Geometric Optics PDF (Batna 2 University)

Geometric Optics 1.0 September 2024 Dr. Soumia LEBBAL Batna 2 University Mustapha Ben Boulaid Faculty of Medicine Department of Medicine Email: [email protected] Attribution - NonCommercial : http://creativecommons.org/licenses/by-nc/4.0/fr/ Table of contents I...

Geometric Optics 1.0 September 2024 Dr. Soumia LEBBAL Batna 2 University Mustapha Ben Boulaid Faculty of Medicine Department of Medicine Email: [email protected] Attribution - NonCommercial : http://creativecommons.org/licenses/by-nc/4.0/fr/ Table of contents I - Principles 3 1. Rectilinear Propagation of Light Principle....................................................... 4 1.1. Light as an Electromagnetic Wave........................................................................................ 5 1.2. Refraction Index of a Medium................................................................................................ 7 1.3. Cauchy's Law.......................................................................................................................... 7 2. Snell-Descartes' Laws....................................................................................... 8 2.1. Snell's Laws............................................................................................................................. 9 2.2. Deviation of Light.................................................................................................................. 10 2.3. Total Internal Reflection - Critical Angle.............................................................................. 10 2.4. Dispersion of Light................................................................................................................ 13 3. Notions of Objects/Images and Stigmatism................................................ 13 3.1. The Optical System............................................................................................................... 13 3.2. Notion of Objet / Image........................................................................................................ 14 3.3. Stigmatism and Approximate Stigmatism.......................................................................... 16 Complementary resources 17 Glossary 19 Abbreviation 20 2 Dr. Soumia LEBBAL I Principles 1. Introduction What is Light? The first serious theories regarding the nature of light were formulated in the 17th century. Two seemingly contradictory theories emerged: one based on the corpuscular aspect and the other on the wave mechanism. These theories sparked a controversy that lasted until the early 20th century, as each explained some phenomena but left others unexplained or even contradicted by them. In the 17th century, Newton's corpuscular theory proposed that light was composed of particles emitted by luminous objects. According to this theory, color differences were due to variations in particle size. However, it failed to explain phenomena like interference and diffraction, where light waves overlap or bend around obstacles. Hooke introduced the wave theory of light in 1665 to explain interference (cf. p.3), and Huygens expanded this theory, proposing that each point on a light wave acts as a source of spherical waves. Later, Young and Fresnel (early 19th century) developed it further, explaining interference patterns and associating wave frequency with color. In 1864, Maxwell revolutionized the understanding of light by describing it as an electromagnetic wave, composed of oscillating electric and magnetic fields traveling at the speed of light. His theory laid the foundation for the electromagnetic spectrum, which ranges from radio waves to gamma rays, with visible light occupying a small portion. Despite the success of wave theory, it couldn't explain certain phenomena, such as the photoelectric effect, where light ejects electrons from a metal. This led to the early 20th-century development of quantum theory. In 1905, Einstein introduced the concept of photons—light as discrete energy quanta. Compton later confirmed this idea in 1921, completing the understanding of light as both a wave and a particle. Light interference Dr. Soumia LEBBAL 3 Principles 2. Rectilinear Propagation of Light Principle At the heart of geometric optics is the principle of rectilinear propagation, which states that:  Fundamental In a transparent, homogeneous p.19, and isotropic medium (like air or vacuum p.19), light propagates p.19 in the form of straight-line rays. This behavior can be easily observed when a flashlight is shone in a dark room, producing a straight beam of light. Light Piercing Through Darkness  Definition A medium is said to be: Transparent: if it allows light to pass through without attenuation; otherwise, it is called opaque. Homogeneous: when the refractive index is the same at all points; otherwise, it is called inhomogeneous. Isotropic: when the refractive index is the same in all directions; otherwise, it is called anisotropic.  Note Other old principals that are worth mentioning which are based on simple common observations such as the formation of shadows or solar eclipses include1: Principle of the Reversibility of Light: If light follows any path from point A to point B (including through an optical system), then light can follow exactly the same path in reverse from B to A. It is said that the path followed by the light is independent of the direction of its propagation. Principle of Independence of Light Rays: When two light rays meet, they do not interact: one light ray cannot be deflected by another light ray. 4 Dr. Soumia LEBBAL  Principles 2.1. Light as an Electromagnetic Wave  Definition An EM p.20 wave is a self-propagating transverse wave6consisting of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of wave propagation. These waves travel through space, carrying energy and momentum, at the speed of light, in vacuum, and at a much less speed in a material medium. An EM wave Light is considered, within this approach, as an EM p.20 vibration that propagates in vacuum at a speed C = 3 × 108 m/s and is characterized by a wavelength λ, which is the distance traveled by the vibration during a time period T , and a frequency f , the number of cycles per second. The frequency and wavelength are related by the equation: C 1 λ = = f T where: The wavelength λ (spatial period) is the distance between two successive peaks (or troughs) of the wave and is a key measure of the wave's spatial extent. The (time) period T is the time it takes for one complete cycle of the wave to pass a given point, representing the temporal aspect of the wave's oscillation. Properties of an EM Wave  Note By definition, the wavelength is related to the frequency by the formula: the propagation speed λ = , where the speed of the propagation of the wave depends on f requency the medium it travels. In vacuum, v = C. Dr. Soumia LEBBAL 5 Principles  Fundamental In a material medium, the frequency of light always remains constant, but its wavelength changes with the medium due to the change in propagation speed v. v λ = f In the MKS system p.20, these physical quantities are expressed in: Period T : in s. Frequency f : in s− 1 ≡ Hz. Wavelength λ: in m. a) The Electromagnetic Spectrum The EM p.20 spectrum is almost entirely invisible to the human eye (Figure 1) (cf. p.6), except for a small portion called the visible spectrum, which corresponds to the range 400 nm < λ < 750 nm (1 nm = 10 9 m). This latter consists of the colors that make up a − rainbow, i.e. ROYGBIV p.20. The EM spectrum  Definition Light is said to be polychromatic when it consists of multiple wavelengths, and monochromatic when it consists of a single wavelength. White light contains all the wavelengths of visible light.  Note Color is the visual and mental perception of the appearance of light, based on the spectral distribution of the light. 6 Dr. Soumia LEBBAL  Principles 2.2. Refraction Index of a Medium While the rectilinear propagation of light principle generally holds, an intriguing question arises: does light always stick to a straight path? Answer: Yes and No! Light does follow a straight path as long as the medium’s properties remain unchanged. Let's explore the situations where the straight-line path of light changes. Light exhibits fascinating behaviors when it encounters different surfaces or media. In fact, the electromagnetic wave, and thus light, propagates in a vacuum at the speed C = 3 × 10 m/s. Its propagation in a medium, however, gets affected by the refractive index n 8 of that medium, due to which, the velocity p.19of the EM wave slows down to a value v < C. As a consequence, light changes its direction: it either reflects or refracts.  Definition The refractive index of a medium is defined as the ratio between the speed of light in a vacuum C and its speed in that medium v: C n = v  Fundamental The refractive index is a dimensionless physical parameter that is greater than or equal to 1 (n ≥ 1).  Example Values of n in some media 2.3. Cauchy's Law While the behavior of a light ray is governed by the medium it passes through, another intriguing question arises: Do all wavelengths of light behave uniformly in a given medium? And does the refractive index interact uniformly with all wavelengths of light? The answer is revealed through Cauchy's Law, which states that the refractive index of a medium ( n ) varies with wavelength. Dr. Soumia LEBBAL 7 Principles  Definition Cauchy Law is an empirical relation that gives the refractive index n as a function of the wavelength of light for a transparent medium B n(λ) ≃ A + +... 2 λ 0 Where: A and B are experimental constatnts that depend on the medium. λ0 is the wavelength in the vacuum. Specifically, it indicates that the refractive index decreases as the wavelength increases, leading to different degrees of bending for various colors of light. This results in a phenomenon, known as dispersion, in the case of polychromatic radiation (see below). 3. Snell-Descartes' Laws when encountering a surface separating two media (with two different refractive indices), Snell's Law, or Snell-Descartes Law, explains how light rays bend as they transition from one medium to another.  Definition Reflection occurs when light bounces off a surface, following the law of reflection. This phenomenon is easily observed when looking into a mirror.  Definition Refraction occurs when light moves from one medium to another, like from air to water or glass. The change in speed as light enters a new medium causes it to bend, following the law of refraction, with the degree of bending determined by the medium's refractive index. A common example of refraction is a straw looking bent or broken when it's partly submerged in a glass of water.  Definition A diopter is the surface separating two transparent media with different refractive indices, through which, light can either be refracted or reflected. It is more often of a form planar or spherical. 8 Dr. Soumia LEBBAL  Principles 3.1. Snell's Laws  Note Light's behavior depends on the type of the surface (optical system) it encounters: In a diopter, which is a surface separating two transparent media with different refractive indices, light can undergo both reflection and refraction. Refraction occurs as light passes through the diopter, bending according to the material's refractive index, while some portion of light is also reflected off the surface. A diopter is more often of a form planar or spherical. In contrast, mirrors operate solely through reflection. When light strikes a mirror, it bounces off the surface without passing through it. allowing mirrors to form clear, accurate images based on the reflected light.  Fundamental We consider a light ray propagating in a transparent, homogeneous and isotropic medium, and falling on a surface (diopter or mirror):2 (Figure 2) (cf. p.9) The ray incident on the surface is called the incident ray. The plane of incidence is the plane containing the incident ray and the normal to the surface at the point of incidence I. The angle of incidence (i or i1 ​) is the angle between the incident ray and the normal to the surface. The angle of reflection (i′ or i′1 ​) is the angle between the reflected ray and the normal to the surface. The angle of refraction (i2 ​) is the angle between the refracted ray and the normal to the surface. n1and n2 ​ are the refractive indices, respectively, of the first medium and the second medium. Refraction and refraction of light through: a) a planar dioptre, b) a plane mirror Law of Reflection: " The angle of reflection is equal to the angle of incidence. "i′ and i1 ′. ′ = i = i1 Law of Refraction: Dr. Soumia LEBBAL 9 Principles " The angles of incidence and refraction are related by the equation ": n1 sin i1 = n2 sin i2 3.2. Deviation of Light  Definition The angle of deviation is the angle between the direction of the (initial) incident ray, and the "bent" ray (whether is is by reflection or refraction).  Example The angle of deviation(d) is the angle between the direction of the incident ray and the reflected ray (see (Figure 4)) (cf. p.18). Deviation of light through a plane mirror d = π − 2i where i is the angle of incidence and i. = r 3.3. Total Internal Reflection - Critical Angle When a ray of light passes from one medium n1 to another n2 , most of it undergoes a refraction. From Snell's law: sini n1 , 2 n1 sin i1 = n2 sin i2 ⇒ = sini1 n2 10 Dr. Soumia LEBBAL  Principles we distinguish two cases: Case 1: Let us suppose that n1 < n2 (see (Figure 4)) (cf. p.17). From the Snell-Descartes' relation , we get: i2 < i1 ​, which means that the refracted ray approaches the normal. We know that ∀α, |sinα| ≤ 1, so if we vary i1 ​ from 0 to 90°, it is clear that i2 will vary from 0 to a limiting (critical) angle ℓ = i2c ,​such that: Critical angle n1 n1 ∘ n1 sin(90 ) = n2 sin(i2c ) ⇒ n1 = n2 sin(i2c ) ⇒ sin(i2c ) = l ≡ i2c = arcsin( ) n2 n2 π This angle is called the critical angle of refraction. It is the angle of refraction for which the angle of incidence grazes p.19 the surface (i1 = ). 2 Dr. Soumia LEBBAL 11 Principles Case 2: Let us suppose now that n1 > n2 ​(see (Figure 5)) (cf. p.18). In this case, the Snell-Descartes relation n1 sin i1 = n2 sin i2 shows that i2 > i1 , meaning that the refracted ray moves away from the normal. π For a grazing refraction (i2 = ): 2 n2 i1 ≡ i1c = arcsin(. ) n1 i1c is called the critical angle of incidence. Let us vary i1 ​starting from 0: n2 As long as i1 < i1c = arcsin ​, the ray undergoes both a refraction and a reflection. n1 If i1 > i1c ​, the refracted ray can no longer exist because we would obtain sin i1 > 1 ! Total reflection In fact: n2 i1 > i1c ⇒ sin(i1 ) > sin(i1c ) ⇒ sin(i1 ) > > 1 n1 In this case, the light ray undergoes only a reflection at point I. This reflection is called total reflection, and the critical reflection angle i1c is also referred to as the critical angle of incidence. Its value depends on the nature of the diopter. 12 Dr. Soumia LEBBAL  Principles 3.4. Dispersion of Light Dispersion of light refers to the refraction phenomenon where white light (polychromatic) is separated into its constituent colors or wavelengths when it passes through a medium. This occurs because different colors of light have different wavelengths (frequencies) and, as a result, they refract (bend) by different amounts when they enter or exit the medium, according to Cauchy's formula. When light enters a medium like glass (prism) or water, it slows down and bends. The amount of bending depends on the wavelength of the light; shorter wavelengths (blue and violet) bend more than longer wavelengths (red and orange). Because of this varying degree of bending, white light (which contains all visible wavelengths) spreads out into a spectrum of colors. This effect is famously demonstrated with a prism, which can split white light into a rainbow of colors (ROYGBIV p.20). Dispersion of white light through a prism Dispersion is the reason why a rainbow forms when sunlight passes through raindrops, and it’s also a key principle behind the operation of prisms and diffraction gratings in optical devices. 4. Notions of Objects/Images and Stigmatism To understand how we see and interpret the world, it is crucial to study how light interacts with objects and surfaces to create visual representations. This involves exploring the core concepts of objects and images in geometric optics. 4.1. The Optical System  Definition An Optical System, (OS) refers to any element or set of elements that can alter the path of light. These systems may include lenses, mirrors, prisms, or more complex assemblies that refract, reflect, or diffract light to modify its direction, focus, or intensity. This system enables the formation of an image, which may be real or virtual, of a luminous object. It consists of a set of surfaces, generally of revolution p.19 (centered systems with a common axis of revolution), that separate (except in the case of mirrors) transparent media, which are usually homogeneous and isotropic, with varying refractive indices (Figure 6) (cf. p.14). It is called dioptric if it contains only diopters (e.g., the eye). It is called catadioptric if it contains a combination of diopters and mirrors (e.g., a telescope). It is called centered if it has a symmetry axis of revolution: the principal optical axis. Dr. Soumia LEBBAL 13 Principles Schematic representation of an optical system 4.2. Notion of Objet / Image A point source (light point) is placed at A. This point emits a diverging conical p.19 beam onto the entrance face. Some rays pass through the system and form an emergent beam. If this beam is conical with its apex p.19at A′ , we say that A′ is the image of A through the system4.4 A and A′ are referred to as the conjugates of one another (conjugate pointsin the optical system).  Definition We call the source of light rays, whose propagation through a given OS p.20 is studied the object. It either emits light, such as a lamp or a candle, or reflects it.  Definition An image is defined with respect to a given optical system. It is the visual reproduction of an object created by light rays after they pass through it p.20. In practice, the image of a point object is point-like, and the image of an extended object is extended. a) Real / Virtuel Object If the incident beam on the optical system is divergent, the object point is the intersection of the incident rays. Therefore, a real object is the source of light for the system. The real object is located before the entrance of the optical system (see (Figure 7).) (cf. p.17) 14 Dr. Soumia LEBBAL  Principles If the incident beam on the optical system is convergent, the object point is the intersection of the extensions of the incident rays. The virtual object is located after the entrance of the optical system (see (Figure 8)) (cf. p.17). b) Real / Virtuel Image If the beam emerging from the OS is convergent, meaning that the rays actually pass through point A′ : the image is real (see (Figure 9)) (cf. p.17). If the beam emerging from the OS is divergent, it is the extensions (which are virtual) of the rays that pass through point A′. In this case, the image is virtual (see (Figure 10)) (cf. p.17). Regardless of the optical system, whether dioptric or catadioptric, four spaces can be defined (Figure 11) (cf. p.15): 1. A real object space located before the entrance of the optical system. 2. A virtual object space located after the entrance of the optical system. 3. A real image space located after the exit of the optical system. 4. A virtual image space located before the exit of the optical system. Object space and image space c) Focal Points The focal points (foci p.19) of an optical system are special points and can be defined as follows: The image focus is the image of an object located at infinity. The object focus is the object of an image located at infinity. Dr. Soumia LEBBAL 15  Principles d) Magnification For an extended object AB, which is perpendicular to the optical axis, and whose image through the OS is A′B′, the transverse magnification is defined as the following ratio: ¯ ′ ′ A B γ = ¯ AB If γ > 1, the image is upright and magnified. If 0 < γ < 1 , the image is upright and reduced. If γ < 0, the image is inverted. 4.3. Stigmatism and Approximate Stigmatism  Definition An optical system is rigorously p.19 stigmatic if all the rays emitted from a point object A converge to a single point image A′ after passing through the optical system.  Note In practice, the only optical system that is rigorously stigmatic is the plane mirror. a) Gauss' Condition For other centered systems, there is only approximate stigmatism for all rays slightly inclined relative to the optical axis. The Gauss' approximation (or Gauss' condition) is referred to the paraxial approximation p.19 that is usually applied when using a centered optical system by limiting it to paraxial rays, i.e., rays that are: slightly inclined relative to the optical axis, close to this axis. This means that the Snell- Descartes relation n sin θ = n′ sin θ′ is replaced by nθ ≈ nθ′ (see (Figure 12)) (cf. p.18) , where: Gauss' approximation sin θ ≈ θ. To achieve Gauss' conditions, it's usually sufficient to place a diaphragm at the entrance of the system. 16 Dr. Soumia LEBBAL Complementary resources Critical angle Critical angle Dr. Soumia LEBBAL 17 Complementary resources Deviation of light through a plane mirror Deviation of light through a plane mirror Gauss' approximation Gauss' approximation Total reflection Total reflection 18 Dr. Soumia LEBBAL Glossary Apex Peak or Summit Conical cone-like shaped Foci Plural of focus Grazes Causes the light to travel along the boundary between two media. Homogeneous Uniform paraxial approximation small angle approximation Propagates Travels Revolution Symmetry rigorously strictly Vacuum Void, Emptiness Velocity Speed Dr. Soumia LEBBAL 19 Abbreviation EM : Electromagnetic MKS : The MKS system stands for "Meter-Kilogram-Second" system. In this system, the fundamental units are the meter for length, the kilogram for mass, and the second for time. OS : Optical System ROYGBIV : An acronym for the sequence of colors that make up the rainbow: Red, Orange, Yellow, Green, Blue, Indigo and Violet. 20 Dr. Soumia LEBBAL  Abbreviation Dr. Soumia LEBBAL 21