Lambert’s Cosine Law

Diffuse Reflection

Diffuse reflection can be explained as an incident ray or the light’s reflection that appears as it is scattering from many angles.

Lambertian reflection is the illustration retrieved by the diffuse reflection. Lambertian reflection is illustrated as the presentation of the object which is luminous in such a way that all the angles of observations are equal.

This is the vital inequality between diffuse reflection and specular reflection.

A plane made up of non-absorbing powder like plaster, or paper made up of fibers, or white marble-like polycrystalline material, reflects diffused light at great efficiency. Many common materials can produce both diffuse and specular reflection.

Lambert’s Cosine Law

As per this law, the radiant intensity about the surface/radiator of ideal diffusely reflecting mechanism is directly proportional to the cosine of angle θ between the direction of the surface plain & emitted light.

This law is coined by Johann Heinrich Lambert, available in optics. This law is also known as cosine emission law or Lambert’s emission law.

Examples of Diffuse Reflection

• Frosted light bulbs.

• The matte finish paints used in home paintings generate diffuse reflection, whereas the glossy paints exhibit specular as well as diffused reflection.

• The human eyes’ vision mechanism is based on diffuse reflection as they reflect the incident ray in different angles.

Regular and Diffuse Reflection

When the surface is smooth, just like a mirror, the image obtained by reflection is vivid and sharp. This is known as regular reflection, also known as specular reflection.

However, if there is little rough surface or bumpy, the chance of image formation may be less. The nature of the image is fuzzy or blurry. This is known as diffuse reflection.

In the figure given below, we can observe how these reflections happen. Arrow rays represent the waves of light. The arrows (rays) impact the surface are treated as incident rays, and the arrows that take off (reflect) the surface are treated as reflected rays.

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In general, the reflected rays are in the same direction. This illustrates why regular reflection can form a clear image. In diffuse reflection, the rays are scattered to many directions so that the image formed is blurry. 

This diffused reflection generated image is more noisy and unclear than regular reflection.

How Does Diffuse Reflection Occur?

When the rays are reflected after the propagation upon the rough surfaces, it causes diffuse reflection in various directions.

We encounter diffuse reflection mostly in our day-to-day lives. It happens only when there are little imperfections on the surface of the material.

Let’s consider a piece of paper as its surface may look smooth. Although appearance is smooth, the tiny microscopic deformation may make it rough. So it will propagate diffuse reflection.

Even a leaf has that roughness because of its cell infrastructure. In diffuse reflection, each ray collides upon a part surface that is diverted into a different direction. This follows the law of reflection, but the normal is distinct for each ray where the reflected ray will travel to all directions.

We know, a single incident ray travels in one direction after reflection, the second incident ray reflects in any direction, but both don’t merge at all. This is why we cannot visualize the clear image in the diffuse reflection.

Diffuse Reflection of Light

Diffuse reflection of light is the reflection of light that incident ray falls on a plane that is reflected at many angles. This reflection is not like specular reflection because the reflection angles are always the same here.

A surface that is made of non-absorbing powder such as paper from fibers, or plaster, or from white marble, reflects light with great efficiency called diffuse reflection.

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As shown in fig. a, when a beam of light strikes upon a smooth surface at a particular angle, almost the total amount of light will be reflected from the surface at the same and narrow direction.

As shown in fig. b, an observer can see the reflected images at certain observing angles on the surface. The rough surface upon which the light is reflected can travel at many different angles.

It happens because the non-symmetric surface consists of different angles.

As shown in fig. c, the reflected light, as per the observer's point of view, is so diffused that he can’t view these images on the surface. A glossy surface will reflect a considerable amount of diffused as well as specular light.

Few images will be visible on the reflected surfaces, but the appearance of the image will be blurry.

Luminous Flux Derivation

Luminous flux is also named as luminous power. It is the measurement of the power of the light spotted.

Luminous flux derivation can be written as follows;

F\[_{tot}\] = \[\int_{0}^{\frac{π}{2}}\] \[\int_{0}^{2π}\] Cos Cos(θ)I\[_{max}\] Sin Sin(θ) dØ dθ = 2π ∗ I\[_{max}\] \[\int_{0}^{\frac{π}{2}}\] Cos Cos θ Sin Sin θ dθ

= 2π ∗ I\[_{max}\] \[\int_{0}^{\frac{π}{2}}\] \[\frac{\text{Sin Sin(2θ)}}{2}\] dθ = πsr ∗ I\[_{max}\]

Where,

Imax = Luminous flux

Sin θ = Jacobian matrix determinant