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Module 7 - Geometric Optics
Learning Goals
- Understand the Law of Reflection a. Investigate the relationship between the angle of incidence and the angle of reflection on flat and curved mirrors. b. Examine how light rays behave upon reflection from concave and convex surfaces.
- Explore Refraction and Snell’s Law a. Observe how light bends when transitioning between media of different optical densities. b. Verify Snell’s law by measuring angles of incidence and refraction, and use these measurements to calculate the refractive index.
- Determine the Refractive Index of Transparent Materials a. Devise methods to measure and analyze refraction angles. b. Accurately determine the index of refraction for a half-cylindrical (or semicircular) acrylic block.
Key Concepts
- Reflection of Light a. Law of Reflection: The angle of incidence (θᵢ) equals the angle of reflection (θᵣ) as measured from the normal to the surface. b. Specular vs. Diffuse Reflection: Smooth surfaces yield mirror-like (specular) reflection, whereas rough surfaces scatter light in many directions. c. Curved Mirrors: Concave mirrors can converge light rays, whereas convex mirrors diverge them.
- Refraction of Light a. Snell’s Law: 𝑛 1 sin𝜃 1 = 𝑛 2 sin𝜃 2 , relating the angles of incidence (θ₁) and refraction (θ₂) and the refractive indices (n₁, n₂) of the two media.
b. Index of Refraction: 𝑛 = 𝑐 / 𝑣, where 𝑐 is the speed of light in a vacuum, and 𝑣 is the speed of light in the material. c. Critical Angle & Total Internal Reflection: At sufficiently large angles, refraction can cease, and light is totally reflected back into the original medium.
- Ray Model of Light a. Light travels in straight lines (rays) when passing through uniform media. b. Reflection and refraction can be analyzed geometrically by drawing ray diagrams.
Pre-lab Questions
Questions to think about before the lab
- Reflection Basics a. Explain the difference between the angle of incidence and the angle of reflection. Why are these angles measured relative to a normal line? b. Predict how a light ray behaves when it strikes a flat mirror at a 20° incidence angle.
- Curved Mirrors a. Identify two everyday examples of concave and convex mirrors. What effect do each have on the images they form? b. How would parallel rays striking a concave mirror differ from those striking a convex mirror?
- Refraction and Snell’s Law a. State Snell’s law. If a beam of light travels from air (𝑛 ≈ 1.0) into a block of unknown material, what measurements would you need to determine that material’s index of refraction? b. Conceptually, how does the path of light change if it moves from a less dense to a more dense medium?
- Index of Refraction a. Define refractive index. Why is it always greater than or equal to 1.0 for transparent materials? b. Name some approximate refractive indices for everyday materials (e.g., water, glass).
- Experimental Precautions
Experiment 1: Reflection on a Plane Mirror
Objectives
- Verify that the angle of incidence equals the angle of reflection.
- Observe how a single laser beam reflects off a plane mirror at various angles.
Background
Reflection is one of the most fundamental interactions of light, in which light rays strike a boundary and change direction without passing through the surface. A plane (flat) mirror is the simplest reflective surface, typically consisting of a thin reflective coating (silver or aluminum) on smooth glass. When a light ray encounters this surface:
- Angle of Incidence and Reflection a. The angle of incidence (𝜃𝑖) is measured between the incoming ray and the normal (an imaginary line perpendicular to the mirror surface at the point of incidence). b. The angle of reflection (𝜃𝑟) is similarly measured between the reflected ray and the same normal. c. The Law of Reflection states 𝜃𝑖 = 𝜃𝑟. This law is derived from considerations of wavefront geometry but can be observed simply with a geometric ray approach.
Figure 1. Angle of Reflection in a Plane Mirror
- Plane Mirror Images (Conceptual, not always measured here) a. Plane mirrors form virtual images located behind the mirror at a distance equal to the object’s distance in front. Although not the primary focus of this experiment (which measures angles, not images), this concept is closely tied to the geometry of reflection.
Experiment Procedure
Figure 2. Experiment Setup for Plane Mirror
- Draw a normal line on paper at the point where the mirror will intersect.
- Set the mirror so that it coincides with a reference axis on your paper.
- Aim the laser at angles of incidence (e.g., 10°, 20°, 30°, 40°, 50°, 60°).
- Measure and record the corresponding reflection angles in Table 1.
Datasheet
Table 1. Plane Mirror Experiment Results Laser Angle of Incidence ( 𝜃𝑖 ) Angle of Reflection ( 𝜃𝑟 ) 1 2 3 4 5 6
- Ray Parallel to the Principal Axis: Reflects through (concave) or away from (convex) the focal point.
- Ray Through the Focal Point: Reflects parallel to the principal axis.
- Ray Through the Center of Curvature: Reflects back on itself (if the surface is truly spherical). Table 2. Convex Mirror Image Object's position (S), focal point (F)
Image Diagram
S > F, S = F, S < F ● Virtual ● Upright ● Reduced (diminished/smaller)
Table 3. Concave Mirror Image Object's position (S), focal point (F)
Image Diagram
S < F
(Object between focal point and mirror)
● Virtual ● Upright ● Magnified (larger)
S = F
(Object at focal point)
● Reflected rays are parallel and never meet, so no image is formed. ● In the limit where S approaches F, the image distance approaches infinity, and the image can be either real or virtual and either upright or inverted depending on whether S approaches F from its left or right side.
F < S < 2F
(Object between focus and centre of curvature)
● Real image ● Inverted (vertically) ● Magnified (larger)
Experiment Procedure
Part 1: Concave Mirror
- Place the concave mirror template on the table.
- Place the laser and combination mirror on the concave mirror template as shown in Figure 3.
Figure 3. Reflection of Light by a Concave Mirror
- Turn on the laser to produce three beams. Adjust the position of the laser and mirror beams so that they exactly follow the image on the concave mirror template.
- Change the beams to single beams
- Direct the single beam at each special ray line on the concave mirror template
- Photograph the results in each direction of the beam
Part 2: Convex Mirror
- Place the convex mirror template on the table
- Place the laser and combination mirror on the convex mirror template as shown in Figure
Figure 4. Reflection of Light by a Convex Mirror
- Turn on the laser to produce three beams. Adjust the position of the laser beam and mirror so that they exactly follow the image on the convex mirror template.
- Change the beam to a single beam
- Direct the single beam at each special ray line on the convex mirror template.
- Photograph the results in each direction of the beam
Conclusion
Describe and summarize what you see in the experiment.
- Photograph the results of the rays in the reflection experiment on a concave mirror
- Photograph the results of the rays in the reflection experiment on a convex mirror
- Compare the results with the properties of special rays on a concave mirror?
- Do the incident rays and reflected rays exactly follow the image on the incident ray template in the reflection experiment on a concave mirror on a single beam of rays and three beams of rays?
- Do the incident rays and reflected rays exactly follow the image on the incident ray template in the reflection experiment on a convex mirror on a single beam of rays and three beams of rays?
- Compare the results with the properties of special rays on a convex mirror?
- Index of Refraction (𝑛) a. Defined as 𝑛 = 𝑐𝑣, where 𝑐 is the speed of light in vacuum and 𝑣 is the speed of light in the material. b. Most transparent materials have 𝑛 > 1 because light travels more slowly in those media than in vacuum.
- Snell’s Law a. Mathematically, 𝑛 1 sin𝜃 1 = 𝑛 2 sin𝜃 2 , where 𝜃 1 is the angle of incidence (relative to the normal) in the first medium (index 𝑛 1 ) and 𝜃 2 is the angle of refraction in the second medium (index 𝑛 2 ). b. Conceptually, the wavefronts slow down or speed up upon entering the new medium, bending the ray accordingly.
- Flat vs. Curved Interfaces a. In a flat interface, the geometry is simpler: a single plane boundary. b. The angle of incidence and the angle of refraction are measured at the same point on this flat boundary.
- Total Internal Reflection (TIR) (Related, but not always measured) a. If a light ray in a higher-index medium approaches the boundary at a steep angle, it might reflect entirely back into the medium (if 𝜃𝑖 > critical angle). b. This effect is used in fiber optics and prisms, though it may or may not be part of this particular lab.
During this experiment, shining a laser beam at varying angles onto the flat side of a semicircular acrylic block allows you to measure 𝜃𝑖 and 𝜃𝑟, from which you can compute sin𝜃𝑖/sin𝜃𝑟. Since 𝑛air ≈ 1.0, the ratio helps identify the block’s refractive index.
Figure 6. Types of refraction at the interface between two media
Experiment Procedure
- Direct a ray of light towards the center of a flat surface of a semicircular object.
- Direct the beam at a 90° angle to the flat surface of the object, at the center of the plate (Point C in Figure 7).
- Draw the path of the light and also draw the position of the plate.
Figure 7 Light paths perpendicular to the flat surface of a semicircular block
- Change the direction of the incident beam by rotating the laser (or beam with C as the center), so that the point of incidence of the light remains at C. Make the angle between the incident beam and the normal of 10°.
- Move the blocks from the white paper.
Figure 8 Refraction of Light
- Draw the path of incident light and the path of refracted light. The angle between the normal and the incident ray is called the angle of incidence, usually marked with the letter i , while the angle between the normal and the refracted ray is called the angle of refraction, usually marked with the letter r (see Figure 8).
- With a protractor, measure the angle of incidence i and angle of refraction r of the light. Write down the results you get in Table 4 and 5.
- Put the semicircular block back in place, repeat Steps 3 to 8 four times, increasing the angle of incidence by 10° each time.
Table 5. Flat Interface Refraction Results (Derivations) Laser (^) Ratio (^) 𝑟𝑖 Comparison 𝑙/𝑟 1 𝑚/𝑟 2
sin i sin r sin 𝑖 sin 𝑟
Average
Conclusion
- Observe the path of light as it passes through the object. Is the path straight or curved in Figure 8 according to your experiment? If it is curved, in which direction does it curve?
- How does the value of i compare to r in your experimental data sheet?
- Can you deduce the rules for the values of i-r , 𝑖𝑟, and (^) sin 𝑟^ sin 𝑖 in the data sheet? If so, what rules do you find?
- What can you conclude about the values of the ratio (^) 𝑚^ 𝑙? Are they approximately the same (constant), or are there a wide range of values?
- What is the refractive index ( n ) of the semicircular block material?
Experiment 4: Determining Index of Refraction
(Curved Interface)
Objectives
- Confirm the same (or similar) refractive index is obtained when the beam enters through the curved side of the half-cylindrical block.
- Compare these results to those of the flat-interface method.
Background
Continuing the study of refraction, one can also send the beam into the curved side of a half-cylindrical block. The underlying physics is the same—Snell’s law, however:
- Curved (Cylindrical) Surface Entry a. The normal at the point of incidence is no longer perpendicular to a flat plane; instead, it is perpendicular to a tangent on the curved surface. b. By carefully positioning the laser to pass through the block’s center (point 𝐶), you ensure symmetrical incidence, simplifying geometry.
- Ray Diagram Complexity a. If the laser enters near the curved edge but not through the center, the angles can become more complicated to track. b. Centering the beam on 𝐶 ensures the incident angle is measured consistently relative to the local normal.
- Confirming Index of Refraction a. The same material (acrylic, perspex, or another plastic) should yield the same 𝑛 whether measured via the flat or curved side. b. Differences might indicate experimental misalignments or surface imperfections.
This final experiment consolidates your understanding of Snell’s law in a scenario where the
boundary is not flat, reaffirming that (^) sin 𝑟^ sin 𝑖 remains constant for a given pair of media. Comparing
these measurements to those from the flat interface offers insight into the robustness of your methods and the fundamental consistency of the refractive index concept.
- Draw 12 chords for each incident ray and the refracted ray as in Figure 11. Measure the 12 chords and fill in the remaining blank table.
Figure 11. Half chords of incident and refracted rays (curved interface)
Datasheet
Table 6. Curved Interface Refraction Results (Measurements) Laser Incidence Angle ( i )
Refraction Angle ( r )
Difference in Angle ( i - r )
Length of half chords of incidence ( l )
Length of half chords of refraction ( m ) 1 2 3 4 5
Table 7. Curved Interface Refraction Results (Derivations) Laser (^) Ratio (^) 𝑟𝑖 Comparison 𝑙/𝑟 1 𝑚/𝑟 2
sin i sin r sin 𝑖 sin 𝑟
Average
Conclusion
- How does the value of i compare to r on your experimental data sheet?
2. What is the value of 𝑛𝑝𝑢according to this experiment?
3. Can you find a relationship between the 𝑛𝑢𝑝 obtained in experiment 3 and the 𝑛𝑝𝑢value
obtained in experiment 4? If not, try comparing the values of 𝑛^1 𝑢𝑝 and 𝑛𝑝𝑢(or between
𝑛𝑢𝑝 and 𝑛^1 𝑝𝑢). How are they related?
- What is the refractive index, and how does its nature affect the bending of light in transparent materials?
- How does the difference in chord length of the incident ray (𝑙) and the refracted ray (𝑚) affect the results of the refractive index calculation?
