Compound Lenses

What is A Lens?

A lens is a portion of a transparent refracting medium bound by two spherical surfaces or one spherical surface and the other plane surface.

Lenses are used to focus light so that a person can get a clear picture of the objects.


The Commonly Used Lenses are of Two Types:

  1. Convex (Converging lens) 


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  1. Concave (Diverging lens)          


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There are Certain Uses of Lens In our Life Such as:

We see many people do observe nearby things and find it difficult in observing far objects (myopia) for which they use spectacles or contact lenses that are concave in shape.


Many people can see far objects while they find irt hard in observing nearby objects (hypermetropia) for which they use convex lenses while some people facing astigmatism are recommended to use cylindrical lenses..

Compound Lenses Thin Lenses in Contact

In various optical instruments like microscopes,telescopes, two or more lenses are combined to get the following requirements:

  1. Increase the magnification of the image.

  2. Obtain the erect image of an object.

  3. Reduce aberrations or defects caused by using a single lens.


The position, size and nature of the final image produced by the combination of thin lens can be represented by a ray diagram as shown below:

                                 

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In Figure.1, we have used two thin convex lenses placed on the common principal axis.

Let C₁, C₂ be the optical centers of two thin lenses L₁ and L₂.

These lenses are held co-axially with each other in the air.

Suppose, f₁ and f₂ be their respective focal lengths.

Let’s place an object at point O on the principal axis at a distance OC₁ = u.

The Lens  L₁  would alone form its image at I’.

Here, C₁I’ = v’.


From the Lens formula, we obtain our first equation:

                 1/v’ - 1/u = 1/f₁ -   (1)

Looking at Figure.1, this I’ would serve as a virtual object for lens L₂.

Thus, image is formed by this virtual object at I.

Where the distance C₂ I = v.

Since these lenses are thin.

Therefore, we can say for lens L₂ i.e., 

                u = C₂ I’  ≈ C₁I’ = v’


Now, let’s obtain an equation for lens L₂,

              1/v - 1/v’ = 1/f₂…(2)

Adding (1) and (2), we get

               

               1/v’ - 1/u + 1/v - 1/v’ = 1/f₁ + 1/f₂ 

                    1/v - 1/u = 1/f₁  + 1/f₂….(3) 

Here,  eq(3) is similar to the lens formula for the focal length in a combination of two lenses.


Now, if we replace these two lenses by a single focal length F which forms image I at a distance v of object at distance u. 

Then,  

                      1/v -1/u =1/F…(4)

                 

    Which means,

                              1/F =1/f₁ + 1/f₂                       


F  = f₁  + f₂/ f₁ f₂


If we consider taking one lens as a convex lens of focal length f₁  and the other of concave lens with focal length f₂ then,


                                    1/F =  1/f₁ + 1/ -f₂ 

                            

F =  f₁f₂/ f₂  - f₁


If we take ‘n’ no of lenses, then effective focal length of the combination will be:        


1/F = 1f₁ + 1/ f₂ + 1/f₃ +......+1/fₙ


Total magnification of the combination is the product of magnification of individual lenses, given by,

                  m  = m₁  x m₂ x m₃ x ….x mₙ

Afocal Definition

A system that outputs parallel rays with input rays. 

The prominent examples for the same are telescopes, beam expanders, etc.

Afocal

Afocal is an optical system that has an infinite focus.

Afocal projection is a method of photography in which a lens attached with a camera is mounted over the eyepiece of another image forming an optical system such as a telescope or microscope.

Here, the lens of a camera acts as a human eye.

Afocal System

Afcoal system is an optical system that produces no net convergence or divergence of the beam.


Let us discuss the combination of thin lenses that produces afocal systems.


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       Fig.2 : Thin lenses in contact 


This system is formed by the combination of two focal systems.

In Fig.2, the rear focal point of lens F₁’ is coincident with the focal length of the second lens F₂ .

Since object and image rays are parallel to the axis. 

Therefore,magnification will be constant given by, 

                                 h₀/ - f₁  =  hᵢ / f₂ 

 Magnification (m) = height of image /height of object  = hᵢ /  h₀


m  = hᵢ /  h₀ = - f₂ / f₁


Here, the system magnification is negative.

The sum of the individual focal lengths is equivalent to: 

                 

t  = f₁ + f₂


Such a configuration forms the basis for a Keplerian telescope.

Therefore, imaging equations for focal systems do not apply to afocal systems as there are no focal points. However, focal systems can form images.


FAQ (Frequently Asked Questions)

1. What is achromatic doublet?

Ans: Achromatic means ‘free of color.’

Achromatic lens or achromat is a type of lens that is made up of two optical components (lenses) cemented together, and are designed to limit the effects of chromatic and spherical aberrations inherent in singlet lenses.

Applications of achromatic lenses:

  1. Used to parallelize and focus laser beams.

  2. Useful for high-quality imaging on axis.

  3. Range of wavelengths can be brought into focus in the same plane.

2. What is the difference between achromatic and apochromatic lenses?

Ans: Achromatic lenses correct the lens error for two wavelengths; typically, red and blue to bring them into focus in the same plane while apochromatic is used to bring three wavelengths, typically, red, blue and green into focus in the same plane.

3. How does the thickness of a lens affect refraction?

Ans: A thick lens has more bending power as they have shorter focal length because they alter the path of light rays.