Unit 1: Review of Geometrical optics | Visual Optics-I | 3rd Semester of Bachelor of Optometry

 

Visual Optics - I

Unit 1: Review of Geometrical Optics

1. Vergence and Power

Vergence is a compact way to describe how convergent or divergent a bundle of paraxial rays is at a given reference plane. In clinical optics it is especially convenient because it links object distance, image distance and the refractive index of the medium into a single scalar quantity that is additive across refracting surfaces. Vergence (commonly denoted L) is defined as the refractive index n divided by the distance d (measured in metres) from the reference plane to the point where the rays converge or appear to diverge from:

Vergence: L = n / d (units: diopters, D, where 1 D = 1 m⁻¹)

When the rays are converging to a real focus in front of the reference plane, the vergence is positive; when they are diverging as if from a virtual source, the vergence is negative. For example, light from an object at infinity in air has d → ∞, so L → 0 D. Light from an object at 1 m in air has vergence +1.00 D (if rays are converging) or −1.00 D depending on sign convention; the clinical convention used here assumes objects in front of the surface produce positive vergence when they produce converging wavefronts.

Optical power of a thin lens or refracting surface (denoted Φ or P) is the change in vergence produced by that surface. For a thin lens in air, the power is simply:

Power (thin lens): Φ = L_after − L_before (units: diopters)

For a refracting spherical surface separating media of refractive indices n₁ and n₂ with radius of curvature R (positive if centre of curvature is on the outgoing side), the paraxial formula for surface power is:

Φ = (n2 − n1) / R (units: diopters) (This assumes paraxial approximation and R in metres.)

Clinically, the concept of vergence simplifies understanding how spectacle lenses, contact lenses and the cornea alter the vergence incident on the retina. When teaching or writing equations in blogs, providing vergence tables (object distance → vergence → required corrective power) helps students quickly internalize relationships between working distance, accommodation and lens prescription.

2. Conjugacy — Object Space and Image Space

Conjugacy is a geometric relationship between two points (or planes) such that rays emanating from one point are focused to the other by an optical system. In simple terms, two points are conjugate if one becomes the image of the other. This idea underpins refraction and imaging: an object at an object point will have a corresponding image point determined by the lens or refracting surfaces.

We commonly speak of object space and image space. Object space refers to the region and distances measured from the object-side reference plane (for example, the spectacle plane or the first principal plane of a system), while image space refers to distances measured on the image side (for example, internal to the eye or beyond the spectacle lens). Using vergence, conjugacy is expressed compactly: if an optical element of power Φ transforms an incident vergence L_in to an outgoing vergence L_out, and if L_in corresponds to an object at distance d_o, then the image distance d_i is found from L_out = n / d_i.

In paraxial optics the thin lens imaging formula is often written as: 1/f = 1/v + 1/u where u and v are object and image distances and f is focal length. In vergence notation this becomes simpler because vergences add algebraically. 

Conjugacy also clarifies the concepts of principal planes and nodal points of complex systems: the effective object and image distances are measured to these principal planes, not always to the physical surfaces.

For the human eye, common conjugate pairs of interest are the spectacle plane ↔ retina (when considering spectacle correction) and external object ↔ retinal image (when considering imaging performance). Understanding which space you measure or quote distances in (object space vs image space) is key to avoiding sign and unit errors when converting between spectacle, contact lens and retinal vergences.

3. Sign Convention

Sign convention is essential in geometrical optics to ensure consistent algebra when working with distances, radii, vergences and powers. Multiple sign conventions exist (Gaussian, Cartesian, clinical/ophthalmic). For clinical optics and many optometry problems the refractive (or clinical) sign convention is commonly used because it matches how spectacle lens power and vergence are tabulated in practice.

Key points of a commonly used clinical sign convention:

• Distances measured in the direction of light propagation are taken as positive when they refer to image distances; object distances in front of the surface are often considered positive for vergence calculations.
• Radii of curvature (R) are positive if the centre of curvature is on the outgoing side of the surface (i.e. towards the transmitted light), negative if on the incident side.
• Vergence is positive for converging waves (which meet to form a real focus) and negative for diverging waves (as if coming from a virtual source).
• Lens power is positive for converging (convex) lenses and negative for diverging (concave) lenses, consistent with clinical prescriptions (+ for plus lenses, − for minus lenses).

It is important to state the adopted sign convention in any worked example. A mismatch of conventions between textbooks is a common source of exam errors: e.g., some physics texts treat object distances as negative when the object is on the incoming side of a surface, while clinical optics texts take them as positive for vergence calculation. When chaining multiple refracting surfaces, it is safest to work with vergence values (which already incorporate the refractive index and signed distance) and to consistently apply the same sign rules for radii and distances

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✅ Unit 5