Lateral Displacement Calculator: Formula & Examples

A lateral displacement calculator works out the sideways shift of a ray of light (or any wave) passing through a transparent parallel-sided block, such as glass or acrylic. You enter the block’s thickness, the angle of incidence, and the refractive index of the material, and it returns the perpendicular distance between the incident ray’s original path and the emergent ray’s path. This distance is the lateral displacement, typically measured in millimeters or centimeters.

Formula

The lateral displacement d is given by:

d = t · sin(i) · (1 − cos(i) / √(n² − sin²(i)))

Where:

• t = thickness of the block (in mm, cm, or m – any unit, as long as you use the same for the result)
• i = angle of incidence (in degrees, measured from the normal to the surface)
• n = refractive index of the block material (dimensionless, typically 1.5 for common glass, 1.49 for acrylic, 2.42 for diamond)

The formula comes from Snell’s law, which gives the angle of refraction r as sin(r) = sin(i) / n. The geometry of the two parallel surfaces then produces the displacement expression above. Note that the refractive index of air is taken as 1.00, which is accurate enough for most practical purposes.

Worked Example

Using the default values from the calculator: block thickness t = 10 mm, angle of incidence i = 45°, refractive index n = 1.5.

1. First, find sin(i): sin(45°) = 0.7071.
2. Compute sin²(i) = 0.5.
3. Compute n² − sin²(i) = 2.25 − 0.5 = 1.75.
4. Take the square root: √1.75 = 1.3229.
5. Compute cos(i) / √(…) = 0.7071 / 1.3229 = 0.5345.
6. Now 1 − 0.5345 = 0.4655.
7. Multiply by t · sin(i) = 10 · 0.7071 = 7.071.
8. Finally, d = 7.071 · 0.4655 = 3.29 mm.

The calculator gives 3.29 mm, matching the manual calculation. The angle of refraction is asind(0.7071/1.5) = asind(0.4714) = 28.13°.

When to Use This Calculator

This tool is useful in physics labs, optics classes, and engineering work where you need to predict how much a beam of light shifts when it passes through a window, a glass slide, or a protective cover. It’s also relevant in photography (thinking about filters), in laser alignment, and in any situation where you have a transparent plate between a source and a target.

The calculator assumes the two faces of the block are parallel and the block is surrounded by air (or vacuum). It also assumes the material is homogeneous and isotropic – meaning the refractive index is the same in all directions. For most common glasses and plastics, that’s true.

How to Read the Result

The lateral displacement is the perpendicular distance between the original line of the ray (if it had continued straight) and the line of the ray after it exits the block. It is always less than the thickness of the block, and it increases with a thicker block, a larger angle of incidence, and a higher refractive index – but not linearly. For small angles (under about 10°), the displacement is roughly proportional to the angle, but for larger angles the relationship becomes more curved.

If you are working with a block that is not surrounded by air – for example, glass in water – the formula changes because the external refractive index is not 1. In that case, you would need a more general version of Snell’s law, and this calculator would not be accurate.

Common Mistakes and Misconceptions

• Mixing up angles: The angle of incidence is measured from the normal (the line perpendicular to the surface), not from the surface itself. A common error is using the angle from the surface, which would give a completely different result.
• Using the wrong refractive index: For air, n = 1.0003, which is usually taken as 1. For water, n = 1.33, for crown glass n ≈ 1.52, for flint glass n ≈ 1.62. Using 1.5 for all glass is a rough approximation; check the specific material.
• Forgetting the units: The thickness and the displacement will be in the same unit, so if you enter thickness in mm, the result is in mm. If you need cm, convert before entering or after.
• Thinking the ray exits at the same angle: It does – the emergent ray is parallel to the incident ray, but shifted. The calculator gives the shift, not the exit angle (which is the same as the incident angle).
• Ignoring the wavelength: Refractive index varies slightly with wavelength (dispersion). The calculator uses a single value, so for very precise work with lasers of different colors, you need the index at that specific wavelength.

Limitations of the Formula

The formula is derived for a ray of light, not a beam with finite width. In reality, a beam of light has a diameter, and the displacement applies to the central ray. The edges of the beam will also experience the same shift, so the whole beam moves sideways without changing its direction.

Another limitation: if the angle of incidence is very large (close to 90°), the displacement approaches the thickness of the block, but the formula still works. However, at the critical angle for total internal reflection (which occurs when the ray is trying to go from a denser to a rarer medium), the formula breaks down. For a block in air, the ray is going from air into glass, so total internal reflection does not happen at the first surface, but it can happen at the second surface if the angle inside the block exceeds the critical angle. For a block with n = 1.5, the critical angle is about 41.8°, so if the angle of refraction exceeds that, the ray will not exit – it will be totally internally reflected. In practice, for a parallel-sided block, this only occurs if the incident angle is extremely high (above about 90°), which is not physical, so the calculator is safe for all real incident angles.

Alternatives and Related Concepts

If you are dealing with a prism (non-parallel surfaces), the ray is not just displaced but also deviated, and you would need a different calculation for the deviation angle. For a glass slab with non-parallel faces, the problem becomes more complex and depends on the wedge angle.

If you need to know the optical path length (the distance the light actually travels inside the block), that is different from lateral displacement. The path length inside the block is t / cos(r), where r is the angle of refraction. The lateral displacement only gives the perpendicular shift.

For a thin lens, the concept of displacement is not used; instead, you deal with refraction at curved surfaces. The lateral displacement calculator is specifically for parallel-sided plates.

Frequently Asked Questions

What is lateral displacement in simple terms?

It’s the sideways shift of a light ray after it passes through a transparent plate with parallel sides. The ray comes out parallel to its original direction, but offset by a small distance. Imagine looking through a glass window at an angle – objects appear shifted slightly to the side.

Does the lateral displacement depend on the color of light?

Yes, slightly. The refractive index of a material changes with wavelength, so different colors have slightly different refractive indices. This causes a small difference in displacement, which is the same phenomenon that creates rainbows in prisms, though for a parallel plate the effect is much smaller and not usually noticeable.

What happens if the angle of incidence is 0 degrees (perpendicular)?

If the ray hits the block straight on (normal incidence), the angle of refraction is also 0, and the ray passes straight through without any lateral displacement. The formula gives zero, which is correct.

Can I use this for a glass window in a building?

Yes, but the displacement is usually very small – for a typical 6 mm window at a 45° angle, the shift is about 2 mm. That’s why you don’t notice it in everyday life. For thicker glass, like aquarium walls, it becomes more noticeable.

What is the difference between lateral displacement and optical path length?

Lateral displacement is the perpendicular distance between the incident and emergent rays. Optical path length is the physical distance the light travels inside the material, which is longer than the thickness because the ray goes at an angle. They are different quantities.

Why does the formula have a square root in it?

The square root comes from converting sin(r) to cos(r) using the identity cos²(r) + sin²(r) = 1. Since sin(r) = sin(i)/n, you get cos(r) = √(1 − sin²(i)/n²), and that simplifies to the expression with the square root.

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in t | Block thickness | mm | 10 | 0
in a | Angle of incidence | degrees | 45 | 0
in n | Refractive index | - | 1.5 | 1
out disp | Lateral displacement | mm | t * sind(a) * (1 - cosd(a) / sqrt(n^2 - sind(a)^2)) | 2
out refr | Angle of refraction | degrees | asind(sind(a) / n) | 2