The **geometric optics** It is the branch of Physics that focuses on studying the way in which light is propagated and reflected when it passes from one medium to another, without taking diffraction effects into account.

In this way, light is represented geometrically by rays, imaginary lines perpendicular to the luminous wavefronts.

Light rays emerge from light sources such as the Sun, a flame or a light bulb, spreading out in all directions. The surfaces partly reflect these rays of light and that is why we can see them, thanks to the fact that the eyes contain light-sensitive elements.

Thanks to the treatment of rays, geometric optics does not take into account so much the wave aspects of light, but rather explains how images are formed in the eye, mirrors and projectors, where they do and how they appear.

The fundamental principles of geometric optics are the reflection and refraction of light. The rays of light fall at certain angles on the surfaces they meet, and thanks to this a simple geometry helps to keep track of their trajectory in each medium.

This explains everyday things like looking at our image in the bathroom mirror, seeing a teaspoon that seems to bend inside a glass full of water, or improving vision with proper glasses.

We need light to relate to the environment, for this reason, its behavior has always amazed observers, who have wondered about its nature.

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**What does geometric optics study? (Object of study)**

Geometric optics studies the propagation of light in a vacuum and in various media, without explaining what its true nature consists of. For this, it makes use of the model of rays and simple geometry.

A ray is the path that light follows in a certain transparent medium, which is an excellent approximation as long as the wavelength is small compared to the size of the objects.

This is true in a good part of everyday cases, such as those mentioned at the beginning.

There are two fundamental premises of geometric optics:

-Light travels in a straight line.

-While it propagates through various media, light does so by following empirical laws, that is, obtained from experimentation.

**Basic concepts in geometric optics**

**Refractive index**

The speed of light in a material medium is different from that of a vacuum. There we know it’s 300,000 km/s, but in air it’s only slightly less, and even more in water or glass.

The refractive index is a dimensionless quantity, which is defined as the ratio between the speed with which light travels in a vacuum *co* and the speed *c* in said medium:

*n=co/c*

**optical path**

It is the product between the distance traveled by light to pass from one point to another, and the refractive index of the medium:

L = yes. no

Where L is the optical path, s is the distance between the two points and n represents the refractive index, assumed to be constant.

Using the optical path, light rays traveling in different media are compared.

**Angle of incidence**

It is the angle that the light ray makes with the normal line to a surface that separates two media.

**Laws of geometric optics**

**Fermat’s principle**

The French mathematician Pierre de Fermat (1601-1665) noted that:

*When a ray of light travels between two points, it follows that path in which it takes the least amount of time.*

And since light moves with constant speed, its trajectory must be straight.

In other words, Fermat’s principle states that the path of the light ray is such that the optical path between two points is minimal.

**law of reflection**

When incident on the surface that separates two different media, a part of the incident ray -or all- is reflected back and does so with the same angle measured with respect to the normal to the surface with which it was incident.

In other words, the angle of incidence is equal to the angle of reflection:

* θi = θi’*

**Snell’s Law**

The Dutch mathematician Willebrord Snell (1580-1626) carefully observed the behavior of light as it passes from air to water and glass.

He saw that when a ray of light falls on the surface separating two mediums, making a certain angle with it, one part of the ray is reflected back towards the first medium and the other continues its way through the second.

Thus he deduced the following relationship between both media:

*n1 ⋅ sin θ**1** = n2 ⋅ sin θ**2*

where n1 and n2 are the respective *refractive indices*while *θ1 *and * θ2 * are the angles of incidence and refraction, measured with respect to the normal to the surface, according to the figure above.

**Applications**

**mirrors and lenses**

Mirrors are highly polished surfaces that reflect light from objects, allowing images to be formed. Flat mirrors are common, such as those in the bathroom or those that are carried in the wallet.

A lens consists of an optical device with two very close refractive surfaces. When a bundle of parallel rays passes through a converging lens, they converge to a point, forming an image. When dealing with a diverging lens, the opposite occurs: the rays of the beam diverge sharply.

Lenses are frequently used to correct refractive errors in the eye, as well as in various magnifying optical instruments.

**optical instruments**

There are optical instruments that allow magnifying images, such as microscopes, magnifying glasses and telescopes. There are also those to look above eye level, such as periscopes.

To capture and preserve images, there are cameras, which contain a system of lenses and a recording element to save the image formed.

**fiber optic**

It is a long, thin, transparent material based on silica or plastic, which is used for data transmission. It takes advantage of the property of total reflection: when light reaches the medium at a certain angle, refraction does not occur, therefore the ray can travel long distances, bouncing inside the filament.

**solved exercise**

Objects at the bottom of a pool or pond appear to be closer than they really are, due to refraction. At what apparent depth does an observer see a coin at the bottom of a 4-m-deep pool?

Suppose that the ray emerging from the coin reaches the observer’s eye at an angle of 40º with respect to the normal.

Fact: the refractive index of water is 1.33, that of air is 1.

**Solution**

The apparent depth of the coin is s´ and the depth of the pool is s = 4 m. The coin is at point Q and the observer sees it at point Q’. The depth of this point is:

s´ = s – Q´Q

From Snell’s law:

*nb ⋅ sin 40º = na ⋅ sin θ**r*

*sin θ**r** = (nb ⋅ sin 40º)÷ na = sin 40º /1.33 = 0.4833*

*θ**r** = arcsin (0.4833) = 28.9º*

Knowing this angle, we calculate the distance d=OV from the right triangle, whose acute angle is *θ**r*:

tan 28.9º = OV/4 m

OV = 4m × tan 28.9º = 2.154 m

Besides:

tan 50º = OQ´/OV

Therefore:

OQ´= OV × tan 50º = 2.154 m × tan 50º = 2.57 m.

**References**

Bauer, W. 2011. Physics for Engineering and Science. Volume 2. Mc Graw Hill.

Figueras, M. Geometric optics: optics without waves. Open University of Catalonia.

Giancoli, D. 2006. Physics: Principles with Applications. 6th. Ed Prentice Hall.

Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 2. 7ma. Ed. Cengage Learning.

Tippens, P. 2011. Physics: Concepts and Applications. 7th Edition. McGraw Hill.