Section 1: Diffraction Gratings - The Optics of Spectroscopy

Diffraction gratings are manufactured either classically with the use of a ruling engine by burnishing grooves with a diamond stylus or holographically with the use of interference fringes generated at the intersection of two laser beams. (For more details see Diffraction Gratings Ruled & Holographic Handbook, Reference 1.)

Classically ruled gratings may be plano or concave and possess grooves each parallel with the next. Holographic grating grooves may be either parallel or of unequal distribution in order that system performance may be optimised. Holographic gratings are generated on plano, spherical, toroidal, and many other surfaces.

Regardless of the shape of the surface or whether classically ruled or holographic, the text that follows is equally applicable to each. Where there are differences, these are explained.

1.1 Basic Equations

Before introducing the basic equations, a brief note on monochromatic light and continuous spectra must first be considered.

Monochromatic light has infinitely narrow spectral width. Good sources which approximate such light include single mode lasers and very low pressure, cooled spectral calibration lamps. These are also variously known as "line" or "discrete line" sources.

A continuous spectrum has finite spectral width, e.g. "white light". In principle all wavelengths are present, but in practice a "continuum" is almost always a segment of a spectrum. Sometimes a continuous spectral segment may be only a few parts of a nanometre wide and resemble a line spectrum.

The equations that follow are for systems in air where m₀ = 1. Therefore, l = l₀ = wavelength in air.

Definitions	Units
a - angle of incidence	degrees
b - angle of diffraction	degrees
k - diffraction order	integer
n - groove density	grooves/mm
D_V - the included angle or deviation angle	degrees
m₀ - refractive index
l - wavelength in vacuum	nanometres
l₀ - wavelength in medium of refractive index, m₀, where l₀ = l/m₀
1 nm = 10^-6 mm; 1 micrometer = 10^-3 mm; 1 A = 10^-7 mm

The most fundamental grating equation is given by:

(1-1)

In most monochromators the location of the entrance and exit slits are fixed and the grating rotates around a plane through the centre of its face. The angle, D_V, is, therefore, a constant determined by:

(1-2)

If the value of alpha and beta is to be determined for a given wavelength, lambda, the grating equation (11) may be expressed as:

(1-3)

Assuming the value of D_V, is known, alpha and beta may be determined through equations (1-2) and (1-3). See Figs. 1 and 2 and Section 2.6.

L_A = Entrance arm length
L_B = Exit arm length at l_n
b_H = Angle between the perpendicular to the spectral plane and the grating normal
L_H = Perpendicular distance from the spectral plane to grating

Table 1 shows how alpha and beta vary depending on the deviation angle for a 1200 g/mm grating set to diffract 500 nm in a monochromator geometry based on Fig. 1.

Table 1: Variation of Incidence, alpha, and Angle of Diffraction, beta, with Deviation Angle, D_V, at 500 nm in First Order with 1200 g/mm Grating.

Deviation	alpha	beta
0	17.458	17.458 (Littrow)
10	12.526	22.526
20	7.736	27.736
24	5.861	29.861
30	3.094	33.094
40	-1.382	38.618
50	-5.670	44.330

1.2 Angular Dispersion

(1-4)

db = angular separation between two wavelengths (radians)

dl = differential separation between two wavelengths (nm)

1.3 Linear Dispersion

Linear dispersion defines the extent to which a spectral interval is spread out across the focal field of a spectrometer and is expressed in nm/mm, Å/mm, cm^-1/mm, etc. For example, consider two spectrometers: one instrument disperses a 0.1 nm spectral segment over 1 mm while the other takes a 10 nm spectral segment and spreads it over 1 mm.

It is easy to imagine that fine spectral detail would be more easily identified in the first instrument than the second. The second instrument demonstrates "low" dispersion compared to the "higher" dispersion of the first. Linear dispersion is associated with an instrument's ability to resolve fine spectral detail.

Linear dispersion perpendicular to the diffracted beam at a central wavelength, l, is given by:

(1-5)

where L_B is the effective exit focal length in mm and dx is the unit interval in mm. See Fig. 1.

In a monochromator, L_B is the arm length from the focusing mirror to the exit slit or if the grating is concave, from the grating to the exit slit. Linear dispersion, therefore, varies directly with cos b, and inversely with the exit path length, L_B, order k, and groove density n.

In a spectrograph, the linear dispersion for any wavelength other than that wavelength which is normal to the spectral plane will be modified by the cosine of the angle of inclination (gamma) at wavelength l_n. Fig. 2 shows a "flat field" spectrograph as used with a linear diode array.

Linear Dispersion

(1-6)

(1-7)

(1-8)

1.4 Wavelength and Order

Figure 3 shows a first order spectrum from 200 to 1000 nm spread over a focal field in spectrograph configuration. From Equation (1-1) with a grating of given groove density and for a given value of alpha and beta:

(1-9)

so that if the diffraction order k is doubled, lambda is halved, etc.

If, for example, a light source emits a continuum of wavelengths from 20 nm to 1000 nm, then at the physical location of 800 nm in first order (Fig. 3) wavelengths of 400, 266.6, and 200 nm will also be present and available to the same detector. In order to monitor only light at 800 nm, filters must be used to eliminate the higher orders.

First order wavelengths between 200 and 380 nm may be monitored without filters because wavelengths below 190 nm are absorbed by air. If, however, the instrument is evacuated or N₂ purged, higher order filters would again be required.

1.5 Resolving "Power"

Resolving "power" is a theoretical concept and is given by

(dimensionless) (1-10)

where, dl is the difference in wavelength between two spectral lines of equal intensity. Resolution is then the ability of the instrument to separate adjacent spectral lines. Two peaks are considered resolved if the distance between them is such that the maximum of one falls on the first minimum of the other. This is called the Rayleigh criterion.

It may be shown that:

(1-11)

l = the central wavelength of the spectral line to be resolved

W_g = the illuminated width of the grating

N = the total number of grooves on the grating

The numerical resolving power "R" should not be confused with the resolution or bandpass of an instrument system (See Section 2).

Theoretically, a 1200 g/mm grating with a width of 110 mm that is used in first order has a numerical resolving power R = 1200 x 110 = 132,000. Therefore, at 500 nm, the bandpass

In a real instrument, however, the geometry of use is fixed by Equation (1-1). Solving for k:

(1-12)

But the ruled width, W_g, of the grating:

(1-13)

(1-14)

after substitution of (1-12) and (1-13) in (1-11).

Resolving power may also be expressed as:

(1-15)

Consequently, the resolving power of a grating is dependent on:

* The width of the grating
* The centre wavelength to be resolved
* The geometry of the use conditions

Because band pass is also determined by the slit width of the spectrometer and residual system aberrations, an achieved band pass at this level is only possible in diffraction limited instruments assuming an unlikely 100% of theoretical. See Section 2 for further discussion.

1.6 Blazed Gratings

Blaze: The concentration of a limited region of the spectrum into any order other than the zero order. Blazed gratings are manufactured to produce maximum efficiency at designated wavelengths. A grating may, therefore, be described as "blazed at 250 nm" or "blazed at 1 micron" etc. by appropriate selection of groove geometry.

A blazed grating is one in which the grooves of the diffraction grating are controlled to form right triangles with a "blaze angle, w," as shown in Fig. 4. However, apex angles up to 110° may be present especially in blazed holographic gratings. The selection of the peak angle of the triangular groove offers opportunity to optimise the overall efficiency profile of the grating.

1.6.1 Littrow Condition

Blazed grating groove profiles are calculated for the Littrow condition where the incident and diffracted rays are in auto collimation (i.e., alpha = beta). The input and output rays, therefore, propagate along the same axis. In this case at the "blaze" wavelength l_B.

(1-16)

For example, the blaze angle (w) for a 1200 g/mm grating blazed at 250 nm is 8.63° in first order (k = 1).

1.6.2 Efficiency Profiles

Unless otherwise indicated, the efficiency of a diffraction grating is measured in the Littrow configuration at a given wavelength.

% Absolute Efficiency = (energy out /energy in) X (100/1) (1-17)

% Relative Efficiency = (efficiency of the grating / efficiency of a mirror) X (100/1) (1-18)

Relative efficiency measurements require the mirror to be coated with the same material and used in the same angular configuration as the grating.

See Figs. 5a and 5b for typical efficiency curves of a blazed, ruled grating, and a nonblazed, holographic grating, respectively.

As a general approximation, for blazed gratings the strength of a signal is reduced by 50% at two thirds the blaze wavelength, and 1.8 times the blaze wavelength.

1.6.3 Efficiency and Order

A grating blazed in first order is equally blazed in the higher orders Therefore, a grating blazed at 600 nm in first order is also blazed at 300 nm in second order and so on.

Efficiency in higher orders usually follows the first order efficiency curve.

For a grating blazed in first order the maximum efficiency for each of the subsequent higher orders decreases as the order k increases.

The efficiency also decreases the further off Littrow (alpha ¹ beta) the grating is used.

Holographic gratings may be designed with groove profiles that discriminate against high orders. This may be particularly effective in the VUV using laminar groove profiles created by ionetching.

Note: Just because a grating is "nonblazed" does not necessarily mean that it is less efficient! See Fig. 5b showing the efficiency curve for an 1800 g/mm sinusoidal grooved holographic grating.

1.7 Diffraction Grating Stray Light

Light other than the wavelength of interest reaching a detector (often including one or more elements of "scattered light") is referred to as stray light.

1.7.1 Scattered Light

Scattered light may be produced by either of the following:

(a) Randomly scattered light due to surface imperfections on any optical surface.

(b) Focused stray light due to nonperiodic errors in the ruling of grating grooves.

1.7.2 Ghosts

If the diffraction grating has periodic ruling errors, a ghost, which is not scattered light, will be focused in the dispersion plane. Ghost intensity is given by:

(1-19)

where,

I_G = ghost intensity

I_P = parent intensity

n = groove density

k = order

e = error in the position of the grooves

Ghosts are focused and imaged in the dispersion plane of the monochromator.

Stray light of a holographic grating is usually up to a factor of ten times less than that of a classically ruled grating, typically non-focused, and when present, radiates through 2pi steradians.

Holographic gratings show no ghosts because there are no periodic ruling errors and, therefore, often represent the best solution to ghost problems.

1.8 Choice of Gratings

1.8.1 When to Choose a Holographic Grating