Planar scribing grating

Diffraction grating refers to an optical element composed of a large number of parallel, equal width and equal spacing parallel slits, which can be carved on the surface of glass with a diamond knife, or can be obtained by holographic photography. Compared with prisms, grating spectroscopy has the characteristics of wide spectral range, large role divergence, dispersion linearity, and high spectral resolution.

Fig.1 Prism separates wavelengths by refraction Fig.2 Diffraction gratings separate wavelengths by diffraction due to their surface structure

According to the shape of the grating surface, gratings are divided into planar gratings and spherical gratings. According to the working mode, gratings are divided into transmission gratings that use transmitted light diffraction and reflection gratings that use reflected light diffraction, as shown in Figure 3.

(a) Transmission grating (b) Reflection grating

Fig.3. Diffraction grating

As shown in Figure 4, there is a parallel beam of coherent light incident at the angle θ₀ to the grating slit, the diffraction angle is θ, and the slit spacing (grating constant) is d, according to the grating equation, the optical path difference between the two parallel rays emitted by the adjacent slit when they reach the focal plane of the converging lens is ∆

$∆ = (a + b) (\sin θ_{0} \pm \sin θ) = d (\sin θ_{0} \pm \sin θ) = k λ$

where a is the width of the light-transmitting part of the grating, b is the width of the light-opaque part of the grating, and k is the interference order of the grating，k=0，±1，±2，…

Fig.4 Schematic diagram of transmission grating

1. Transmission grating

The transmission grating is to carve a series of equal-width and equal-spaced nicks on the optical flat glass, the notch is not transparent, and the unengraved part is a light transmission slit, and the slit produces transmission grating stripes through the diffraction effect of light. For transmissive diffraction gratings, the optical path difference is ∆

$d (\sin θ_{0} - \sin θ) = k λ$

When the incident light and the diffracted light are on either side of the normal of the grating surface, both θ₀ and θ are positive values, as shown in Figure 5(a); When the incident light and the diffracted light are on the same side of the normal of the grating surface, θ₀ is positive and θ is negative, as shown in Figure 5(b).

2. Reflection grating

The reflection grating is to carve a series of equal-width and spaced marks on the metal mirror, and diffuse reflection occurs on the notch, and diffraction occurs in the direction of the reflected light when it is not engraved. For reflective diffraction gratings, the optical path difference is ∆

$d (\sin θ_{0} + \sin θ) = k λ$

When the incident light and the diffracted light are on either side of the normal of the grating surface, θ₀ is a positive value and θ is a negative value, as shown in Figure 5(c); When the incident and diffracted light are on the same side of the normal of the grating surface, both θ₀ and θ are positive, as shown in Figure 5(d).

Fig.5. Symbolic illustration in the grating equation

3. Shining grating

The biggest disadvantage of diffraction gratings is that most of the energy passing through the grating is concentrated in the zero-order spectrum without dispersion, and the energy of other levels that can divide light is very weak. The reason for this phenomenon is that the zero-order principal maxima of single-slit diffraction and the zero-order principal maxima of inter-slit interference of ordinary diffraction gratings coincide completely, and the grating of sparkling gratings comes into being. The grating is generally a reflection grating, and the proper selection of the groove shape can concentrate the energy to a certain desired spectral level while attenuating the rest of the spectrum. Fig. 6 (a) is a transmissive grating with saw-shaped grooves carved into a flat glass, and Fig. 6 (b) is a reflective grating with toothed grooves carved into the surface of a metal plate.

Fig.6. Flash grating

The notched surface of the grating is not parallel to the grating surface, and there is an angle between the two. Such a structure can separate the central maximum of diffraction of a single groove surface (equivalent to a single slit) from the interference zero-order principal maximum between each groove surface (interslit), and transfer and concentrate the light energy from the interference zero-order principal maximum (zero-order spectrum) to a certain order spectrum, so as to realize the sparkle of the spectrum of this stage, which is the basic working principle of the grating.

As shown in Fig. 7, taking the reflective grating as an example, the incident light is set to be incident at the angle of incidence α relative to the trough surface, diffracted at an angle of β, incident at the angle i relative to the grating surface, diffracted at the angle of θ, and the angle between the trough surface and the grating plane is the flare angle γ.

Fig.7. Principle of glitter grating

For single-groove diffraction, the intensity of the diffracted light from the single-groove surface reaches the principal maximum when α=β. According to the grating equation (incident and diffracted light on the same side of the grating surface normal)，

$d (\sin i + \sin θ) = k λ$

$2 d \sin \frac{i + θ}{2} \cos \frac{i - θ}{2} = k λ$

According to the geometric relationship,

i=α+γ

θ=γ-β

From this, the relationship between the angle of incidence, the angle of diffraction and the angle of flare can be obtained.

i+θ=2γ

i-θ=2α

According to the above formulas,

$2 d \sin γ \cos α = k λ$

The above equation is the relationship between the central principal maxima direction of single-groove diffraction and the k-order interferometric principal maximum. The diffraction fringes of the grating are the result of a combination of diffraction from a single groove and interference between the reflected beams from multiple grooves.

When a plane light wave is emitted along the normal of the groove surface, α=β=0, i.e

$2 d \sin γ = k λ$

The above equation is the flare condition, where λ is the flare wavelength of the grating, k is the corresponding flare order, and γ is the flare direction (the spectral line with a wavelength λ in this direction has the maximum light intensity). Taking k=1 as an example, the wavelength λ₁ satisfying 2dsinγ=λ₁ is the first-order shining wavelength, and the formation principle of the shining stage is shown in Figure 8.

Fig. 8(a) Single-slot diffraction (b) Slot-to-multi-beam interference (c) The central principal maxima of single-slot diffraction coincides with the first-order principal maxima of multi-beam interference

The flare grating produces the maximum light intensity for the flare wavelength in the same spectrum, but because the center of the groove diffraction is very large to very small and has a certain width, the spectral lines in a certain wavelength range near the flare wavelength also have a considerable light intensity, so the flare grating can be used in a certain wavelength range.

4.Holographic grating

In the past, the grating was made by using a scoring machine to scribble a parent grating and then reproduce it. With the development of holographic technology, the production and application of holographic gratings have developed. According to the shape of the base billet and the interference wave surface, the holographic grating is divided into planar grating, concave grating, etc.

The production process of holographic grating is as follows:

Blank processing and pretreatment - coating of photosensitive layer - exposure recording interference fringes - development - (vacuum aluminization)

The ground grating blank is evenly coated with a layer of light-sensitive material, which is then placed in two laser interference fields of the same monochromatic light source and exposed, and then the interference fringes between light and dark are recorded on the light-sensitive layer. The exposed billet base is immersed in a special solvent, and each part of the coating is subjected to different degrees of dissolution (equivalent to "development") due to the different exposure received, so that a groove line equivalent to the interference fringe appears on the billet base, and a transmissive holographic grating can be obtained; When the surface is vacuum-coated with a protective aluminum film, a reflective holographic grating can be obtained.

Compared with the scoring grating, the holographic grating has the following characteristics,

1. There is no ghost line. Holographic gratings are made using photochemical methods, so ghost lines are not actually visible (mechanical errors cause some untrue spectral lines to appear in the spectrum of the scribed grating).

2. The diffraction efficiency is low. The groove shape of the scoring grating is usually zigzag, and the diffraction efficiency is highest when the flare condition is met. However, the groove of the holographic grating is usually an approximate sinusoidal waveform, which does not have the flare conditions and has no obvious flare characteristics, so the diffraction efficiency is low.

3. High resolution. According to the formula, $\frac{λ}{∆ λ} = k N$ (where the fraction increment $\frac{λ}{∆ λ}$ is the resolution of the grating,k is the order of the spectrum, and N is the total number of graduations of the grating), and the resolution will increase due to the increase of the total number of graduations N.

4. Short production cycle.

5. The applicable range of the spectrum is wider than that of the scoring grating.

According to the principle of light interference, two coherent monochromatic lights (with the same amplitude, frequency, and vibration direction, and a constant phase difference) are projected onto the screen at a certain angle at the same time, and there will be straight interference fringes on the screen with alternating light and dark, parallel and equidistant from each other. The spacing of the interference fringes (i.e., the grating constant d of the holographic grating) can be obtained by the following formula,

$d = \frac{λ}{2 \sin θ}$

where λ is the wavelength of the laser light source used, and θ is half of the angle between the two monochromatic lights. When the magnitude of the angle 2θ between two coherent monochromatic lights is changed, diffraction gratings with different d values can be obtained, as shown in Figure 9.

Fig.9. Principle of light interference