30Chromatic abberation in Lenses, Optical instruments and Human eye

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CHROMATIC ABERRATION IN LENSES, OPTICAL INSTRUMENTS AND HUMAN EYE
▢ Optical Instruments:
❖ Main Instruments:
  • •
    Simple microscope
  • •
    Compound microscope
  • •
    Telescope
  • •
    Human eye
❖ Important Symbols:

Table 1: Symbols Used

Symbol
Meaning
\(M\)
Magnifying power
\(D\)
Least distance of distinct vision
\(f_o\)
Focal length of objective
\(f_e\)
Focal length of eyepiece
\(u_o\)
Object distance from objective
\(v_o\)
Image distance from objective
\(v_e\)
Image distance from eyepiece
▢ Simple Microscope:
❖ Definition: Single convex lens used to see magnified image of small object

Table 1: Simple Microscope Formulae

Condition
Magnifying power
General
\(M=\frac{D}{v_e}\left(1+\frac{v_e}{f_e}\right)\)
Final image at infinity
\(v_e\to\infty\)
Final image at infinity
\(M=\frac{D}{f_e}\)
Final image at least distance of distinct vision
\(M=1+\frac{D}{f_e}\)
▢ Compound Microscope:
❖ Definition: Optical instrument with objective and eyepiece used to obtain highly magnified image of very small object
❖ Magnification: Total magnification = objective magnification × eyepiece magnification

Table 1: Compound Microscope Formulae

Condition
Formula
Objective magnification
\(m_o=\frac{v_o}{u_o}\)
Eyepiece magnification
\(m_e=\frac{D}{v_e}\left(1+\frac{v_e}{f_e}\right)\)
General magnifying power
\(M=\frac{v_o}{u_o}\times\frac{D}{v_e}\left(1+\frac{v_e}{f_e}\right)\)
Final image at infinity
\(M=\frac{v_o}{u_o}\times\frac{D}{f_e}\)
Final image at least distance of distinct vision
\(M=\frac{v_o}{u_o}\left(1+\frac{D}{f_e}\right)\)
❖ Image Formation:
  • •
    Objective forms real, inverted and magnified intermediate image
  • •
    Eyepiece acts as simple microscope
  • •
    Final image is virtual, inverted and magnified
▢ Telescope:
❖ Definition: Optical instrument used to see distant objects

Table 1: Telescope Formulae

Condition
Magnifying power
General
\(M=\frac{f_o}{f_e}\left(1+\frac{f_e}{v_e}\right)\)
Final image at infinity
\(M=\frac{f_o}{f_e}\)
Final image at least distance of distinct vision
\(M=\frac{f_o}{f_e}\left(1+\frac{f_e}{D}\right)\)
❖ Important Points:
  • •
    Objective focal length should be large
  • •
    Eyepiece focal length should be small
  • •
    Magnifying power increases if \(f_o\) increases or \(f_e\) decreases
▢ Microscope vs Telescope:

Table 1: Comparison

Feature
Microscope
Telescope
Used for
Very small nearby objects
Very distant objects
Objective focal length
Small
Large
Eyepiece focal length
Small
Small
Objective aperture
Small
Large
Final image
Virtual, magnified
Virtual, magnified
Main magnification factor
\(\frac{v_o}{u_o}\times\frac{D}{f_e}\)
\(\frac{f_o}{f_e}\)
▢ Chromatic Aberration:
❖ Definition: Defect of lens in which different colours focus at different points due to different refractive indices
❖ Cause:
  • •
    Refractive index depends on wavelength
  • •
    Violet has higher refractive index than red
  • •
    Violet deviates more than red
  • •
    Focal length is different for different colours

Table 1: Colour and Focal Length

Colour
Refractive index
Deviation
Focal length
Violet
Maximum
Maximum
Minimum
Red
Minimum
Minimum
Maximum
❖ Longitudinal Chromatic Aberration: \(f_R-f_V\)
❖ Dispersive Power: \(\omega=\frac{\mu_V-\mu_R}{\mu_Y-1}\)
❖ Correction:
◉ Method: Use achromatic doublet
◉ Combination: Convex lens of crown glass + concave lens of flint glass
◉ Condition: \(\omega_1P_1+\omega_2P_2=0\)
◉ Also: \(\frac{\omega_1}{f_1}+\frac{\omega_2}{f_2}=0\)
▢ Human Eye:
❖ Important Parts:

Table 1: Parts of Eye

Part
Function
Cornea
Main refraction of light
Iris
Controls size of pupil
Pupil
Regulates amount of light entering eye
Eye lens
Fine focusing by accommodation
Retina
Image formation screen
Yellow spot
Region of sharpest vision
Blind spot
No photoreceptors; no image perception
Ciliary muscles
Change focal length of eye lens
❖ Least Distance of Distinct Vision: \(D=25\ cm\)
❖ Normal Eye:
  • •
    Near point = 25 cm
  • •
    Far point = infinity
  • •
    Image forms on retina
▢ Defects of Vision:

Table 1: Eye Defects and Correction

Defect
Cannot see
Image forms
Correction
Myopia / short-sightedness
Distant objects
In front of retina
Concave lens
Hypermetropia / long-sightedness
Nearby objects
Behind retina
Convex lens
Presbyopia
Near objects due to ageing
Accommodation decreases
Convex lens / bifocal lens
Astigmatism
Lines in different planes clearly
Unequal curvature of cornea/lens
Cylindrical lens
▢ Correction of Eye Defect:
❖ General Formula: \(\frac{1}{f}=\frac{1}{\text{can't see}}-\frac{1}{\text{can see}}\)

Table 1: Correction Formula Application

Defect
Meaning of formula
Myopia
Image of distant object should be formed at far point of defective eye
Hypermetropia
Object at normal near point should appear at near point of defective eye
Power of correcting lens
\(P=\frac{1}{f}\) in metre
▢ High-Yield Recall:

Table 1: Optical Instruments and Human Eye One-Liners

Fact
Answer
Simple microscope infinity image
\(M=\frac{D}{f_e}\)
Simple microscope near point image
\(M=1+\frac{D}{f_e}\)
Compound microscope general
\(M=\frac{v_o}{u_o}\times\frac{D}{v_e}\left(1+\frac{v_e}{f_e}\right)\)
Compound microscope infinity image
\(M=\frac{v_o}{u_o}\times\frac{D}{f_e}\)
Compound microscope near point image
\(M=\frac{v_o}{u_o}\left(1+\frac{D}{f_e}\right)\)
Telescope infinity image
\(M=\frac{f_o}{f_e}\)
Telescope near point image
\(M=\frac{f_o}{f_e}\left(1+\frac{f_e}{D}\right)\)
Telescope objective
Large focal length
Microscope objective
Small focal length
Chromatic aberration
Different colours focus at different points
Chromatic aberration cause
\(\mu\) depends on wavelength
Violet focal length
Minimum
Red focal length
Maximum
Correction of chromatic aberration
Achromatic doublet
Achromatic doublet condition
\(\omega_1P_1+\omega_2P_2=0\)
Least distance of distinct vision
25 cm
Normal far point
Infinity
Myopia correction
Concave lens
Hypermetropia correction
Convex lens
Astigmatism correction
Cylindrical lens
Correction formula
\(\frac{1}{f}=\frac{1}{can\ not\ see}-\frac{1}{can\ see}\)