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** Introduction to Diffraction Grating**:

Diffraction Grating is the optical tool employed to learn the different wavelengths or colors contained in a light beam. The device generally comprises of thousands of narrow, closely spaced parallel slits (or grooves). Due to interference, the intensity of the light getting passed via the slits based on 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 preferred directions depend on the wavelength, and therefore the light beams having different wavelength will propagate in various directions. The condition for the maximum intensity is similar as that for the double slit or multiple slits; however having a huge number of slits the intensity maximum is extremely sharp and narrow, giving the high resolution for the spectroscopic applications. The peak intensities are as well higher and based proportionally to the second power of amount of the slits illuminated.

** Arrangement of diffraction experiment**:

Whenever coherent monochromatic light, like that from a laser, passes via a narrow slits an interference pattern is made. A diffraction grating is comprised of a large number of narrow evenly spaced slits. Whenever laser light passes via the grating, a regular pattern of sharp bright maxima in the intensity of the light can be made on a screen. The position of the m^{th} maximum in the pattern is represented by the relationship:

m λ = d sin (θ_{m}), m = 1, 2, 3,....

Here, 'm' is the order of the diffraction maximum, 'd' is the separation among the slits, 'λ' is the wavelength of the light, and θ_{m} is the angular displacement from the center of the Zeroth order maximum (that is, the center of pattern) to the center of the m^{th} order maximum.

Whenever the linear distance between the center of the Zeroth order maximum and the m^{th} bright maximum is represented by x_{m}, then sin (θ_{m}) is approximately represented by:

Sin (θ_{m}) = x_{m}/L

Here, 'L' is the distance from the grating to the screen where the pattern is represented and L >> x_{m}. By joining the equations, we get:

x_{m} = mλL/d

Or λ = dx_{m}/mL, for m = 1, 2, 3...

If the diffraction angle, θ_{m}, is not small, then the wavelength should be computed from the formula:

λ = r/m sin [tan^{-1} (xm/L)]

** The grating equation**:

The monochromatic light is incident on the grating surface; it is diffracted to a discrete directions. We can picture each and every grating groove as being an extremely small, slit-shaped source of diffracted light. The light diffracted by each and every groove combines to form a diffracted wavefront. The worth of a grating depends on the fact that there exists a unique set of discrete angles all along which, for a specific spacing 'd' between grooves, the diffracted light from each and every facet is in phase having the light diffracted from any other facet, therefore they join beneficially.

*Diffraction Gratings - Ruled and Holographic*

The Diffraction gratings can be categorized into two fundamental categories: holographic and ruled. A ruled grating is generated via physically forming grooves on a reflective surface by employing a diamond tool mounted on the ruling engine. The distance among the adjacent grooves and the angle they form by the substrate influence both the dispersion and effectiveness of the grating.

A holographic grating, by contrary, is produced by employing a photolithographic process where an interference pattern is produced to expose preferentially portions of a photo resist coating. The general grating equation might be represented as:

m λ = d (sin θ + sin θ')

Here, 'm' is the order of diffraction, 'λ' is the diffracted wavelength, 'd' is the grating constant (that is, the distance between the grooves), 'θ' is the angle of incidence measured from the grating normal, and θ' is the angle of diffraction evaluated from the grating normal.

The total efficiency of the gratings based on some application-specific parameters like wavelength, polarization and angle of incidence of the incoming light. The efficiency is as well influenced by the grating design parameters like blaze angle for the ruled gratings and profile depth for the holographic gratings.

The Ruling Process:

Ruling an original or master grating needs a suitable substrate (generally glass or copper), polishing the substrate to a tenth wave (λ/10), and coating it by a thin layer of aluminum through vacuum deposition. Parallel, equally spaced grooves are ruled in a groove profile. The ruling engine should be capable to retrace the accurate path of the diamond forming tool on each and every stroke and to index (advance) the substrate a predetermined amount after each and every cut. Numerous test gratings are made and measured. After testing, a new original grating is ruled on a large substrate. The original grating is extremely expensive, and as an outcome, ruled gratings were rarely employed until after the growth of the replication procedure.

The Holographic Process:

The substrate for a holographic grating is coated by a photosensitive (or photo resist) material instead of the reflective coating employed in ruled gratings. The photo resist is exposed via positioning the coated blank between the intersecting, monochromatic, coherent beams of light from a laser (example: an argon laser at 488 nm). The intersecting laser beams produce a sinusoidal intensity pattern of parallel, uniformly spaced interference fringes in the photo resist material. As the solubility of the resist is based on its exposure to light, the intensity pattern becomes a surface pattern after being immersed in the solvent. The substrate surface is then coated by a reflective material and can be replicated through the similar procedure employed for ruled originals. As holographic gratings are generated optically, groove form and spacing are very uniform that is why holographic gratings do not show the ghosting effects observed in ruled gratings. The outcome is that holographic gratings produce significantly less stray light than ruled gratings.

The Replication Process:

In the year 1940, White and Frazer developed the procedure for precision replication, allowing a huge number of gratings to be generated from a single master, either ruled or holographic. This method outcomes in the transfer of the three dimensional topography of a master grating to the other substrate. Therefore, the master grating is reproduced in full relief to extremely close tolerances. This method led to the commercialization of gratings and has resulted in the current widespread utilization of gratings in spectrometers.

Transmission Grating:

Transmission gratings simplify optical designs and can be valuable in fixed grating applications like spectrographs. Thorlabs offering of blazed transmission gratings is designed for the optimum performance in the visible, UV, or near IR spectrum, having varying dispersiveness. In most of the conditions, the efficiency is comparable to that of reflection gratings in general used in the similar area of the spectrum. By necessity, transmission gratings need relatively coarse groove spacing to maintain the high efficiency. As the diffraction angles rise by the finer spacing, the refractive properties of the materials used limit the transmission at the higher wavelengths and performance drops off. The grating dispersion characteristics, though, lend themselves to compact systems employing small detector arrays.

Moreover, the transmission gratings are relatively insensitive to the polarization of the incident light and are very forgiving of some kinds of grating alignment errors.

Visible Transmission Gratings:

The line of replicated gratings is particularly designed for the transmission use. Transmission gratings let for linear optical designs which can be advantageous in fixed grating applications like spectrographs. The incident light is dispersed on the opposite side of the grating at a fixed angle. Transmission gratings give low alignment sensitivity that minimizes the errors.

These blazed transmission gratings were designed for the optimum performance in the visible spectrum, and in most of the cases, the efficiency is comparable to that of reflection gratings usually employed in the similar area of the spectrum.

By need, transmission gratings need relatively coarse groove spacing to maintain the high efficiency. As the diffraction angles rise by the finer groove spacing, the refractive properties of the substrate materials utilized limit the transmission at the higher wavelengths and performance drops off. The grating dispersion features, though, lend themselves to compact systems employing small detector arrays. The gratings are as well relatively polarization insensitive.

** Parameters of Diffraction Gratings**:

Efficiency:

Grating efficiency can be deduced as either absolute efficiency or relative efficiency. The absolute efficiency of a grating is the percentage of incident monochromatic radiation on a grating that is diffracted to the desired order. This efficiency is found out by both the groove profile (that is, blaze) and the reflectivity of the grating's coating. In contrary, relative (or groove) efficiency compares the energy diffracted into the desired order by the energy reflected through a plane mirror coated by the similar material as the grating. Every efficiency curves in this catalog are deduced as absolute.

Blaze Angle and Wavelength:

The grooves of a ruled grating encompass a saw-tooth profile by one side longer than the other. The angle made via a groove's longer side and the plane of the grating is the blaze angle. Changing the blaze angle focuses the diffracted radiation of a specific area of the spectrum, raising the efficiency of the grating in that spectral area or region. The wavelength at which the maximum efficiency takes place is the blaze wavelength. Holographic gratings are usually less efficient than ruled gratings as they can't be blazed in the classical sense. There are as well special cases (example: if the spacing to wavelength ratio is near one) where a sinusoidal grating has virtually the similar efficiency as a ruled grating. A holographic grating having 1800 lines/mm can encompass the similar efficiency at 500 nm as a blazed, ruled grating.

Resolving Power:

The resolving power of a grating is the product of the diffracted order in which it is employed and the number of grooves illuminated through the incident radiation. It can as well be represented in terms of grating width, groove spacing and the diffracted angles. Resolving power is the property of the grating, and thus, unlike resolution, it is not based on the optical and mechanical features of the system in which it is employed.

System Resolution:

The resolution of an optical system, generally found out by the examination of closely spaced absorption or emission lines for adherence to the Rayleigh criteria (R = λ/?λ), based not only on the grating resolving power however as well on focal length, slit size, 'f' number, the optical quality of all components and system alignment. The resolution of an optical system is generally much less than the resolving power of the grating.

Dispersion:

The angular dispersion of a grating is a function of the angles of incidence and diffraction, the latter of which is based on the groove spacing. Angular dispersion can be raised through increasing the angle of incidence or by reducing the distance between the successive grooves. A grating having a large angular dispersion can generate good resolution in a compact optical system. Angular dispersion is the slope of the curve represented by λ = f (θ). In auto collimation, the equation for dispersion is represented through:

dλ/dθ = λ/2 tan θ

The formula might be employed to find out the angular separation of the two spectral lines or the bandwidth that will be passed through a slit subtending a given angle at the grating.

Diffracted Orders:

In the given set of angles (θ, θ') and groove spacing, the grating equation is valid at more than one wavelength, giving mount to some orders of diffracted radiation. Constructive interference of diffracted radiation from adjacent grooves takes place whenever a ray is in phase however retarded through a whole integer number of wavelengths. The number of orders generated is limited through the groove spacing and the angle of incidence, which naturally can't surpass 90°. At higher orders, efficiency and free spectral range reduces whereas angular dispersion rises. Order overlap can be compensated for by the judicious utilization of sources, detectors and filters and is not a main problem in gratings employed in the low orders.

Free Spectral Range:

The free spectral range is the maximum spectral bandwidth which can be acquired in a specified order devoid of spectral interference (overlap) from the adjacent orders. As grating spacing reduces, the free spectral range rises. It reduces with higher orders. If λ_{1 }and λ_{2} are the lower and upper limits, correspondingly, of the band of interest, then

Free Spectral Range = λ_{2} - λ_{1} = λ_{1}/m.

Ghosts and Stray Light:

Ghosts are stated as the spurious spectral lines occurring from periodic errors in groove spacing. Interferometrically controlled ruling engines minimize ghosts, whereas the holographic method removes them.

On ruled gratings, stray light makes from random errors and irregularities of the reflecting surfaces. Holographic gratings produce less stray light as the optical method that transfers the interference pattern to the photo resist, is not subject to mechanical irregularities or inconsistencies.

Sizes:

Gratings are available in some standard square and rectangular sizes ranging from the 12.5 mm square up to 50 mm square. Nonstandard sizes are available on request. Unless or else specified, rectangular gratings are cut by grooves parallel to the short dimension.

** Experimental Setup & Derivation**:

A diffraction grating that comprises of a very large number of parallel, uniformly spaced slits in an opaque sheet.

A typical grating would encompass 10,000 slits in 1 cm and therefore the slit separation is much smaller than that employed in the double-slit experiment. Whenever a beam of monochromatic light is permitted to pass via the grating positioned in a spectrometer, images of the sources can be seen via the telescope at various angles.

As diffraction grating is a multiple-slit plate, the maxima take place at accurately the similar position as a double slit interference. Though different a double slit, the bright fringes are brighter and sharper.

Assume that monochromatic light is directed at the grating parallel to its axis as represented. Assume that the distance between successive slits be d.

The diffraction pattern on screen is the outcome of the combined effects of diffraction and interference. Each and every slit causes diffraction, and the diffracted beams in turn interfere with one other to generate the pattern. The path difference between waves from any two adjacent slits can be determined by dropping a perpendicular line between the parallel waves. Through geometry, this path difference is d sin θ. When the path difference equals one wavelength or a few integral multiple of a wavelength, waves from all slits will be in phase and a bright line will be noticed at that point. Thus, the condition for maxima in the interference pattern at angle θ is:

d sin θ = m λ

Here m = 0, 1, 2, 3.....

As 'd' is extremely small for diffraction grating, a beam of monochromatic light passing via a diffraction grating is divided into very narrow maxima (that is, bright fringes) at large angle θ.

The maximum number of orders which can be determined by letting maximum θ = 90^{o} and finding 'n' by using equation:

m ≤ (d sin 90^{o})/λ

White Light Spectrum:

Whenever a narrow beam of white light is directed at a diffraction grating all along its axis, rather than a monochromatic bright fringe, a set of colored spectra are observed on both sides of the central white band.

For a diffraction grating, the condition for the n^{th} order maximum is represented by:

d sin θ = mλ

As 'θ' rises with wavelength 'λ', red light that consists of the longest wavelength is diffracted via the biggest angle. Violet light consists of the shortest wavelength and is diffracted the least. Therefore, white light is divided into its component colors from violet to red light. The spectrum is repeated in the different orders of diffraction. Just the Zeroth order spectrum is pure white.

Overlapping of Colors:

The two colors of different orders might overlap if their angles of diffraction θ are equivalent. As d and θ are similar, the condition for overlapping of spectra of two dissimilar colors is

m_{1} λ_{1 }= m_{2} λ_{2}

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