Optics Tutorial
Section 1: Diffraction Gratings Ruled & Holographic
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).
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 to optimize system performance. 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; explanations are provided where there are differences.
1.1 Basic Equations
Before introducing the basic equations, a brief note on monochromatic light and continuous spectra must first be considered.
Note: 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.
Note: 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 µ_{0} = 1. Therefore, λ = λ_{0} = wavelength in air.
Definitions | Units |
---|---|
α - (alpha) angle of incidence | degrees |
β - (beta) angle of diffraction | degrees |
k - diffraction order | integer |
n - groove density | grooves/mm |
D_{V} - the included angle or deviation angle | degrees |
µ_{0} - refractive index | dimensionless |
λ - wavelength in vacuum | nanometres |
λ_{0} - wavelength in medium of refractive index, µ_{0}, where λ_{0} = λ/µ_{0} | |
1 nm = 10^{-6} mm; 1 mm = 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 α and β is to be determined for a given wavelength (λ), the grating equation may be expressed as:
(1-3)
Assuming the value of D_{V}, is known, α and β may be determined through Equations 1-2 and 1-3 (see Figures 1 and 2 and Section 2.6).
Figure 1. Monochromator Configuration
Figure 2. Spectrograph Configuration
L_{A} = Entrance arm length
L_{B} = Exit arm length at λ_{n}
β_{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 a and β vary depending on the deviation angle for a 1200 g/mm grating set to diffract 500 nm in a monochromator geometry based on Figure 1.
Table 1: Variation of α (angle of incidence) and β (angle of diffraction), with DV (Deviation Angle), at 500 nm in First Order with 1200 g/mm Grating.
DV | α | β |
---|---|---|
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
rad/nm (1-4)
dβ = angular separation between two wavelengths (radians)
dλ = 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, λ, is given by:
nm/mm (1-5)
where L_{B} is the effective exit focal length in mm and dx is the unit interval in mm. See Figure 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 β, and inversely with the exit path length, L_{B}, diffraction 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 or tilt angle (γ) at wavelength λ_{n}. Figure 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 α and β:
(1-9)
so that if the diffraction order k is doubled, λ 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 (Figure 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 N_{2} 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, dλ 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)
λ = 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 is equal to:
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 bandpass is also determined by the slit width of the spectrometer and residual system aberrations, an achieved bandpass 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 is defined as 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, ω," as shown in Figure 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., α = β). The input and output rays, therefore, propagate along the same axis. In this case at the "blaze" wavelength λ_{B}.
(1-16)
For example, the blaze angle (ω) for a 1200 g/mm grating blazed at 250 nm is 8.63° in first order (k = 1).
Figure 4. Littrow Condition for a Single Groove of a Blazed Grating
1.6.2 Efficiency Profiles
Unless otherwise indicated, the efficiency of a diffraction grating is measured in the Littrow configuration at a given wavelength.
(1-17)
(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 Figures 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.
Figure 5a. Efficiency Curve of a Blazed, Ruled Grating
Figure 5b. Efficiency Curve of a Non-blazed, Holographic Grating
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 conditions in which the grating is used (e.g. α, β).
Holographic gratings may be designed with groove profiles that discriminate against high orders. This may be particularly effective in the UV and VIS using laminar groove profiles created by ionetching.
Note: Just because a grating is "nonblazed" does not necessarily mean that it is less efficient! See Figure 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:
- Randomly scattered light due to surface imperfections on any optical surface.
- 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 2 pi 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
- When grating is concave.
- When laser light is present, e.g., Raman, laser fluorescence, etc.
- Any time groove density should be 1200 g/mm or more (up to 6000 g/mm and 120 mm x 140 mm in size) for use in near UV, VIS, and NIR spectral regions.
- When working in the UV below 200 nm down to 3 nm.
- For high resolution when high groove density will be superior to a low groove density grating used in high order (k > 1).
- Whenever an ionetched holographic grating is available.
1.8.2 When to Choose a Ruled Grating
- When working in IR above 1.2μm, if an ionetched holographic grating is unavailable.
- When working with very low groove density, e.g., less than 600 g/mm.
Remember, ghosts and subsequent stray light intensity are proportional to the square of order and groove density (n^{2} and k^{2} from Equation 1-18). Beware of using ruled gratings in high order or with high groove density.