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# Geometrical and Physical Optics

Did you know that roughly 90 million years ago our early ancestors used to have ultraviolet vision? As time went on, our eyesight shifted into the current ability to see the visual spectrum, which is the primary light we focus on in optics. Considering light's dual nature of behaving like a wave or a particle depending on the circumstances, two different field of optics developed. Geometric optics is a simple model of light which treats light consisting of rays that travel in straight lines, assuming that light travels in rays and can be reflected and refracted the boundaries between optical media. Physical optics, on the other hand, deals with the wave properties and phenomena of light such as interference and diffraction. In this article, we will establish the key differences between the two branches of optical physics!

Contents

## Geometric and Physical Optics Similarities

As the title implies, geometric and physical optics both stem from the same branch of science - optical physics, but they describe different properties of light, and so as you might expect, the situations in which they are applied differ depending on the aspect of the physical system under consideration. To begin with, let's define what exactly optics is before delving into the specifics of each field.

Optics is the study of the behavior and characteristics of light.

Understanding how light behaves in different media and developing various instruments, such as mirrors, lenses, and interferometers, to further apply this knowledge to our everyday lives is a critical aspect of this physical science.

There are a number of different properties of light, but usually they can be categorized into two main types: particle phenomena and wave phenomena. Let's recall a critical property that light possesses - wave-particle duality.

Wave-particle duality states that the behavior of light can be described as either a particle or a wave.

Depending on the circumstances, sometimes it makes more sense to describe light as electromagnetic radiation that propagates through a medium in the form of a wave. Meanwhile, other phenomena, such as the photoelectric effect, can only be explained by considering light as a particle which we call a photon. So it makes sense to separate optics into two separate branches, focusing on different properties of light and with different applications.

## Geometric and Physical Optics Differences

We have already established that besides the fact that geometric and physical optics deal with light, they will have varying applications. Let's look at each field separately so we can illuminate the distinction between the two closely related models of light phenomena.

### Geometrical Optics

Geometric optics, also known as ray optics, is used when light waves interact with objects much greater than the wavelength of the visible light ($$400-700 \, \mathrm{nm}$$), therefore, utilizing the particle nature of light.

Geometrical optics are used when dealing with the transmission of light in rays.

An important aspect of geometrical optics is the image formation using light rays and devices such as lenses, mirrors, and prisms.

A ray of light is a hypothetical line representing a path along which light energy is transferred.

Rays realistically represent the observations made by scientists, but considering it is an abstraction, certain assumptions have to be made:

1. In a consistent environment, the light rays travel in a straight line.

2. If the light rays approach some optical aperture, they are stopped.

3. Light rays can be absorbed, reflected or refracted.

4. If light rays cross one another, no changes occur.

The main properties to focus on are reflection and refraction, whilst interference and diffraction are disregarded as we ignore the wave properties of light and instead are described using physical optics.

### Physical Optics

In contrast to geometric optics, physical optics focuses on the wave nature of light.

Physical optics is used when dealing with the inherent nature and properties of light.

As a result, another term used to describe physical optics is wave optics.

A wave is a disturbance that moves through a medium or a vacuum and transmits energy.

In this context, light is a transverse electromagnetic (EM) wave in which the oscillations of the electric and magnetic field are perpendicular to the direction of travel of the wave. While mechanical waves such as sound require a medium to travel through, light is an EM wave; therefore it can travel through a vacuum as well, with a maximum speed of $$c=3\times10^8\, \frac{\mathrm{m}}{\mathrm{s}}$$. The speed can be determined in other mediums, all we need to know is the frequency $$f$$ and the wavelength $$\lambda$$ of the wave.

The main properties studied by physical optics are interference, diffraction, and polarization, which we'll discuss in greater detail in the next section.

## Geometrical and Physical Optics Applications

We already touched upon certain aspects each branch of optics focuses on, let's look at each of these concepts in more depth.

### Geometrical Optics Applications

Imagine turning on a torch and pointing it at a flat mirror perpendicular to its surface. In this case, we'll be able to trace the overall path of the light as a straight line approaching the surface of the mirror and then reflecting off of it.

Reflection is the change in direction of a light ray, as it strikes a surface made up of a different medium to the medium it is initially travelling in.

So the light was initially traveling through air, and then bounced off of the mirror. Based on the law of reflection, the angle of reflection is always the same as the angle of incidence with respect to the normal, which is the line perpendicular to the surface. All of this occurs in the same plane.

Depending on the curvature of the mirror (i.e. flat, concave, or convex), the path of the rays will differ!

Now, let's say we direct the ray of light toward a still body of water. Some of the light will reflect off the surface, following the law of reflection stated previously. However, some of the light will transmit through the water. Water is much denser than air, so the path of the light will become redirected or refracted.

Refraction is the bending of light as it passes from one transparent medium into another.

The ability of certain materials to bend light is described using Snell's law and the refractive index $$n$$, with the mathematical expression for both defined in the formula table later in the article.

The refractive index is a quantity associated with a specific optical medium and measures the amount of bending of light that occurs as the rays pass from one medium into another.

It's a dimensionless quantity, which depends on the velocity of light $$v$$ in the specific medium. The more 'optically dense' the medium is, the more particles that get in the way of the light ray, resulting in a larger refractive index.

The refractive index for vacuum is $$n_\text{vacuum}=1$$, while that of water is $$n_\text{water}=1.33$$.

Both of these properties are visualized in Figure $$1$$ below.

Fig. 1 - The reflection and refraction of a ray of light as it coincides with the boundary between a less dense and more dense region, as described by Snell's law.

In this diagram, the incident ray of light is passing from a less dense medium $$n_1$$ (e.g. air) into a more dense medium $$n_2$$ (e.g. water).

Similarly, geometrical optics is used to analyze the rays of light interacting with various lenses, mirrors and prisms. There are two types of lenses and mirrors - concave and convex, with different properties and applications. Mathematically, they can be analyzed using the thin lens equation and the magnification equation, while graphically, we can determine the types of images formed in each case by drawing a ray diagram. More in-depth information on the various properties of different lenses is available in the thin lenses and the reflection in spherical surfaces explanations here on StudySmarter!

### Physical Optics Applications

When light waves hit a bubble, we sometimes observe colorful patterns on it's surface. This is a result of interference, one of the properties studied in physical optics.

Interference occurs when two or more waves superimpose to form a new wave.

The resultant wave and its amplitude will depend on the relative phase of the waves. Constructive interference will occur if the waves meet in phase, and destructive interference if they meet out of phase. Mathematically, these can be expressed using a path difference equation, shown in the equation table below. Both of these occurrences are visualized in Figure $$2$$ below. In these examples, the interfering waves have the same frequency so that the effects of constructive and destructive interference can be visualized more easily, but interference can occur between any two waves of different frequencies, amplitudes and relative phases.

Fig. 2 - Constructive interference occurs when waves are in phase with one another, whereas destructive interference results from waves out of phase.

Interference is demonstrated in Young's double slit experiment and is also demonstrated by diffraction gratings. As the light passed through two narrow slits, instead of seeing two bright spots corresponding to the slits, an interference pattern of waves overlapping and canceling one another is observed. The process of light encountering the slits and spreading out is known as diffraction.

Diffraction is the bending of light as it reaches an obstacle.

Diffraction will only occur if the slit width is similar in size to the wavelength of the light source. The equation used to mathematically describe a diffraction grating can be found in the formula table below.

## Geometrical and Physical Optics Formula

All the most commonly used formulas in both, geometrical and physical optics, are compiled in the table below. For the purposes of this article, we will not go into too much detail about each equation. They will be discussed in more detail in other explanations here on StudySmarter.

 Name Equation Variables Snell's law $$n_1\sin\theta_1=n_2\sin\theta_2$$ $$n_1$$ - incident index$$\theta_1$$ - incident angle$$n_2$$ - refracted index$$\theta_2$$ - refracted angle Refractive index $$n=\frac{c}{v}$$ $$n$$ - index of refraction$$c$$ - speed of light $$v$$ - velocity in a substance Thin lens equation $$\frac{1}{s_\mathrm{i}}+\frac{1}{s_\mathrm{o}}=\frac{1}{f}$$ $$s_\mathrm{i}$$ - image distance $$s_\mathrm{o}$$ - object distance $$f$$ - focal length Magnification of a thin lens $$M=\frac{h_\mathrm{i}}{h_\mathrm{o}}=\frac{-s_\mathrm{i}}{s_\mathrm{o}}$$ $$M$$ - magnification$$h_\mathrm{i}$$ - image height$$h_\mathrm{o}$$ - object height$$s_\mathrm{i}$$ - image distance$$s_\mathrm{o}$$ - object distance Wave equation $$v = f \lambda$$ $$v$$ - velocity $$f$$ - frequency$$\lambda$$ - wavelength Interference (double slit, diffraction grating) $$d\sin\theta=m\lambda$$ $$d$$ - separation$$\theta$$ - angle$$m$$ - an integer$$\lambda$$ - wavelength Path difference equation - constructive interference $$\Delta L=m\lambda$$ $$\Delta L$$ - distance $$m$$ - an integer$$\lambda$$ - wavelength Path difference equation - destructive interference $$\Delta L=\left (m + \frac{1}{2} \right )\lambda$$ $$\Delta L$$ - distance $$m$$ - an integer$$\lambda$$ - wavelength

Example problems applying some of these formulas can be found later in this explanation.

## Geometrical and Physical Optics Examples

Now that we've established what the main differences between geometric and physical optics are, let's apply this knowledge to some example problems!

One of the applications of geometric optics is obtaining and analyzing the images formed by thin lenses.

Firstly, let's look at the converging lens visible in Figure $$3$$ below, and find the image this lens would form!

Fig. 3 - Geometrical optics applied to a converging lens.

Solution

To locate the image formed by this convex lens, we must draw a ray diagram. A minimum of two rays must be used to find their point of intersection, indicating the highest point of the image (the lowest point will be on the principal axis as that's where the object is rested). First of all we need to know some basic principles which we can follow which will tell us how to draw the path of light rays passing through the lens:

1. Rays parallel to the principal axis will always pass through the lens in such a manner that they pass through the focus of the lens.
2. Rays that are colinear with the principal axis will not be refracted by the lens as they are incident perpendicular to the surface of the lens. These rays pass in a smooth straight line, through the centre of the lens, intercepting the optical axis at a right-angle, before passing through the focus of the lens.

So, let's draw:

1. Ray 1 parallel to the principal axis until it reaches the optical axis and then straight through the focus $$F$$.
2. Ray 2 passing straight through the center of the lens, where the optical and principal axis cross.

as visualized in Figure $$4$$ below.

When a light ray passes through the center of a lens, it experiences no refraction; hence the ray travels in one smooth straight line. For the remaining rays, in this case, we chose to draw a parallel ray of light, which refracts at the optical axis and passes through the focus. However, we also could've drawn a ray going through the focus on the object side, where light would refract and continue parallel to the principal axis on the other side of the lens.

Fig. 4 - The ray diagram of a converging lens.

This ray diagram provides us with a lot of information about the image obtained. For instance, we can see that it is flipped upside down or inverted, as well as appears smaller than the object itself, or in other words, it is diminished. Since the image is inverted, we can conclude that it is also real.

Real images can be projected onto a screen as they form on the opposite side of the object, while virtual images cannot as they are formed behind a lens.

All of these characteristics depend on the type of lens used and the location of the object regarding the focus of the lens. So, if this was a concave lens, the image would be virtual and upright.

Secondly, let's say that the object in Figure $$3$$ is $$3\,\mathrm{cm}$$ high and is located $$16.0 \, \mathrm{cm}$$ away from the lens with a focal length of $$6.00 \, \mathrm{cm}$$. Calculate the height of the image and how far away from the lens it forms!

Solution

To find the image distance $$s_\mathrm{o}$$ we can use the following equation:

$$\frac{1}{s_\mathrm{i}}+\frac{1}{s_\mathrm{o}}=\frac{1}{f}.$$

By rearranging the thin lens equation and plugging in our values for object distance and focal length, we get

\begin{align} \frac{1}{s_\mathrm{i}}&=\frac{1}{f}-\frac{1}{s_\mathrm{o}} \\ \frac{1}{s_\mathrm{i}}&=\frac{1}{6.00 \, \mathrm{cm}}-\frac{1}{16.0 \, \mathrm{cm}} \\ \frac{1}{s_\mathrm{i}}&= 0.104\,\mathrm{cm} \\ {s_\mathrm{i}}&= 9.62 \,\mathrm{cm}. \end{align}

The positive value for image distance confirms our ray diagram results: the image is located on the opposite side of the lens and is inverted.

Now we can use the magnification equation

$$M=\frac{h_\mathrm{i}}{h_\mathrm{o}}=\frac{-s_\mathrm{i}}{s_\mathrm{o}}$$

to find the height of this image. We were given the height of the object and the object distance, and the image distance was calculated using the thin lens equation, so we get

\begin{align} h_\mathrm{i}&=\frac{-s_\mathrm{i} \, h_\mathrm{o}}{s_\mathrm{o}} \\ h_\mathrm{i}&= -\frac{(9.62 \, \mathrm{cm})(3 \, \mathrm{cm})}{(16.0 \, \mathrm{cm})} \\ h_\mathrm{i}&=-1.80 \, \mathrm{cm},\end{align}

where the negative sign implies that the image is inverted (as we established earlier using the ray diagram).

Now, let's look at an example problem applying the concepts of physical optics.

Monochromatic light with a wavelength of $$610 \, \mathrm{nm}$$ falls normally on a diffraction grating with $$100$$ lines per $$\mathrm{mm}$$. Find the angle of the third order maximum.

Solution

First, we must calculate the separation distance $$d$$ between the slits. We are told that there are $$N=100$$ lines in every millimeter which converted to SI units is equivalent to $$10^{5}$$ lines in every meter, so we can use the following equation

\begin{align} d&=\frac{1}{N} \\ d&=\frac{1}{10^5 \, \frac{\text{lines}}{\mathrm{m}}}=10^{-5} \, \mathrm{m}. \end{align}

To find the angle $$\theta$$ of the third order ($$m=3$$) we rearrange the interference equation

$$d\sin\theta=m\lambda$$

and plug in our values:

\begin{align} \sin\theta_3 &= \frac{m\lambda}{d} \\ \sin\theta_3 &= \frac{(3) (610\times 10^{-9} \, \cancel{\mathrm{m}})}{\left (10^{-5} \, \cancel{\mathrm{m}} \right )} \\ \sin\theta_3 &=0.183 \\ \theta_3 &= \sin^{-1}(0.183)=10.5^{\circ}. \end{align}

So, the angle of the third order maximum is equal to $$10.5^{\circ}$$.

## Geometrical and Physical Optics - Key takeaways

• Optics is the study of the behavior and characteristics of light.
• Wave-particle duality states that the behavior of light can be described as either a particle or a wave.
• Geometric optics treats light as consisting of abstract rays which are perpendicular to the wavefronts of a wave and travel in straight lines in homogenous media.
• Physical optics are used when dealing with the inherent nature and properties of light.
• Some geometrical optics applications include explaining reflection, refraction, and behavior of light in lenses and mirrors.
• Reflection is the change in direction of a light ray, as it strikes a surface made up of a different medium.
• Refraction is the bending of light as it goes from one transparent medium into another.
• Some physical optics applications are interference and diffraction.
• Interference occurs when two or more waves superimpose to form a new wave.
• Diffraction is the bending of light as it reaches an obstacle.

## References

1. Fig. 1 - The reflection and refraction of light in different mediums, StudySmarter Originals.
2. Fig. 2 - Constructive and destructive interference, StudySmarter Originals.
3. Fig. 3 - Converging lens ray diagram setup, StudySmarter Originals.
4. Fig. 4 - Converging lens ray diagram, StudySmarter Originals.

#### Flashcards in Geometrical and Physical Optics 69

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What is geometrical and physical optics?

Geometrical and physical optics is the study of behavior and characteristics of light, where geometrical optics explain and use various devices to form images, and physical optics focus on the properties of light.

What are the applications of geometrical and physical optics?

The applications of geometrical and physical optics are explaining the properties and behavior of light such as reflection, refraction, interference, and diffraction.

What is the formula for calculating geometrical and physical optics?

The formulae for calculating geometrical and physical optics include the Snell's law, the lens equation, magnification of thin lenses, wave equation, and interference equations.

What is the difference between physical and geometrical optics?

The difference between physical and geometrical optics is that geometric optics are used when dealing with the transmission of light in rays, and physical optics are used when dealing with the inherent nature and properties of light.

What are the similarities between geometrical and physical optics?

The similarities between geometrical and physical optics are that they are both branches of optics studying the behavior and characteristics of light.

## Test your knowledge with multiple choice flashcards

The amplitude of the resulting wave in a destructive interference with the phase difference of $$180^{\circ}$$ is larger than the amplitude of the individual waves.

When light travels from water into air, it will bend ...

A monochromatic light with a wavelength of $$610 \, \mathrm{nm}$$ falls normally on a diffraction grating with $$200$$ lines per $$\mathrm{mm}$$. Find the angle of the second order maximum.

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