Optics Lecture Notes Full Set PDF

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Optics Full Module Lecture Notes...

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School of Chemical & Physical Sciences Physics & Astrophysics PHY-20027 Optics and Thermodynamics 2020 − 2021 Optics Lecture Notes Tutor: Dr. Joana M. Oliveira ([email protected], LJ 1.48)

Module Information Aims: This module aims to develop a deep understanding of the theoretical principles and applications of the key physical topics of geometrical and physical optics. It also aims to develop problem solving and experimental skills in the context of these topics, and further develop a critical approach to the physical and natural world. Learning outcomes: At the end of this module you should have enhanced your knowledge of geometric and physical optics by: • developing a comprehensive and contextualised understanding of key theoretical and experimental concepts; • using a wide range of established techniques for critical analysis of numerical calculations and problem solving; • extending your abilities in the execution and reporting of laboratory work, increasing communication skills in the context of scientific work. Recommended texts: The lectures and the accompanying lecture notes should cover all the required topics. Your standard Physics textbook: • “University Physics with Modern Physics”, Young H.D. and Freedman R.A. (Pearson/Addison Wesley, 14th edition 2016) includes many useful examples and exercises (the lecture materials will signpost the appropriate sections). To complement these, you can look at the following books, all available in the library: • “Introduction to Optics”, by Pedrotti F.L., Pedrotti L.M. and Pedrotti L.S. (PrenticsHall, 3rd edition 1993) • “Optics”, by Hecht E. (Addison-Wesley, 4th edition 2002) • “Fundamentals of Optics”, by Jenkins F.A. and White H.E. (McGraw Hill, 1981); for an historical perspective. Assessment: Examinable material includes all material covered in the lectures, lecture notes on Blackboard and material in the problem classes. There are 3 assessment components: • Unseen exam (60%): This course will be examined during one two hour paper, one half covering Optics and the other half Thermodynamics. • Assessed problem sheets (20%): These are to be issued on weeks 2, 5, 8 and 10, deadlines respectively weeks 4, 7, 10, 12. Separate sheets with different deadlines are issued for Optics and Thermodynamics. • Laboratory component (20%): Laboratory book and reports contribute to this grade component. These reports should be submitted on weeks 8 and 12. They refer to two key experiments and more details will be provided in the first Laboratory session. i

Problem classes: Four problem classes are devoted to the Optics/Thermodynamics module, on weeks 3, 6, 9 and 11. Half of each session is dedicated to Optics. Syllabus: You will find here a brief listing of topics covered in this module. Some of these topics are covered in depth in the lectures, while others are touched upon in the associated laboratory sessions. • Geometric Optics: reflection, refraction, refractive index, dispersion, Snell’s law, total internal reflection, thin lens, focusing and ray tracing, eyepieces, microscope and telescope; • Interference: Young’s double-slit experiment, Michelson interferometer, multiple beam interference and Fabry-Perot interferometer; • Properties of Electromagnetic Waves: temporal and spatial coherence, polarisation and Huygens’ principle; • Fraunhofer diffraction: single slit, multiple slits, diffraction grating and resolving power; • Lasers: spontaneous and stimulated emission, population inversion and pumping, optical cavities.

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Contents 1 Introduction

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2 Geometrical optics 2.1 Reflection, refraction and refractive index . . . . . . . . . . . . . . . . . . . 2.2 Huygens’ principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Huygens’ construction applied to the laws of reflection and refraction 2.2.2 Extensions to Huygens’ principle . . . . . . . . . . . . . . . . . . . 2.3 Fermat’s principle applied to reflection and refraction . . . . . . . . . . . . 2.4 Optical path length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Lenses, focusing and ray tracing . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Useful definitions and sign rules . . . . . . . . . . . . . . . . . . . . 2.5.2 Focusing by an aspherical refracting interface . . . . . . . . . . . . 2.5.3 Focusing of paraxial rays by a spherical interface . . . . . . . . . . 2.5.4 Lenses and the lensmaker’s equation . . . . . . . . . . . . . . . . . 2.6 Thin lenses in common optical systems . . . . . . . . . . . . . . . . . . . . 2.6.1 The eye and the magnifying glass . . . . . . . . . . . . . . . . . . . 2.6.2 The microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 The telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Interference 3.1 Diffraction and Interference . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Young’s double slit experiment . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . .

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4 The properties of electromagnetic waves 4.1 Maxwell’s equations of electromagnetism . . . . . . . . 4.2 Polarisation . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Methods for producing polarised light . . . . . . . . . . 4.4 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Temporal coherence. . . . . . . . . . . . . . . . 4.4.2 Spatial coherence. . . . . . . . . . . . . . . . . . 4.5 A more detailed look at Young’s double slit experiment

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5 Diffraction 5.1 Diffraction from a single slit: Fraunhofer diffraction pattern 5.1.1 Intensity of the diffraction pattern for a single slit . . 5.1.2 Intensity minima and maxima . . . . . . . . . . . . . 5.2 Two slits of finite width . . . . . . . . . . . . . . . . . . . . 5.3 Multiple slits and the diffraction grating . . . . . . . . . . .

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Introduction

The understanding of the nature of light eluded mankind until the early twentieth century. Isaac Newton regarded light rays as streams of very small particles emitted by a source of light travelling in straight lines. Christiaan Huygens on the other hand considered light to be a disturbance propagating in a medium called “ether”. Many experiments continued to demonstrate this wave-like nature of light, until James Clerk Maxwell postulated that light was an electromagnetic wavefront that propagates by means of mutually supporting magnetic and electric fields (Electromagnetism). Of course, light interacts with matter and this can only be understood if the energy carried by the light waves is packaged in discrete bundles called photons or quanta (Quantum theory), i.e. light also has particlelike behaviour. Finally, in the early 30s the theory of Quantum Electrodynamics is able to reconcile the dual (wave and particle) nature of light. Light is thus a transverse electromagnetic wave which propagates through space at a speed c = 2.997925 × 108m s−1 (in vacuum). Light possesses a wavelength λ and a frequency ν which are related by c = ν × λ. What we call light is just one part of the electromagnetic spectrum, i.e. radiation with λ ∼ 0.5 µm. You are aware of other forms of radiation, for instance X-ray (λ ∼ nm) and radio (λ ∼1 mm–1 km) waves. Optics is the branch of physics that deals with the behaviour of light and other electromagnetic waves. A full treatment of light propagation can be achieved with Maxwell’s electromagnetic theory, but in many instances such approach is not necessary and simplifications can be used. Geometrical optics can be used for instance to explain the rectilinear propagation of light and the formation of shadows, i.e. the behaviour of a light beam as it travels through apertures and around obstacles. However, if one examines the interface between dark and light at a shadow’s edge, a fine structure can be seen that suggests that light “bends and spreads” around the edges of obstacles (an effect called diffraction). It is at this level of detail that geometrical optics fails and we enter the domain of physical optics that deals with the wave behaviour of light.

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Geometrical optics

Geometrical optics (also called ray optics) is thus a simplified model of light that is useful for describing the formation of the image of an object. It generally gives a reasonable description of light propagation through any system whose physical size is large compared to the wavelength of the light. The basic principles of geometrical optics are: in free space light travels along rays, that are imaginary lines along the direction of travel of the wave (perpendicular to the wavefront1 ); rays are emitted in all directions from (infinitesimal) point sources; extended objects can be considered as comprising infinitely many point sources; rays striking a surface may change direction by reflection or refraction.

2.1

Reflection, refraction and refractive index

Let us consider medium 1 and medium 2 on either side of a transparent smooth surface or interface (Fig. 1). Light rays travel through medium 1 to strike the interface, and are reflected and refracted (or transmitted) according to the laws formulated below. 1

A wavefront is a surface of constant phase.

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i r i

t

t

Figure 1: Laws of reflection and refraction (left) and total internal reflection (right) (figures from Serway and Jewett).

Law of reflection. When a ray of light is reflected from a surface the reflected ray remains within the plane of incidence (plane normal to the surface) and the angle of reflection θr is equal to the angle of incidence θi (Fig. 1) — angles are normally measured with relation to the normal to the surface. Law of refraction. The law of refraction or Snell’s law states that when a ray of light travels from one medium to another with a different refractive index, it is deflected in such a way that it remains in the plane of incidence and the angle of refraction (or angle of transmission) θt is given by: ni sin θi = nt sin θt , where ni and nt are the refractive indices of media 1 and 2. The refractive index of an optical material n is the ratio of the speed of light in vacuum c and the speed of light in the material v, i.e., c n= . v Light always travels more slowly in a material than in vacuum for which n = 1. Snell’s law was formulated long before the discovery of the relation between n and c. The refractive index n depends on the wavelength of the light, since the speed of light in a medium other than vacuum also depends on the wavelength of the light. This effect is called dispersion. The wavelength of a wave is also changed upon refraction. Having described how light is partially reflected and partially transmitted at the interface between two materials, under particular circumstances all light can be reflected back from the interface, with none being transmitted even though the material is transparent. The critical angle θc is the incidence angle for which the angle of refraction is θt = 90◦ (Fig. 1). From Snell’s law this means that sin θc = 3

nt . ni

For incidence angles θi > θc , the incident rays experience total internal reflection. Note that this only occurs if nt < ni , for instance in a reflection from glass to air. There are many interesting applications of this phenomenon, for instance in the interior of optical fibres. Optical fibres have a core of glass or plastic surrounded by a jacket of a material with low refractive index; within this core light is subject to numerous events of total internal reflection, assuring that light propagates without (or with minimal) loss of intensity (or energy) due to refraction out of the core.

2.2

Huygens’ principle

Huygens assumed that every point on a “wavefront” disturbance could be considered as a “vibrating particle” which is itself a point source for further waves. Although this was done well in advance of Maxwell’s theory, it still remains a useful concept in optics. Huygens proposed that “All points on a wavefront can be considered as point sources for the production of spherical secondary wavelets. After a time t the new position of the wavefront will be the surface of tangency to these secondary wavelets.” Figure 2 illustrates the basic idea for the production of a new wavefront A′ B ′ after time t from an old front AB, from what can be thought of as an infinite number of spherical wavelets (with radius v t). The shape of the new wavefront at a position v t is the line tangential to all the wavelets. Huygens’ principle provided an explanation of the laws of reflection and refraction and the behaviour of wavelets accounted for the failure of the principle of rectilinear propagation through small apertures. However, Huygens’ construction allows for the contribution only from the tangential point of the preceding wavelets, all other points on the wavelets were assumed to make no contribution to new wavefronts. Thus Huygens’ principle neglected the possibility of diffraction of light into the region of shadow for instance. It also arbitrarily excludes the existence of “backwaves” travelling from the wavefronts towards the source. These weaknesses in the formulation were later addressed in the Fresnel and Kirchhoff extensions to Huygens principle. 2.2.1

Huygens’ construction applied to the laws of reflection and refraction

Law of reflection. This law states that the angles of incidence and reflection are the same and the incident and reflected rays, and the normal to the surface all lie in the plane of incidence. This can be simply illustrated by drawing a Huygens construction as shown in Fig. 3 (left). Huygens’s principle needs to be modified slightly to accommodate the case in which a wavefront AC encounters a plane surface X Y at an angle. The rays AD, BE and CF (perpendicular to the wavefront) have an angle of incidence relative to P D (the normal to XY at D) that is θi . Note that points along the wavefront do not arrive at the surface at the same time (the speed of light is constant in the medium and the path length travelled by the rays is different), and allowances need to be made for these differences when building the wavelets that define the reflected wavefront. If the reflective surface was not present, the wavefront AC would produce the wavefront GI at the instant ray CF reaches the interface at I. The presence of the surface means that in the time it takes ray CF to travel from F to I, ray BE has progressed from E to J and then travel a distance equivalent to JH after reflection. We then can draw a wavelet of radius J H = J N centred at J above 4

Figure 2: Propagation of plane (left) and spherical (right) waves using Huygens’ construction.

i

=

r

i

r

ni

nt t i t i t

i

Figure 3: Huygens’ constructions illustrating the laws of reflection (left) and refraction (right).

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the reflective surface. Similarly, ray AD would have travelled a distance corresponding to DG above the surface, i.e. we draw a wavelet centred at D with radius DM = DG. The new wavefront must be tangent to all new wavelets (i.e., must include points M and N ) and must include point I, the new wavefront is then I K. The reflected rays are perpendicular to the wavefront — DL is a representative ray of the reflected wavefront. What about the angle of reflection θr ? If we look at Fig. 3, we can see that the angles ADX and IDG are identical and ADX = IDG = (90◦ −θi ). We also see that the triangles DIG and DIM are also identical (DG = DM ). This implies the following angles are the same:, IDM = IDG = ADX = (90◦ − θi ). θr is just (90◦ − IDM) and thus θr = θi . Law of refraction. This law relates the angles of incidence and refraction (or transmission) to the refractive indices of the materials. To apply Huygens’ principle to the law of refraction we must take into account the fact that the speed of light varies depending on the medium it is travelling through: the speed of light in a medium of refractive index n is given by v = c/n. In the construction in Fig. 3 (right) we consider refraction at an interface between materials having refractive indices ni and nt . The speeds of light in these two materials are thus vi = c/ni and vt = c/nt respectively. As before the rays AD, BE and CF have an angle of incidence θi . The points D, E and F arrive at D, J and I on the surface XY at different times. In the absence of the refraction surface, the wavefront GI is formed when ray CF reaches point I (ray AD would have travelled the distance DG). As ray CF progresses from F to I in time t, ray AD has penetrated a different medium, with a different speed: instead of travelling DG = vi t it travels DM = vt t and DM = vt t = vt

DG ni = DG, vi nt

thus we draw a wavelet of radius DM centred at D. In the same way we draw a wavelet of radius nnit JH centred at J. As before the new wavefront KI is tangential to the wavelets, i.e. it includes points M, N and I . From the diagram the angle IDF = θi . Similarly the angle DIM = (90◦ −MDI) = θt . This implies that: sin θi = sin IDF =

FI DM and sin θt = sin DIM = DI DI

or

DG nt sin θi FI = = = DM sin θt DM ni that is just Snell’s law. 2.2.2

Extensions to Huygens’ principle

In the previous sections we have seen how Huygens’ principle was used to explain the reflection and refraction processes. Huygens’ principle however contained many deficiencies: it did not account for the diffraction effects observed when light rays encounter apertures or obstructions, and it provided no really convincing explanation for the absence of “backwaves” which in principle would allow the generation of wavefronts going back towards the source of light but are actually not observed. Huygens’ principle was extended by Fresnel who attempted to formulate a more general theory based on the superposition of secondary waves, taking into account the amplitude and the phase of the wavelets. Fresnel introduced an “obliquity factor” which made 6

the contributions from any wavefront source fall off with the angle from the normal to the wavefront. This superposition also provided an explanation of observed diffraction fringes. The Fresnel extension to Huygens’ principle suffered however in that: i) there was no particular physical basis for choosing the form of the obliquity factor and therefore of excluding the presence of radiation proceeding backwards from the primary wavefront; ii) there were difficulties in predicting the correct amplitude of the secondary wavelets; iii) the theory gave an incorrect phase for the resultant disturbance. Huygens’ principle was further developed by Kirchhoff who used a rigorous treatment based on differential wave equation to produce an expression giving the superposition of secondary wavelets which removed the difficulties inherent in the Huygens-Fresnel formulation. In its application to spherical waves the Kirchhoff formula contains an obliquity factor2 of the form 1 + cos θ f= , 2 which is clearly f = 1 in the forward direction and f = 0 in the backwards direction.

2.3

Fermat’s principle applied to reflection and refraction

The laws of geometrical optics can also be derived using Fermat’s principle. Fermat’s principle3 states that the path...