference
between partial waves reflected by any two adjacent grooves is
S=dsin, and the corresponding phase difference is given by the
total amplitude of the partial waves reflected from all N grooves
in the direction is (4.10) The Littrow grating acts as a
wavelength-selective reflector because light is only reflected if
the incident light wavelength satisfies the condition (4.8). Laser
Spectroscopy/SJX Chap. 4 Components of Spectroscopic Instruments 15
where R is reflectivity of the grating, which depends on the
reflection angle , and A g is the amplitude of the partial wave
incident onto each groove. Because the intensity of the reflected
wave is related to its amplitude A R by I R = 0 cA R A R *, we find
from(4.10) (4.11) 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 The intensity
distribution I R is plotted in Fig4.6 for two different gratings
with different total groove number N. The principal maxima occur
for =2m, which is, according to (4.9) equivalent to dsin =m (4.12)
with Laser Spectroscopy/SJX Chap. 4 Components of Spectroscopic
Instruments 16 is an integer multiple of the wavelength . The
integer m is called the order of the interference. The function
(4.11) has N-1 minima with Fig Intensity distribution I( ) for
different numbers of grooves. Laser Spectroscopy/SJX Chap. 4
Components of Spectroscopic Instruments 17 I R =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 I R ( ) at a given wavelength 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( , ) is not only dependent on the reflection
angle but also on the slope of the grooves. In order to achieve the
optimum value of R(, ) the slope of the grooves must be carefully
designed. We define a particular angle (blaze angle) for obtaining
the optimum value of the reflectivity. From Fig. 4.7 one infers in
the case of specular reflection i=r, with i= - and r= + , the
condition for the blaze angle The incident angle is determined by
the particular construction of the spectrometer and the angle for
which the constructive interference occurs =( - )/2, Laser
Spectroscopy/SJX Chap. 4 Components of Spectroscopic Instruments 18
depends on the wavelength. 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 is
correctly chosen to satisfy (4.19) and (4.18) with m=2.
Differentiating the grating equation (4.7) with respect to we
obtain Fig Illustration of blaze angle. Laser Spectroscopy/SJX
Chap. 4 Components of Spectroscopic Instruments 19 (4.13) The
spectral resolving power is the product of the diffraction order m
with the total number N of grooves. This illustrates that the
angular dispersion is determined solely by the angles and and not
by the number of grooves. The resolving power can be derived as
following (4.15) Substituting from (4.7), We find (4.14) the
angular dispersion at a given angle of incident Laser
Spectroscopy/SJX Chap. 4 Components of Spectroscopic Instruments 20
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 m, m is the mth
diffraction order.