Prism Spectrometer and Diffraction Grating: 787550

Questions:

Experiment 2

PRISM SPECTROMETER and DIFFRACTION GRATINGS

Theory

Dispersion of a beam of white light into its component colours by a glass prism is due to the variation in refractive index of the glass with the frequency (colour) of the light. This is a result of the variation in the speed of the light in the medium. Thus blue light (higher frequency) will be refracted more than red light (lower frequency) as it passes through the prism. The angle between the undeviated path of an incident ray and the final path of the ray as it exits the prism is called the deviation angle. When the ray passes symmetrically through the prism, (so that its path in the prism is parallel to the base), minimum deviation occurs.

Referring to the diagram, either increasing or decreasing θ will result in an increase in the deviation angle.

The refractive index of the prism is given by

(1)

Where A = prism angle; D = angle of minimum deviation

The value of n is different for different wavelengths, and a relationship between n and l was given by Cauchy:

(2)where a and b are constants and λ is the free space wavelength.

– 2 –

Procedure

Dispersion

Figure 2 Determination of Minimum Deviation Angle

1. Turn on the sodium lamp and allow to warm up for at least 10 minutes.
1. Position the prism on the prism table with the unpolished side flush with the prism holder and lock into place. (Note: Do not over tighten).
1. Rotate the telescope to the straight through position.
1. Open the slit adjust to give a wide yellow line through the telescope. Note: You will have to physically move the spectrophotometer and the telescope to be able to see the line.
1. Close the slit adjust to give a sharp narrow yellow line. Note: You will probably have to adjust the focus on (a) the collimator and (b) the telescope. While you are at it adjust the focus for the cross hairs, which is at the eyepiece end of the telescope.

Note: All focus controls pull in and out.

1. Ensure that the prism table lock is released (anticlockwise) and rotate the prism table until it is in the position in the diagram. Lock the table in this position.
1. Ensure the telescope lock is released (anticlockwise) and rotate the telescope until spectral lines are observed. Note: You may have to adjust both the prism table and the telescope to achieve this.

– 8 -The position of minimum deviation may be obtained by rotating the prism table in one direction – the spectral lines will appear to move across the field of view, stop and reverse their motion. Note: You will need to adjust both the prism table and the telescope to achieve this.

The point where a particular spectral line changes direction corresponds to the minimum deviation for that particular wavelength in the prism.

1. Line the cross hairs of the telescope up on the red spectral line, release the prism table lock and rotate the prism table until that spectral line changes direction. Lock the prism table where this occurs. Note: Do not adjust the prism table until all the measurements have been obtained. (This is not technically correct but for this instrument it is far more accurate than attempting to find the minimum deviation angle for each spectral line).
1. Position the telescope near the spectral line, lock it, and use the telescope fine adjust to line up the cross hairs on that line.
2. When this is done, note the reading on the scale. This is angle A for that spectral line. (Note: It is a vernier scale and should be read to at least 0.1o)
3. Repeat parts 10-11 of the procedure for the other five (5) strong lines.
1. Note the colour, and the wavelength of each of the lines you have measured.
1. Repeat steps 9-13 and find the average value.

NOTE: To do the next section you will have to move the prism to Position B and the telescope to the ‘B’ angle position as per Figure 2

1. Repeat Parts 3-14 of the procedure.
1. Tabulate all results.
1. Determine the angle of minimum deviation, D, for each line by subtracting the mean of the smaller angle from the mean of the larger angle and halving the result.
1. Construct a calibration curve of minimum deviation angle versus wavelength.

Spectra

1. Use the prism spectrometer to examine the spectra of the other light sources available in the laboratory.

For one other light source, determine the angular positions of at least three spectral lines at minimum deviation and using your graph of D vs λ obtained for the sodium lamp, calculate the wavelengths of these lines.

Compare the results with tabulated values.

Diffraction Grating

Diffraction of monochromatic light in a Young’s double slit experiment produces an intensity pattern which consists of a double slit “interference” pattern which is modulated by a single slit diffraction pattern.

– 4 –

The position of the maxima in the double slit interference pattern is given by

 dsin θ = nλ (3)

where d is the separation of the slits, θ is the angular position of a given maximum with respect to the x-axis, n is the order of a given maximum and λ is the wavelength of the light.

A diffraction grating consists of a system of many slits (in fact, grooves ruled on a transparent material) and equation (3) may be used to describe the position of the principal maxima produced by a diffraction grating. If non-monochromatic light is incident on the grating, the different wavelengths of light will be dispersed (ie occur at different values of θ) within each order of diffraction.

Procedure

1. Replace the prism with the transmission diffraction grating supplied. Fix the grating to the surface of the prism table using the clamp provided. You will have to remove the prism clamp and replace it with the diffraction grating clamp.
1. With the grating aligned approximately perpendicular to the collimator, find (by eye) the sodium spectra produced on either side of the straight-through position.
1. Rotate the telescope to the position of the first-order spectrum on one side and align the cross hairs with a spectral line. Note the angular setting of the telescope.
• Comment on any differences between the sodium spectrum you observe with the diffraction grating and that observed with the prism.
1. Repeat the procedure for as many lines as possible, for the first-order spectra.
1. If the angle is increased past the first order spectral lines another set of spectra lines, known as the second order, will be observed. Note: You may need to increase the intensity of the light by opening the slit adjust. Note the angular setting of the telescope for as many lines as possible.

NOTE: To do the next section you will have to position the telescope in the ’B’ angle position.

1. Repeat Parts 3-5 of the procedure.
1. Draw up a table showing the angular settings for each line (wavelength) on the right and left of the straight through position. Determine θ for both first-order and second-order lines by subtracting the smaller angle from the larger angle and dividing by two.
• Explain how, and why, the angular separation between the spectral lines varies between the first-order and second order spectra.
1. Plot λ vs sin θ for both first-order and second-order spectra, and determine the slope of each line. Hence obtain an average value for the spacing, d, of the lines on the diffraction grating. Give an estimate of the error involved in the value you obtain.
• Compare your values of d with the value printed on the diffraction grating.
• 5 –
• Simulation software

If time permits, use the simulation software provided to compare your value of the slit separation with that predicted by the program.

1. Choose Applications of Interference and Diffraction from the menu.
1. Choose Gratings from the menu at the top of the screen.
1. Choose the Transmission Grating – Spectrum option from the pull-down menu.
1. Enter the wavelengths of two of the lines you have measured, a slit width of 0.0001 mm and the slit separation you determined.
1. Compare the predicted angular positions of the lines with the values you measured. Comment on the result.
1. Investigate the effect of changing the slit width, slit separation or wavelength of the radiation.

– 6 –

 Sodium Line Spectra Neon Line Spectra λ (nm) Colour λ (nm) Colour 614.8 red 701 red 589.5 yellow 1 691 red 588.9 yellow 2 671 red 568.6 green 666 red 515.0 light blue green 658 red 498.0 blue green 652 red 466.6 light indigo 649 red 454.3 indigo 639 red 442.1 light purple 637 red Helium Line Spectra 632 red 629 red λ (nm) Colour 626 red 667.8 red 621 red 587.5 yellow 615 red 501.5 blue green 614 orange 492.2 blue green 609 orange 471.3 blue 607 orange 447.3 purple 602 orange 438.9 purple (faint) 597 yellow 594 yellow Cadmium Line Spectra 588 yellow λ (nm) Colour 585 yellow 644 red 576 green (faint) 510 blue green 540 green 481 blue 534 blue green 468 blue 442 purple (faint) Mercury Line Spectra λ (nm) Colour 579.1 yellow 577.0 yellow 546.2 green 491.7 blue 436.0 purple 407.9 purple 404.8 purple

Experimental Aim

This experiment aims at estimating the refractive index of a prism for different wavelengths of the Sodium Spectrum and then plotting calibration and dispersion curves through the use of Prism Spectrometer.

Introduction

A spectrometer is an instrument used in the analyses of the spectra of radiations. The glass-prism spectrometer is ideal in taking measurements of ray deviations as well as refractive indices. At times, diffraction grating may be used instead of the prims in the study of optical spectra. A prism is capable of refracting light into one spectrum while diffraction grating spreads the available light in numerous spectra (Duarte 2015). Due to this, slit images that are formed using a prism are mostly brighter as compared to the ones formed through grating. The only challenge in this is that the enhanced brightness of the spectral lines is often offset through a decreased resolution as the prism cannot effectively separate the various lines as the case of grating. However, these brighter lines permit a slit width that is narrow in shape to be used which is partially able to compensate for the lowered resolution (Guanter et al., 2015).

Theory

There is no direct proportionality between the angle of refraction and the wavelength of light in a prism. For this reason, the measurement of the wavelengths using a prism is achieved through the construction a calibration graph of the deviation angle against the wavelength and using the source of light with a known spectrum. The wavelength of the unknown spectral lines can thus be interpolated from the obtained graph (Hadni 2016). Future determinations of wavelengths is validated upon the creation of  a calibration graph for th prism and this is only possible if they are made from prism that is aligned precisely just the same it was at the time of production of the graph. To achieve the reproduction of such an alignment, all the measurements must be made when the prism is aligned to enable refracting the light at the angle of the lowest possible deviation.

The light that is studied is rendered parallel using a collimator that is composed of a tube that has a slit of adjustable width at an end and a convex lens at the other end. The collimator must be maintained in a highly focused through the adjustment of the position of the slit to the point at which it is at the focal point of the lens (Hartmann et al., 2014). The parallel beams that originate from the collimator are made to pass through a glass prism that is on a prism table which is rotatable, levelized, lowered or even raised. The prism then deviates the components colours of the released light through various amounts and spectrum so generated is examined through the use of a telescope that is mounted on a rotating arm and oscillates over the divided angular scale.

The theory of the prism spectrometer illustrates that a spectrum that has maximum definition is achieved when the light ray angular deviation of the light ray that goes through the prism is least. Under such conditions it can be demonstrated that they ray goes through the prism is a symmetrical manner. For a specific wavelength of light that is traversing a certain prism, that exists a characteristic incidence angle for which the deviation angle is least. This angle is dependent in the refractive index of the prism and the angle that is formed between the two sides of the prism that have been traversed by light (Hossain et al., 2015). The equation below is used in illustrating the relationship between the two variables

in which n is the refractive index of the prism,  the angle formed between the two sides of the prism that has been traversed by light and A the angle of minimum deviation.

Methods

Dispersion

Figure 2: Determination of Minimum Deviation Angle

1. The sodium lamp was turned on and allowed to warm up for more than 10 minutes
2. The prism was positioned on the prism table having the unpolished side flush with the holder of the prism and then locked into place (Leedle et al., 2015)
3. The telescopes was rotated to the straight through position
4. The slit adjust was open to provide a wide yellow line through the telescope
5. The slit adjust was then closed to provide a sharp narrow yellow line
6. The prism table lock was ascertained to be released in an anticlockwise manner and then the prism table rotated until it was in the position as illustrated in the diagram.
7. The telescope lock was ascertained to be released in an anticlockwise directed and then the telescope rotated until the spectral lines were noticed.
8. The position of the minimum deviation was obtainable through the rotation of the prism table in one direction only where the spectral lines would seem to move across the field of view, stop and the move in a reverse direction (Mouroulis et al., 2014)
9. The cross hairs of the telescope were lined up on the red spectral line and the prism table lock released and the prism table rotated until there was a change in position of the spectral line. The prism table was then locked when it occurred
10. The telescope was position close to the spectral line and the telescope fine adjust was then used in lining up the cross hairs on the line
11. The reading on the scale was noted which was the angle A of the spectral line
12. The parts 10-11 of the procedure were repeated for the other five string lines
13. The wavelength and the colour were noted for each of the lines measured
14. Steps 9-13 were repeated to estimate the average value

Note: The prism has to be moved to position B and the telescope moved to the B angle position as illustrated in figure 2 in order to perform the next section

1. Parts 3-14 of the method were repeated
2. The results were tabulated
3. The minimum angle of deviation, D, was then determined for every line through subtracting the mean of the smaller angle from the mean of the greater angle and then halving the result
4. A calibration curve was constructed of the minimum deviation angle against the wavelength (Piascik et al., 2014)
5. The prism spectrometer was used in the examination of the spectra of the other sources of light that were available in the laboratory. Comparison was made with the tabulated values

Diffraction Grating Procedure

1. The prims were substituted with the transmission diffraction grating that was supplied in which the grating was fixed to the prism table surface with the clamp given.
2. The sodium spectra generated on either side of the straight-through position was determined using the eye while the grating was aligned about perpendicular to the collimator
3. The telescope was rotated to the position of the first order spectrum on one of the side and then the cross hairs aligned with the spectral line. The angular setting of the telescope was taken care of.
4. The procedure was repeated for as numerous lines as possible for the first order spectra
5. Another set of spectral lines known as second order would be observed upon an increase in the angle beyond the first order spectra lines

Note: Performing the next section of this experiment required moving the telescope to B angle position

1. The Parts 3-5 of the method were repeated
2. A table illustrating the angular setting for every line on the right as well as left of the straight line through position was then drawn. The Ɵ was determined for both the first order and second order lines through finding the difference between the smaller angle and the larger angle and the final answer divided by two (Squires, Constable & Lewis (2015)
3. Graphs of λ versus sin Ɵ were plotted for both the first order and second order spectra and then the slope of each of the lines determined. The averaged value of the spacing, d, of each of the lines on the diffraction grating was then determined and an estimate of the error incurred determined.

Results

Prism Spectrometer Experiment

 colour Deviation angle (degree) Lemda (nm) Red 133.9 614.8 orange 133.5 589.5 green 133 568.6 Dark green 132 498 light blue 131.5 466.6 violet 130.4 442.1

Table 1: Sodium calibration results

Figure 3: Sodium calibration plot

Diffraction Grating

 Diffraction grating colour deviation angle (sintheta) lemda (nm) violet 0.282 442.1 light blue 0.3 466.6 Dark green 0.312 498 lime green 0.344 515 orange 0.357 588.9 red 0.371 614.8

Table 2: Diffraction Grating results

Figure 3: Diffraction Grating plot

Discussion and Conclusion

The prism spectrum that was obtained for the sodium lamp that could be seen with the resolution of the prism was provided as shown in the table from top to bottom. The measured angles i.e. 2A= and thus the angle of the prismA= (Vaughan 2017). The behavior of the dispersion curve was observed that there is no rapid fall over the range of the wavelengths thus it can be concluded that there is no heavy sloping line meaning that the dispersion of the different spectral lines do not vary so much from each other which is illustrated by the closeness of the refractive index of the provided wavelength range.

For the calibration curve, it is almost a straight line illustrating that the impact of the wavelength of the Angle of Minimum Deviation tends to being linear (Vaughan 2017). This curve can be used in establishing the wavelength of the spectral line that has an unknown wavelength but the Angle of Minimum Deviation is determined using the very apparatus. The aims and objectives of this experiment were thus met with the results illustrating high correlation with the theoretical values as recorded in literature.

References

Duarte, F. J. (2015). Tunable laser optics. CRC Press

Guanter, L., Kaufmann, H., Segl, K., Foerster, S., Rogass, C., Chabrillat, S., … & Straif, C. (2015). The EnMAP spaceborne imaging spectroscopy mission for earth observation. Remote Sensing7(7), 8830-8857

Hadni, A. (2016). Essentials of Modern Physics Applied to the Study of the Infrared: International Series of Monographs in Infrared Science and Technology (Vol. 2). Elsevier

Hartmann, N., Helml, W., Galler, A., Bionta, M. R., Grünert, J., Molodtsov, S. L., … & Bostedt, C. (2014). Sub-femtosecond precision measurement of relative X-ray arrival time for free-electron lasers. Nature photonics8(9), 706

Hossain, M. A., Canning, J., Ast, S., Cook, K., Rutledge, P. J., & Jamalipour, A. (2015). Combined “dual” absorption and fluorescence smartphone spectrometers. Optics letters40(8), 1737-1740

Leedle, K. J., Pease, R. F., Byer, R. L., & Harris, J. S. (2015). Laser acceleration and deflection of 96.3 keV electrons with a silicon dielectric structure. Optica2(2), 158-161

Mouroulis, P., Van Gorp, B., Green, R. O., Dierssen, H., Wilson, D. W., Eastwood, M., … & Loya, F. (2014). Portable Remote Imaging Spectrometer coastal ocean sensor: design, characteristics, and first flight results. Applied optics53(7), 1363-1380

Piascik, A. S., Steele, I. A., Bates, S. D., Mottram, C. J., Smith, R. J., Barnsley, R. M., & Bolton, B. (2014, July). SPRAT: spectrograph for the rapid acquisition of transients. In Ground-based and Airborne Instrumentation for Astronomy V(Vol. 9147, p. 91478H). International Society for Optics and Photonics

Squires, A. D., Constable, E., & Lewis, R. A. (2015). 3D printed terahertz diffraction gratings and lenses. Journal of Infrared, Millimeter, and Terahertz Waves36(1), 72-80

Vaughan, M. (2017). The Fabry-Perot interferometer: history, theory, practice and applications. Routledge