Optics

Optics was an event for divisions B and C during the 2012 season. This event deals with geometric and physical optics, such as reflection, refraction, critical angle, electromagnetic and visible spectrum, lenses, and mirrors. Competitors for this event may bring any type of calculator and should have knowledge of SI units, as all answers will require a student to know and understand them. Some aspects of optics are similar to the current event Crave the Wave, though Optics focuses on light while Crave the Wave was about waves in general.

Introduction
Optics is a science which studies light. Optics is usually divided into two subfields: geometric optics and physical optics. Geometric optics studies how light travels, and includes concepts of reflection and refraction. In geometric optics, light is thought of as rays which travel according to certain principals.

Physical optics studies the nature of light, as well as phenomena which cannot be explained by the ray approximation used in geometric optics. Physical optics includes the visible and electromagnetic spectrum, and concepts such as interference.

Geometric optics
When waves of light (or electromagnetic waves in general) hit the boundary between two media, there are several ways that the light can react. Namely, it can reflect, refract, or be absorbed into the second medium. These will be explained below.

Reflection
The law of reflection states:


 * The angle of incidence of a wave or stream of particles reflecting from a boundary, conventionally measured from the normal to the interface, is equal to the angle of reflection, measured from the same interface.

This may seem very complicated, but it really isn't that hard to understand. For reflection to occur, you need a "wave or stream of particles" and an interface. Light, of course, takes the former role. For our purposes, let's assume that the interface is a mirror.

Figure 1 is a convenient example of reflection. In the diagram, $$PO$$ is an incoming ray of light about to reflect off a mirror. $$OQ$$ is the ray after it has hit the mirror. You also can see a new line, called the "normal". This is an imaginary line through point $$O$$, which is perpendicular to the mirror. The angle of incidence is the angle labeled $$\theta_i$$, and the angle of reflection is the one labeled $$\theta_r$$.

From this, we can represent the law of reflection by the following simple equation: $$\theta_i=\theta_r$$

You can also conclude that the angles that are complements to $$\theta_i$$ and $$\theta_r$$ will be congruent.

Refraction
The speed of light is always constant when in a vacuum. However, light can change speeds while traveling through different media, such as air, water, and glass. The optical density of a medium measures how well it can transmit light. Water has a higher optical density than air, so light travels slower in water than it does in air. Note that optical density is different than the actual density of a medium.

Another way to characterize the refraction of light between media is with the index of refraction, which is useful in determining the angle at which light refracts at a boundary. The index of refraction is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. When light strikes a boundary between media at an angle, the refractive index can be calculated by using Snell's Law: n1sinθ1= n2sinθ2 where θ1 is the angle the light is traveling at in the first medium relative to the normal, and θ2 is the angle the light is at after it passes through the second medium relative to the normal. By dividing the sines of those two angles, you have the index of refraction. This equation is known as Snell's law and can also be used backwards to find the angle at which light leaves if you know both the index of refraction of the second medium and the angle at which the light entered. The refractive index of a medium is constant for every angle at which light can enter, but it is different for every wavelength of light. This is why light disperses in a prism.

Refraction is calculated by using Snell's Law, which is: $$n_1\sin\theta_1=n_2\sin\theta_2$$

Lenses and mirrors
There are two main types of lenses and mirrors: convex and concave. Convex mirrors are known as diverging mirrors because the incident rays diverge upon reflection. Similarly, concave mirrors are called converging mirrors and lenses because the incident rays converge upon reflection.

In lenses, the opposite is true. Convex lenses are converging lenses, and concave lenses are known as diverging lenses. There are six types of lenses that fall under those two categories. The six lenses types are: plano-convex, plano-concave, double convex, double concave, concavo-convex, and convexo-concave.

Also remember that mirrors reflect, and lenses refract.

Vocabulary

 * Aperture: Describes how much light will be intercepted by the mirror
 * Center of curvature: Useful when locating images, it is the center of the imaginary sphere upon which a curved mirror rests. The flatter the mirror, the farther away the center of curvature
 * Concave mirror: Converging mirror which forms either real or virtual images which may be magnified
 * Convex mirror: Diverging mirror which forms virtual images which may be magnified
 * Focal length: The distance from the lens or mirror to the principal focus
 * Principal axis: The main line drawn through the center of the mirror or lens upon which information such as center of curvature or principal focus is given
 * Principal focus: The point where light rays parallel and close to the principal axis converge or appear to diverge
 * Ray diagram: A tracing of the light rays to show where the image forms after being reflected off a mirror or refracted through a lens
 * Real image: An image that can be projected onto a screen at its location, inverted relative to the object, and can be magnified
 * Secondary axis: Any line drawn through the center of curvature to the mirror or lens
 * Vertex: The center of the lens or mirror
 * Virtual image: An image that cannot be projected onto the screen, is not inverted relative to the object, and can be magnified

Image location in mirrors
It may be required of you to find the approximate location of the image of an object reflected off a mirror. The following is a diagram of the possible locations for an image:



Trial 1: (Concave Mirror) The object is located beyond the center of curvature. The image is located between the center of curvature (C) and the principal focus (F). It is reduced in size and inverted. The image is real.

Trial 2: (Concave Mirror) The object is located at the center of curvature. The image is located at the center of curvature. It is inverted. The image is real.

Trial 3: (Concave Mirror) The object is located between the center of curvature and the principal focus. The image is located beyond the center of curvature. It is enlarged in size and inverted. The image is real.

Trial 4: (Concave Mirror) The object is located at the principal focus. No image is formed. All rays are reflected from the mirror as parallel rays.

Trial 5: (Concave Mirror) The object is located between the principal focus and the mirror. The image appears to be located behind the mirror. It is enlarged in size. The image is virtual.

Trial 6: (Convex Mirrors) All convex mirrors form a virtual image reduced in size.

NOTE: On Trial 4, the ray that does not go through the center of curvature should go through the principal focus.

Image location in lenses
Finding approximate image locations with lenses is fairly similar to finding image locations with mirrors. The only difference is instead of having a center of curvature, instead, the important point is listed as "2F" or twice the focal length



Trial 1: (Convex Lens) The object is located beyond twice the focal length. The image is located between the focal length and twice the focal length on the opposite side of the lens. It is reduced in size and inverted. The image is real.

Trial 2: (Convex Lens) The object is located at twice the focal length. The image is located at twice the focal length on the opposite side of the lens. It is inverted. The image is real.

Trial 3: (Convex Lens) The object is located between twice the focal length and the focal length. The image is located beyond twice the focal length on the opposite side of the lens. It is enlarged in size and inverted. The image is real.

Trial 4: (Convex Lens) The object is located at the principal focus. Just like mirrors, no image is formed at this position. All rays are refracted from the lens as parallel rays.

Trial 5: (Convex Lens) The object is located between the principal focus and the lens. The image appears to be located behind the object on the same side of the lens. It is enlarged in size. The image is virtual.

Trial 6: (Concave Lenses) All concave lenses from a virtual image reduced in size.

The eye
You may be asked several questions concerning the eye. Here's a simple definition of what different parts of the eye do.
 * Cones: The color receptors of your eyes; they sense color. They are used more during daytime. There are 7 million cones in the eye.
 * Cornea:The transparent part of the eye covering the iris and pupil. It refracts light and accounts for about two-thirds of the eye's optical power.
 * Iris:The iris controls the size and diameter of the pupils, and in turn, the amount of light that enters the retina.
 * Lens:The lens helps refract light to focus it onto the retina. It changes shape to adjust its focal distance so that the eye can focus on objects at different distances
 * Optic Nerve:Sends information from the retina to the brain
 * Retina:The retina lines the inner surface of the eye and creates images via the optics of the eye, sent to the brain via the optic nerve
 * Rods: The photoreceptors in your eyes; they sense brightness and are more used during nighttime. There are 120 million rods in the eye.



Colors
During the competition, you may also be asked different questions concerning colors. Here's what you may need to know.

The primary colors of light are red, blue and green. The secondary colors are yellow, magenta, and cyan. You can mix colors in two different ways; additive or subtractive. When you 'add' colors together you are mixing red, blue, and green together to make the three secondary colors, just like in a computer. Yellow is made of green and red, magenta is made of red and blue, and cyan is made of green and blue. Think of adding colors like shining two lights onto the same spot. You also have subtractive mixing, which is what happens in printers. When you mix cyan, magenta, and/or yellow, only the primary colors in common show through. Therefore, when using the subtractive method, yellow + magenta = red, cyan + yellow = green, and cyan + magenta = blue.



You may be asked questions about filters. A colored filter will absorb its complement and transmit the two other colors. For example, a cyan filter would absorb its complement, red light, and transmit the other two colors, blue and green light. The complementary colors are Red and Cyan, Orange and Greenish-Blue, Yellow and Blue, Green and Magenta, Blue and Yellow, and Violet and Greenish-Yellow.

Sample questions

 * 1) Measure the focal length of convex and/or concave lenses and/or mirrors.
 * 2) Using ray-tracing techniques, find the image/object locations of one and/or two lens and/or mirror systems. Specify the image/object characteristics (real/virtual, magnification, erect/inverted, object/image distances, lens/mirror focal lengths).
 * 3) Set up mirrors and/or lenses to direct a beam of light on to a target around an obstacle. The object is to have the student set up the problem, but the supervisor turns on a light such as a Maglite (or laser). For the first year prisms are not to be used!
 * 4) The object is to align a beam from a light source provided by the officials to bounce off all given mirrors (Division C only may also include lenses and refraction) to strike a given target. Students will begin with a set number of points and then points will be deducted for the time it takes to set up the devices and the distance that the light ends up from the center point of the target.

Snell's Law
$$n_1\sin\theta_1=n_2\sin\theta_2$$
 * where $$n_1$$ = refractive index of material 1, and $$\theta_1$$ = angle between ray and normal to surface.

Wave velocity
$$v=\lambda f$$
 * where $$v$$ = wave velocity, $$\lambda$$ = wavelength, and $$f$$ = frequency.

Images in mirrors and lenses
$$\frac{h_i}{h_o} = \frac{d_i}{d_o}$$
 * where $$h_i$$ = image height, $$h_o$$ = object height, $$d_i$$ = image distance, and $$d_o$$ = object distance.

Focal length of mirrors and lenses
$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$
 * where $$f$$ = focal Length, $$d_o$$ = object distance, and $$d_i$$ = image distance.

Energy of a Photon
$$E=hf$$
 * where E=energy of photon, h=Planck's Constant (6.62606957×10−34 J/s), f=frequency

$$E=hc/\lambda $$
 * where E=energy of a photon (in Joules), h=Planck's Constant (6.626×10^-34 J/s), c=the speed of light (2.998×10^8), and λ=wavelength (in meters).

Simplified and giving the units of λ in microns (µm) and E in electron volts (eV), the above espression can be simplified and rounded to: $$E=1.2398/\lambda $$
 * where 1.2398 is the rounded product of h, c, eV (1.602×10^-19 J), and µm (1×10^-6) so that the value of lambda can be plugged into the equation in µm without the entire hassle of converting into meters and giving the answer in eV.

Laser Shoot
A portion of your score will be made up by the Laser Shoot. In the Laser Shoot, a box will be set up with a laser on one side of the box and a target on the opposing side. This box is known as the "Laser Shoot Surface", or "LSS". You are allowed to use anywhere from 1-5 mirrors to direct the laser to the target, with 4 minutes of set-up time. This year in B division there will be one obstacle blocking the laser from hitting the target, and in C division, there will be three obstacles, with at least one directly blocking the laser from the target. You may use whatever resources you have brought with you to the event. By no means should you ever touch the laser, target, or obstacle. You should only touch the mirrors when participating in the LSS. Your score will not be based on time at all, but rather accuracy and number of mirrors used. The more mirrors you use, the higher score you will receive. You can set the laser to reflect off of anywhere from 1-5 mirrors; therefore, you will need to calculate the angles and such yourself. The laser will only be shot once, by the event supervisor, when you notify them that you have finished setting up the mirrors.

Resources

 * Optics resources created by hftf
 * Virtual laser shoot