FINDING OPTICAL DISPERSION OF A PRISM WITH APPLICATION OF MINIMUM DEVIATION ANGLE MEASUREMENT METHOD

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1 Warsaw University of Technology Faculty of Physics Physics Laboratory I P Joanna Konwerska-Hrabowska 6 FINDING OPTICAL DISPERSION OF A PRISM WITH APPLICATION OF MINIMUM DEVIATION ANGLE MEASUREMENT METHOD. Funamentals Optical ispersion D n of a material is the phenomenon in which refraction inex n varies with ifferent wave frequencies (sometimes, base on the epenence = c/λ, one speaks of epenence of n on the wavelength λ. However, one has to remember that wavelength epens on the meia where the wave travels, by contrast, the frequency is the unique feature of the wave): n = f() or n = f(λ) () To unerstan the mentione above ispersion efinition, we have to learn what the refraction inex is why it epens on the wave frequency an what the measure of material ispersion is. Moreover, the experiment requires the knowlege on the prism an how it works an how to apply the metho of fining the minimum eviation. Fig.. Refraction an reflection of rays at the bounary of two isotropic meia. Light refraction phenomenon is the change in irection of beam path (in the language of geometric optics) or the change in irection of light wave propagation (in the language of wave optics) when it crosses a bounary of two meia. This phenomenon an relate to it reflection phenomenon are manage by the laws known as geometrical optics laws. Let s recall them: When the light hits the bounary of two isotropic material meia ), the transmitte (refracte) wave an the reflecte wave appear. Three vectors escribing wave propagation irections are in the same plane calle plane of incience (see fig. ) an irection of propagation of these waves fulfills following formulas:. The angle of reflection α 0 is equal to the incience angle α: α = α 0 (). The ratio of the sine of the angle of incience an the sine of the angle of refraction is equal to the ratio of velocities v an v in two meia an is constant for a given pair of two meia an for a given wavelength. This value n / is calle refractive inex of one meium with respect to another: Isotropic meium has ientical physical properties in all irections. Properties of anisotropic meia epen on the analyze irection; in particular, light refraction inex varies with orientation of incient light with respect to the optical axis of a meium an results in phenomenon calle birefringence". Anisotropy can be cause artificially, for example by applying an external force this phenomenon is a basis for a wie range of practical applications so-calle elastooptics (real construction moels, sensors).

2 sin sin V V n /, (3) where α, β an α 0 are the angles between irection of the waves: incient, refracte an reflecte respectively an the normal of the bounary between meia an (see fig. ). The law escribe with the formula (3) is known as Snell s law. If a light of a wavelength λ comes out from vacuum, where the spee of light has the known value c, inepenent of wave frequency, into the meium in which the spee of light is equal v(λ), the formula (3) can be rewritten in a form expressing efinition of an absolute light refractive inex n(λ): c n( ) (3a) V ( ) Explanation of light refraction an reflection phenomena an eriving the laws managing these phenomena (geometrical optics law) can be one in various ways, for example: - base on Fermat s principle, - base on Huygens Fresnel principle, - base on Maxwell s electromagnetic theory. In all these ivagations we take the assumption that the light propagation spee varies with neighboring meia. Because of technical ifficulties, the experimental confirmation of this assumption was impossible. It was eventually proven by Foucault in 850. In this manual, we are going to erive the laws of geometrical optics base on Fermat s principle. Fermat s principle is usually state as the path taken by a ray of light in traveling between two points requires either a minimum, a maximum or the same time (in a stationary case). Fermat s principle is a particular case of a very general principle ruling in the nature, stating that all the processes follow optimum paths. With the reference to ray paths, this can be escribe with a formula: ns extremum (4) where: n light refraction inex for a chosen meium, s geometrical path. The prouct L=n s is calle an optical path. Thus, accoring to Fermat s principle, when a light ray travels, its optical path is being optimize. With application of simple geometrical constructions one can show that optical paths travele by a ray being reflecte or refracte in a homogeneous meia, are the shortest of possible paths linking two chosen points A an B. Let s analyze this base on the example of refraction law (see fig. ). We have points A an B given in two meia an an a ray APB linking both of them. Base on known mechanics formula we can write that the time require for traveling A-P-B path is given with a formula: s s t. (5) V V Once we introuce the optical path concept an we consier the epenency (3a), formula (5) is now: n s n s L t, (5a) c c Where L = L +L is the overall optical path of wave front traveling from A to B, an L an L are optical path travele in meium an. We shoul not confuse the optical path with the geometrical path equal to s = s + s. Optical path is equal to geometrical path multiplie by the refraction inex of the meium.

3 3 A n a S x P x x n S B b Fig.. The ray originating from point A is refracte on the meia bounary at point P an reaches point B. Refraction inexes n an n are absolute light refraction inexes for a wave of a frequency ω in meium an. Base on Fermat s principle, we know that optical path has to be optimum, so P point has to be locate in such a place on X axis so that erivative of optical path L with respect to x coorinate was equal to zero, so: L 0 (6) x With application of geometrical epenencies shown in fig. we can write: L n L x s n s n a x n b ( x). Differentiating, we obtain: n ( a x ) x n[ b ( x) ] ( x)( ), what can be written as: n a x x n b x ( x). (7) Formula 7, with respect to the trigonometric epenencies, is the equation escribing the refraction law (3): n sinα = n sinβ. (7a) There are as many ways of eriving geometrical optics law as many there are recognize light theories. Any light theory can only be recognize uner conition it can explain the experimentally verifie laws of reflection an refraction. Light refraction inex in a meium epens on the light wave frequency (see formula ()). This phenomenon, as it was state above, is calle light ispersion, characteristic for a chosen material. In spectral ranges, for which a chosen material is translucent, an increase of light refraction inex is being observe as light frequency is being increase. In the ranges where a meium absorbs light so-calle abnormal light ispersion is being experience, it means ecrease of light refraction inex with light frequency is being increase. To explain light ispersion, we make an assumption that electrons, atoms an particles of a meium through which the light wave is traveling, react in a various way to magnetic fiel that forces them to vibrate, as they have specific frequencies of their own vibrations 0 which may be ifferent from the electromagnetic wave frequency v.

4 4 Let s analyze the example of interaction of electromagnetic waves with valence electrons of a meium through which these waves pass. These waves belong to a visible range. Electrons have their own specific frequencies 0. Incient wave forces vibration of frequency v, but the amplitue an phase of force vibrations of electrons (an in consequence the seconary waves they sen) epen on the ifference between the own electron vibrations frequency 0 an the frequency of the incient wave. If a wave hits perpenicularly the plane meium bounary, amplitues an phases of vibration of all electrons with which wave interacts are the same in a very thin layer (comparable to wavelength) ahering to the bounary of the meium. The vibrations of the analyze layer create a seconary plane wave, coherent with incient wave, but shifte with respect to it in phase of φ angle expresse with a formula: o ctg (8) ( ) o where enotes the amping ratio. The output wave, create in this thin layer as a result of superposition of the incient an the seconary wave, is shifte in phase with respect to the incient wave at a certain angle epenent on the amplitue of the seconary wave an the phase shift between it an the primary wave. In each subsequent layer, sorte out in min in the same way, there is a similar phase shift of the output wave with respect to the incient wave. Thus, as the output wave propagates in the meium, the phase changes with respect to the primary incient wave by an angle proportional to the path travele by the wave in the meium. In consequence, we speak of the velocity of the phase shift or the phase velocity v of a wave. Taking into consieration the escribe above properties of wave propagation in meia, we can say that any wave propagates in any meium with the phase velocity ifferent from the phase velocity c in the absence of the meium. As we see from the formula (8), the phase change, an therefore the phase velocity, epens on the ifference between the frequency of incient wave an the frequency of proper vibrations of electrons of the meium. The phase velocity of waves of a ifferent frequency will be therefore ifferent. As the white light is a mixture of waves of various frequencies, each component wave will propagate at a ifferent velocity. Accoring to the formula (3a), each component of the white light will refract with a refraction inex of a ifferent value. This phenomenon is calle ispersion. The commonly use ispersion measure D n of any meium is a ifference of refraction inexes of K (violet color) an A (re color) lines (Fraunhofer s efinition): [D n ] = n F - n C (9) so this is the ifference of light refraction inexes for a specific ifference of wavelengths. The ispersion phenomenon can be experience when a white light beam is passing through a prism (see Appenix). As each component of the white light has a ifferent refraction inex an the angle at which the prism bens a ray epens on the light refraction inex therefore the prism bens the light rays of ifferent wavelength in a ifferent way. Light of longer wavelengths for example re is less ben by the prism than the light of shorter wavelengths for example violet. In consequence, we will see a characteristic rainbow in the screen locate behin the prism which is a result of separation of waves of ifferent frequencies.

5 5. Experiment The goal of this experiment is to etermine the apex angle of the prism an to fin its optical ispersion an its resolving power with application of the minimum eviation metho... Fining the apex angle of the prism The metho of fining the apex angle of the prism applie in this experiment uses geometrical optics law referring to light reflection phenomenon (see formula ). The principle is shown in fig 3. a) K L L b) a-b b a Fig. 3. Fining the apex angle of the prism: a) reflection of rays from the walls of the prism; b) illustration of geometrical ivagations. We set the prism in such a way that the apex angle φ is locate in front of the collimator an it is illuminate with a parallel beam. We observe two beams reflecte from the prism walls an we etermine angle location of the viewfiner a an b corresponing to these beams. As it is seen in the fig. 3b: a b = α - β, α = 90 - φ, β = 90 - φ.

6 6 Then we obtain a b = (90 - φ ) (90 - φ ), an then: a b = φ + φ = φ, so: a b. (0) Application of formula (0) enables us to figure out the value of the apex angle φ of the prism at known angular locations a an b of the viewfiner through which we observe the beams reflecte from the prism walls.. Fining the refraction inex with application of the minimum eviation metho Depenence of eviation angle ε of a beam passing through the prism on the angle of incience α at the prism wall is erive base on the following ivagation (see fig. 4a). Let s think of behavior of a parallel beam of monochromatic (single color) light. This passing can be illustrate in a vertical cross-section. The ray hits the sie wall I of the prism at an angle α, refracts at an angle β (see formula (3)), hits the sie wall II at an angle β an goes out of the prism at an angle α with respect to the normal to the wall II, creating the angle ε with the irection of the incient ray. This angle is calle a eviation angle. a) C A I II B L K 90 o D 90 o b) K min Fig.4. Path of monochromatic light in a simple prism: a) geometrical ivagations; b) setting the viewfiner at an angle of minimum eviation ε min. Having in min the fact that the external angle in the ABD triangle is equal to the apex angle φ of the prism (this angle has arms perpenicular to prism walls), we can easily erive the following geometrical epenencies:

7 7 φ = β + β ε = (α β ) + (α β ) ε = α + α φ (a) (b) (c) The eviation angle ε epens on the value of the angle of incience α. If we observe the spot of light eviate by the prism an we rotate the prism to change angle α, we will be able to notice that the spot reaches the point near to the location that woul be reache if the prism was remove. Then the spot regresses espite the fact that the prism is still being rotate in the same irection. There exist therefore such an angle of incience α at which the angle of eviation is minimum ε it happens [] (see appenix) when we have a so-calle symmetric course, for which ε = α = α an β = β = β. Base on () relations, we can write for a symmetric course: min min ; ; ; () Plugging the above epenencies to the formula (3), we obtain the formula which is important for the presente metho: sin min n (3) sin This formula enables us to etermine refraction inex when one knows the apex angle of the prism φ an the minimum eviation angle ε min for a given wavelength λ. These values can be measure with spectrometers. 3. Measurements 3.. Preparation of the spectrometer for measurements Before starting the main measurement one has to ajust the spectrometer accoring to the instructions locate at the experimental setup or accoring to the supervisor s explanations. 3.. Measurement of the apex angle of the prism We set the prism is such a way that the apex angle was locate in front of the collimator an we observe through the viewfiner L images of the slit create by the rays reflecte from the prism wall (fig.3a). The angle between irections L of the reflecte light rays will be equal to φ. Therefore, to etermine the apex angle of the prism, we set the viewfiner for observation of the beam reflecte from first wall of the prism an we rea the viewfiner location ( a egrees), an then we observe the image of rays reflecte from the secon wall an we write own b location. The apex angle is equal to the half of the ifference of these two reaouts. During the measurement you have to make sure that the crossing of the ruler lines in the viewfiner was going through the center of the slit image that shoul be as narrow as possible. The results have to be written own in Table, prepare accoringly in your laboratory log. The precision of the measurement has to be etermine uring the experiment as they are necessary for preparation of the measurement ata an etermination of the calculation uncertainties. Apart from the instrument precision one has to consier the precision of setting the cross lines in the centre of the slit. The accuracy of the measurement of the apex angle of the prism can be estimate as: φ = reaout precision + ½ of the angular with of the slit image. The results of estimations (in raians) have to be written own in the laboratory log Measurement of the minimum eviation angle Ajust the table an viewfiner position to set them at the minimum eviation angle of the re fringe (fig. 4b) for the apex angle φ that has been foun previously. Rotate the table to change the light incience angle. At a certain position of the table (at a specific light incience angle) fringe stops an returns with a further table rotation. Set the table position at the turning point as

8 8 precisely as possible (using the table ajustment knob). The table position correspons now to the minimum angle ε min of the eviation of the beam passing the prism. At this table position, set the viewfiner in this way that the cross lines in the viewfiner were locate in the mile of the fringe. Write own the location of the viewfiner rea out from the nonius. Repeat the measurement three times. Once the escribe above measurements are complete, remove the prism (the table has to be locke), set the viewfiner in front of the collimator an rea the noniuses once again. The rotation angle from the position corresponing to the minimum eviation to the location in front of the collimator equals to the angle of the minimum eviation ε min. Repeat the measurement three times. The results of all the measurements are to be written in the log in table. Next, perform the analogical measurements for the brightest lines. Wavelengths in [nm] corresponing to this lines can be foun in the chart at the measurement setup or in physical tables. The results are to be written in the log, in table. When measuring the minimum eviation angle, one can notice that in the proximity of turning point, regarless the table rotation, the fringe seems to be stationary the eye oes not notice changes in its location. The table rotation between the moment when the fringe stops an the moment when it starts returning is calle a ea range. The uncertainty of the minimum eviation can be estimate as: Δε min = reaout precision + of the angular with of the slit image. Write own the estimation results (in raians) in the laboratory log. 4. Results. Basing on the formula (0), calculate the apex angle of the prism φ an report the result in table.. Calculate angles of minimum eviation ε min for subsequent neon lines as ifferences in angular location of the viewfiner (the average value of three measurements) corresponing to the turning point for a given line an to the location in front the collimator (the average value of three measurements). Write own the results in table. 3. Basing on the formula (3), figure out light refraction inexes n λ an their uncertainties u(n λ ) for subsequent lines. 4. The measurement uncertainty n epens on the uncertainties of measurements φ an ε min. This is the uncertainty that has to be calculate base on the knowlege of stanar uncertainties u(φ) an u(ε min ). 5. When performing calculation, o not forget to use raians [r] to express the angle uncertainties. 6. Draw a curve of epenence of light refraction inex n from the wavelength λ, a so-calle ispersion curve an calculate ispersion [D n ] (formula 9). 7. In the conclusion part, compare obtaine results to the table values. 5. Questions. What is the light refraction phenomenon?. How are the relative absolute an the absolute light refraction inexes efine? 3. How oes light acts on the simple prism? 4. What is the ispersion of a material meium? 5. How can the apex angle of the prism be measure? 6. Describe the metho of fining light refraction inex with application of the minimum eviation angle? 6. References. D. Halliay, R. Resnick, J. Walker, Funamentals of Physics, Wiley (0), part 4, Chapter 33-8.

9 9 APPENDIX The prism is mae of a transparent meium limite by two flat surfaces that have an angle φ between them. The angle is calle the apex angle of the prism. Complex prisms, create for special purposes, can be use to break a light beam up, to change a light irection without splitting it or to polarize light. Influence of a prism on a white light beam A simple single prism always breaks white light up an changes its path. Its action is base on the light ispersion phenomenon or ifferent values of light refraction inexes for various light wavelengths ν (see efinition ). Pr Fig.A. Breaking up of white light in a simple prism: white light ray, re light ray, 3 violet light ray; φ prism apex angle, ε ray eviation angle. 3 f 3 cz S violet re 3 f 3 cz Each of components of a parallel white light beam (which is a mixture of waves of ifferent frequencies ν), hitting one of the prism s walls creating the apex angle, refracts on it at a ifferent angle an travels with a ifferent velocity insie the prism, passes the prism an while leaving it, it refracts once again, creating a colorful ivergent beam (see fig. A.). A path of a monochromatic light ray in a prism A monochromatic light ray, while travelling along a path insie a prism refracts from the primary path at an angle ε. The angle epens on the angle of incience on the prism wall α an light refraction inex for a given wavelength n λ (or frequency n ν ). There exists an angle α (for a given prism) for which ε angle reaches a minimum value ε min. It can be proven that for this specific angle monochromatic light ray travels in the prism perpenicularly to the bisector of the apex angle (it means in a symmetric way, which results in the fact that the angle of incience α an the angle of emergence α are equal (α = α )). This property of the prism is a base for the metho of fining the optical ispersion of material D n known as the metho of measurement of the angle of minimum eviation ε min. The proof for conition of reaching by the eviation angle ε the minimum value can be one both on the geometrical an analytical way. Let s recall here the analytical proof []: We ifferentiate the formula efining the epenence of the eviation angle ε on the angle of incience α (see fig.4 an formula c: ε = α +α φ) with respect to α :, an for minimum: 0, so: 0 (A) Basing on the conition efining epenence between the apex angle of the prism an refraction angle β an angle of incience on the exit wall β (see fig. 4 an formula b): β + β = φ, we fin, by calculating the erivative with respect to α :

10 0 0 (A) But sinα = n sinβ an sinα = n sinβ (see formula (3)). By ifferentiating these epenences with respect to α, we fin: (a) cos n cos an (b) cos n cos. (A3) From the formulas (D) an (D) we have: an. (A4) Plugging the obtaine epenencies (A4) into formula (A3(b)), we fin: cos n cos. (A5) By iviing sie by sie obtaine epenencies (A5) an (A3(a)), we obtain: cos cos, (A6) cos cos where, after raising both sies to the power of two, we have: sin sin. (A7) sin sin By transforming the formula (A7), taking into consieration epenence (3), we obtain: n sin sin n sin sin, (A8) then, after the transformation we have: sin sin. (A9) Both angles are positive an acute. Thus we have: an (A0) By iviing both formulas (A3), taking into consieration (A0), we have: cos cos (A) cos cos By calculating the n erivative of the formula (A) with respect to α an taking into consieration (A3) an (c) we fin that: For α = α an β = β secon erivative 0 which means that we eal with minimum of eviation ε of the light right with the prism. The resolving power of the prism R λ i.e. the capability of resolving neighboring spectral lines of a wavelengths λ an λ +δλ is efine as: R, (A)

11 is relate to the iffraction phenomenon. We know that if a light beam hits a slit, it is being iffracte. When the beam hitting the slit is parallel, then for the wavelength λ the angle of refraction, at which the first minimum is present is given by a formula: sin, (A3) where is the slit with. Pr L h E Fig.A.. A setup for figuring out a resolving power of a prism: L light source, S, S lenses, Pr prism, E screen Let s analyze the beam of a light hitting the prism at an angle of minimum eviation (fig. A). The beam has a imension limite by the imension of the prism the beam with plays a role of a slit, at which refraction takes place. Moreover, let s recall that the light incient on the prism consists of two beams: one of a wavelength λ an refraction inex n an the secon one of a wavelength λ + δλ an refraction inex n + δn. For beams travelling near the base of the prism the ifference of optical paths travelle by these two beams is δs = h δ n, where h is the length of the prism base. The wave fronts corresponing to these beams will create the angle τ, an, as it is seen in fig A.3, a following formula is vali: h n sin s (A4) There is a minus sign, because δn<0. The slit images create by these two beams will be separate when τ angle is at least equal to the υ angle, efining the istance between the central an the first minimum, create thanks to the light iffraction on the prism, which is a iaphragm of a with. Thus, the conition of separation of two images will be obtaine by comparison of (A3) an hn, (A5) where we fin the formula for the resolving power of the prism R (see A): R n h h Dn (A6) Base on formula (A6), one can see that the resolving power of a prism R λ is proportional to the length of a prism h an the spee of change of refraction inex with a change of a wavelength, or socalle meium ispersion or material ispersion D n.