Optical Instruments

For compound microscope:

Normal adjustment (image at infinity): $m = \frac{v}{u}\frac{D}{f_e} \approx -\frac{l}{f_o}\frac{D}{f_e}$

Image at near point: $m = \frac{v}{u}\left(1+\frac{D}{f_e}\right) \approx -\frac{l}{f_o}\left(1+\frac{D}{f_e}\right)$

Where:

• l = tube length
• f₀ = focal length of objective
• f₉ = focal length of eyepiece
• D = least distance of distinct vision

The difference (the added term D/f₉) occurs because:

• At near point, the eyepiece forms a virtual image at distance D
• This produces greater angular magnification due to increased angular subtension
• The eye must accommodate to view this closer image

The sign difference in magnifying power relates directly to image orientation:

Astronomical telescope:

• Uses two converging lenses (positive focal lengths)
• Magnifying power: $m = -\frac{f_o}{f_e}$ (negative)
• Produces inverted final image
• Negative sign occurs because angles β and α are on opposite sides of axis

Galilean telescope:

• Uses converging objective and diverging eyepiece
• Magnifying power: $m = -\frac{f_o}{f_e}$ (positive, since f₉ is negative)
• Produces erect final image
• Angular subtension maintains same sign because rays don't cross axis

The diverging eyepiece in Galilean design intercepts rays before they form a real image, preserving orientation but reducing field of view.

Compound Microscope:

• Object placed slightly beyond focal point of objective
• Objective forms real, magnified, inverted image
• Linear magnification by objective: $\frac{v}{u}$ (typically large)
• Eyepiece provides angular magnification: $\frac{D}{f_e}$
• Total magnification: $m = \frac{v}{u}\frac{D}{f_e}$
• Works with nearby objects (<25 cm)

Astronomical Telescope:

• Object at effectively infinite distance
• No linear magnification by objective (only angular)
• Objective forms real image at its focal plane
• Total magnification purely angular: $m = -\frac{f_o}{f_e}$
• Works with distant objects

Key difference: Microscope combines linear and angular magnification; telescope only uses angular magnification due to infinite object distance.

Optical path changes:

• Adds converging lens between objective and eyepiece
• Positioned so the focal plane of objective is 2f away from this lens
• Inverts the first image, making final image erect relative to object
• Ray path: object → objective → first image → intermediate lens → second image → eyepiece → eye

Magnifying power:

• Basic formula remains similar: $m = \frac{f_o}{f_e}$ (positive)
• Sign changes from negative (astronomical) to positive due to extra inversion
• Intermediate lens ideally has magnification = 1 (just inverts, doesn't magnify)

Overall length:

• Astronomical: $L = f_o + f_e$
• Terrestrial: $L = f_o + 4f + f_e$ (where f = focal length of intermediate lens)
• Significantly longer due to the added 4f term
• Trade-off: correct image orientation vs increased length and complexity

Derivation of length:

• For normal adjustment (image at infinity)
• Objective focal point must coincide with eyepiece focal point
• Objective: converging lens with focal length f₀ (positive)
• Eyepiece: diverging lens with focal length f₉ (negative)
• Length: $L = f_o + f_e = f_o - |f_e|$

Why shorter than astronomical telescope:

• Astronomical telescope: $L_{ast} = f_o + f_e$
• Galilean telescope: $L_{gal} = f_o - |f_e|$
• For same magnifying power: $|m| = \frac{f_o}{|f_e|}$ in both cases
• Difference: $L_{ast} - L_{gal} = 2|f_e|$ (a significant reduction)

Advantages:

• More compact design (useful for opera glasses, binoculars)
• Produces erect images without need for additional lenses
• Less light loss (fewer optical elements)

Disadvantage:

• No real intermediate image, limiting use of reticles or measuring scales

Myopia (Nearsightedness):

• Mechanism: Eyeball too long or cornea too curved
• Image forms in front of retina for distant objects
• Correction: Diverging lens (negative power)
• Formula: $P = -\frac{1}{d_{far}}$ where d₍₍ᵣ is furthest clear vision point
• Key measurement: Far point distance

Hypermetropia (Farsightedness):

• Mechanism: Eyeball too short or cornea too flat
• Image would form behind retina for close objects
• Correction: Converging lens (positive power)
• Formula: $P = \frac{1}{d_{near}} - \frac{1}{D}$ where D is normal near point (25 cm)
• Key measurement: Near point distance

Presbyopia (Age-related):

• Mechanism: Decreased lens flexibility reduces accommodation
• Similar effect to hypermetropia but age-related
• Correction: Converging lens for reading
• Formula: Same as hypermetropia
• Key measurement: Near point distance

Additional considerations:

• Astigmatism requires cylindrical lenses
• Bifocals/progressive lenses address combined defects
• Diopter (D) = 1/focal length (meters) describes lens power

When viewing objects:

• With naked eye at near point: maximum visual angle θ₀ = h/D (where h is object height, D is near point distance ~25 cm)
• With simple microscope (normal adjustment): angle θ = h/f (where f is focal length)

The magnifying power equals θ/θ₀ = D/f

This shows that using a converging lens with focal length less than the eye's near point distance creates greater angular magnification, making objects appear larger by increasing the visual angle.

Key insight: The magnification increases as focal length decreases, with maximum theoretical magnification of D/f.

Object placement affects magnification and eye strain:

1. Normal adjustment (object at focal point F):

   • Final image forms at infinity
   • Magnifying power = D/f
   • Minimal eye strain as ciliary muscles are relaxed
   • Parallel rays enter eye

2. Maximum magnification adjustment (object closer than F):

   • Image forms at near point
   • Magnifying power = 1 + D/f
   • Greater eye strain as ciliary muscles are contracted
   • Object distance u₀ relates to near point by: D/u₀ = 1 + D/f

This relationship demonstrates the trade-off between higher magnification and increased eye strain when using a simple microscope.

For any lens system including the eye: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$

In the eye:

• $v$ is fixed at $v_o$ (distance from lens to retina)
• Clear vision requires image to form precisely on retina
• Rearranging: $\frac{1}{u} = \frac{1}{f} - \frac{1}{v_o}$

Vision defects explained:

• Myopia: $f_{max} < v_o$ → even with relaxed muscles, parallel rays from infinity focus before reaching retina
• Hyperopia: $f_{min} >$ required focal length to focus at normal reading distance → cannot reduce focal length enough to focus on near objects

Normal Eye:

• Far point: Infinity
• Near point: ~25 cm
• Functionality: Full range from reading to distant viewing without correction

Myopic Eye:

• Far point: Finite distance (e.g., 1-2 meters)
• Near point: Often closer than 25 cm
• Functionality: Excellent close vision (may read at closer distances), but struggles with driving, recognizing people at distance, viewing presentations

Hyperopic Eye:

• Far point: Infinity (with accommodation strain)
• Near point: Further than 25 cm (e.g., 50-100 cm)
• Functionality: May see distant objects clearly (with muscle strain), but struggles with reading, phone use, detailed close work, leading to eye fatigue and headaches

In practical terms, myopia affects navigation and recognition at distance, while hyperopia impacts reading, writing, and precision tasks at normal working distances.