3Trig SeriesTrigonometric functions can be expanded in power series, which facilitates approximations of the functions in extreme cases. The angle x must be in radians.
4Small Angle Approximation... One of the most important applications of trigonometric series is for situations involving very small angles. For such angles, the trigonometric functions can be approximated by the first term in their series. This gives the useful small angle approximations:Examples of the use of the small angle approximation are in the calculation of the period of a simple pendulum, and the calculation of the intensity minima in single slit diffraction. This approximation is used in most of the common expressions of geometrical optics which are built on the concept of surface power for lenses.This approximation sin x = x reaches a 1% error at about 14 degrees
5Maths puzzling....DegreesTheta RadSin ThetaDifference0.0010.0220.0330.0540.0750.0960.1070.1280.1490.16100.17200.350.34-0.01300.520.50-0.02400.700.64-0.06500.870.77-0.11601.05-0.18701.220.94-0.28801.400.98-0.41Construct an excel spreadsheet to explore and prove the concept that for the situation where <10 and measured in radians;
613.1 Refraction of lightSpecification link-up 3.2.3: Refraction at a plane SurfaceWhat do we mean by ‘rays’?What is Snell’s law?Does refraction differ for a light ray travelling from a transparent substance into air?
10Snells Law Θ1 = angle of incidence Θ2 = angle of refraction Incident raynormal1Refracted ray21. The incident ray, the refracted ray and the normal all lie in thesame plane.2. For two given media, the ratio of the sine of the angle of incidenceto the sine of the angle of refraction is constant.
11Verification of Snells Law Why not try it out for a glass prism?Look at the example results first…irsin isin rn106.60.1740.1151.52013.20.3420.2283019.50.50.3344025.40.6430.4295030.70.7660.5116035.30.8660.5787038.80.940.627
12Arsenic trisulfide and sulfur in methylene iodide 1.9 Materialλ (nm)nVacuum1 (per definition)STP0 °C and 1 atmAir589.29Carbon dioxide20 °CWater1.3330Arsenic trisulfide and sulfur in methylene iodide1.9room temperatureDiamond2.419Strontium titanate2.41Amber1.55Water ice1.31cornea (human)1.3375
13Equilateral Prisms Sometimes it is better to think of Snells law as... Where you have two mediums. It does not matter which way they are around or what they are as long as you plug in the refractive mediums number.The equation simplifies to n = 1 for air.For an equilateral prism you can use this construct to work out the refraction as unlike a rectangular prism i1 i2In essence what we mean is that every angle of reflection & refraction is different for the double refraction in an equilateral prism. You have to draw it out to work it out. Q4 page 189.Hint:Draw a scale diagram 5cm each side.60 angles.First refracted angle 22Measure 2nd incident 38Calc 2nd refracted 72.6
15Homework - Fishes & Straws Diving birds automatically adjust for refraction effects when hunting fish.This straw shows it clearly.Try it out with a pencil and glass at home then try and explain it with a ray diagram and text…
1613.2 More about refractionSpecification link-up 3.2.3: Refraction at a plane SurfaceWhat happens to the speed of light waves when they enter a material such as water?How is refractive index related to the speed of light waves?Why does a glass prism split sunlight into the colours of a spectrum?
18REFRACTIVE INDEX FROM 1 TO 2 Explaining it...REFRACTIVE INDEX FROM 1 TO 2Optically less dense medium (1)Optically denser medium (2)Waves travel SLOWERWavelength REDUCEDFrequency UNCHANGEDNB: If c = f if speed is less as light finds it harder to get through glass the must go down as well
19Problem drawn out...If we try and think about the problem by drawing out the wave front and looking at triangles..Then considering the constant velocity of the wave which then changes to a new constant velocity.
20r Triangles again... ct=YY’ i XY’ XY’ cst=XX’ Think about how the wave slows when it reaches the medium.We can express the distance it travels YY’ or XX’ simply by s/v=t (constant vel)Then similar triangles result in a sin function for the angle calculations and an expression for a ratio of velocities.Finally we can express this ratio as what we call a refractive index or “how much the light slows down”. Higher is slower!
21Speed of Light...The speed of light, usually denoted by c, is a physical constant, so named because it is the speed at which light and all other electromagnetic radiation travels in vacuum. Its value is exactly 299,792,458 ms-1The speed at which light propagates through transparent materials, such as glass or air, is less than c.The ratio between c and the speed v at which light travels in a material is called the refractive index n of the material (n = c / v).For example, for visible light the refractive index of glass is typically around 1.5, meaning that light in glass travels at c / 1.5 ≈ 200,000 km/s; the refractive index of air for visible light is about , so the speed of light in air is very close to c.Wavelength REDUCEDWaves travel SLOWERFrequency UNCHANGED
2613.3 Total Internal Reflection Specification link-up 3.2.3: Refraction at a plane surfaceWhat are the conditions for total internal reflection?How is the critical angle related to refractive index?Why do diamonds sparkle?
27TIR As the angle of incidence increases towards the critical weak reflectedrayAs the angle of incidenceincreases towards the criticalangle ( glass = 420 ) therefracted ray gets weakerand the reflected ray getsstronger.NB: When light travels from an optically denser medium to a less dense medium, rays are bent away from the normal. The incident substance has a larger refractive index than the other substance
28Critical angle depends upon the refractive indices of the media 12CHENCEIf medium 2 is air, n2 = 1, and so
30Diamonds Sparkle the most! Diamond has a refractive index ofThis means that colours are spread out more.TIR occurs many times inside the diamond before emerging.Can you work out the critical angle?24.44
31Applications of Fibre Optics… End probe containing coherent bundle, incoherent bundle, lens and surgical instrumentsControlsEyepieceLight injected hereENDOSCOPEThis is an endoscope image of the inside of the throat. The arrows point to the vocal chords
32Applications of Fibre Optics… The endoscope is insertedinto a body cavity,which is then illuminatedthrough an in coherentbundle of fibresA lens over the endof the other bundleis used to form animage of the body
33Applications of Fibre Optics… A coherent bundle means that the fibre ends at each endmust be in the same relative positions
34Applications of Fibre Optics… A COHERENT BUNDLE: A bundle of optical fibres in which the relative spatial coordinates of each fibres are the same at the two ends of the bundle. Such a bundle are used for the transmission of images.A NON-COHERENT FIBRE bundle, as you would expect, does not have this precise matrix alignment since they need only transmit light for illumination purposes. They are cheaper to produce.
35Applications of Fibre Optics… PROBLEM: Input light rays cannot be precisely parallel.corecladdingRays taking different paths will take different times to travel along the fibre, resulting in the jumbling of the signal.Solution : Monomode fibre only 5μm in diameter
36Applications of Fibre Optics… Optical fibresin communicationsPurpose?
39Pulse broadening with spectral dispersion Pulse out with spectral dispersionPulse in - white light
40TIR uses….1) Endoscopes (a medical device used to see inside the body):2) Binoculars and periscopes (using “reflecting prisms”)
41Mirage…..Hot Desert Sandair layersWaterfor my hump!Air layers closer to the sand are hotter and less dense. Light from the sky is successively bent until a critical angle is reached and then total internal reflection occurs.A mirage "water" illusion is seen because the mind initially interprets the light rays reaching our eyes as having come along a straight path originating from the ground. Thus, the image of that patch of sky we see "on the ground" is interpreted as a surface "pool of water."
4713.4 Double Slit Interference Specification link-up 3.2.3: InterferenceWhat is the general condition for the formation of a bright fringe?Are Young’s fringes equally spaced?What factors could be (i) increased, or (ii) decreased, to increase the fringe spacing?
48Interference Rules…. For an interference pattern to be observable…. The waves must be of the same type and must meet at a point.The waves are coherent, i.e. the waves from each source maintain a constant phase difference.The waves must have the same wavelength and roughly the same amplitude.The waves must be both either unpolarised or have the same plane of polarisation.NB: Two sources are said to be coherent if waves from the sources have a constant phase difference between them.
49Young's Double Slit Interference - Introduction In 1801, an English physicist named Thomas Young performed an experiment that strongly inferred the wave-like nature of light.Because he believed that light was composed of waves, Young reasoned that some type of interaction would occur when two light waves met.Young passed sunlight through a red colour filter then single slit in a screen to produce coherent light. Then another screen that has a double slit.The results of interference between the diffracted light beams can be visualised as light intensity distributions on the dark film.His hypothesis was that if light were wave-like in nature, then it should behave in a manner similar to ripples or waves on a pond of water. They should either cancel or add at different points depending on the distance from the slit.
51Wavefront View…Single source passes through single then double slits (a few apart). The waves diffract then interference occurs where the beams cross.1st brightfringecentralfringe1st brightfringe
52Results….Here are some possible ideas of what could happen….
53Young's Double Slit Interference - Results Young observed that when the slits were large, spaced far apart and close to the screen, then two overlapping patches of light formed on the screen.However, when he reduced the size of the slits and brought them closer together, the light passing through the slits and onto the screen produced distinct bands of colour separated by dark regions.Young coined the term interference fringes to describe the bands and realised that these coloured bands could only be produced if light were acting like a wave.Pattern is wider as slit gets smaller…
54Young's Double SlitsYoung actually worked out a formula for the theory of how far the fringes are separated (w). It related the distance from S1 or S2 to screen (D) the wavelength of the light () and s (slit spacing – distance between centres of slits S1 or S2)W
55Factors affecting fringe separation Another way to express this idea is that Fringe separation….Which is the same thing!Δx is increased if distance to the screen D is increased.Δx is decreased if slit separation a is increased.Δx is increased as wavelength of light λ is increased
58Test yourself….Use the example data to test out the theory. Try several versions where you can see a change in…Slit separationDistanceWavelengths = wavelength = 590nmD = slit to screen distance = 1mS = slit separation = 0.4 mmW = fringe width = 1.48mmW
59Complex TheoryConsider the diagram above where at P the fringe is observed. Light emitted from S1 arrives later than S2 as it travels further.The difference in the travel is called the path difference.We find that for a wave to add constructively the path difference must be a whole wavelength i.e. m = 1 when S1P-S2P = mHence: light emitted at same time from S1, S2 will arrive in phase at P if reinforcement occursWe find that for a wave to cancel the path difference must be a ½ wavelength i.e. m = 1 when S1P-S2P = (m+0.5)Hence: light emitted at same time from S1, S2 will arrive out of phase at P if cancellation occurs.
61For constructive interference at the screen: ( ie a bright fringe )Wave fronts from S1 and S2 must arrive at the screen in phasewith a path difference of a whole number of wavelengthsFor destructive interference at the screen:( ie a dark fringe )Wave fronts from S1 and S2 must arrive out of phasewith a path difference of half a wavelength
62Complex Theory Point P has been selected so that QP = S2P This means that path difference is S1P – S2P or S1P – S2P = S1QConsider the triangles S1S2Q and MOP.M is midpoint between slits and O the midpoint of brightest fringeThe two triangles are similar in terms of angles PMO and QS2S1We also know that sin = tan when <5So we can say that for triangle OMP and S1S2Q that they equal the same ratios (see eqs)This works for each fringe where P is the m’th bright fringe (m = 0, 1,2,3)w is fringe separation, s spacing, D distance to screen perpendicular.For this to be true s << D.
6313.5 More about Interference Specification link-up 3.2.3: InterferenceWhat are coherent sources?Why are slits used, rather than two separate light sources, in Young’s experiment?What are the roles of diffraction, and interference, when producing Young’s fringes?
64What a great use for a LASER Obi Wan or Obi Non!What a great use for a LASERGreat idea for cutting metals, fighting and generally chopping up any undesirables!However, what the Jedi Knights did not reckon on is E = hf.TASKCan you describe using a Quantum Physics explanation why this is a load of “Hoki Magic” and what would happen if light could behave this way?
66Rhodamine 6G dye (tunable) Wavelength examples...A ruby laser is a solid-state laser and emits at a wavelength of 694 nm. Other lasing mediums can be selected based on the desired emission wavelength (see table below), power needed, and pulse duration. Some lasers are very powerful, such as the CO2 laser, which can cut through steel. The reason that the CO2 laser is so dangerous is because it emits laser light in the infrared and microwave region of the spectrum. Infrared radiation is heat, and this laser basically melts through whatever it is focused upon. Other lasers, such as diode lasers, are very weak and are used in today’s pocket laser pointers. These lasers typically emit a red beam of light that has a wavelength between 630 nm and 680 nm. Lasers are utilised in industry and research to do many things, including using intense laser light to excite other molecules to observe what happens to them.Laser TypeWavelength (nm)Argon fluoride (UV)193Krypton fluoride (UV)248Xenon chloride (UV)308Nitrogen (UV)337Argon (blue)488Argon (green)514Helium neon (green)543Helium neon (red)633Rhodamine 6G dye (tunable)Ruby (CrAlO3) (red)694Nd:Yag (NIR)1064Carbon dioxide (FIR)10600
67SummaryE = hfP = nhfn = number of photons arriving per second
7013.6 Diffraction Specification link-up 3.2.3: Diffraction Why is diffraction of light important in the design of optical instruments?How does the single slit diffraction pattern compare with the pattern of Young’s fringes?What is the effect of the single slit pattern on the brightness of Young’s fringes?
71Laser LightThe most common and inexpensive gas laser, the helium-neon laser is usually constructed to operate in the red at nm.One of the excited levels of helium at eV is very close to a level in neon at eV, so close in fact that upon collision of a helium and a neon atom, the energy can be transferred from the helium to the neon atom.Helium-neon lasers can still be dangerous! An unfocused HeNe laser is dangerous to stare at directly and will blind you. You must be careful to avoid refracted beams whilst conducting experiments.
74Single SlitThis is an attempt to more clearly visualise the nature of single slit diffraction. The phenomenon of diffraction involves the spreading out of waves past openings which are on the order of the wavelength of the wave.
75Slit WidthOne of the characteristics of single slit diffraction is that a narrower slit will give a wider diffraction pattern as illustrated in the images.
76Or can view as a water wave (single slit diffraction) Semi circularwave frontsFirst minima & maximabecome visibleDiffraction is the spreading of wavefronts around corners and obstacles.If the slit gets narrower diffraction increases.If the wavelength increases diffraction increases.
77Single Slit Diffraction The diffraction pattern here is taken with a helium-neon laser and a narrow single slit. We can see….This means that if you project a light source i.e. laser, measure the width of the central blob you can work out the wavelength of the light.The formulae you need for the exam is called fringe spacing…
78Single Slit Diffraction The diffraction pattern here is taken with a helium-neon laser and a narrow single slit. We can see….This means that if you project a light source i.e. laser, measure the width of the central blob you can work out the wavelength of the light.The formulae you need for the exam is called fringe spacing…But remember that the central fringe is twice this!a = w in this case
80It can be shown that the first minima occurs when sin Ə = λ/a . Central maximaQ1 Find the angle at which the first minima occurs using microwavesof wavelength 3 cm when directed towards a gap of:1) 6cm2) 4cmQ2 Find the angle at which the first minima occurs using lightwavesof wavelength 500 nm when directed towards a pupil of diameter:1) 6mm2) 4mm
81* central fringe is twice as wide as the other fringes Visit :Blue light has narrower fringesSo cameras and microscopescan see more detailusing blue filtersPoints to note:* central fringe is twice as wide as the other fringes* intensity decreases from the centre* Central Fringe width W = λ/a x 2D
84Experimental observations from the double slit i) For a pair of slits 0.5 mm apart:λ red » λ blueii) Using white light, fringes appear from all the various wavelengths presentand do not overlap exactly, hence coloured fringes* Inner fringes are tinged with blue on the insideand red on the outsideDiffraction is the spreading of wavefronts around corners and obstacles.If the slit gets narrower diffraction increases.If the wavelength increases diffraction increases.
85Experimental observations from the double slit cont’d iii) Fringes obtained using slits 0.5 mm apart drawn with different widthsThe double slit interferencepattern is modulatedby the single slit pattern(a)(b)(b) thick slitsMissing fringes(a) thin slitsDiffraction is the spreading of wavefronts around corners and obstacles.If the slit gets narrower diffraction increases.If the wavelength increases diffraction increases.
86Double & Single Slit Interference Same double-slit assembly (0.7mm between slits); in top image, one slit is closed. Note that the single-slit diffraction pattern — the faint spots on either side of the main band — is also seen in the double-slit image, but at twice the intensity and with the addition of many smaller interference fringes.
87NB: The double slit fringes are still in the same place Summary of PatternsThe double slit pattern is superimposed on the much broader single slit diffraction pattern.The bright central maximum is crossed by the double slit interference pattern, but the intensity still falls to zero where minima are predicted from single slit diffraction.The brightness of each bright fringe due to the double slit pattern will be “modulated” by the intensity envelope of the single slit pattern.NB: The double slit fringes are still in the same placeSingle slit patternDouble slit pattern
88Single Slit and Double Slit? A diffraction pattern formed by a real double slit. The width of each slit is much bigger than the wavelength of the light. This is a real photo.This idealised pattern is not likely to occur in real life. To get it, you would need each slit to be comparable in size to the wavelength of the light, but that's not usually possible. This is not a real photo.A real photo of a single-slit diffraction pattern caused by a slit whose width is the same as the widths of the slits used to make the top pattern.
9513.7 The Diffraction Grating Specification link-up 3.2.3: DiffractionWhy does a diffraction grating diffract monochromatic light in certain directions only?If a coarser grating is used, what is the effect on the number of diffracted beams produced and on the speed of each diffracted beam?How can we determine the grating spacing for any given grating, if it is not known?Derive the Generic Diffraction Formulae
96Laser Diffraction!Model = 1 x 10-6m1.6 microns ,dvd is .74 micronsExperiemental 5.4 micronsCan you work out the spacing of lines on a CD if light is nanometres. (0.5 to 0.9mW Power)L = 229.5cmOpp =55.5cm57.1cm58.3cm(three repeats 2nd order)
97Light through a diffraction grating... If we pass light rays through a diffraction grating. Which is made up of very small slits in a regular pattern we find that there is a pattern of dots formed on a screen. This is because the wave fronts interfere with each other destructively and constructively.Simply put a wave front which emerges at point P is reinforced by another wave front acting in phase at point Q and then R. The result is a wave front along Y &Z
98Light through a diffraction grating... If we consider the triangle as shown in the zoomed in diagram we can see that an angle can be worked out from knowing QP and QY and then similar triangles rule used to find out the angle of the rays as they appear on a screen.We can use the idea of constructive interference and take the idea that from trigonometry &But we also know that the length QY should be the length of a whole wavelength and also that QP is the slit spacing.
99Light through a diffraction grating... We can also now think about the idea that this happens not only once but several times or orders.Each time we have a whole wavelength we get the construction of a new wave front or “n” wave front. Our formulae can be changed to;d = slit spacing = angle from normaln = order of diffraction = wavelength of the light
100PracticalSetup a laser light source and use the formula below to check the wavelength of the light. Your quoted value is 632.8nm(632.8 x 10-9m).Your teacher will give you a slit which has 300 lines per mm or a spacing d = 3.33 x 10-6m. Use this one at first to check the calculation. Then try and take some other readings.d = slit spacing = angle from normaln = order of diffraction = wavelength of the light
102Use of Emission Spectrums? How do we know which elements stars are made up from?How do we know the age of stars?Scientists can begin to answer these questions once they have an understanding of line spectra.
103What are we talking about? When we talk about “Line Spectra” for an atom we simply mean that an atom can absorb or emit radiation at certain frequencies. If we look at the frequencies emitted or absorbed we can see a spectrum with omissions.656nm
104There are three types that you might see.... How are they made?There are three types that you might see....
106MSpectral GasesImage we look towards a star and then pass its light through a cold gas then then a prism to make an absorption spectrum what elements would be present?Star X
107M Spectral Gases & Shift Here are three spectral diagrams from a star. The first one is a reference diagram.The second two are shifted across to the red end of the spectrum. Hence, they are moving away from us. We can tell how fast they are moving by the shift. The middle band is actually shifted by a distance of 100Å (angstrom) or 100 x m. (10nm).This translates to a speed of 24,000 km/hour or 15,000 mph. The bottom band is shifted by 760Å or (76nm) which translates to a speed of 136,000km/hour or 84,000 mph.
108Spectral Gases Conclusion MSpectral Gases ConclusionSo these pictures show us that the lower stars are moving away from usAlso the further they are away the faster they travel.