Basic Notions of Optics:
To begin, let's establish a few basic concepts that will help us understand the optical phenomena we will highlight through the experiments.
Note: If you encounter words or expressions you do not understand, no problem! It is enough that you've made first contact with some new concepts. They will become clearer along the way as you advance in exploring this branch of physics.
The experiments are described below, starting with the yellow background section.
What is light?
In physics, light is a form of electromagnetic radiation visible to the human eye. Light propagates in the form of electromagnetic waves, which are synchronous oscillations of electric and magnetic fields, oriented perpendicular to each other, and both perpendicular to the direction of propagation, as illustrated in figure 2.
The interpretation and explanation of light have evolved significantly throughout history. In Antiquity, thinkers like Empedocles and Euclid had various theories about light. Empedocles believed that light was made of particles emitted by the eye, while Euclid studied the reflection of light and formulated geometric laws to describe this phenomenon.
In the 17th century, two major theories began to dominate the debate about the nature of light. Isaac Newton presented the Corpuscular theory of light in 1704, stating that light is composed of small particles. In contrast, Christiaan Huygens proposed the Wave theory, suggesting that light propagates as waves.
In the 19th century, James Clerk Maxwell unified electricity, magnetism, and light within the Electromagnetic theory, demonstrating that light is an electromagnetic wave.
Today, it is known that one of the fundamental characteristics of light is its duality, manifesting both as a wave and as a particle. This duality was confirmed in the 20th century through the contributions of Max Planck and Albert Einstein, which led to the development of the quantum theory of light.
Einstein explained the photoelectric effect, showing that light can be thought of as consisting of photons, discrete particles of energy (a theory for which he received the Nobel Prize).
As a wave, light has properties such as wavelength and frequency, and it obeys certain laws that we will discuss below and analyze through experiments. Visible light has wavelengths ranging between approximately 400 and 700 nanometers.
In the corpuscular theory, light is composed of photons, which are quanta of energy with no rest mass but which carry energy and momentum. These photons travel through a vacuum at a constant speed of approximately 300,000 km/s.
The physical properties of light include reflection, refraction, dispersion, and diffraction.
- Reflection is the phenomenon by which light is repelled back into the medium from which it came (also called the "medium of incidence") when it encounters a separation surface between this medium of incidence and another medium with a different refractive index, such as the transition from air to glass or water.
- Refraction refers to the change in the direction of light propagation when it passes through the interface between two optical media with different refractive indices, such as the transition from air to glass or water.
- Dispersion occurs when different wavelengths of light are deflected at different angles, as in the decomposition of white light into a spectrum of colors by a prism (formation of a rainbow).
- Diffraction is the bending of light waves around obstacles or through small openings, a phenomenon highlighted in various experiments such as Young's experiment or the interaction of light with a diffraction grating.
The evolution of our understanding of light, from ancient theories to the most modern discoveries, reflects the remarkable progress of scientific knowledge and the human capacity to decipher the complexity of nature.
Determining the focal length of a converging lens
As is known, a converging lens is an optical instrument that has the property of concentrating the light rays of a parallel beam, incident on one of its surfaces, into a single point, called the focus.
A diverging lens is exactly the opposite of a converging lens. A diverging lens "spreads out the rays" of a parallel beam incident on one of its surfaces, creating the illusion that they originate from a point located behind the lens. This imaginary point is also called the focus.
The focal distance or focal length of a lens is the distance between the center of the lens and the focal point, where the light rays meet (or appear to meet) after passing through the lens. The focal length varies depending on the shape of the lens and the material it is made of. In short, converging (convex) lenses gather light into a point, while diverging (concave) lenses make light appear to come from a point. The image below illustrates these definitions:
We keep this property of lenses in mind to describe the simplest method of determining the focal length (Method No. 1), to which we will return shortly.
The phenomenon that generates this behavior of lenses is light refraction. This phenomenon depends on the angle at which the light ray hits the separation surface between the two optical media (air and glass, in our case), according to Snell's Law:
Where:
- θ1 and θ2 are the angles, relative to the perpendicular to the separation surface between the two media (called the surface normal), at which the light ray enters from medium no. 1, also called the medium of incidence, and exits into medium 2, called the medium of emergence.
- n1 and n2 are the refractive indices of the two media, and each represents the ratio between the speed at which light propagates in a vacuum and the speed at which light propagates in that medium (n=c/v, where "c" is the speed of light in a vacuum and "v" is the speed of light in the medium with the refractive index "n").
Thus, based on Snell's formula, the formula calculating the focal length of a lens was determined depending on the material it was made of, the radii of curvature of the two surfaces, and the thickness of the lens. The formula below is called the Lensmaker's equation:
This formula also has a simplified form for the case where the radii of curvature of the two surfaces are much larger than the thickness of the lens. The simplified equation is called the thin lens formula:
Through these two formulas, we arrived at (Method No. 2), by which we can determine the focal length of a lens.
To recap, the first two methods for determining the focal length of a lens are as follows:
Method No. 1
For this method, we need a parallel light beam. The most common parallel light sources are laser sources and the Sun. Having a parallel light beam, all we have to do is place the lens so that it is intersected perpendicularly by the beam and measure the distance between the lens and the screen placed at the distance where the smallest spot is obtained.
Method No. 2
For method 2, we need to know certain constructive data of the lens, namely the refractive index (n) of the optical material from which it was manufactured and the radii of the two curved surfaces of the lens; we apply the Lensmaker formula and obtain the focal length.
Method No. 3
The method that we will, however, put into practice with the help of optical experiment kit No. 1 is Method 3.
Kit 1 - Experiment 1
Method No. 3 for determining the focal length of a converging lens
This method is based on the property of lenses to form images, and the formula describing this property is as follows:
Where d0 is the distance from the object whose image is being formed to the lens; di is the distance from the lens to the screen on which the clear image of the object was formed; and f is, obviously, the focal length of the lens, which we are going to determine.
We therefore need the following three elements to perform the experiment:
- an object whose image we will have to form on a screen;
- the lens whose focal length is to be determined;
- a screen on which the object's image will be formed.
To choose the easiest way to perform the experiment, the object whose image we will form on the screen will be the LED itself from the experiment kit's light source. Specifically, on the screen, we will have to observe elements of the internal structure of the LED, namely, its four electrodes, which we will see as thin black wires against the bright background of the image, as shown in the figure below:
The experimental setup will be carried out as follows:
1. place the light source at the left end of the base plate, between the two guides;
2. place the screen at the right end of the base plate, also between the two guides;
3. place one of the lenses on the base plate, between the two guides and between the light source and the screen, approximately halfway between the two elements;
4. power the light source. You can plug directly into the jack on the base plate if you have installed batteries in the two sockets on the base plate or, using the adapter cable you found in the kit box, you can connect it to the USB port of a phone charger, laptop, or desktop;
5. gradually move the lens closer to the light source. You will notice that the light spot on the screen begins to take shape, and at some point, the 4 fine lines we talked about earlier (the electrodes inside the LED) will become distinguishable;
6. make a fine adjustment by gently moving the lens back and forth until you consider you have obtained the best image of the 4 little black lines;
7. measure the distance from the light source to the lens (measure between the side lines, which have been drawn next to the optical element assembled in its mount). Note the value obtained as do (distance from the object to the lens);
8. measure the distance from the lens to the screen where the image of the 4 electrodes is seen. Note the value obtained as di (distance between the lens and the image);
9. having the two values, d0 and di, you can enter them into the formula above and thus calculate the focal length of the lens;
9bis. ...or, more simply, open the page from the button below where you will find the focal length calculator. Enter the two values you determined during the experiment and obtain the focal length value of the lens.
10. Note the value obtained for the focal length of each lens. You will need these values in the following experiments.
Kit 1 - Experiment 2
Obtaining a light spot from a divergent beam, using converging lenses
This is a simple experiment, but one that will help us introduce a few notions regarding the shape of light beams.
Classification of Light Beams by Shape
1. Divergent Beam - is characterized by the fact that the light rays move away from each other as the beam propagates (it widens with the distance traveled). It propagates in the form of a light cone with uniform transverse distribution.
Examples: Light emitted by a point source (ideal) or small sources (as they are in reality), such as a light bulb or an LED.
2. Convergent Beam - is characterized by the fact that the light rays come closer together as the beam propagates, finally intersecting at a common point, usually called the "focal point" or "focus".
Examples: Light passing through a converging lens, such as a biconvex or plano-convex lens.
3. Parallel Beam is characterized by the fact that the light rays remain at the same distance from one another. Parallel beams do not meet and do not scatter. These are ideal beams. In reality, a beam is considered parallel if its divergence or convergence is extremely small (negligible).
Examples: Laser light (ideal), a light beam after collimation.
4. Other types of beams: - these will only be listed here, but you can read more about them on Wikipedia.
Gaussian beam; Bessel beam; Hermite-Gaussian or Laguerre-Gaussian beam.
Collimating a light beam
Description: Collimation is the process of transforming a divergent beam into a parallel beam. It is achieved through an optical system, such as a lens or a system of lenses.
The simplest way to collimate a beam is by placing a converging lens so that the light source is in its focal plane and on the optical axis, as shown in the figure below:
This is, however, only an ideal, hypothetical situation, unachievable from a practical point of view, for at least two reasons:
1 - in reality, there is no true point light source. Real light sources are physical objects that emit light from an extended surface. The larger this emitting area is, the more the emitted beam deviates from the ideal (mathematical) shape of a light cone with a uniform transverse distribution, as the ideal divergent beam is described.
2 - Lenses, in turn, are not ideal optical components. They introduce a series of aberrations, such as: chromatic aberrations, spherical aberrations, coma, astigmatism, etc. These accumulated aberrations prevent the lens from functioning exactly as it results from mathematical calculations and primary optical theories.
This is also the reason why we did not call this experiment "Collimating a divergent light beam" but rather "Obtaining a light spot from a divergent beam, using converging lenses".
The experimental setup will be carried out as follows:
1. place the light source at the left end of the base plate, between the two guides;
2. power the light source. You can plug directly into the jack on the base plate if you have installed batteries in the two sockets on the base plate or, using the adapter cable you found in the kit box, you can connect it to the USB port of a phone charger, laptop, or desktop;
3. place the screen as close to the light source as possible and visually evaluate its illumination level;
4. move the screen to the right end of the base plate and observe how the screen's illumination level decreases as it moves further away. This happens because when it is near the light source, almost all the light emitted by the source reaches the screen. In our situation, where the light source emits a divergent beam, which, as we defined above, propagates as a light cone, the further we move the screen from the source, the more light passes beside the screen, and consequently, less light will reach the screen, so it will be more dimly lit. Also, it can be observed that the screen is uniformly illuminated over its entire surface, even though the light source is only 5mm in diameter. This happens, obviously, because the beam is divergent and propagates as a "light cone".
In conclusion, the further we move from the source, the larger the illuminated surface will be, but the illumination of a given surface (for example, our screen) will be lower.
The initial setup should look like the figure below:
5. Place one of the lenses halfway between the light source and the screen and start moving it towards the light source. Observe the evolution of the light spot on the screen. The moment you reach a distance from the source equal to the focal length of the lens, theoretically, if the source were a point and the lens "ideal," without aberrations, the light beam passing through the lens would become parallel, which would mean that a collimated beam was obtained. In reality, by placing the lens in an optimal position, where its focus is near the light source, we will obtain an "almost parallel" beam, but you will notice that although the illumination in the central area of the spot is uniform, towards the edges, it will be lower.
At this stage, the setup should look like the figure below:
6. Check the degree of collimation of the beam by moving the screen back and forth to see if the diameter of the light spot remains approximately constant. You can move the screen further away than the mounting base plate allows, or if the setup is pointing towards a wall, located at a distance of about 1 - 2 meters, you can move the screen aside and check if the size of the spot projected on the wall remains relatively constant. If the spot on the wall is significantly larger, make a fine adjustment by moving the lens closer and further from the source until you achieve an acceptable spot size. At that moment, it can be said that the best possible collimation of the beam passing through the lens has been obtained.
Kit 1 - Experiment 3
Analysis of image formation using a converging lens
We saw in the first experiment that lenses have the property to form images, and we used the formula that calculates the distance at which the image appears, depending on the focal length of the lens and the distance at which the object is located relative to the lens.
To see more clearly how things stand, we can rewrite the formula as:
di =
f do
do - f
We recall that do is the distance from the object whose image is formed to the lens; di is the distance from the lens to the image, and f is the focal length of the lens.
In this experiment, we will analyze what happens to the image if:
1. the object is at a distance greater than the focal length of the lens;
2. the object is at a distance smaller than the focal length of the lens (the object is between the focal plane of the lens and the lens);
These are the two situations that we will analyze experimentally and explain using the formula above.
Case 1. Object at a distance greater than the focal length of the lens - notice in the sketch below how the image is formed in this case:
- light rays that are parallel to the optical axis are directed by the lens to pass through the focus;
- light rays passing through the optical center "O" of the lens travel along the path undeviated;
- the image is formed at the intersection of the two rays.
The first observation is that in this case, an "inverted" (upside-down) image is obtained relative to the object's position.
The second observation is that the image obtained has a different size than the object. Consequently, we will introduce the concept of "Magnification" (of the image relative to the object). The formula for the magnification obtained with a lens is, obviously, the ratio between the image height (hi) and the object height (ho), but observing in the figure above the two triangles that have formed and which are "similar", we notice that this ratio is also equal to the ratio between the two distances (di) and (do). The magnification formula is, therefore:
M =
hi
ho
= -
di
do
It can be seen from this formula (using the standard Cartesian sign convention) that the math handles the image orientation automatically. If the image is real (di > 0), the magnification M becomes negative, indicating an inverted image. The absolute value of M tells us if the image is larger (|M| > 1) or smaller (|M| < 1) than the object.
Case 2. Object at a distance smaller than the focal length of the lens (object between the lens and the focus) - we trace, similarly, the two rays, one parallel to the optical axis, which the lens sends into the focus, and a ray passing through the optical center "O" of the lens, which will not be deviated. This time, we notice that the two rays passing through the lens no longer meet to form a real image, so no matter where we place the screen, a focused image of the object will no longer be obtained.
We notice, however, that although the rays passing through the lens do not meet, their extensions (drawn in the sketch with a dashed line) do meet, and for this reason, the image formed at their intersection is called a virtual image. Virtual images are those images we see when using lenses as a magnifying glass, but these images cannot be seen on a screen. You might wonder: if it cannot be projected onto a screen, how can we see it with our naked eye by looking through the lens? The answer is that the human eye acts as a second converging lens. The crystalline lens takes the virtual image created by the magnifying glass and transforms it into a real image on the retina (which acts as a screen).
In this experiment, we will therefore not be able to visualize the virtual image that formed from the extensions of the rays passing through the lens, but the surprise will come in the next experiment, where you will see how easily this image can also be brought onto the screen (essentially replicating how our eye works).
The second observation is that the formed image is upright (unlike the previous case), and for this reason, the magnification (M), which is calculated with the same formula as in the first case, will naturally result in a positive value (+).
The experimental setup:
The experimental setup will be carried out as follows:
1. place the light source at the left end of the base plate, between the two guides;
2. place the screen at the right end of the base plate, also between the two guides;
3. choose one of the lenses and place it on the base plate, between the two guides and between the light source and the screen, roughly halfway but a bit closer to the light source;
4. secure the transparent film with the printed arrow in the "crocodile" clip and place it between the light source and the lens, so that the drawing (arrow) is in front of the lens and approximately centered with it. This arrow represents the "object" in the setup;
5. power the light source. You can plug directly into the jack on the base plate if you have installed batteries in the two sockets on the base plate or, using the adapter cable you found in the kit box, you can connect it to the USB port of a phone charger, laptop, or desktop;
6. knowing the focal length of the lens and guided by the photo above, place the transparent film initially at a distance from the lens greater than the focal length of the lens;
7. open the page with the calculator for calculating the image formation distance and magnification from the button below. Enter the focal length value of the lens you chose and the distance at which you placed the object relative to the lens. Click the "Calculate" button.
Thus, you obtain the distance at which the image is formed (di) and the image magnification value (M) -
You can continue to enter different values for the object distance and analyze the result obtained:
For example, if the lens has a focal length of 73mm and the object is placed 100mm from the lens, applying the two formulas we get, for the distance at which the image is formed relative to the lens, di = 270.37mm and for the image magnification M = -2.7
How do we interpret these results?
- the fact that the distance at which the image is formed has a positive value means that the image is real, so it can be focused on the screen at a distance of approximately 270mm;
- the fact that the magnification has a minus sign (-) means that the obtained image is inverted;
- the fact that the absolute value of the magnification is 2.7 means that the dimensions of the image are 2.7 times larger than those of the object. In other words, a magnified image of the object is obtained.
8. continuing to use the calculator you opened from the button above, you can analyze the results obtained if you enter values for two particular situations:
8.1 - the object placed exactly in the focal plane of the lens (do = F) - In this case, you will notice that both the image formation distance (di) and the system magnification (M) are infinite. Obviously, these values have no physical meaning, but as we discussed before, if from this plane the object moves closer to the lens, the resulting images will be virtual, whereas if it moves further away, the images will be real. The greater the distance do, the smaller the distance di, as well as the magnification (M), will be.
8.2 - the object placed at twice the focal distance of the lens (do = 2F) - In this case, you will notice that the image distance (di) is equal to the object distance (do), and the magnification M = -1 (the image is inverted but has the same dimensions as the object).
9. calculate by entering values for do smaller than the focal length of the lens. You will notice that the value for di becomes negative, meaning the image is virtual, thus impossible to obtain a focused (clear) image on the screen, regardless of where we place it. On the other hand, because of the standard sign convention, the mathematical result for the magnification will automatically have a plus sign (+), indicating that the virtual image will be upright.
10. create experimental setups reproducing the distances you entered and/or that resulted from calculations. Observe the formed images where applicable, measure the distances, measure and calculate the ratio between the image dimensions and the object dimensions to obtain the magnification of the optical system. Compare the experimentally obtained results with those obtained by calculation.
Kit 1 - Experiment 4
Real image formation using two lenses (slide projection)
In the previous experiment, we saw that if we place the object between the lens and its focal plane, we obtain a virtual image, which we cannot view on the screen. Now it's time to "bring to light" this virtual image, by transforming it into a real image that can be projected on the screen.
Let's repeat this schematic in the figure below, which represents:
- the light source (S), placed on the left side of the setup;
- the lens (L1), with focuses at F1 and F1';
- the object (O), which can be a slide or a transparent film on which an image is drawn or printed;
At this moment, we are in the second case of the previous experiment, a case in which a converging lens forms a virtual image of an object placed between the lens and its focus. This virtual image is represented in the sketch below by the bi-colored arrow (red + green).
Although the virtual image obtained this way cannot be seen directly, if we continue the setup with another converging lens (L2), placed at a distance from the virtual image greater than its focal length (F2), this lens will treat the virtual image as if it were an object and will form a new, real image at a certain distance that emerges from the same lens formula presented in the previous experiment:
di =
f do
do - f
The setup will therefore continue with:
- the lens (L2), with focuses at F2 and F2';
- the screen (E), placed on the right side of the setup.
The experimental setup will be carried out as follows:
1. as usual, place the light source at the left end of the base plate, between the two guides;
2. place the screen at the right end of the base plate, also between the two guides;
3. choose the assembly with the lens that has the smaller focal length (it is the one with a black band mounted on the back, just below the lens). Secure the film between this band and the lens holder, as in the image below, so that the pattern printed on the film is roughly centered with the lens;
4. place the assembly with the attached film about 10 cm from the light source.
5. power the light source. You can plug directly into the jack on the base plate if you have installed batteries in the two sockets on the base plate or, using the adapter cable you found in the kit box, you can connect it to the USB port of a phone charger, laptop, or desktop;
6. look at the image formed on the screen. You will find that wherever the screen is placed, a focused image cannot be obtained;
7. place the second lens (L2) between the first lens (L1) and the screen, as seen in the figure below.
8. find the position of L2 for which the clearest image is obtained.
9. move the screen until the image becomes blurry and subsequently find the new position for L2 for which the clearest image is obtained. You can continue to move the lenses and the screen until you discover an empirical relationship between the distances of L1, L2, and the screen, correlated with the size of the obtained image.
10. open the calculator related to this experiment from the button below:
11. Enter values for the object distance relative to the first lens (8 mm, if the slide was placed directly on the lens mount, or the distance you measure if it was held in the "crocodile" clip and placed in another position). Also, enter values for the focal lengths of the lenses involved in the experiment and obtain the distances and magnifications for the two images: the intermediate image (the virtual one) and the final image, which will be viewed on the screen.
12. Also, enter imaginary values for the object distance and the focal lengths of the two lenses to conduct your own analysis of the image evolution based on these parameters.
