Lenses and Optical Instruments - Study Summary

Units Covered:

• Thin Lenses; Ray Tracing
• The Thin Lens Equation; Magnification
• Combinations of Lenses
• Lensmaker's Equation
• Cameras: Film and Digital
• The Human Eye; Corrective Lenses
• Magnifying Glass
• Telescopes
• Compound Microscope
• Aberrations of Lenses and Mirrors

33-1 Thin Lenses; Ray Tracing

• Definition: Thin lenses are those whose thickness is small compared to their radius of curvature.
• Types:
  • Converging Lenses: Thicker in the center than at the edge. Bring parallel rays to a focus.
  • Diverging Lenses: Thicker at the edge than in the center. Make parallel light diverge; the focal point is where the diverging rays would converge if projected back.
• Lens Power (P): The inverse of its focal length.
  • Formula: P = 1/f
  • Measured in Diopters (D). 1 D = 1 m-1.
  • Positive for converging lenses, negative for diverging lenses.
• Ray Tracing for Thin Lenses (Three Key Rays):
  1. Ray parallel to the axis exits through the focal point.
  2. Ray through the focal point exits parallel to the axis.
  3. Ray through the center of the lens is undeflected.

33-2 The Thin Lens Equation; Magnification

• The Thin Lens Equation: Relates object distance (do), image distance (di), and focal length (f).
  • Formula: 1/do + 1/di = 1/f
• Magnification (m): Ratio of image height (hi) to object height (ho), also related to object and image distances.
  • Formula: m = hi/ho = -di/do
• Sign Conventions:
  1. Focal Length (f): Positive for converging lenses, negative for diverging lenses.
  2. Object Distance (do): Positive if the object is on the same side as the light entering the lens.
  3. Image Distance (di): Positive if the image is on the opposite side from the light entering the lens; otherwise, negative (virtual image).
  4. Height of Image (hi): Positive if the image is upright, negative otherwise (inverted image).
• Problem Solving Steps:
  1. Draw a ray diagram.
  2. Solve for unknowns using the equations.
  3. Follow the sign conventions carefully.
  4. Check consistency with the ray diagram.

33-3 Combinations of Lenses

• Optical instruments often use multiple lenses.
• To analyze: The image formed by the first lens acts as the object for the second lens. (Object distances for subsequent lenses can be negative).
• Total Magnification (Mtotal): The product of the magnifications of each lens. Mtotal = m1 * m2 * ...
• Power of Combined Lenses (in contact): The powers add up. Pco = PA + PB (This is only for thin lenses held in contact).
  • Consequently, 1/Fco = 1/fA + 1/fB

33-4 Lensmaker's Equation

• Relates the focal length (f) of a thin lens to its index of refraction (n) and the radii of curvature of its two surfaces (R1 and R2).
  • Formula: 1/f = (n - 1)(1/R1 + 1/R2)
• Sign Conventions for Radii of Curvature:
  • R is positive if the surface is convex (bulges out towards the incoming light).
  • R is negative if the surface is concave (caves in towards the incoming light).
  • R is infinite for a flat (plano) surface.
• The equation is symmetrical in R1 and R2, meaning the focal length is the same even if the lens is turned around.

33-5 Cameras: Film and Digital

• Basic Parts of a Camera:
  • Lens: Forms the image.
  • Light-tight Box: Prevents unwanted light from reaching the sensor.
  • Shutter: Controls the duration of light exposure.
  • Film or Electronic Sensor (CCD): Captures the image.
• Digital Cameras: Use CCD (Charge-Coupled Device) sensors instead of film. The image is digitized and processed.
• Camera Adjustments:
  • Shutter Speed: Controls the amount of time light enters. Faster speed = sharper picture (less motion blur).
  • f-stop (Aperture): Controls the maximum opening of the shutter, regulating the amount of light and depth of field. A higher f-number means a smaller opening.
  • Focusing: Adjusts the position of the lens to ensure the image is positioned sharply on the film/sensor.
• Depth of Field: The range of distances over which objects appear in focus. Stopping down (increasing f-stop number) increases depth of field.
• Types of Lenses:
  • Telephoto Lens: Longer focal length, produces magnified images.
  • Wide-angle Lens: Shorter focal length, wider field of view, produces smaller images.
  • Zoom Lens: Adjustable focal length.
  • Digital Zoom: Enlarges pixels, resulting in a loss of resolution.

33-6 The Human Eye; Corrective Lenses

• The human eye functions like a camera with an adjustable lens, iris, and retina.
• Refraction in the Eye: Most of the refraction occurs at the cornea surface. The eye's lens provides fine adjustments for focusing at different distances (accommodation).
• Accommodation: The muscles contract to thicken the lens for focusing on nearby objects, shortening its focal length.
• Near Point: The closest distance at which the eye can focus clearly (approx. 25 cm for normal vision).
• Far Point: The farthest distance at which an object can be seen clearly (infinity for normal vision).
• Nearsightedness (Myopia):
  • Far point is too close (less than infinity).
  • Caused by cornea/lens being too powerful or eyeball being too long.
  • Distant objects focus before reaching the retina.
  • Correction: A diverging lens is needed.
• Farsightedness (Hypermetropia):
  • Near point is too far away (greater than 25 cm).
  • Caused by cornea/lens combination not being powerful enough or eyeball being too short.
  • Nearby objects would focus behind the retina.
  • Correction: A converging lens is needed.
• Vision Underwater: Appears blurry because light rays are bent less than in air due to the similar refractive indices of water and the cornea. Can be corrected with goggles (creating an air interface).

33-7 Magnifying Glass

• A magnifying glass (simple magnifier) is a converging lens.
• Allows focusing on objects closer than the near point to create a larger, clearer image on the retina.
• Angular Magnification (M):
  • For eye relaxed (image at infinity, N = near point distance, typically 25 cm for normal eye): M = N/f
  • For eye focused at near point: M = N/f + 1

33-8 Telescopes

• A refracting telescope consists of two lenses:
  • Objective Lens: Closest to the object.
  • Eyepiece: Closest to the eye.
• Magnification (M) for a Refracting Telescope:
  • Formula: M = -fo/fe (where fo is objective focal length, fe is eyepiece focal length). The negative sign indicates an inverted image.
• Astronomical Telescopes: Use large objectives (often mirrors instead of lenses) to gather as much light as possible for faint objects. Mirrors can be made larger and with greater precision.
• Terrestrial Telescopes: Designed to produce an upright image for viewing objects on Earth (e.g., Galilean type, spyglass, which often include an additional lens or prism system).

33-9 Compound Microscope

• Also has an objective and an eyepiece, but the object is placed very close to the objective lens, unlike a telescope where the object is effectively at infinity.
• The objective lens forms a real, inverted, magnified image, which then acts as the object for the eyepiece.
• Magnification (M) for a Compound Microscope:
  • General formula: M = Memo = (N/fe) * (l - fe)/do (where N is near point, fe is eyepiece focal length, l is tube length, do is object distance for objective).
  • Approximate formula (when fo and fe << l): M ≈ Nl / (fefo)

33-10 Aberrations of Lenses and Mirrors

• Spherical Aberration:
  • Problem: Rays far from the lens axis do not focus at the same point as rays closer to the axis.
  • Solutions: Use compound-lens systems or restrict light to the central part of the lens (by stopping down the aperture).
• Distortion:
  • Problem: Caused by variation in magnification with distance from the lens center.
  • Types:
    • Barrel Distortion: Magnification decreases with distance, making straight lines bow outwards.
    • Pincushion Distortion: Magnification increases with distance, making straight lines bow inwards.
• Chromatic Aberration:
  • Problem: Light of different wavelengths (colors) has different indices of refraction, causing them to focus at different points. Results in color fringes around images.
  • Solution: An achromatic doublet, made of two lenses of different materials (e.g., crown and flint glass) with different dispersion properties, designed to bring at least two colors to the same focus.

Summary of Chapter 33 (Key Takeaways)

• Lenses use refraction to form real or virtual images.
• Converging lenses bring parallel rays to a focal point.
• Diverging lenses cause parallel rays to appear to diverge from a focal point.
• Lens power is given by P = 1/f in diopters (m-1).
• The Thin Lens Equation: 1/do + 1/di = 1/f.
• Magnification: m = hi/ho = -di/do.
• Cameras focus images using a lens, with adjustable opening (f-stop) and exposure time (shutter speed).
• The human eye adjusts its lens shape and pupil size for focusing.
• Nearsightedness is corrected by a diverging lens.
• Farsightedness is corrected by a converging lens.
• Magnification of a simple magnifier (angular magnification for eye relaxed): M = N/f.
• Telescope magnification (refracting): M = -fo/fe. (Astronomical telescopes often use mirrors for the objective).
• Common Aberrations include spherical aberration, distortion, and chromatic aberration. Chromatic aberration is often corrected using an achromatic doublet.