Diffraction I Version 1.0 : March 12, 2019

Transcription

1 Diffraction I Interference an Diffraction of Light Through Slit: The Jut Give Me The Formulae Verion! Steve Smith March 12, 2019 Content 1 Introuction Huygen Principle The Mathematic of Diffraction The Difference Between Interference an Diffraction The Phyic of Diffraction Document Structure One Slit 5 3 Two Slit Two Slit Screen Ditance The Difference Between Interference an Diffraction Many Slit The Diffraction Grating Summary of Equation More Than One Slit Obervation Example Quetion Example Example Appenice 11 A in(θ) an tan(θ) for Small Angle θ 12 B The Approximation For in(θ) 12 1

2 Prerequiite None. Note In orer to try an give you an iea of what topic are require on which A-Level yllabi, I have put inicator in the margin by ection heaer. So, for example, if you ee thi in the margin, it i eigne to inicate that the topic i require in the Eexcel an yllabi (green for require), but not require in the an yllabi (re for not require) (all concerning the 2015 pecification). Document Hitory Date Verion Comment 12 March Initial creation of the ocument. 2

3 1 Introuction 1.1 Huygen Principle Thi principle wa evelope by Chritian Huygen ( ). It a practical way of working out what i going to happen to a wave in the future, if we know where it i now. The principle tate that each point of a meium (iturbe by a paing wave) become a ource of a iturbance (i.e. a new wave) which propagate from thi point in all irection. Huygen i not upply a rigorou mathematical proof of hi principle. It wa imply a ueful tool to help to explain a variety of wave phenomena, uch a reflection, refraction an iffraction. An it work The Mathematic of Diffraction So what ha Huygen Principle got to o with iffraction? Check out Figure 1. A B Figure 1: Huygen Principle P In thi picture a plane wave approache the gap from the left. In the gap, there i a wave cret (AB). Huygen Principle ay that every point on thi wave cret act a a ouce of pherical wave emanating from the point. I ve hown ome of the point on the wave cret in the gap, in ifferent colour, together with a few of the pherical wave front going out from the point to the right, in matching colour. So to fin the brightne of the light at a point P on the creen, all we have to o i to a together the iplacement of all the infinite number of wave (taking into account how far they travel, which etermine their phae at P, of coure) that emanate from the infinite number of point in the gap. Tricky, huh! An here, there only one gap! If at the point P, the infinite number of wave a together contructively, then there will be a bright pot at P. I ve trie to pick P o that wave cret from all the hown point in the picture coincie there. So there houl inee be a bright pot at my choice of P! On the other han, if at a point on the creen, the infinite number of wave a together in uch a way that they cancel each other out, then there will be a ark pot there. An other point will be omewhere in between. An when there are two gap, you have two lot of infinitely many iplacement to a together. For three gap, three lot of infinitely many iplacement. For a iffraction grating... Thi i iffraction. Pretty horrible! So, in the ret of thi ocument, whenever I ay omething like...it turn out that... when I introuce an equation, the equation reult from thi aing-up-an-infinite-number-of-wave thing. 1 The principle can be obtaine from Maxwell equation (a et of equation evelope by Jame Maxwell in which explain how electromagnetic wave propagate), o it oe have a firm founation in theory. It i intriguing that Huygen work precee Maxwell by about two centurie, an yet eeme to anticipate it, without the oli theoretical bai that Maxwell provie. Ampere law an Faraay law preict that every point in an electromagnetic wave act a a ource of the continuing wave, which i perfectly in line with Huygen principle. 3

4 1.3 The Difference Between Interference an Diffraction... 1 I think that there i a lot of confuion about the term uperpoition, interference an iffraction. I ve certainly been very confue. I expect you have been too. During the coure of thi ocument, I m going to try an clear it all up! A a firt tantaliing tep, I m going to offer thi. There i no generally accepte itinction between the efinition of thee term, but in my view we houl ue them like thi: uperpoition i the proce that we ue to a two (or more) wave together. An we o that by aing the iplacement of the wave (at each point in pace an at each point in time), to get the total iplacement of the reulting wave (at that point in pace an time). interference i when we uperpoe a finite number of wave. Uually only two! iffraction i when we uperpoe an infinite number of wave. So a you might imagine, iffraction i lightly harer than interference!! So when book claim that two-ource interference i the ame thing a two-lit iffraction, then I woul beg to iffer. More of thi later! 1.4 The Phyic of Diffraction Diffraction i the preaing out of wave a they pa through a gap in a barrier, or if they encounter an obtacle. (a) Through a gap (b) Aroun an obtacle Figure 2: Diffraction Through a Gap an Aroun an Obtacle Figure 2 how plane wave approaching a gap in a barrier (a), an an obtacle (b), from the left. A the wave goe through the gap, it prea out. A the wave goe pat the obtacle, it prea out behin it. Diffraction oen t alway happen, however: the amount of iffraction that occur when a wave encounter a gap or an obtacle epen entirely on the ize of the gap/obtacle an the wavelength of the wave. It turn out that for ignificant iffraction to occur, the ize of the gap/obtacle ha to be aroun the ame ize a the wavelength of the wave. 1.5 Document Structure To implify thing a bit, in the ret of thi ocument I m jut going to be talking about wave that go through gap. An in the ret of thi ocument you will fin: Section 2 look at the ituation where there i only one gap; Section 3 look at the ituation where there are two gap: the o-calle Young ouble-lit experiment; Section 4 look at the ituation where there increaing number of gap, leaing up to the iffraction grating, where there are loa of gap! Section 5 provie a ummary of the iffraction equation that are require at A-Level. Finally, Section 6 tackle a few example quetion. 4

5 2 One Slit If you hine plane wave of light at a hole in a wall (thi i what phyicit call a lit ) you get (if you chooe the wavelength of the light an the with of the lit appropriately) an intereting pattern of light on a creen place on the other ie of the lit. Figure 3: Diffraction Pattern with Different Number of Slit You might expect that the light jut goe traight through the lit in a traight line, o you woul jut ee a ingle pot of light on the creen, the with of the lit. An mot of the time that exactly what you o ee. But if you chooe the with of the lit to be about the ame ize a the wavelength of the light (which i very mall: about m), then omething very intereting happen. To ee what happen, check out Figure 3, where the one-lit pattern appear at the top of the picture. In Figure 4 I ve trie to illutrate how the brightne of the light on the creen varie with poition on the creen. θ The blue line repreent the intenity (brightne) of the light on the creen. There i a bright ban at the centre of the creen: a you move away from the centre, there are ucceive ark an bright ban; the bright ban becoming le an le bright the further from the centre. Figure 4: Single Slit Diffraction [Note that in the figure in thi ocument, the ize of the lit an the itance between lit (ee later) are hown much larger than they woul appear in practice. A I ay, for ignificant iffraction to occur, the lit with nee to be aroun the ame ize a the wavelength λ of the light. Uually alo, the itance between lit (when we have more than one lit) i much greater than the iniviual lit with.] Hopefully, you houl be able to ee a number of intereting feature of the pattern of light on the creen: there i a bright area on the creen oppoite the lit. Thi area i wier than the lit an the ize of thi area can be change by changing the lit with: narrowing the lit with wien the central maximum; on either ie of the central bright area i a region of arkne(!); outie thoe ark area there i a pair of bright region, but the brightne of thoe region i much le than that of the central bright area; continuing out on either ie, we have alternating bright an ark area that eem to go on forever. However, each bright area i much le bright than the one next to it (towar the centre of the creen). 5

6 3 Two Slit OK, o what if you have two lit? Thi i the very famou Young ouble-lit experiment, an play a very big part in the hitory of Phyic! Well, if you hine plane wave of light at a pair of hole in a wall (thi i what phyicit call a ouble-lit ) you get (if you chooe the wavelength of the light an the with of the lit appropriately) an intereting pattern of light on a creen place on the other ie of the lit. Check out Figure 3 an 5. The re line repreent the intenity (brightne) of the light on the creen. There i a bright ban at the centre of the creen: a you move away from the centre, there are ucceive ark an bright ban; the bright ban becoming le an le bright the further from the centre. Figure 5: Double Slit Diffraction 1 You might expect that the light jut goe traight through the lit in traight line, o you woul jut ee a pair of pot of light on the creen. An actually, mot of the time, that exactly what you o ee. But if you chooe the with of the lit to be about the ame ize a the wavelength of the light (which i very mall: about m), then omething very intereting happen. The pattern of light on the creen now conit of two part: the one-lit pattern that we aw in ection 2; an within that a new pattern that ha many more peak an trough in it. Exactly how many peak there will be epen on the value of the three variable that are involve: the with of the lit, ; the itance between the lit, ; the wavelength of the light, λ. θ 1 in(θ 1 ) = λ θ 2 in(θ 2 ) = 2λ The ahe blue line repreent the one-lit intenity (brightne) pattern; the re line repreent the two-lit intenity pattern. The two-lit pattern ha more peak an trough than the one-lit pattern, an i contraine within it. Figure 6: Double Slit Diffraction 2 6

7 It turn out that for the n th orer bright pot, in(θ n ) = nλ [To cover myelf, I houl a a note here that thi i only true if i much bigger than, which i uually the cae in practice. Alo notice that, poibly urpriingly, the lit with oen t appear in thi equation!] 3.1 Two Slit Screen Ditance Now if you check out Appenix B, we can ue the mall angle approximation again in thi ituation, o long a the angle θ i mall. Thi mean, for example, that the itance on the creen from the centre of the iffraction pattern to the firt bright pot, x, woul be given by: in(θ 1 ) x D = λ or x = λd An the itance on the creen from the centre of the iffraction pattern to the econ bright pot, y, will be: in(θ 2 ) y D = 2λ or (ee Figure 7). y = 2λD in(θ 1 ) = λ D θ 1 θ 2 x y in(θ 2 ) = 2λ Figure 7: Double Slit Screen Ditance 3.2 The Difference Between Interference an Diffraction... 2 Earlier I ai that I iagree when book claim that two-ource interference i the ame thing a two-lit iffraction. That becaue in orer to obtain the equation for thee iffraction intenitie, we have to a together the infinite number of (Huygen ) wave that emanate from the infinite number of point in the lit. two-ource interference i the proce where we a together two wave. two-lit iffraction i the proce where we a together an infinite number of wave: the wave that emanate from the infinite number of point in the lit. I mut emphaie that you on t nee to know any of thi aing-together-the-infinite-number-of-wave thing at A-Level. All you nee are the equation that reult from the proce! 7

8 4 Many Slit When you have five lit the picture look like that in Figure 8. A the number of lit increae, certain peak in the re pattern become prominent, an other peak eem to cancel out. An the more lit you have, the more prominent the remaining peak become, an the ret eem to winle away to nothing. θ 1 in(θ 1 ) = λ θ 2 in(θ 2 ) = 2λ The ahe blue line repreent the one-lit intenity (brightne) pattern; the re line repreent the five-lit intenity pattern. The five-lit pattern ha more peak an trough than the one-lit pattern, an i contraine within it. Figure 8: Multiple Slit Diffraction An all the time, no matter how many lit you have, all thee re peak never ecape the blue one-lit iffraction pattern. An jut like the cae of two lit, it turn out that the bright pot on the creen obey the relationhip: in(θ n ) = nλ [Covering myelf again, thi i actually only true if the number of lit i very large, or the lit eparation i much bigger than the lit with.] 4.1 The Diffraction Grating A iffraction grating i jut a thing with loa of lit in it. I mean, hunre of thouan of lit. All evenly pace. But i oen t matter how many lit there are, the equation that applie i till that above, where thi time the in the equation i the itance between any pair of ajacent lit. The only real ifference between having an infinite number of lit a in a iffraction grating, an two lit, i that becaue of all thoe maxima that hrivel to nothing in the many-lit cae, the maxima that are left are all very narrow. So again, in(θ n ) = nλ [An again, thi i actually only true if the number of lit i very large, or the lit eparation i much bigger than the lit with. For a iffraction grating, we have a very large number of lit, o we re covere.] 8

9 5 Summary of Equation 5.1 More Than One Slit For multiple lit (which inclue the iffraction grating, of coure), each pair of ajacent lit eparate by a itance, the angle to the bright pot in the pattern are given by: in(θ n ) = nλ (1) where λ i the wavelength of the light. Thi equation applie whatever the ize of θ n. So long a the angle θ n i mall, the itance x n from the center of the pattern to the n th bright pot i given by where D i the lit-creen itance. x n = nλd Now here a curiou thing: o long a the angle θ n to the n th bright pot i mall, thi fringe pacing formula for bright pot on the creen turn out to be the ame formula that applie for two-ource interference!! That hany! But again only a long a θ n i mall. (2) 5.2 Obervation From Equation (1) if the lit with an eparation are fixe, then θ n λ, o re light will be iffracte more than blue light at each fringe; ince the maximum value of in(θ n ) = 1, then there will be a maximum n: that mean only ome of the fringe will be viible. From Equation (2) if the lit with an eparation are fixe, an the lit-creen itance i fixe, then x n λ for a given fringe. 9

10 6 Example Quetion 6.1 Example 1 Monochromatic light of wavelength 500 nm i incient normally on a iffraction grating having 600 line per mm. Calculate the angular poition θ of the firt two orer on one ie of the normal. What i the maximum number of bright image that can be een on one ie of the normal? For a iffraction grating we have the formula in(θ n ) = nλ We know the λ; we have information to fin ; an n will be 1 an 2. So, let get going. Firt, let fin. If there are 600 line per mm, then there will be line per m, an o So, for n = 1, = m in(θ 1 ) = 1 λ = ( 3 ) θ 1 = in = (to 4 f) An for n = 2, ( ) θ 2 = in = (to 4 f) The maximum value in(θ n ) can be i 1. So that mean that the maximum value of o n max λ = 1 n max λ = = 3.33 So that maximum value n can be i 3, ince n ha to be a whole number. 10

11 6.2 Example 2 A plane iffraction grating i illuminate by a ource which emit two pectral line of wavelength 420 nm an 600 nm. Show that the thir orer line of one of thee wavelength will be iffracte through a greater angle than the fourth orer of the other wavelength. For a iffraction grating, in(θ n ) = nλ We on t know thi time; we know the two λ an the two n involve; we are ake to fin omething out about θ. Now from thi formula we know that the bigger the wavelength, the bigger the angle. So they mut want u to how that the thir orer angle for the longer wavelength i bigger than the fourth orer angle for the horter wavelength. Here goe. For the fourth orer of the horter wavelength: in(θ n ) = nλ For the thir orer of the longer wavelength: in(θ 4 ) = = which i bigger than the previou value. in(θ n ) = nλ in(θ 3 ) = = One thing left: for angle le than 90, the bigger the angle the bigger the in() of the angle, an vice vera. So we are one! 11

12 Appenix A in(θ) an tan(θ) for Small Angle θ Have a look at the value of in(θ) an tan(θ) for ome mall angle of θ in Table 1: Table 1: Value of in(θ) an tan(θ) for Small Angle θ θ( ) in(θ) tan(θ) % Error Table 1 how that if the angle θ i mall, in(θ) ha a very imilar value to tan(θ). An the maller the θ, the cloer the two value are. So, if you are face with a ituation where your angle θ are mall, then in(θ) tan(θ) an wherever you have a in(θ), you coul replace it with tan(θ), an vice vera. Appenix B The Approximation For in(θ) D θ x Now, check out Figure 9. From baic trigonometry, Figure 9: The approximation for in(θ) tan(θ) = x D But, if the angle θ i mall, then in(θ) will be approximately the ame a tan(θ) (ee Appenix A). That mean that in(θ) x D 12