Reflection by a Spherical Mirror
Spherical mirrors are curved mirrors formed from the surface of a sphere. They can be concave (converging) or convex (diverging). Understanding how light reflects off these mirrors is crucial in geometrical optics, especially in image formation and clinical applications like indirect ophthalmoscopy, retinoscopy, and mirror-based diagnostic instruments.
1. Types of Spherical Mirrors
- Concave Mirror: Inner surface is reflective; converges light rays
- Convex Mirror: Outer surface is reflective; diverges light rays
Light rays follow the law of reflection at every point on the mirror’s surface: angle of incidence equals angle of reflection. However, due to curvature, we apply some approximations and conventions to make calculations practical.
2. Paraxial Approximation
The paraxial approximation is used to simplify calculations by assuming that only rays very close to the principal axis (called paraxial rays) are considered. These rays make small angles with the axis and do not suffer from spherical aberration.
Why is this important?
- It allows accurate image predictions without distortions.
- It helps derive the mirror formula for precise optical calculations in clinical lenses and systems.
Assumption: For paraxial rays, sinθ ≈ tanθ ≈ θ in radians. This simplifies geometry in curved mirror problems.
3. Cartesian Sign Convention
In geometrical optics, a standard sign convention is used to assign directions and values:
- The pole (P) of the mirror is taken as the origin.
- The principal axis is taken as the horizontal x-axis.
Rules of Cartesian Convention:
| Quantity | Sign |
|---|---|
| Object distance (u) | Always negative (object in front of mirror) |
| Image distance (v) | Positive for real images (in front of mirror) Negative for virtual images (behind mirror) |
| Focal length (f) | Positive for convex mirror Negative for concave mirror |
| Height above axis | Positive |
| Height below axis | Negative |
4. Derivation of the Mirror Formula (Vergence Equation)
This formula relates object distance (u), image distance (v), and focal length (f):
1/f = 1/v + 1/uDerivation:
Consider a concave mirror:
- Let O be the object, v the image distance, and u the object distance (all from pole P).
- Draw a ray parallel to the axis from O — it reflects through the focus (F).
- Another ray passes through the center of curvature (C) and reflects back on itself.
- Using similar triangles, and applying sign convention:
From geometry:
ΔABP ~ ΔA'B'P ⇒ h/u = h'/vBy using similar triangles and focal point relationships, we arrive at:
1/f = 1/v + 1/uThis is known as the **mirror formula**, which works for both concave and convex mirrors when proper signs are applied.
5. Clinical Relevance in Optometry
- Retinoscopy: Reflection from a spherical mirror inside the retinoscope helps evaluate refractive errors.
- Indirect ophthalmoscopy: Uses concave mirrors to project real inverted images of the retina.
- Vision testing tools: Mirrors help direct and control light for illumination and observation.
- Vergence understanding: This formula is used in lens calculations and ocular vergence corrections.
Conclusion
Reflection by spherical mirrors combines geometry and physics to predict where and how images form. By applying paraxial approximation and sign conventions, the mirror formula becomes a powerful tool for understanding real and virtual image formation. For optometry students, mastering this is key to understanding diagnostic tools and corrective systems based on mirrors.
Imaging by Concave and Convex Mirrors
Spherical mirrors — especially concave and convex mirrors — form images by reflecting light rays. Understanding how and where images form is essential for optometry students, especially for mastering diagnostic instruments and concepts like vergence, accommodation, and mirror-based assessments.
1. Concave Mirror (Converging Mirror)
A concave mirror has a reflective surface curved inward, like the inside of a spoon. It converges light rays to a real focus point (if rays are parallel).
Image Formation Depends on Object Distance:
| Object Position | Image Position | Image Nature | Size |
|---|---|---|---|
| At infinity | At focus (F) | Real, inverted | Point-sized |
| Beyond center of curvature (C) | Between F and C | Real, inverted | Smaller than object |
| At center of curvature (C) | At C | Real, inverted | Same size |
| Between F and C | Beyond C | Real, inverted | Larger than object |
| At focus (F) | At infinity | No image (rays parallel) | Infinitely large |
| Between pole (P) and F | Behind mirror | Virtual, erect | Magnified |
Ray Diagram for Concave Mirror:
- Ray parallel to axis → reflects through focus (F)
- Ray through center of curvature (C) → reflects back on itself
- Ray through focus → reflects parallel to axis
Use in Clinical Optics: Concave mirrors are used in indirect ophthalmoscopy to focus light on the retina and form real, magnified, inverted images.
2. Convex Mirror (Diverging Mirror)
A convex mirror has a reflective surface that bulges outward. It diverges rays, meaning they appear to come from a virtual point behind the mirror.
Image Characteristics (regardless of object position):
- Location: Behind the mirror
- Nature: Virtual and erect
- Size: Diminished (smaller than object)
Ray Diagram for Convex Mirror:
- Ray parallel to axis → reflects as if it came from focus behind mirror
- Ray aimed at center → reflects back as if diverging from center
Use in Clinical Optics: Convex mirrors are used in visual field testing and rear-view arrangements in instruments for wider field of view.
4. Clinical Relevance in Optometry
- Indirect Ophthalmoscope: Uses concave mirrors for magnified, real images of the fundus
- Retinoscopy: Relies on real vs. virtual image shifts to determine refractive error
- Vision therapy: Convex mirrors are sometimes used to expand peripheral vision fields
- Mirrors in tools: Used for manipulating image direction and magnification in phoropters, keratometers, and perimeters
Conclusion
Understanding how concave and convex mirrors form images helps optometry students accurately predict image size, type, and location — skills essential for using mirror-based instruments and analyzing optical systems of the eye.
Reflectivity, Transmissivity, Snell’s Law, and Refraction at a Plane Surface
When light encounters the surface between two different media (like air and glass), three things can happen:
- Some light is reflected back into the first medium.
- Some light is transmitted (refracted) into the second medium.
- A small part may be absorbed (in opaque or semi-transparent materials).
In optometry, understanding how light behaves at interfaces (cornea–air, lens–aqueous, lens–vitreous) is critical for diagnosing vision problems and designing lenses.
1. Reflectivity (R)
Reflectivity refers to the fraction or percentage of light that is reflected at the boundary between two media.
Formula (Fresnel's Approximation for Normal Incidence):
R = ((n₁ - n₂) / (n₁ + n₂))²Where:
- n₁ = Refractive index of the first medium
- n₂ = Refractive index of the second medium
Example: At the cornea-air interface:
n₁ (air) ≈ 1.00, n₂ (cornea) ≈ 1.376
R = ((1 - 1.376) / (1 + 1.376))² ≈ 0.026 or 2.6%That means about 2.6% of incident light is reflected off the cornea’s surface — the rest goes into the eye.
Clinical Relevance:
- Ocular reflections: Seen during slit-lamp exams.
- Spectacle lenses: Use anti-reflective coatings to reduce glare (minimize R).
- Retinoscopy: Depends on reflections to estimate refractive error.
2. Transmissivity (T)
Transmissivity refers to the fraction of light that passes through the interface into the second medium.
Basic Relationship:
T + R = 1 (assuming no absorption)From the above example:
R = 0.026 → T = 1 - 0.026 = 0.974 or 97.4%So about 97.4% of the light entering the eye is transmitted through the cornea into deeper layers.
Clinical Relevance:
- Contact lenses: Should have high transmissivity for comfort and clarity.
- Intraocular lenses (IOLs): High transmission is critical for post-cataract vision quality.
- Optical coatings: Adjust transmission properties to reduce harmful blue light or UV.
3. Snell’s Law and Refraction at a Plane Surface
When light enters a different medium (like from air into water or cornea), it changes speed and direction. This change in direction is called refraction.
Snell’s Law:
n₁ × sin(θ₁) = n₂ × sin(θ₂)Where:
- n₁ = Refractive index of the first medium
- n₂ = Refractive index of the second medium
- θ₁ = Angle of incidence (with normal)
- θ₂ = Angle of refraction
If light travels from a less dense to more dense medium (e.g., air to cornea), it bends towards the normal. If it goes from denser to rarer medium, it bends away from the normal.
Example: Air to Cornea
- n₁ = 1.00 (air)
- n₂ = 1.376 (cornea)
- Light bends inward (toward normal)
Graphical Representation:
A ray entering at an angle bends toward the normal if n₂ > n₁. Ray diagrams help visualize this.
4. Refraction at a Plane Surface
In many optical systems — especially in the human eye — refraction occurs at curved surfaces, but understanding plane surface refraction is foundational.
Key Concept:
Apparent depth = Real depth / nThis is observed when looking at an object submerged in water — it appears closer than it is. The same principle helps in understanding how retinal image positions may vary with refractive error.
Clinical Relevance:
- Keratometry: Refraction at corneal surface is key in measuring curvature and calculating dioptric power.
- Ophthalmic lens design: Based on refraction at both plane and curved surfaces.
- Slit lamp exam: You observe layers of the eye based on refracted light paths.
Conclusion
Understanding reflectivity, transmissivity, and refraction at surfaces helps us explain how light behaves at ocular boundaries. These principles are critical for interpreting how lenses and the eye form images, diagnose optical issues, and improve vision using corrective lenses. Snell’s Law forms the mathematical basis for refraction, while reflectivity and transmissivity impact clarity and brightness in clinical and real-world vision scenarios.
Glass Slab: Displacement without Deviation & Displacement without Dispersion
A glass slab is a rectangular, flat, transparent medium with parallel sides. When light passes through a glass slab, it undergoes refraction twice — once when entering the slab and once when exiting. This causes the light ray to be shifted from its original path, but without changing its direction.
This topic explains two important phenomena:
- Lateral displacement (without deviation of angle)
- Absence of dispersion (no color separation)
1. Refraction Through a Glass Slab
When light enters the glass slab from air:
- It slows down and bends toward the normal (due to higher refractive index)
- Inside the glass, it travels in a straight line
- On exiting the glass, it speeds up and bends away from the normal
Due to the opposite and equal bending at both surfaces, the emerging ray is parallel to the incident ray, but it is shifted sideways. This is known as lateral displacement.
2. Lateral Displacement (Displacement without Deviation)
Lateral displacement refers to the sideways shift of the emergent ray compared to the original ray, even though both rays are parallel.
Formula for Lateral Displacement (d):
d = t × sin(θ₁ - θ₂) / cos(θ₂)Where:
- d = Lateral displacement
- t = Thickness of the slab
- θ₁ = Angle of incidence
- θ₂ = Angle of refraction
Key Observations:
- The ray comes out parallel to its original direction → no angular deviation
- The shift increases with:
- Greater thickness (t)
- Larger angle of incidence
Clinical Relevance:
- Prism effect in spectacle lenses: High-powered lenses may cause lateral shift, similar to slab displacement.
- Corneal edema: May act like a slab with slightly different refractive index, causing distorted vision.
- Lens centration: Important in frame fitting to minimize unwanted displacement of visual axis.
3. Displacement without Dispersion
Dispersion occurs when white light separates into colors due to different refractive indices for different wavelengths (like in a prism).
However, in a parallel-sided glass slab:
- All colors undergo equal and opposite refractions at entry and exit
- So, even though they bend differently inside the glass, they exit together
Thus, no dispersion occurs, and the emerging light remains white (no rainbow effect).
Why no dispersion?
- Equal path lengths for all rays
- Same angle of emergence (parallel rays)
- All components recombine perfectly
Clinical Relevance:
- Ophthalmic lenses: Designers minimize dispersion to avoid color fringes (chromatic aberration)
- Slit-lamp observations: Transparent ocular media like cornea/lens act like mini slabs; no color fringing occurs if optics are healthy
- Optical instruments: Use plane parallel glass (e.g., cover slips, lens surfaces) for minimal optical disturbance
4. Summary of Key Concepts
| Concept | Description | Optometry Application |
|---|---|---|
| Lateral Displacement | Parallel shift of light ray without angular change | Lens decentration, prism effects |
| No Angular Deviation | Emergent ray is parallel to incident ray | Minimizing distortion in corrective lenses |
| No Dispersion | White light remains undivided | Prevents chromatic blur in vision systems |
Conclusion
The glass slab is a simple yet powerful model in optics. It demonstrates how light can be displaced laterally without changing direction or splitting into colors. For optometry students, it introduces fundamental ideas used in designing and understanding corrective lenses, optical devices, and ocular media transparency.
For more geometrical optics unit click below 👇
👉 Unit 1
👉 Unit 3
👉 Unit 4
👉 Unit 5
