Premed · Premed · Physics 2
Lecture 17: Optical Instruments and the Eye
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe the optics of the human eye, including accommodation and common refractive errors
- Explain how corrective lenses work for myopia, hyperopia, and astigmatism
- Calculate the angular magnification of a simple magnifying glass
- Describe the optical design and magnification of compound microscopes and telescopes
- Explain the concept of resolving power and the diffraction limit of optical instruments
Lecture Content
I. The Human Eye
The eye is a remarkably sophisticated optical instrument. The cornea provides about two-thirds of the eye's total refractive power (approximately 43 D) and has a fixed curvature. Behind the cornea lies the aqueous humor (n approximately 1.336), followed by the iris and pupil, which control the amount of light entering the eye by adjusting the aperture between 2 and 8 mm. The crystalline lens is a variable-focus element that provides approximately 15-25 D of power and changes shape through a process called accommodation. The interior of the eye is filled with the vitreous humor (n approximately 1.337), a gel-like substance. The retina at the back of the eye is the light-sensitive layer containing rods (for dim light) and cones (for color and detail). The fovea is the central region of highest visual acuity, densely packed with cones.
The total optical power of the relaxed eye is about 60 D, corresponding to a focal length of approximately 17 mm. Accommodation is the process by which the ciliary muscles change the shape of the lens to focus at different distances. The near point is the closest distance at which the eye can focus clearly, about 25 cm for a young adult (increasing with age). The far point is the farthest distance for clear focus, which is infinity for a normal eye.
II. Refractive Errors and Correction
Myopia (nearsightedness) occurs when the eye focuses images in front of the retina, either because the eyeball is too long or the cornea is too curved. The far point is at a finite distance, so distant objects appear blurred. It is corrected with a diverging (concave) lens that moves the focal point back onto the retina. The required correction power is P = -1/far point (in meters).
Hyperopia (farsightedness) occurs when the eye focuses images behind the retina, either because the eyeball is too short or the cornea is too flat. The near point is farther than normal. It is corrected with a converging (convex) lens that adds focusing power, with the correction chosen so that the near point becomes 25 cm.
Presbyopia is the age-related loss of accommodation caused by the lens becoming less flexible. The near point recedes with age, often requiring reading glasses or bifocals. The correction is similar to that for hyperopia.
Astigmatism occurs when the cornea or lens has different curvatures in different planes, causing images to be blurred in one direction. It is corrected with cylindrical (toric) lenses that have different powers along different axes.
<image>Four panels showing eye conditions and corrections. Panel A: Normal eye — parallel rays from a distant object focus precisely on the retina. Panel B: Myopic eye — the image forms in front of the retina; a diverging lens placed before the eye shifts the focal point back onto the retina. Panel C: Hyperopic eye — the image forms behind the retina; a converging lens shifts the focal point forward onto the retina. Panel D: A diagram showing the near point receding with age (presbyopia), with a graph of near point distance versus age from 10 to 70 years.</image>
III. The Simple Magnifying Glass
A converging lens used with the object placed inside the focal point (d_o < f) produces a virtual, upright, enlarged image and functions as a magnifying glass. The relevant quantity is the angular magnification M, defined as the ratio of the angle subtended by the image to the angle subtended by the object when viewed at the near point (25 cm): M = theta_image / theta_unaided.
With the image formed at the near point (maximum magnification), M = 1 + 25 cm / f. With the image at infinity (a more comfortable viewing condition for the relaxed eye), M = 25 cm / f. For example, a lens with f = 5 cm provides M = 5x with the image at infinity, or 6x with the image at the near point.
IV. The Compound Microscope
The compound microscope uses two converging lenses for high magnification. The objective lens has a short focal length (f_o) and is placed close to the specimen. It creates a real, inverted, enlarged intermediate image inside the tube of the microscope. The eyepiece (ocular) acts as a magnifying glass to view this intermediate image, producing a virtual, further-enlarged final image.
The total magnification is M_total = m_objective x M_eyepiece. The objective magnification is m_objective = -L/f_o, where L is the tube length (the distance between the focal points of the two lenses). The eyepiece magnification is M_eyepiece = 25 cm / f_e for the image at infinity. The total magnification is therefore M_total = -(L x 25 cm) / (f_o x f_e), with typical values ranging from 40x to 1000x.
The resolving power of a microscope, its ability to distinguish two closely spaced points, is limited by diffraction. The Rayleigh criterion gives delta_min = 1.22 lambda / (2 n sin(alpha)), where n sin(alpha) is the numerical aperture (NA). Oil immersion increases the NA and thereby improves resolution.
<image>A ray diagram of a compound microscope. The object (small arrow) is placed just beyond the focal point of the objective lens. The objective produces a real, inverted, enlarged intermediate image inside the tube (between the two lenses). The eyepiece, positioned so the intermediate image is within its focal length, produces a virtual, further-enlarged final image at a great distance (or at the near point). The tube length L (distance between the rear focal point of the objective and the front focal point of the eyepiece) is labeled. Key distances f_o, f_e, and L are marked. The final image is shown as a large virtual image to the left, viewed by an eye at the right.</image>
V. Telescopes
Telescopes are designed to view distant objects, which are effectively at infinity. A refracting telescope (Keplerian) uses a large-diameter converging lens as the objective (with long focal length f_o) to collect light and form a real image at its focal point. A short-focal-length converging lens serves as the eyepiece to magnify this intermediate image. The angular magnification is M = -f_o / f_e, where the negative sign indicates the image is inverted. The total length of the telescope is L = f_o + f_e.
A reflecting telescope (Newtonian) uses a concave mirror as the objective instead of a lens. This design offers several advantages: it is free of chromatic aberration, and large mirrors are easier and cheaper to manufacture than large lenses. A small flat secondary mirror redirects the converging light to an eyepiece at the side of the tube.
Several key design considerations govern telescope performance. Light-gathering power is proportional to the area of the objective, so a larger aperture allows fainter objects to be observed. Resolving power is limited by diffraction, with angular resolution theta_min = 1.22 lambda / D, where D is the aperture diameter. Magnification is determined by the focal length ratio of the objective to the eyepiece.
VI. Aberrations and Resolution Limits
Spherical aberration occurs when rays far from the optical axis focus at a different point than rays close to the axis (paraxial rays). It is corrected with aspherical surfaces or aperture stops. Chromatic aberration arises because different wavelengths focus at different points due to dispersion; it affects lenses but not mirrors and is corrected with achromatic doublets combining crown and flint glass. Other off-axis aberrations include coma, astigmatism, field curvature, and distortion.
The fundamental resolution limit for any optical system is set by diffraction. The Rayleigh criterion states that two point sources are just resolved when the central maximum of one coincides with the first minimum of the other, giving a minimum angular separation of theta_min = 1.22 lambda / D. Larger apertures and shorter wavelengths yield better resolution.
The f-number (f/#) is the ratio of focal length to aperture diameter: f/# = f/D. A smaller f-number means a larger aperture, which admits more light and provides better resolution. In photography, each stop doubles or halves the light reaching the sensor.
<image>A comparison diagram. Left: A refracting telescope showing two converging lenses. Parallel rays from a distant object enter the objective (large lens, focal length f_o), converge to a focal point, and diverge into the eyepiece (small lens, focal length f_e), which collimates them into the eye at a larger angle. The angular magnification M = f_o/f_e is labeled. Right: A reflecting telescope (Newtonian design) showing a concave primary mirror at the base of the tube, which reflects incoming parallel light to a focus. A small flat diagonal mirror near the focus redirects the converging beam out the side of the tube to the eyepiece. Both designs show the path of light rays clearly.</image>
