Multiple Reflections and Periscopes

Introduction

When we glance into a shiny shop window and see countless copies of ourselves stretching into infinity, or when a submarine captain peers safely above the ocean surface, we’re witnessing the remarkable phenomenon of multiple reflections.
Multiple reflections occur when a beam of light reflects more than once between two or more surfaces before reaching our eyes. This simple process is at the heart of numerous optical devices—most famously the periscope, a tool that allows us to see around obstacles or above water while remaining hidden or protected.

Understanding multiple reflections not only deepens our grasp of basic optics but also opens doors to practical technologies in science, engineering, and everyday life. In this article we will explore:

The physics of reflection and how repeated bounces of light create striking effects
Mathematical relationships for the number of images formed
The design and working of periscopes, both simple and complex
Real-world applications from funhouse mirrors to cutting-edge optical instruments

Let’s begin by revisiting the essential concept of reflection.

1. A Quick Review of the Laws of Reflection

The laws of reflection form the foundation of all mirror-based devices:

Incident ray, reflected ray, and the normal (a line perpendicular to the surface at the point of incidence) all lie in the same plane.
Angle of incidence equals angle of reflection: i=ri = ri=r

Whenever light meets a smooth surface, these rules hold whether the light reflects once or a hundred times.

2. What Is Multiple Reflection?

Multiple reflection occurs when a ray of light undergoes successive reflections from two or more reflective surfaces before it reaches the observer.

Examples include:

The endless corridor effect between two parallel mirrors
Light bouncing inside a diamond, producing brilliant sparkle
The zig-zag path of light inside a periscope or kaleidoscope

Because each reflection obeys the two fundamental laws, we can predict the path of the light by carefully tracing angles and surfaces.

3. Infinite Images Between Two Mirrors

One of the most striking demonstrations of multiple reflection is the formation of multiple images when two plane mirrors face each other.

3.1 Parallel Mirrors

Place two large mirrors parallel and opposite each other. A single candle or object placed between them creates an apparently infinite series of images receding into the distance. This is because each mirror reflects the images formed in the other mirror repeatedly.

3.2 Mirrors at an Angle

If two plane mirrors meet at an angle θ\thetaθ, the number of images NNN is given by: N=360∘θ−1N = \frac{360^\circ}{\theta} – 1N=θ360∘−1

Example:
If the mirrors are at 60∘60^\circ60∘, N=36060−1=6−1=5 images.N = \frac{360}{60} – 1 = 6 – 1 = 5 \text{ images}.N=60360−1=6−1=5 images.

If 360θ\frac{360}{\theta}θ360 is not an integer, the integer part is taken. This formula explains the beautiful symmetrical patterns in a kaleidoscope, which uses three mirrors arranged in a triangular tube to create intricate designs.

4. Applications of Multiple Reflection in Everyday Life

Multiple reflection is far more than a parlor trick; it underlies many familiar technologies:

Barber shop and dressing mirrors – side mirrors create several views of your hair or clothing.
Optical instruments – periscopes, reflecting telescopes, binoculars use multiple mirrors or prisms to fold light paths.
Lighting and architecture – mirrored walls amplify brightness in interior design.
Safety reflectors – bicycle and road reflectors contain tiny corner-cube prisms that return light back toward the source through three successive reflections.

5. From Multiple Reflection to Periscopes

A periscope is a classic device that deliberately exploits multiple reflections to let the observer look over, under, or around an obstacle.

5.1 Historical Background

Early periscopes were used in the mid-19th century by sailors.
During World War I, trench soldiers used simple periscopes to see over parapets without exposing themselves to enemy fire.
Submarines in the 20th century advanced the technology, adding lenses and prisms for magnification and wider fields of view.

6. Working Principle of a Simple Periscope

The simplest periscope consists of a long tube with two plane mirrors set at 45° to the length of the tube and facing each other.

Light from the object strikes the upper mirror, reflects downward at a 90° angle.
It then strikes the lower mirror, which reflects it horizontally into the observer’s eye.

Because each reflection follows i=ri = ri=r, the final image is upright and virtually unchanged in size.

Key Characteristics

Image orientation: Upright and laterally inverted only once (so left–right are correct).
Magnification: None in a basic periscope; the image is the same size.
Distance: Equal to the physical length of the tube.

7. Ray Diagram (Described in Words)

Draw a vertical tube.
At the top, place a plane mirror inclined at 45°.
Draw an incident ray from a distant object striking this mirror.
Reflect it downward along the tube.
At the bottom, add a second 45° mirror.
Reflect the ray horizontally into an eye symbol.
The two reflections allow the observer to see the object while remaining hidden.

8. Variations and Improvements

While the simple periscope is excellent for teaching, practical devices incorporate refinements:

8.1 Prism Periscopes

Modern periscopes often replace mirrors with right-angle prisms. Advantages:

Prisms provide total internal reflection with nearly 100 % efficiency, avoiding the slight loss from mirrored glass.
Prisms stay aligned better than mirrors and are easier to clean.

8.2 Magnifying Optics

Submarine and tank periscopes may contain:

Objective lenses for collecting more light
Eyepiece lenses for magnification
Rotatable tops to scan the horizon without moving the entire vessel.

8.3 Periscopic Binoculars

Combine binocular magnification with periscope geometry, used in observation posts and armored vehicles.

9. Mathematical Treatment of Periscope Geometry

For a periscope of length LLL:

The light path length is 2L2L2L if mirrors are perfectly aligned.
If the top opening is at height hhh above the observer’s eye, the observer can see objects at least that height higher than their own position.

When magnifying lenses are added, the angular magnification MMM is given by: M=fobjectivefeyepieceM = \frac{f_{\text{objective}}}{f_{\text{eyepiece}}}M=feyepiecefobjective

where fff represents focal lengths of the lenses.

10. Everyday and Scientific Uses of Periscopes

Periscopes appear in diverse contexts:

Submarines – to observe ships or coastlines while remaining submerged.
Tanks and armored vehicles – to monitor the battlefield while staying protected.
Theater and Concert Halls – handheld periscopes help people see over crowds.
Nature Observation – wildlife enthusiasts use compact periscopes to view animals without disturbance.
Architectural Design – to bring natural light to lower floors via periscopic light tubes.

11. Multiple Reflection in Periscope Design

Although a basic periscope requires only two reflections, complex military periscopes may incorporate several sets of prisms, causing the light to zig-zag many times before reaching the eye. Each reflection must maintain the law i=ri = ri=r to ensure the image remains undistorted.

To minimize brightness loss after many reflections:

High-quality optical glass and coatings are used.
Total internal reflection inside prisms is preferred to metallic mirrors.

12. Related Optical Instruments

The concept of folding a light path through multiple reflections appears in many instruments:

Reflecting telescopes – secondary mirrors redirect light to a convenient viewing position.
Periscopic cameras – modern smartphone “periscope lenses” use prisms to create long focal lengths in slim bodies.
Endoscopes – medical instruments that guide light and images through flexible tubes with mirrors or fibers.

13. Experiments You Can Try

Here are simple activities to bring the topic alive:

DIY Parallel Mirror Infinity Tunnel
Two large mirrors facing each other create an endless corridor of reflections.
Angle-Dependent Images
Place two mirrors at different angles (90°, 60°, 45°) and count the images to confirm the formula N=360θ−1N = \frac{360}{\theta} – 1N=θ360−1.
Build a Cardboard Periscope
Use two small plane mirrors and a rectangular tube to see over fences or walls.

These hands-on experiments reinforce theoretical concepts through direct observation.

14. Reflection Loss and Brightness Considerations

Each reflection from a standard mirror typically loses about 5–10 % of the light intensity. With many reflections the image may appear dim. That is why professional devices prefer prisms utilizing total internal reflection, where almost no light is lost.

15. Environmental and Safety Aspects

Mirror manufacturing historically used silvering with mercury, which posed environmental hazards. Modern mirrors employ safer aluminum coatings and protective layers, ensuring that optical instruments like periscopes remain environmentally friendly.

16. Artistic and Architectural Impact

Artists and architects exploit multiple reflections for dramatic visual effects:

Infinity rooms in modern art galleries use mirrored walls to immerse visitors in a seemingly endless space.
Skyscrapers with mirrored glass façades bounce light across cityscapes, creating dynamic reflections that change with the sun.

17. Future Directions

With the advent of augmented reality (AR) and miniaturized optics, the periscope concept is evolving:

Compact periscopic smartphone cameras achieve 10× optical zoom in thin bodies.
Robotics and drones integrate micro-periscopes for stealth surveillance.
Space telescopes may fold light paths through multiple reflections to fit large apertures into compact spacecraft.

18. Summary of Key Equations and Facts

Concept	Formula / Key Point
Law of Reflection	i=ri = ri=r
Images between two mirrors at angle θ\thetaθ	N=360θ−1N = \frac{360}{\theta} – 1N=θ360−1
Magnification of a periscope with lenses	M=fobjectivefeyepieceM = \frac{f_{\text{objective}}}{f_{\text{eyepiece}}}M=feyepiecefobjective
Brightness loss per mirror	~5–10 % for metallic mirrors