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Thin Lens :-

The thin lens equation is stated as follows:

where

d_o is the distance (measured along the axis) from the object to the center of the lens
d_i is the distance (measured along the axis) from the image to the center of the lens
f is the focal length of the lens

The expression 1/f in called the power of a lens. It is measured in Diopters, where 1 D = 1 m^-1.

When using this equation, signs are very important:

d_o	positive	when the object is placed "in front of the lens"

d_i	positive	when real images are formed (inverted, "behind the lens")
d_i	negative	when virtual images are formed (upright, "in front of the lens")

f	positive	when the lens is converging
f	negative	when the lens is diverging

Remember that d_o, d_i, and f must be measured in the same unit - usually meters is preferred.

The following formula is used to calculate the magnification of an image:

If a problem states that a real image is formed that is twice as large as an object, then you would use the relationship d_i = +2d_o in the thin lens equation. If a problem states that a virtual image is formed that is twice as large as the object, then you would use the relationship that d_i = ?2d_o.

Double Lens System :-

When two or more lenses are used in an optical system, the formula used to calculate the magnification of the final image produced by the system is:

M_system = (M_{lens #1})(M_{lens #2})(M_{lens #3}) ....

When an optical system uses two lens, work each lens separately. Remember that the image of lens #1 will serve as the object for lens #2. If the image of lens #1 falls "in front of" lens #2, you can calculate the object distance for lens #2 by subtracting the value of d_i for lens #1 from the total distance separating the two lenses.

In this example, two converging lens produce a final image which is real and upright. A scaled diagram would be necessary to determine how the height of the final image, I₂, compares to the height of the original object, O.

A similar process will produce the image for a double lens system involving a converging and a diverging lens.

In this second example, a converging lens followed by a diverging lens produce a final image which is virtual and inverted. A scaled diagram would be necessary to determine how the height of the final image, I₂, compares to the height of the original object, O.

Lens Maker Eqn :-

The following formula, called the Lensmaker Equation, is used to determine whether a lens will behave as a converging or diverging lens based on the curvature of its faces and the relative indices of the lens material [n₁] and the surrounding medium [n₂].

Usually the expression

is treated as a constant (K_shape ) allowing us to work more often with the following second form of the equation:

Remember that K_shape represents the shape of the lens which remains constant regardless of the type of surrounding medium [n₂] into which the lens is used.

If this expression yields a negative value for 1/f, then the lens is diverging; a positive 1/f means that the lens is converging.

converging lenses are lenses that are "thicker in the center" than on the edges (convex)

geometry
	r₁ > 0, r₂ < 0 therefore K_shape > 0	Since n₁ > n₂ and K_shape > 0 1/f > 0 and these lenses will be converging.
	r₁ = , r₂ < 0 therefore K_shape > 0

diverging lenses are lenses that are "thinner in the center" than on the edges (concave)

geometry
	r₁ < 0, r₂ > 0 therefore K_shape < 0	Since n₁ > n₂ and K_shape < 0 1/f < 0 and these lenses will be diverging.
	r₁ = , r₂ > 0 therefore K_shape < 0

Power

To calculate the power of a lens, we use the relationship that

Lenses in close combination

When two or more lenses are nested or used in close combination, that is, with no space in between them, the equation to calculate the effective power of the combination is

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