Geometric Optics PDF

Geometric Optics 1.0 September 2024 Dr. Soumia LEBBAL Batna 2 University Mostafa 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 Mostafa 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 - Elements 3 1. Planar Systems................................................................................................. 3 1.1. Refractive Systems: Planar Diopters..................................................................................... 3 1.2. Reflective Systems: Plane Mirrors......................................................................................... 5 2. Spherical Systems: Dioptrers........................................................................... 5 2.1. Geometrical Construction Of an Image................................................................................. 6 2.2. The Conjugation Relation....................................................................................................... 7 2.3. Magnififaction......................................................................................................................... 8 3. Lenses............................................................................................................... 8 3.1. The Optical Center.................................................................................................................. 9 3.2. Thin Lenses........................................................................................................................... 10 3.3. Types of Lenses.................................................................................................................... 10 3.4. The Conjugation Relation..................................................................................................... 12 3.5. Focal Length.......................................................................................................................... 13 3.6. Magnification........................................................................................................................ 13 3.7. Geometric Construction of an Image.................................................................................. 13 3.8. Adjacent Lenses................................................................................................................... 14 2 Dr. Soumia LEBBAL I Elements 1. Introduction In this chapter, we will delve into the foundational optical systems that shape our understanding of light behavior, focusing on how various optical systems manipulate light to create and transform images. Our journey will cover three essential components: 1. Planar Systems: which involve flat surfaces that reflect or refract light. 2. Spherical Diopters: where we deal with curved refractive surfaces that alter light paths based on their shape and curvature. 3. Lenses: which are composed of both systems that are crucial for manipulating light to form images. 2. Planar Systems  Definition We call planar systems all systems that have an infinite curvature. The Conjugation Relation  Definition It is an algebraic relation that links the positions of the two conjugates: the image and the object. 2.1. Refractive Systems: Planar Diopters The planar dioptre is not a stigmatic system. For each object point, there are multiple image points. Under Gauss conditions, we can consider the planar diopter to be stigmatic. It is said that the planar dioptre possesses an approximate stigmatism (Figure 13) (cf. p.3). Image points of an object point through a planar dioptre Dr. Soumia LEBBAL 3  Elements a) The Conjugation Relation In Gauss conditions, the conjugation relation for plane diopters is written as follows: n1 n2 = ¯ ¯ H A1 H A2 A2 is the image of A1 through the plane diopter see (Figure 14)). Image through a plane diopter for n1 > n2 Demonstration: At the incidence point (I): Gauss n1 sin(i1 ) = n2 sin(i2 ) ⟶ n1 i1 = n2 i2 , where: IH IH i1 ≈ tan(i1 ) = , and, i2 ≈ tan(i2 ) = H A1 H A2 i) Magnification The virtual image A2 B2 ​ of the object A1 B1 ​ is formed on the normal to the diopter see (Figure 15)). Applying the conjugation formula of the planar diopter to the conjugate point pairs (A1 A2 ) and (B1 B2 ) allows us to establish that the magnification is: ¯ A2 B2 n2 γ = = ¯ n1 A1 B1 A1 B1 Image of an object ​ perpendicular to the surface of the dioptre. Explanation: ¯ A2 B2 γ = , ¯ A1 B1 where, using Chasles' relation: ¯ ¯ ¯ ¯ ¯ A1 B1 = A1 H + H B1 = −H A1 + H B1 , and ¯ ¯ ¯ ¯ ¯ A2 B2 = A2 H + H B2 = −H A2 + H B2. Knowing that, from the conjugation relations of both conjugates A1,2 and B1,2 : n1 n2 = ¯ ¯ H A1 H A2 4 Dr. Soumia LEBBAL  Elements and n1 n2 =. ¯ ¯ H B1 H B2 We get: n2 ¯ ¯ (−H A1 + H B1 ) n1 n2 γ = =. ¯ ¯ n1 (−H A1 + H B1 ) 2.2. Reflective Systems: Plane Mirrors As stated earlier, the plane mirror is the only optical system that is strictly stigmatic. a) The Conjugation Relation To construct the image A′ of an object A, we use two incident rays and apply the law of reflection. The image point A′ is the symmetrical counterpart of the object point A with respect to the plane of the mirror. The conjugation relation for the plane mirror is written as: ¯ ¯ ′ H A = −H A Where: H is the orthogonal projection of A onto the plane mirror mirror see (Figure 16)). Image point of an object point through a plane mirror. b) Magnification Through a plane mirror, the image has always the same characteristics and of the same size as the object see (Figure 17)): γ = 1 Image of an extended object 3. Spherical Systems: Dioptrers We call spherical systems all systems that have a finite curvature. Dr. Soumia LEBBAL 5 Elements  Definition A spherical diopter is a spherical interface with a radius of finite curvature R, allowing light to be refracted from one medium to another. The sign of the radius of curvature of the diopter, and thus the type of this latter, is interpreted by the incident beam at the diopter. We can distinguish two types: A spherical diopter is convex if the light first passes through the apex (S) and then through the center (C) (i.e., if R > 0) see (Figure 18)). ¯ Convex diopter (R ≡ SC > 0) A spherical dioptre is concave if the light first passes through the center (C) and then through the apex (S) (i.e., if R < 0) see (Figure 19)). ¯ Concave dioptre (R ≡ SC < 0) 3.1. Geometrical Construction Of an Image In order to geometrically construct the image of an object, at least two rays are needed. This construction must obey to the following rules (figure 20 (cf. p.7)): The ray that passes through the center C do not deviate. (in red) The ray that is parallel to the optical axis has to pass through the image focus F ′. (in blue) The ray that passes through the object focus F must emerge parallel to the optical axis. (in green) 6 Dr. Soumia LEBBAL  Elements  Example Examples of geometric constructions of an image through a spherical dioptre 3.2. The Conjugation Relation The conjugation relation of the spherical diopter is given (in Gauss conditions) by the following equation: ′ ′ n n n − n − = = ν ¯ ¯ ¯ ′ SA SA SC Where: n is the refraction index of the object space. n ′ is the refraction index of the image space. A ′ is the image of A through the spherical diopter of apex S and center C. ¯ SC ≡ R is the curvature of the diopter. ν is the vergence (the focal power) of the diopter. a) The Focal Power  Definition It's an algebraic quantity, and it reflects the optical system's ability to converge light rays. ′ n − n ν = ¯ SC It is measured in (m−1 ) which is also called the dioptre, symbolized by δ, where 1δ = 1m −1. Its value can tell the type of the dioptre in question. We distinguish two cases: If ν > 0 , the dioptre in convergent. If ν < 0 , the diptre is then divergent. Dr. Soumia LEBBAL 7  Elements i) Object and Image Foci  Definition ¯ ′ The object of an image that appears at the location OA → ∞ would be located at the object focus. From the conjugation formula (cf. p.7), we get: ¯ ¯n SF = SC. ′ n − n ¯ For an object that is located at OA → ∞, the image would be located at the image focus. From the conjugation formula (cf. p.7), we get: ′ ¯ ¯ n ′ SF = ′ SC. n − n  Note We can tell, also, by the falues of the foci the type of the dioptres: ¯ ¯ ′ If SF > 0 and SF , the dioptre is convergent. < 0 ¯ ¯ ′ If SF < 0 and SF , the dioptre is divergent. > 0 3.3. Magnififaction For the spherical dioptre, the magnification is given by the equation: ¯ ¯ ′ ′ ′ A B n SA γ = = ¯ ¯′ n AB SA Where A′ B′ is the image of AB through the dioptre. 4. Lenses Lenses are a widely used optical system that can be found in many devices as varied as a microscope, a camera, a pair of glasses, or an overhead projector. 8 Dr. Soumia LEBBAL  Elements  Definition A Lens is a homogeneous, transparent and isotropic medium that is constructed by the association of two diopters, of which al least one has to be spherical (Figure 21) (cf. p.9). Schematic cuts of a bi-convex (right) and a plane-concave ( left) lenses 4.1. The Optical Center  Definition The Optical Center is a position on the optical axis of the lens. It is designed by the letter O (Figure 22) (cf. p.9). Optical Centers of a Concerging Lens (a) and a Diverging Lens (b) Dr. Soumia LEBBAL 9 Elements  Fundamental A ray of light travelling through the optical center of a lens emerges parallel to its incidence. 4.2. Thin Lenses  Definition A lens is thin if its diameter (D) is much larger than its thickness (e) see (Figure 23)). If e