Optics Lecture 1: Geometrical Optics
Recommended article : 【Physics】 Physics Table of Contents
2. Image
1. Fermat’s Principle
⑴ Fermat’s principle
① The law that light travels along the path that takes the least time
② Theory that encompasses the laws of reflection and refraction
③ To understand its profound significance, the concept of quantum optics is required
⑵ Law 1. Law of rectilinear propagation
① Proof 1. Fermat’s principle
② Proof 2. Huygen’s principle
⑶ Law 2. Law of Reflection
① Definition: The angle of incidence is equal to the angle of reflection, and the incident ray and the reflected ray lie in the same plane.
○ Normal line: A line perpendicular to the boundary surface at the point where light is reflected.
○ Incidence: The process in which light approaches a boundary surface.
○ Reflection: The process in which light returns from a boundary surface.
② Proof 1. Fermat’s Principle
③ Proof 2. Huygens’ Principle
⑷ Law 3. Snell’s Law
① Definition: The incident ray and the refracted ray lie in the same plane with the following formula:
○ $c$: Speed of light in a vacuum
○ $v$: Speed of light in a medium
○ $n = c / v$
② Proof 1. Fermat’s Principle
③ Proof 2. Huygens’ Principle
⑸ Application 1. Total Internal Reflection
① Implication of the law of refraction
② Critical angle (θc)
○ Note that the critical angle of diamond is 24.5°, the smallest known critical angle to date.
③ How fish see the world
Figure 2. How fish see the world
④ Optical Fiber
Figure 3. Structure of Optical Fiber
○ Definition : A bundle of glass, fused silica, or plastic that can transmit light over hundreds of meters or more
○ Diameter ranges from 0.05 μm to 0.6 cm
○ Structure : Core, Cladding
○ The material used for the core has a higher refractive index than that used for the cladding
○ Practical applications
○ Multiple layers of optical fiber are used in practice: Optical fiber is too thin for geometric optics approximation (i.e., Snell’s law) to apply.
○ Overlapping light of various frequencies is used for information transmission: Unlike electrons where Pauli’s exclusion principle applies, light can overlap.
○ Bending of the fiber causes irregular reflections, leading to blurred images → To prevent this, graded index is used.
○ ~0.2 dB/km loss
○ Classification based on material
○ Glass or plastic: Visible light, near-infrared range
○ Fused silica: Ultraviolet range to near-infrared range
○ Classification based on usage
○ Reflecting probes
○ Transmissive probes
○ Dip probes
⑹ Application 2. Negative Refraction Index
① Classification of materials
○ Type 1: ε > 0, μ > 0: Normal optical materials, right-handed propagating wave
○ Type 2: ε < 0, μ > 0: Electromagnetic plasma, decaying electromagnetic waves, many metals (UV - optical), thin wire structures (GHz)
○ Type 3: ε > 0, μ < 0: Magnetic plasma, decaying electromagnetic waves, structured materials, natural magnetism (~GHz)
○ Type 4: ε < 0, μ < 0: Negative refraction index, left-handed propagating wave, artificial metamaterial
② References
○ V. G. Veselago, Usp. Fiz. Nauk 92, 517 (1967)
○ V. G. Veselago, Soviet Physics Uspekhi 10, 509 (1968)
○ R. A. Shelby et al., Science 292, 77 (2001): Experimental evidence
⑺ Application 3. Dispersion
① Prism Experiment
Figure 3. Prism Experiment
② Explanation 1.
○ 1st. The frequency of light remains constant even when the medium changes.
○ 2nd. The speed of light increases as wavelength in the medium increases. As speed of light increases, refractive index decreases.
○ 3rd. Longer wavelengths (shorter frequencies) result in less bending of light.
③ Explanation 2.
○ Short wavelengths correspond to shorter time spent within atoms.
○ Short wavelengths result in more collisions and resonance absorption.
④ Complexity of wavelengths due to various reasons
○ Interaction of atoms results in a variety of resonant wavelength ranges.
○ Motion of atoms also alters the resonant wavelength ranges.
○ To observe specific resonant wavelength ranges, cooled atoms need to be observed.
⑤ Application: Rainbow
○ Refraction → Internal reflection → Refraction lead to dispersion of colors
○ Longer wavelengths (closer to red) result in less bending.
2. Image
⑴ Type 1. Plane Mirror
① Constructed so that the mirror surface is the perpendicular bisector of the object and the image.
⑵ Type 2. Spherical Mirror: Concave Mirror, Convex Mirror
① Spherical and Parabolic Mirrors
○ Spherical mirrors have a constant curvature but parallel rays do not converge exactly to a single point.
○ Parabolic mirrors have parallel rays that converge exactly to a single point.
② Focal Length
○ When the object distance is ∞, the image is at the focal point.
○ For a spherical mirror, f = R / 2
③ Construction Method
○ Parallel rays pass through the focus.
○ Rays passing through the focus travel parallel to the optical axis.
○ Rays heading towards the center of the mirror reflect symmetrically about the optical axis.
○ Rays passing through the center of the lens continue in a straight line.
④ Proof of Concave Mirror 1: A construction using only two rays does not require approximation.
Figure 4. Proof of Concave Mirror 1
⑤ Proof of Concave Mirror 2: A construction using arbitrary rays requires approximation.
Figure 5. Proof of Concave Mirror 2
⑤ Spherical Aberration: When not dealing with paraxial rays
○ When not dealing with paraxial rays, parallel rays do not converge to the focus (because it is not a parabolic mirror).
○ In the case of paraxial rays: Quadratic function, i.e., equation of parabola
⑶ Type 3. Spherical Lens: Convex Lens, Concave Lens
① Construction Method
○ Parallel rays pass through the focus.
○ Rays passing through the focus travel parallel to the optical axis.
○ Rays heading towards the center of the lens reflect symmetrically about the optical axis.
○ Rays passing through the center of the lens continue in a straight line.
⑷ Spherical Boundary
① Proof of Spherical Boundary
Figure 6. Proof of Spherical Boundary
② Apparent Depth
○ The apparent depth appears greater when looking from outside the water into the water (this can be approached using the concept of visual angle).
○ Proof: Negative values imply imaginary.
Figure 7. Apparent Depth
⑸ Thin Lens Formula: Also known as the lens maker’s formula
3. Optical Devices
⑴ Optical Devices
① Converges incoming light from the outside to form an image on the retina.
② Mainly, light is well refracted at the cornea.
③ The lens finely adjusts the refractive index.
⑵ Magnifying Glass ↔ Telescope
① Since we should be able to see comfortably with our eyes, the near point (25 cm) is important.
② Magnification factor m
○ To understand how an optical device magnifies an object, we need to consider the size of the image formed on the retina.
○ The magnification factor or angular magnification m of a device is the ratio of the angular size of an object with the device to the angular size of the same object without the device.
③ Calculation of Magnification
○ The magnification is the greatest when the object is at the near point, i.e., when q = -25 cm.
○ The most comfortable state for the eye is when the object is at an infinite distance.
⑶ Microscope
Figure 8. Microscope
① Placing the object at the focal length gives the highest magnification.
② Visible light has a significantly larger wavelength compared to atoms, making direct observation impossible.
③ When trying to see electrons with light, λ↓ → E↑
⑷ Telescope
Figure 9. Telescope
① Collecting Power: R ↑, N ↑, t ↑
② Resolution: R ↑, λ ↓
③ Refraction: Cannot create a large angular separation
④ Reflection: Reflection can block objects at the position where the image is formed.
⑤ Example 1. A refracting telescope consists of two convex lenses separated by a distance of 100 cm. If the focal length of the objective lens is 20 cm, what is the angular magnification mθ of the telescope? (ref)
⑥ Example 2. The diameter of a collimated laser beam emitted from a commercial He-Ne laser is approximately 1 mm. To convert this beam into a well-collimated beam with a diameter of 10 mm, two convex lenses are to be used. The focal length of the first lens is 1.5 cm, and it is mounted at the laser output. What should the focal length (f) of the second lens be, and how far from the first lens should it be placed? (ref)
⑸ Astronomical Telescope
① Magnification = Focal length of objective lens ÷ Focal length of eyepiece lens
② Aligning the polar axis: To easily track celestial bodies in their daily motion.
③ Types
○ Optical telescopes: Most affected by weather conditions.
○ Radio telescopes: Observing electromagnetic waves from the ground.
○ Space telescopes: Observing short-wavelength electromagnetic waves that can’t be observed from the ground. Suitable for observing high-temperature celestial bodies.
⑹ Aberration
① Spherical Aberration: A phenomenon where not dealing with paraxial rays shortens the focal length.
② Chromatic Aberration: Refractive index varies with wavelength, causing shorter wavelengths to have shorter focal lengths.
Input : 2019.04.11 16:05