Diffraction Gating and Grating Spectrum and Polarization
Diffraction Gating
Diffraction Grating is optical device used to learn the different wavelengths or colors contained in a beam of light. The device usually consists of thousands of narrow, closely spaced parallel slits (or grooves).
When there is a need to separate light of different wavelengths with high resolution, then a diffraction grating is most often the tool of choice. This "super prism" aspect of the diffraction grating leads to application for measuring atomic spectra in both laboratory instruments and telescopes. A large number of parallel, closely spaced slits constitutes a diffractiongrating. The condition for maximum intensity is the same as that for thedouble slit or multiple slits, but with a large number of slits the intensity maximum is very sharp and narrow, providing the high resolution for spectroscopic applications. The peak intensities are also much higher for the grating than for the double slit.
When light of a single wavelength , like the 632.8nm red light from ahelium-neon laser at left, strikes a diffraction grating it is diffracted to each side in multiple orders. Orders 1 and 2 are shown to each side of the direct beam. Different wavelengths are diffracted at different angles, according to thegrating relationship.
1. Introduction and theory.
Diffraction Grating is optical device used to learn the different wavelengths or colors contained in a beam of light. The device usually consists of thousands of narrow, closely spaced parallel slits (or grooves). Because of interference the intensity of the light getting pass through the slits depends upon the direction of the light propagation. There are selected directions at which the light waves from the different slits interfere in phase and in these directions the maximums of the light intensity are observed. These selected directions depend upon wavelength, and so the light beams with different wavelength will propagate in different directions. The condition for maximum intensity is the same as that for the double slit or multiple slits, but with a large number of slits the intensity maximum is very sharp and narrow, providing the high resolution for spectroscopic applications. The peak intensities are also higher and depend proportionally to the second power of amount of the slits illuminated.
This figure shows the dependence of the light intensity on the angle in the case of diffraction on one slit (red curve) and for two slits diffraction (blue curve). We can see in this figure that the maximal intensities of the interference fringes follow the curve for diffraction on one slit.
Talking about "Fraunhofer" diffraction we mean the far-field diffraction, i.e. when the point of observation is far enough from the screen with the slits. Quantitatively the criteria of the Fraunhofer diffraction is described by the formula:
z >> d2/l
where z is the distance from the screen with the slits to the point of observation. In the close proximity to the screen with the slits the diffraction pattern will be described by the Fresnel's equations
Next, we shall consider the diffraction grating, which consists of N parallel slits. In this case the light waves from every slit will interfere each other producing the interference fringes as shown in figure. Because of diffraction the distribution of the light intensity behind of every slit will not be isotropic (see figure for diffraction at one slit). For diffraction gratings both these effects take place, so the resultant intensity of the light on the screen is described by the equation:
It is seen from the figure that d·sinj is the path length difference D between the rays emitted by the slits. If it is equal to the integer number, then the oscillations will interfere in phase magnifying each other. Therefore, we can write the equation for the main maximums of interference pattern: d·sinj= ml, where m = 0, 1, 2,…
 | This animation shows the experiment when the width b of the silts is varied, while the distance d between them is constant. We can see in the figure that for the narrower slits the diffraction pattern is wider and the visibility is lower. The frequency of the interferometric fringes is the same. |
 | This animation shows the experiment when the width b of the silts is constant (1000 nm) and the distance d between them is varied in the range 1000-10000 nm. Wavelength is 600 nm. The frequency of the interferometric fringes is increasing proportionally to the distance d between the slits, while the width of the diffraction pattern is the same and depends only on b. |
 | This animation shows the Fraunhofer Diffraction on one slit. The width b of the silts is varied in the range 500-1500 nm, wavelength equal to 600 nm. |
2. Set of diffraction gratings DG-10.
We produce the set of diffraction gratings DG-10 and can send it to you by mail. Specifications are given below. You can order diffraction grating DG-10 using KAGI online payment processing system.
The vertical axis is normalized to the intensity of the light at the center of the screen. Actually, for the single and double slit experiments the intensity of the light on the screen depends proportionally to the second power of the width of the slit. So, particularly, for the slits of 5 pixels wide the intensity on the screen will be 16 times less than for the slits of 20 pixels wide. For multi-slit gratings the intensity of the light on the screen will be N2 times bigger as compared to one slit (where N is number of illuminated slits) and it also depends proportionally to second power of the width of every slit.
Technical specifications:
Dimensions: 148 x 95 mm (every grating 10 x 10 mm)
Material: transparent film of 110 microns thick Resolution: 4000 dpi (1 pixel = 1/4000 inch = 6,35 microns) |
Click on the appropriate diffraction grating to see the graph for the intensity of the light on the screen.
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The gratings in rows 1, 3, 4 are vertical strips marked as d/b, where d is period of the grating in pixels,b is the width of the transparent strips. The gratings in row 2 are superposition of vertical and horizontal strips. Row 5 consists the single slits and rows 6 and 7 consist of double slits.
Row 1: the width of the white (transparent) strips is equal to the width of the black strips (d = 2b). Row 2: the width of the white squares is equal to the width of the black one (d = 2b). Row 3: the width of the white strips b is constant and equal to 1 pixel, while the period of grating d is varying from 2 to 10 pixels. Row 4: the period of the grating d is constant and equal to 10 pixel, while the width of the white strips bis varying from 1 to 5 pixels. Row 5: one vertical slit of width varying from 80 to 5 pixels. Row 6: the couples of vertical slits marked as 2 x b, where b is the width of every slit in pixels. The separation between the slits is constant and equal 50 pixels. Row 7: the couples of vertical slits marked as b, where b is the separation between the couples of slits in pixels. For this row the width of every slit is constant and equal 10 pixels. |
The diffraction can be observed more easily if you:
- Observe the diffraction pattern in the dark room.
- Use the fresh batteries for the laser pointer.
- Place the grating not closer than for 1 meter from the screen.
3. Spectrum analysis using diffraction grating.
In the next experiment we shall explore the light which consists of the waves of different frequencies. In this case the angle of diffraction depends on wavelength of the light and hence instead of single interferometric lines the spectrums will appear in different orders. This property of diffraction grating can be used for investigation of the spectrums of the light of different optical sources.
In the figure above we can see the classical experimental setup for observation of Frounghofer diffraction. We used luminescent lamp B and diffraction grating GD. Light of lamp B passes through a narrow slotsituated in the focus of the lens L1. As a result the parallel beam of the light is formed behind of the lens. Then this light is incident on a transmitting diffraction grating DG. Because of interference the flat waves with different wavelengths appears at the output of diffraction grating. Light with the wavelength λ will propagate in the direction φ in accordance with the equation dsinφ = mλ, where m is the positive integer, which has a sense of the spectrum order. m is equal to difference of the optical paths of the light from two adjacent slits related to the wavelength. So, in the first order the optical path difference equals λ. The light expanded in a spectrum is incident then to lens L2, which focus it to screen S. In the centre of the screen we can see the white line, which corresponds to image of the slot in zero order of spectrum. Then, up and down of the screen there are colored strips, which correspond a spectral composition of the light. The repeated groups of lines is interference in the first, second, etc. orders. Quality of the grating is defined by the resolution: R=λ/δλ=mN, where λ is the wavelength, δλ is the minimal difference in wavelengths of the lines, which can be resolved, m is the order of spectrum, N is the number of the slots used for interference (which are inside the light spot).
We accepts the orders and can send to you by mail the diffraction gratings DG-2/1-4000 or DG-4/2-4000 of any size. Digit 4000 in the labeling of the grating means that the resolution of the grating is 4000 dpi. 1 pixel = 6.35 microns. Grating consists of alternating black and white (transparent) lines on a thin transparent film. Period of the grating is 2 or 4 pixels, the width of the line is 1 or 2 pixels. This corresponds to 2/1 and 4/2 in the labeling of the grating. The price for 1 sq.inch of grating is $5. Minimal order is $20. Contact us to order this diffraction grating.
A diffraction grating is made by making many parallel scratches on the surface of a flat piece of transparent material. It is possible to put a large number of scratches per centimeter on the material, e.g., the grating to be used has 6,000 lines/cm on it. The scratches are opaque but the areas between the scratches can transmit light. Thus, a diffraction grating becomes a multitude of parallel slit sources when light falls upon it.
A parallel bundle of rays falls on the grating. Rays and wavefronts form an orthogonal set so the wavefronts are perpendicular to the rays and parallel to the grating as shown. Utilizing Huygens' Principle, which is that every point on a wavefront acts like a new source, each transparent slit becomes a new source so cylindrical wavefronts spread out from each. These wavefronts interfere either constructively or destructively depending on how the peaks and valleys of the waves are related. If a peak falls on a valley consistently (called destructive interference), then the waves cancel and no light exists at that point. On the other hand, if peaks fall on peaks and valleys fall on valleys consistently (called constructive interference), then the light is made brighter at that point.
Consider two rays which emerge making an angle
with the straight through line. Constructive interference (brightness) will occur if the difference in their two path lengths is an integral multiple of their wavelength (
) i.e., difference = n where n = 1, 2, 3, ... Now, a triangle is formed, as indicated in the diagram, for which
n = d sin( )
and this is known as the DIFFRACTION GRATING EQUATION. In this formula is the angle of emergence (called deviation, D, for the prism) at which a wavelength will be bright, d is the distance between slits (note that d = 1 / N if N, called the grating constant, is the number of lines per unit length) and n is the "order number", a positive integer (n = 1, 2, 3, ...) representing the repetition of the spectrum.
Thus, the colors present in the light from the source incident on the grating would emerge each at a different angle since each has a different wavelength . Furthermore, a complete spectrum would be observed for n = 1 and another complete spectrum for n = 2, etc., but at larger angles.
Also, the triangle formed by rays to the left of 0o is identical to the triangle formed by rays to the right of 0o but the anglesR and L (Right and Left) would be the same only if the grating is perpendicular to the incident beam. This perpendicularity is inconvenient to achieve so, in practice, R and L are both measured and their average is used as in the grating equation.
A parallel bundle of rays falls on the grating. Rays and wavefronts form an orthogonal set so the wavefronts are perpendicular to the rays and parallel to the grating as shown. Utilizing Huygens' Principle, which is that every point on a wavefront acts like a new source, each transparent slit becomes a new source so cylindrical wavefronts spread out from each. These wavefronts interfere either constructively or destructively depending on how the peaks and valleys of the waves are related. If a peak falls on a valley consistently (called destructive interference), then the waves cancel and no light exists at that point. On the other hand, if peaks fall on peaks and valleys fall on valleys consistently (called constructive interference), then the light is made brighter at that point.
Consider two rays which emerge making an angle
with the straight through line. Constructive interference (brightness) will occur if the difference in their two path lengths is an integral multiple of their wavelength () i.e., difference = n where n = 1, 2, 3, ... Now, a triangle is formed, as indicated in the diagram, for which = d sin( )and this is known as the DIFFRACTION GRATING EQUATION. In this formula
is the angle of emergence (called deviation, D, for the prism) at which a wavelength will be bright, d is the distance between slits (note that d = 1 / N if N, called the grating constant, is the number of lines per unit length) and n is the "order number", a positive integer (n = 1, 2, 3, ...) representing the repetition of the spectrum.Thus, the colors present in the light from the source incident on the grating would emerge each at a different angle
since each has a different wavelength . Furthermore, a complete spectrum would be observed for n = 1 and another complete spectrum for n = 2, etc., but at larger angles.Also, the triangle formed by rays to the left of 0o is identical to the triangle formed by rays to the right of 0o but the angles
R and L (Right and Left) would be the same only if the grating is perpendicular to the incident beam. This perpendicularity is inconvenient to achieve so, in practice, R and L are both measured and their average is used as in the grating equation.PROCEDURE
Calibrating the Spectrometer
Read and follow the procedures for calibrating the spectroscope found in the previous experiment. The calibration can be performed with the grating in place on the table.Measuring
- CAUTION: The diffraction grating is a photographic reproduction and should NOT be touched. The deeper recess in the holder is intended to protect it from damage. Therefore, the glass is on the shallow side of the holder and the grating is on the deep side.
- Place the grating on the center of the table with its scratches running vertically, and with the base material (glass) facing the light source. In this way, one can study diffraction without the complication of refraction (recall from the previous lab how light behaves when traveling through glass at other than normal incidence). Fix the grating in place using masking tape.
- Rotate the table to make the grating perpendicular to the incident beam by eye. This is not critical since the average of
- Affirm maximum brightness for the straight through beam by adjusting the source-slit alignment. At this step, the slit should be narrow, perhaps a few times wider than the hairline. Search for the spectrum by moving the telescope to one side or the other. This spectrum should look much like that observed with the prism except that the order of the colors as you move away from zero degrees is reversed.
- Search for the second- and third-order spectra. Do not measure the higher-order angles, but record the order of colors away from zero degrees.
- For each of the seven colors in the mercury spectrum, measure the angles
A grating disperses light incident on it. Dispersion is the phenomenon by which a spectrum of light is separated in space by wavelength (Figure 1). Prisms also disperse light by wavelength and can be used in spectrometers, though grating-based dispersing instruments offer a number of advantages so few commercially available instruments use prisms.
A diffraction grating is an optical device exploiting the phenomenon of diffraction. It contains a periodic structure, which causes spatially varying optical amplitude and/or phase changes. Most common are reflection gratings, where a reflecting surface has a periodic surface relief leading to position-dependent phase changes. However, there are also transmission gratings, where transmitted light obtains position-dependent phase changes, which may also result from a surface relief.
It is instructive to consider the spatial frequencies of the position-dependent phase changes caused by a grating. In the simplest case of a sinusoidal phase variation, there are only two non-vanishing spatial frequency components with ±2π / d, where d is the period of the grating structure.
An incident beam with an angle θ against the normal direction has a wave vector component k · sin θ along the plane of the grating, where k = 2π / λ and λ is the wavelength. Ordinary reflection (as would occur at a mirror) would lead to a reflected beam having the in-plane wave vector component −k · sin θ. Due to the grating's phase modulation, one can have additional reflected components with in-plane wave vector components −k · sin θ ± 2π / d. These correspond to the diffraction orders ±1. From this, one can derive the corresponding output beam angles against the normal direction:
If the grating's phase effect does not have a sinusoidal shape, one may have multiple diffraction orders m, and the output angles can be calculated from the following more general equation:
Note that different sign conventions may be used for the diffraction order, so that there may be a minus sign in front of that term.
The equations above may lead to values of sin θout with a modulus larger than 1; in that case, the corresponding diffraction order is not possible. Figure 2 shows an example, where the diffraction orders −1 to +3 are possible.
Figure 3 shows in an example case of a grating with 800 lines per millimeter, how the output angles vary with wavelength. For the zero-order output (pure reflection, m = 0), the angle is constant, whereas for the other orders it varies. The order m = 2, for example, is possible only for wavelengths below 560 nm.
− Applications of Diffraction Gratings
Diffraction gratings have many applications. In the following, some prominent examples are given:
- They are used in grating spectrometers, where the wavelength-dependent diffraction angles are exploited. Figure 5 shows a typical setup. Artifacts in the obtained spectra can arise from confusion of multiple diffraction orders, particularly if wide wavelength ranges are recorded.
Figure 5: Design of a Czerny–Turner monochromator. More details are given in the article onspectrometers.
- Pairs of diffraction gratings can be used as dispersive elements without wavelength-dependent angular changes of the output. Figure 6 shows a Treacy compressor setup with four gratings, where all wavelength components are finally recombined [1]. The same is achieved with a grating pair when the light is reflected back with a flat mirror. (Note that such a mirror may be slightly tilted such that the reflected light is slightly offset in the vertical direction and can be easily separated from the incident light.) Such grating setups are used as dispersive pulse stretchers and compressors, e.g. in the context of chirped pulse amplification. They can produce much larger amounts of chromatic dispersion than prims pairs, for example.
Figure 6: A four-grating setup, consisting of two grating pairs. Grating 1 separates the input according to wavelengths (with passes for two different wavelengths shown in the figure), and after grating 2 these components are parallel. Gratings 3 and 4 recombine the different components. The overall path length is wavelength-dependent, and therefore the grating setup creates a substantial amount of chromatic dispersion.
- As described above, diffraction gratings (often in Littrow configuration) are often used for wavelength tuning of lasers.
- In spectral beam combining, one often uses a diffraction grating to combine radiation from various emitters at slightly different wavelengths into a single beam.
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