Optical Instruments¶
Lens¶
- Lens is a piece of transparent medium bounded by two surfaces at least one of which is curved.
- Every lens is a part of some sphere.
Types of Lenses¶
- Convex lens: It is thicker at the middle and thinner at the edges.
- Concave lens: It is thinner at the middle and thicker at the edges.
Important Definitions¶
- Centre of curvatures: Centre of sphere from which spherical surface of lens is obtained. Every lens has two centers of curvatures.
- Radius of curvature: Radius of sphere from which spherical surface of lens is obtained. Every lens has two radii of curvature that may not be equal.
- Principal axis: The line joining its two centers of curvature.
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Principal focus:
- For convex lens, principle focus is a point of convergence of refracted light ray. It is a real point.
- For concave lens, principal focus is a point of divergence of refracted light rays. It is imaginary point.
Every lens has two foci one on each side.
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Optical Center: A point inside the body of lens through which light rays pass undeviated.
- Focal Length: A distance between principal focus and optical center It is positive for convex lens and negative for concave lens.
Note
It is not necessary that two focal lengths are equal for a lens.
- Aperture: The size of diameter of lens.
- Power of lens: Its ability to deviate light ray from its original path. $$ P = \frac{1}{f} $$ where \(I\) is focal length in meter.
- SI unit of power of lens is diopter. \(P = 1D\) if \(f = 1m\)
Combination of Lenses¶
- General rule for combination of 'n' lenses is $$ P=P_1+P_2+....+P_n $$
- For two double convex lenses $$ P=P_1+P_2=\frac{1}{f_1}+\frac{1}{f_2} $$ $$ \implies f=\frac{f_1f_2}{f_1+f_2} $$
- For two concave lenses $$ P=P_1+P_2=-\frac{1}{f_1}-\frac{1}{f_2} $$ $$ \implies f=\frac{-f_1f_2}{f_1+f_2} $$
- For one concave & other convex lenses $$ P=P_1+P_2=\frac{1}{f_1}-\frac{1}{f_2} $$ $$ \implies f=\frac{f_1f_2}{f_1-f_2} $$
Lens Formula¶
- General lens formula is: $$ \frac{1}{p}+\frac{1}{q}=\frac{1}{f} $$
- Sign conversions:
- \(p\) is +ve for real objects and -ve for virtual objects
- \(q\) is +ve for real image and -ve for virtual image
- \(f\) is +ve for convex lens and -ve for concave lens
Info
Positive sign is used where rays actually intersect. If not then we can use negative sign.
Image Formation by Lens¶
Convex Lens¶
| Position of Object | Position of Image | Nature of Image |
|---|---|---|
| Beyond \(2F\) | Between \(F\) and \(2F\) | Real, Inverted, Small |
| At \(2F\) | At \(2F\) | Real, Inverted, Equal |
| Between \(F\) & \(2F\) | Beyond \(2F\) | Real, Inverted, Enlarged |
| At \(F\) | At infinity | No Image Formed |
| Inside \(F\) | On same side beyond object | Virtual, Erect, Enlarged |
Magnification of Lens¶
- Linear magnification is: $$ M=\frac{p}{q}=\frac{h_i}{h_o} $$
- Angular magnification is: $$ M=\frac{q_i}{q_o}=\frac{\text{Angle formed at aided eye}}{\text{Angle former at naked eye}} $$
- For real image $$ M=\frac{q}{p} $$
- For virtual image $$ M=-\frac{q}{p} $$
Info
For very small angles we can equate linear and angular magnifications.
Visual Angle¶
- Visual angle is angle made by object at observer's eye.
- Apparent size of object varies with visual angle. $$ \text{Apparent size}\propto \text{Visual angle} $$
Resolving Power¶
The resolving power of an instrument is its ability to reveal the minor details of the object under examination.
Angle of Resolution¶
- The minimum angle that allows two point sources to appear distinctly separated is called angle of resolution. It is expressed as \(\alpha_{\text{min}}\) $$ \alpha_{\text{min}}= 1.22\frac{\lambda}{D} $$
- Raleigh showed that resolving power of a lens of aperture \(D\), under a light source of wavelength \(\lambda\) is $$ R = \frac{D}{1.22\lambda} $$
Lens Aberration¶
Optical Aberration¶
It is the formation of blurred image due to size of lens because outer rays undergo more refraction than the inner ones.
chromatic Aberration¶
It is the inability of a lens to bring all light rays (all colors) at single focus. Multicolored blurred image is formed.
Remedies¶
- Provide such constraints on the lens that allows only parallel rays to enter the lens.
- Use the combination of convex and concave lenses called as achromatic lenses.
Least Distance of distinct Vision¶
It is the minimum distance at which a normal eye can see an object clearly. \(d =25cm\)
Info
As we get over the years upto old age the value of least distance of distinct vision is drastically affected.
Optical Instruments¶
Optical instruments are those, which use light for their operations.
- Simple microscope
- Compound microscope
- Telescope
- Camera
- Michelson interferometer
- Spectrometer
- Periscope
Simple Microscope¶
- Simple microscope (magnifier) is a double convex lens of short focal length
- Its magnifying power is $$ M=1+\frac{d}{f} $$
- It forms virtual, erect and magnified image at \(d\). $$ M\propto \frac{1}{f} $$
Compound Microscope¶
- Compound microscope consists of two convex lenses.
- Objective lens of small size and short focal length.
- Eyepiece lens of large size and large focal length.
- Magnification of compound microscope in given by: $$ M=\frac{L}{f_o}\left(1+ \frac{1}{f_o} \right) $$ where \(L\) is distance between two lenses.
Info
A wider objective and use of blue light of short wavelength produces less diffraction and allows more details to be seen.
- Objective forms real image while eyepiece forms virtual image.
- Magnification of objective is $$ M_o=\frac{L}{f_o} $$
- Magnification of eyepiece is $$ M_e=1+\frac{d}{f_e} $$
Note
- \(f_o<F_e\) always otherwise it will become astronomical telescope.
- In case of expensive microscopes, eyepiece and objectives are combination of lenses to reduce lens defects.
Telescope¶
Telescope is an optical instrument, which enables the observe to see fine details of a far off object.
- Two major classes of telescope are given below:
- Reflecting telescopes
- Refracting telescope
Astronomical Telescope¶
- It is used to see images of heavenly bodies.
- It consists of two lenses.
- Objective is double convex lens of large focal length and large aperture.
- Eyepiece is a double convex lens of short focal length and small aperture.
- When set for infinity or parallel rays, then \(L=f_o+f_e\)
Info
A good telescope has an objective of long focal length and large aperture.
- High magnifying power: $$ M=\frac{f_o}{f_e} $$
- Large field of view.
- Final image is inverted.
Defects of Vision¶
In case of eye following are the common defects of vision.
- Myopia or short-sightedness or near-sightedness.
- Hyperopia or hypermetropia or long-sightedness or far-sightedness
Myopia¶
Cause¶
In it distant objects are not clearly visible, i.e., far point is at a distance lesser than infinity and hence image of distant object is formed before the retina.
Remedy¶
This defect is remedied by using spectacles having divergent lens (i.e., negative focal length or power) which forms the image of distant object at the far point of patient (which is lesser than \(\infty\)).
Hyperopia¶
Cause¶
In it near objects are not clearly visible, i.e., near point is at a distance greater than 25 cm. image of near object is formed behind the retina.
Remedy¶
This defect is remedied by using spectacles having convergent lens (i.e., positive focal length or power) which forms the image of near objects at the near point of the patient eye (which is more than 25 cm).
SPECTROMETER¶
- It is an optical instrument used to study spectrum of light.
- It has following parts
- Collimator: It has slit at focus of a double convex lens to get a parallel beam of light.
- Telescope: It is an astronomical telescope.
- Turn table: Prism or diffraction grating is placed on table called turn table.
- Leveling screws
Working¶
- Telescope is set for infinity and fixed.
- Source of monochromatic light is placed in front of slit and collimator is adjusted for parallel rays.
- Angle of minimum deviation (\(D_m\)) is experimentally determined. Then refractive index of prism is given as $$ m=\frac{\sin\left(\frac{A + D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} $$ where \(A\) is angle of prism or apex angle.
- For wavelength of monochromatic light diffraction grating is used. $$ d\sin\theta=m\lambda $$ $$ \lambda\frac{d\sin\theta}{m} $$ usually \(m=1\) (first order diffraction), \(d=\)grating element, \(d=1/N\) where \(N\) is the number of ruling per centimeter.
Speed of Light¶
- First attempt to measure the speed of light was made by Galileo.
- First time speed of light measured accurately by Michelson
- The Michelson's formula for determination of speed of light is $$ c=16fd $$ where \(f\) is frequency of rotation of octagonal mirror and \(d\) is distance between plane mirror and octagonal mirror.
Refractive Index¶
The refractive index is defined as the speed of light in vacuum divided by the speed of light in the medium. $$ n=\frac{c}{v} $$
Snell's Law¶
Snell's Law relates the refraction index \(n\) of the two media to the direction of propagation in terms of the angles to the normal. $$ \frac{n_1}{n_2}=\frac{\sin\theta_2}{\sin\theta_1} $$
Introduction to Fiber Optics¶
- Graham Bell invented photo phone to transmit voice massage via beam of light.
- In optical fiber, light is used as a transmission carrier wave.
- The principle of transmitting signals through optical fiber is
- Total internal reflection
- Continuous refraction
- Optical fiber is a fine glass rod having diameter in the range of micrometers
Classification of Optical Fiber¶
- Single mode step index fiber: It is also called mono mode fiber & has very narrow core of diameter about \(5\mu m\).
- Multimode step index fiber: It has larger core (\(50\mu m\)) of constant refractive index. Refractive index change at the boundary of core and cladding. It is useful for a short distance only, The mode of transmission in it is total internal reflection.
- Multimode graded index fiber: In it, core has diameter from \(50\) to \(1000\mu m\) There is no noticeable boundary between core and cladding. The mode of transmission of light in it is continuous refraction. It is suitable for long distance transmission in which white light is used.
Signal Communication¶
- Fiber optic communication system consists of three parts
- Transmitter: To convert electrical signals to light signals
- Optical Fiber: To guide the light signals
- Receiver: To capture the light signal and convert them to electric signals.
- Light signals travel through fiber in 'on' and 'off' fashion called digital modulation. A light pulse represents, 1 and absence of light represents '0'
Advantages of communication through optical fiber¶
- Light signals in glass travel faster than electrical signals in copper cable
- More massages per cable length. (High signal strength)
- Clear sound is received
Power Loses¶
- Power is lost in optical fiber due to scattering of light from impurity atoms in glass
- Due to dispersion of light, signal also get weak.
- Power loss in signal due to dispersion can be reduced by using graded index fiber.
- Repeaters are used at suitable distance to overcome the power loss due to scattering from impurity atoms.