Unit 2 – Visual Optics I
1. Spherical Refracting Surface
A spherical refracting surface is the curved boundary between two transparent media with different refractive indices, where one medium is usually air (n1) and the other is a denser optical medium such as glass or the ocular structures (n2). The surface is part of a sphere with a definite center of curvature and radius (R). Depending on whether the surface is convex or concave relative to the incident light, the refraction properties change significantly.
When light passes from one medium to another at such a spherical boundary, it undergoes bending according to Snell’s Law: n1 sin i = n2 sin r where i is the angle of incidence and r is the angle of refraction. For paraxial rays (those close to the axis and making small angles), this relationship can be simplified to obtain an approximate formula for image location.
The general formula for refraction at a spherical surface is: (n2 / v) − (n1 / u) = (n2 − n1) / R where: u = object distance (measured from pole of surface), v = image distance (measured from pole), R = radius of curvature of surface. This equation forms the basis for understanding image formation in the eye, as the cornea and crystalline lens both act as spherical refracting surfaces.
In optometry, this principle is critical to understanding how light is refracted within the human eye, especially at the anterior and posterior corneal surfaces. Errors such as myopia and hyperopia can be attributed to the misalignment between the refractive power of spherical surfaces and the axial length of the eye. Correction with lenses aims to compensate for this mismatch.
Clinically, the concept helps in designing spectacle lenses, contact lenses, and in evaluating corneal shape in keratometry and corneal topography. A perfect understanding of refraction at spherical surfaces allows the optometrist to predict how images form and to plan refractive corrections accordingly.
2. Spherical Mirror; Catoptric Power
A spherical mirror is a part of a sphere with reflecting properties, unlike refracting surfaces that transmit light. Mirrors can be concave (converging) or convex (diverging). The mirror formula is expressed as: 1/f = 1/u + 1/v where f = focal length, u = object distance, and v = image distance.
The catoptric power refers to the power of reflection of a mirror. Unlike lenses, which depend on refractive indices, the catoptric power (P) of a spherical mirror depends solely on its geometry: P = −2n / R where n is the refractive index of the medium in front of the mirror (usually air), and R is the radius of curvature. The negative sign indicates the convention of reflected light.
In optometry, mirrors are used in diagnostic instruments such as ophthalmoscopes, retinoscopes, and keratometers. The posterior surface of the cornea also acts as a convex mirror, reflecting light back into diagnostic devices. Understanding the catoptric power helps in interpreting Purkinje images, which are the reflections of light from ocular surfaces.
The concept is also critical in explaining how indirect ophthalmoscopy works, as mirrors and lenses are combined to form real, magnified, and inverted retinal images. Thus, catoptric power is an important optical property in both theoretical and applied optometry.
3. Cardinal Points
The concept of cardinal points provides a simplified way to analyze image formation by complex optical systems, such as the human eye or a compound lens system. There are six cardinal points in Gaussian optics: two principal points, two nodal points, and two focal points.
- Principal Points (H and H’): These are the points where the principal planes intersect the optical axis. They act as reference points for measuring object and image distances.
- Focal Points (F and F’): The first focal point is where an object must be placed so that its image is formed at infinity. The second focal point is the image of an object placed at infinity.
- Nodal Points (N and N’): These are points where a ray directed towards the first nodal point emerges from the second nodal point in the same direction. They are particularly useful in describing angular magnification.
In the human eye, the principal planes lie within the crystalline lens, while the nodal points are located near the posterior surface of the lens. These cardinal points simplify calculations of image formation, retinal magnification, and the effect of corrective lenses.
Clinically, knowledge of cardinal points is vital in designing optical instruments and in understanding concepts such as spectacle magnification, retinal image size in high ametropia, and contact lens fitting. For example, when analyzing aphakic or pseudophakic eyes, cardinal points provide essential insights into changes in retinal image quality.
4. Magnification
Magnification describes the ratio of the size of the image to the size of the object. It is a key concept in optics, determining how the eye or optical devices alter visual perception. There are different types of magnification relevant to optometry:
- Lateral (Transverse) Magnification (Mt): Mt = h’ / h = v / u × (n1 / n2) where h = object height, h’ = image height.
- Axial Magnification: Describes the change in image size along the optical axis.
- Angular Magnification: Ratio of the visual angle subtended by the image compared to the object. Important in telescopes and magnifying glasses.
- Spectacle Magnification: Relevant in high ametropia, where spectacle lenses can alter the perceived retinal image size.
In optometry, magnification is clinically significant in conditions like anisometropia, where unequal spectacle magnification between two eyes may cause aniseikonia (unequal image sizes). Contact lenses or intraocular lenses are often prescribed to minimize magnification differences and restore binocular vision.
Magnification is also crucial in low vision devices, where magnifiers, telescopes, and electronic devices are prescribed to enhance the retinal image size and improve visual performance. Thus, understanding magnification is both a theoretical necessity and a clinical tool in visual rehabilitation.
Light and Visual Function
Light plays a central role in the process of vision. When light enters the eye, it is refracted by the cornea and lens to focus onto the retina, where photoreceptors (rods and cones) convert it into neural signals. The intensity, wavelength, and coherence of light affect the quality of vision. Scotopic vision (mediated by rods) is dominant in dim light, while photopic vision (mediated by cones) allows color perception and high acuity in bright conditions. The balance of light sensitivity is described by the V(λ) curve, which peaks around 555 nm in photopic conditions. Optometrically, understanding the visual function of light is important for low vision assessment, photophobia management, and designing optical aids like tinted lenses.
Clinical Relevance of Optical Phenomena
Fluorescence
Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation. In optometry, sodium fluorescein dye is used in slit-lamp examinations to assess corneal integrity, tear film stability, and contact lens fitting. It fluoresces under cobalt-blue light, highlighting epithelial defects or tear break-up time. This phenomenon is also used in fundus fluorescein angiography to detect retinal vascular abnormalities.
Interference
Interference occurs when two or more light waves superimpose, leading to reinforcement or cancellation. In the eye, interference effects explain the colorful patterns seen in thin films like the tear film lipid layer. Clinically, interference helps in tear film evaluation, anti-reflective coating design for spectacles, and diagnostic devices like interferometers used in corneal topography and intraocular lens measurements.
Diffraction
Diffraction is the bending of light around obstacles or apertures. It limits the resolving power of optical systems, including the eye. The size of the Airy disc formed by diffraction at the pupil determines the maximum visual acuity. Small pupils increase diffraction blur, while very large pupils increase aberrations. Thus, an optimal pupil size (about 3 mm) balances diffraction and aberrations. Diffraction is also exploited in diagnostic tools like diffraction gratings.
Polarization
Polarization refers to the orientation of the electric field of light waves. The eye is generally insensitive to polarization, but certain clinical instruments, such as polarized slit-lamp filters and polarized sunglasses, use polarization to reduce glare and enhance contrast. In optometry, polarization is used in stereopsis tests and visual performance evaluations.
Birefringence
Birefringence occurs when a material has different refractive indices along different axes, causing double refraction of light. The cornea and retinal nerve fiber layer exhibit birefringence. Optical Coherence Tomography (OCT) and scanning laser polarimetry exploit this property to measure nerve fiber thickness and detect early glaucoma.
Dichroism
Dichroism is the property of certain crystals or materials to absorb light differently depending on polarization or wavelength. In optometry, dichroic filters are used in special diagnostic devices and therapeutic lenses to selectively transmit certain wavelengths while blocking others, which may improve contrast or protect against harmful UV/blue light.
Aberration and Its Applications
In an ideal optical system such as a lens or the human eye, rays of light entering the system from a point object should converge to a perfect point image. However, in practice, due to the limitations of optical media and physical principles of refraction, light rays do not perfectly converge. This deviation from the ideal image formation is known as aberration. Aberrations cause blurring, distortion, loss of contrast, and reduced image quality, which are important not only in optical instruments but also in clinical optometry where precise imaging is essential for good vision.
Types of Aberrations
Aberrations can be broadly divided into two categories:
- Monochromatic Aberrations: These occur even when light of a single wavelength is used. They are caused by geometrical and physical properties of spherical refracting surfaces. Examples include spherical aberration, coma, astigmatism, curvature of field, and distortion.
- Chromatic Aberrations: These occur because white light consists of multiple wavelengths, and different wavelengths refract differently in a lens or optical medium due to dispersion. This results in colored fringes and loss of clarity.
1. Spherical Aberration
Spherical aberration occurs when rays striking a spherical surface at different heights from the axis fail to meet at the same point after refraction or reflection. Peripheral rays bend more strongly compared to paraxial rays, producing multiple foci. This leads to a blurred image circle rather than a sharp point.
Formula:
Longitudinal Spherical Aberration (LSA) ≈ h² / (2R)
where h is the ray height at the aperture and R is the radius of curvature of the surface.
In the human eye, spherical aberration arises due to the spherical cornea and crystalline lens. Interestingly, the eye naturally reduces spherical aberration through:
- Gradient index of refraction in the crystalline lens (higher refractive index in the center).
- Pupil constriction, which blocks peripheral rays and allows only central rays to enter (aperture stop effect).
Clinical Relevance: In refractive surgery (LASIK, PRK), intraocular lens (IOL) design, and contact lens correction, minimizing spherical aberration is crucial for better image quality and contrast sensitivity.
2. Coma
Coma is an off-axis aberration that occurs when oblique rays of light are imaged asymmetrically. Instead of a sharp point, the image appears comet-shaped with a bright head and tapering tail. It is most evident in peripheral vision or when light enters the optical system at an angle.
Coma is particularly problematic in wide-angle lenses, telescopes, and the human eye during peripheral imaging. Like spherical aberration, it can be minimized by using aspheric surfaces and by restricting aperture size.
Clinical Relevance: Coma contributes to glare and halos in patients after refractive surgeries or with decentered intraocular lenses.
3. Astigmatism of Oblique Incidence
Astigmatism occurs when rays from an off-axis point fail to converge into a single focus but form two line foci at different distances. This happens because the tangential and sagittal rays (perpendicular directions) refract differently. The interval between these two foci is known as the astigmatic difference.
Formula:
Astigmatic Difference (AD) = ft - fs
where ft and fs are the focal lengths in tangential and sagittal planes.
Clinical Relevance: Oblique astigmatism is significant in spectacle lenses, especially high-power lenses, where peripheral vision suffers. Lens designers use techniques like “best form lens design” to minimize oblique astigmatism.
4. Curvature of Field
Even when astigmatism is corrected, an image may still be formed on a curved surface (Petzval surface) instead of a flat plane. This is called curvature of field. As a result, the central and peripheral portions of an image cannot be in focus simultaneously on a flat retina or screen.
Clinical Relevance: The human retina itself is curved, which helps reduce the impact of this aberration. However, in optical instruments, field-flattening lenses are required.
5. Distortion
Distortion occurs when magnification varies across the field of view. There are two types:
- Barrel distortion: Image magnification decreases with distance from the center, making straight lines bulge outward.
- Pincushion distortion: Image magnification increases with distance from the center, making straight lines bend inward.
Clinical Relevance: Distortion is less noticeable in the human eye but is a concern in high-power spectacles (e.g., strong plus lenses), where objects appear warped.
6. Chromatic Aberration
Chromatic aberration arises because the refractive index of a medium varies with wavelength (dispersion). Blue light bends more strongly than red light, so different colors focus at different points.
There are two types:
- Longitudinal Chromatic Aberration (LCA): Different wavelengths focus at different axial positions. For the human eye, the difference between blue (short wavelength) and red (long wavelength) focus is about 1.5–2.0 diopters.
- Transverse Chromatic Aberration (TCA): Different wavelengths are displaced laterally, producing colored fringes, especially in peripheral vision.
Formula:
LCA = f(λ1) - f(λ2)
where f(λ) is the focal length for a specific wavelength.
Clinical Relevance: Chromatic aberration contributes to reduced visual acuity, especially under high-contrast conditions. In spectacle lens design, materials with higher Abbe numbers (lower dispersion) are preferred to reduce chromatic aberration.
Clinical and Optical Applications of Aberration Theory
- Refractive Surgery: Modern wavefront-guided LASIK and PRK aim not only to correct refractive errors but also to reduce higher-order aberrations like spherical aberration and coma, improving night vision and contrast sensitivity.
- Intraocular Lenses (IOLs): Aspheric IOLs are designed to compensate for positive spherical aberration of the cornea, providing sharper postoperative vision.
- Contact Lens Design: Custom contact lenses, especially rigid gas-permeable (RGP) lenses, are manufactured to minimize corneal aberrations and improve retinal image quality.
- Wavefront Aberrometry: Instruments measure the total aberrations of the eye, guiding personalized correction strategies.
- Optical Instruments: Aberration correction is vital in telescopes, microscopes, and cameras to improve image clarity, contrast, and resolution.
