Spectroscopic
Properties of Gratings
Introduction
Diffraction gratings are widely used in spectroscopic instruments, for
creating monochromatic light from a white light source. This is achieved
by utilizing the grating's ability of spreading light of different wavelengths
into different angles. The relation between the incidence and diffraction
angles, and the wavelength is given by the well known grating equation.
(eq. 1 ) Many of the most important spectroscopic properties, such as
dispersion, resolution and free spectral range can be derived from the
grating equation, from fairly straight forward algebraic manipulations.
 |
Fig
1. Diffraction at the surface of a plane grating. The grating
is depicted with the grating grooves perpendicular to the
paper and the collimated monochromatic beams are propagating
in the plane of the paper. |
|
The
Grating Equation
A beam of light which falls on a grating will be diffracted into one
or several beams. The directions of these beams depend on the wavelength
and direction of the incident beam, and on the groove frequency of the
grating.
The grating equation is a good starting point when describing the properties
of gratings. With notations according to fig. 1, the grating equation
can be written:
where:
is the angle of incidence
m is the angle of diffraction. The angles are positive if they are directed
counter clockwise, otherwise negative.
m denotes the order number of the diffracted beam. m is an
integer number, positive, negative or zero.
d is the groove spacing of the grating. Usually, gratings are specified
by their groove frequency given as number of grooves per millimeter.
The groove spacing in nanometers is then found by taking the reciprocal
of the groove frequency, and multiplying by 106.
denotes the wavelength of the light in the medium surrounding the grating,
usually air. 
= 0/n
where 0 = wavelength in vacuum, and n= refractive index.
Diffraction
orders
By considering the case when m=0, the equation reduces to = 0 or the law of reflection. There will always be this solution
and therefore a reflected beam, which usually is not wanted. The reflected
beam is the major cause of light losses in a grating. The diffracted
order with m = -1 is the order normally used in monochromators, spectrographs,
and spectrometers.
Gratings with low groove frequency will generate many diffracted orders.
For monochromatic light, e.g. from a laser, a grating may be used as
a beamsplitter, for generating two or more beams. Two beams may also
be combined at a grating surface.
Dispersion
The angular dispersion is the amount of change of diffraction angle
per unit change of the wavelength. It is a measure of the angular separation
between beams of adjacent wavelengths. An expression for the angular
dispersion can be derived from equation (1) by differentiating, keeping
the angle 
fixed.
High dispersion can be achieved either by choosing a grating with a
high groove frequency, or by using a coarse grating in high diffraction
order.
Generally a fine pitch grating would be preferred because of the larger
free spectral range (see below).
The wavelength dispersion at the exit slit of a spectroscopic instrument
is usually specified as reciprocal linear dispersion, given in nm/mm.
If the focal length of the instrument is f, then the reciprocal linear
dispersion is given by:
The size of the instrument depends on the focal length of the optical
system. By choosing a holographic grating with a high groove frequency,
the instrument could be made more compact.
Free
Spectral Range
As we can see from the grating equation, light of wavelength in the first order is diffracted in exactly the same direction as light
of wavelength /2
in the second order, as well as /3
in the third order etc. When using gratings, it is therefore important
to restrict the wavelength interval in some way, either by using a bandpass
filter, or by making use of the limited wavelength range of the light
source or the detector.
 |
Fig.
2. Diagram visualizing the overlapping of spectral orders
and the free spectral range. |
|
The free spectral range of a grating is the largest wavelength interval
in a given order which does not overlap the same interval in an adjacent
order. If 1 is the shortest wavelength and 2 is the longest wavelength in this wavelength interval, then the free
spectral range may be expressed by:
Evidently the free spectral range is reduced when the grating is used
in higher orders. In -1 order, the free spectral range is 2/2,
i.e. the grating can be used from 1 to 2 x 1 without overlapping from the 2 order diffraction.
Resolution
The spectral resolution of an instrument is determined by the separation
(
)
between two spectral peaks that can just barely be detected as separate
with the instrument.
A theoretical treatment of the instrumental resolution shows that the
properties of the grating sets the ultimate limit for the resolution.
The gratings described by its "resolving power" which is a
dimensionless number, R. The simplest definition is:
where m is the diffraction order and N is the total number of grooves
on the entire grating surface. Note, however, that the grating equation
places restrictions on the possible combinations of m and N.
The measured resolving power of a real grating is less than the theoretical
value if the grating surface or the grating grooves are deviating from
the ideal shape and position.
As an example, a 110 mm wide grating with 1800 grooves/mm, used in first
order diffraction, has a theoretical resolving power of 198 000 which
implies a wavelength resolution of 0.003 nm at 500 nm wavelength.
Efficiency
The absolute efficiency is defined as the amount of the incident flux
that is diffracted into a given diffraction order. The relative efficiency
is related to the reflectance of a mirror, coated with the same material
as the grating, and it should be noted that the relative efficiency
is always higher than the absolute efficiency.
For most applications only one diffraction order is used, and one would
like all the diffracted light to go into that order giving an absolute
efficiency of 100% for all wavelengths. However, the grating efficiency
is generally a rather complex function of wavelength and polarization
of the incident light, and depends on the groove frequency, the shape
of the grooves and the grating material. Especially for TM polarization,
when the electric vector is perpendicular to the grating grooves, one
may observe rapid changes in efficiency for a small change in wavelength.
This phenomenon was first discovered by R.W. Wood in 1902, and the rapid
variations are usually called Wood's anomalies.
Sinusoidal
gratings
Holographically manufactured gratings of standard type have a sinusoidal
groove profile. The efficiency curve is rather smooth and flatter than
for ruled gratings. The efficiency is optimized for specific spectral
regions by varying the groove depth, and it may still be high, especially
for gratings with high frequency. When the groove spacing is less than
about 1.25 times the wavelength, only the -1 and 0 orders exist, and
if the grating has an appropriate groove depth, most of the diffracted
light goes into the -1 order. In this region, holographically recorded
gratings give well over 50 % absolute efficiency.
Figure 3. shows a set of efficiency curves for the most common holographic
grating types. Each grating is denoted P XXXX YY, where P stands for
Plane holographic grating, XXXX is the groove frequency, and YY is the
spectral range where the efficiency is highest.
|
Fig.
3. Typical absolute efficiency curves for aluminum coated holographic
gratings of various groove profiles, groove frequencies; optimized
for unpolarized light in the UV, visible and the near infrared regions,
respectively. Each diagram shows the theoretical efficiencies for
TE and TM polarization, Calculated for a 10 degrees constant deviation
mounting. |
Blazed
gratings
Blazed gratings are optimized for a specific wavelength by varying the
blaze angle of the grating. The efficiency is high for the specified
wavelength, approximately 70% absolute efficiency, but on the other
hand the efficiency is lower for other wavelengths.
The efficiency of two typical blazed holographic gratings is shown in
fig. 4. Blazed gratings for the UV spectral range are denoted B XXXX
UV, where XXXX is the groove frequency (grooves/mm).
|
Fig
4. Efficiency curves for blazed holographic gratings, optimized
for the UV spectral range. |
|
Stray
Light
Stray light is another important aspect of gratings. When working at
the detection limit of an optical instrument, the stray light level
from the grating and other optics will set the ultimate limit of detection.
Holographic gratings are known for their low level of stray light and
the total absence of "ghosts" in the spectral image. This
is due to the very precise spacing between grooves which is achieved
in the interference pattern exposure. However there are sources of stray
light also from holographically recorded gratings, and the stray light
levels may vary considerably between gratings due to differences in
the manufacturing processes used.
In fig. 5. we present stray light curves for three gratings: a ruled
grating, an ordinary holographic grating, and a Spectrogon Low Stray
Light Holographic Grating. It is interesting to note the large difference
between the ordinary holographic grating and the low stray light grating.
It is widely known that holographic gratings have much lower stray light
levels than ruled ones; nevertheless, Spectrogon has been able to improve
the holographic process even further. As the diagram shows, the stray
light levels are about ten times lower, which in a double spectrometer,
implies an improvement by a factor of 100 of the spectral purity.
|
Fig
5. Scattered light per unit solid angle, for three different gratings,
illuminated by a HeNe laser beam
(A) Ruled grating, 1200 gr/mm
(B) Holographic grating, 1000 gr/mm
(C) Spectrogon Low Stray Light grating, 1800 gr/mm |
The
stray light curves have been determined by illuminating the gratings
with a HeNe laser beam, and measuring the scattered light for different
wavelength settings. The stray light is normalized with respect to the
incident flux, and the solid angle subtended by the detector. This method
has the advantage that the specified stray light is not dependent on
the particular measurement configuration used, such as focal lengths
and slit widths, etc.
