Optics Geometry Physics Lesson 35 by Owen Borville 1.13.2026
Light rays entering a converging lens parallel to its axis cross one another at a single point on the opposite side. For a converging lens, the focal point is the point at which the converging light rays cross. For a diverging lens, the focal point is the point from which diverging light rays appear to originate. The focal length f is the distance from the center of the thin lens to its focal point. Optical Power P of a lens is the inverse of its focal length, P = 1/f. The optical power of thin, closely spaced lenses is Ptotal = Plens1 + Plens2 + Plens3...
A diverging lens is a lens that causes the light rays to bend away from its axis. Ray tracing is the geometric technique of graphically determining the paths that light rays take through thin lenses. A real image can be projected onto a screen, but a virtual image cannot be projected onto a screen. A converging lens forms either real or virtual images, depending on the object location. A diverging lens forms only virtual images.
A real image is the image in which light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye.
Lens Equations:
1/do + 1/di = 1/f
hi/ho = -di/do = m (magnification)
The image distance is the distance of the image from the center of the lens. A virtual image is an image that is on the same side of the lens as the object and cannot be projected on a screen.
Images formed by a flat (plane) mirror: (1) the image and the object are the same distance from the mirror (do = -di) (2) the image is always a virtual image (3) the image is located behind the mirror. Image length is half the radius of curvature: f = R/2. The image size is the same as the object size and the image is upright.
A convex mirror is a diverging mirror and forms only one type of image, a virtual image.
Spherical mirrors may be concave (converging) or convex (diverging). The focal length of a spherical mirror is one-half of its radius of curvature: f = R/2.
The magnification of a spherical mirror or object is m = hi/h0 = -di/do. The mirror equation and ray tracing allow for a complete description of an image formed by a spherical mirror. The mirror equation is 1/do + 1/di = 1/f.
Since mirrors form real images on the same side of the object, while lenses form real images on the opposite side of the object, there are different sign rules for distances in the equations: Sign conventions for mirrors:
concave mirror, focal length f is +.
convex mirror, focal length f is -
real object, object distance do is +
virtual object, object distance do is -
real image, image distance di is +
virtual image, image distance di is -
upright image, magnification m is +
inverted image, magnification m is -
So the thin-lens equation is also 1/do + 1/di = 1/f
Aberrations or image distortions can occur due to the finite thickness of optical instruments, imperfections in the optical components, and limitations on the ways in which the components are used. Methods for correcting aberrations include using better components and computational techniques. Spherical aberration is an optical flaw in lenses or mirrors with spherical surfaces where light rays passing through the edges focus at a different point than rays passing through the center, causing image blurriness, particularly around the periphery. Spherical aberration occurs for spherical mirrors but not parabolic mirrors. Comatic aberration occurs for both types of mirrors and occurs when an imperfection causes off-axis point sources causing images to be distorted and appearing to have a tail.
A single refracting interface forms when an object is observed through a plane interface between two media, and it appears at an apparent distance hi that differs from the actual distance ho: hi = (n2/n1)ho. An image is formed by the refraction of light at a spherical interface between two media of indices of refraction n1 and n2. Image distance depends on the radius of curvature of the interface, location of the object, and the indices of refraction of the media. Spherical interface equation: n1/do + n2/di = n2-n1/R
Image formation in the eyes is described by the thin-lens equations: P = 1/do + 1/di = 1/f and hi/ho = -di/do = m. Accommodation is the process where the eye produces a real image on the retina by adjusting its focal length and power.
Nearsightedness (myopia) is the inability to see distant far objects and is corrected with a diverging lens to reduce the optical power of the eye. Farsightedness (hyperopia) is the inability to see near close objects and is corrected with a converging lens to increase the optical power of the eye. In myopia and hyperopia, the corrective lenses produce images at distances that are between the person's near and far points so that images can be seen clearly.
For close vision, the eye is fully accommodated and has its greatest power, whereas for distance vision it is totally relaxed and has its smallest power. The loss of ability to accommodate with age is called presbyopia, which is corrected by the use of a converging lens to add power for close vision.
Color vision is possible in the eye because it has four types of light receptors (rods) and three types of color-sensitive cones. The rods are good for night vision, peripheral vision, and motion changes, while the cones are responsible for central vision and color. The eyes can perceive many hues from light having mixtures of wavelengths. In a simple theory of color vision, there are three primary colors, which correspond to the three types of cones, and that various combinations of the primary colors produce all the hues. The true color of an object is related to its relative absorption of various wavelengths of light. The color of a light source is related to the wavelengths it produces. Color constancy is the ability of the eye-brain system to determine the true color of an object illuminated by various light sources. The retinex theory of color vision explains color constancy by suggesting the existence of three retinexes or image systems, associated with three types of cones that are compared to obtain sophisticated information.
Cameras are devices that use combinations of lenses to create an image for recording. Digital photography is based on charge-coupled devices (CCDs) that break an image into tiny pixels that can be converted into electronic signals.
A simple magnifier is a converging lens and produces a magnified virtual image of an object located within the focal length of the lens. Angular magnification accounts for magnification of an image created by a magnifier and it is equal to the ratio of the angle subtended by the image to that subtended by the object when the object is observed by the unaided eye. Angular magnification is greater for magnifying lenses with smaller focal lengths. Simple magnifiers can produce as much as ten times the magnification. The angular magnification M of a simple magnifier is M = θimage/θobject.
The angular magnification of an object a distance L from the eye for a convex lens of focal length f held a distance l from the eye is M = (25cm/L)(1+L-l/f) The range of angular magnification for a given lens for a person with a near point of 25 cm is 25 cm/f <= M <= 1 + 25 cm/f.
Many optical devices have more than one lens or mirror and these devices are analyzed by considering each element sequentially. The image formed by the first is the object for the second, and so on. The same ray-tracing and thin-lens techniques apply to each lens element.
The overall magnification of a multiple-element system is the product of the linear magnifications of its individual elements times the angular magnification of the eyepiece. For a two-element system with an objective and an eyepiece, M = m(obj) M(eye), where m(obj) is the linear magnification of the objective and M(eye) is the angular magnification of the eyepiece.
The microscope is a multiple-element-component system that contains more than a single lens or mirror that enables humans to see details the unaided eye cannot see. Both the eyepiece and objective contribute to the magnification. The numerical aperature (NA) of an objective is NA = n sin α where n is the refractive index and α is the angle of acceptance.
Immersion techniques are often used to improve the light gathering ability of microscopes, where the specimen is illuminated by transmitted, scattered, or reflected light through a condenser. The f/# describes the light gathering ability of a lens: f/# = f/D ≈ 1/2NA.
The magnification of a compound microscope with the image at infinity is Mnet = -(16cm)(25cm)/(f(obj)f(eye), where 16 cm is the standardized distance between the image-side focal point of the objective lens and the object-side focal point of the eyepiece. 25 cm is the normal near point distance, f(obj) and f(eye) are the focal distances for the objective lens and the eyepiece, respectively.
Simple telescopes can be made with two lenses and they are used for viewing objects at large distances and use the entire range of the electromagnetic spectrum. The angular magnification M for a telescope is M = θ'/θ = -f(obj)/f(eye), where θ is the angle subtended by an object viewed by the unaided eye, θ' is the angle subtended by a magnified image, and f(obj) and f(eye) are the focal lengths of the objective lens and the eyepiece, respectively.
The lens maker's equation: 1/f = (n2/n1 -1)(1/R1 - 1/R2) relates focal length (f) to refractive index (n) and the radii of curvature of its surfaces (R1, R2). This equation helps opticians and engineers design lenses for specific focal points, with positive focal lengths for converging (convex) lenses and negative for diverging (concave) lenses
Light rays entering a converging lens parallel to its axis cross one another at a single point on the opposite side. For a converging lens, the focal point is the point at which the converging light rays cross. For a diverging lens, the focal point is the point from which diverging light rays appear to originate. The focal length f is the distance from the center of the thin lens to its focal point. Optical Power P of a lens is the inverse of its focal length, P = 1/f. The optical power of thin, closely spaced lenses is Ptotal = Plens1 + Plens2 + Plens3...
A diverging lens is a lens that causes the light rays to bend away from its axis. Ray tracing is the geometric technique of graphically determining the paths that light rays take through thin lenses. A real image can be projected onto a screen, but a virtual image cannot be projected onto a screen. A converging lens forms either real or virtual images, depending on the object location. A diverging lens forms only virtual images.
A real image is the image in which light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye.
Lens Equations:
1/do + 1/di = 1/f
hi/ho = -di/do = m (magnification)
The image distance is the distance of the image from the center of the lens. A virtual image is an image that is on the same side of the lens as the object and cannot be projected on a screen.
Images formed by a flat (plane) mirror: (1) the image and the object are the same distance from the mirror (do = -di) (2) the image is always a virtual image (3) the image is located behind the mirror. Image length is half the radius of curvature: f = R/2. The image size is the same as the object size and the image is upright.
A convex mirror is a diverging mirror and forms only one type of image, a virtual image.
Spherical mirrors may be concave (converging) or convex (diverging). The focal length of a spherical mirror is one-half of its radius of curvature: f = R/2.
The magnification of a spherical mirror or object is m = hi/h0 = -di/do. The mirror equation and ray tracing allow for a complete description of an image formed by a spherical mirror. The mirror equation is 1/do + 1/di = 1/f.
Since mirrors form real images on the same side of the object, while lenses form real images on the opposite side of the object, there are different sign rules for distances in the equations: Sign conventions for mirrors:
concave mirror, focal length f is +.
convex mirror, focal length f is -
real object, object distance do is +
virtual object, object distance do is -
real image, image distance di is +
virtual image, image distance di is -
upright image, magnification m is +
inverted image, magnification m is -
So the thin-lens equation is also 1/do + 1/di = 1/f
Aberrations or image distortions can occur due to the finite thickness of optical instruments, imperfections in the optical components, and limitations on the ways in which the components are used. Methods for correcting aberrations include using better components and computational techniques. Spherical aberration is an optical flaw in lenses or mirrors with spherical surfaces where light rays passing through the edges focus at a different point than rays passing through the center, causing image blurriness, particularly around the periphery. Spherical aberration occurs for spherical mirrors but not parabolic mirrors. Comatic aberration occurs for both types of mirrors and occurs when an imperfection causes off-axis point sources causing images to be distorted and appearing to have a tail.
A single refracting interface forms when an object is observed through a plane interface between two media, and it appears at an apparent distance hi that differs from the actual distance ho: hi = (n2/n1)ho. An image is formed by the refraction of light at a spherical interface between two media of indices of refraction n1 and n2. Image distance depends on the radius of curvature of the interface, location of the object, and the indices of refraction of the media. Spherical interface equation: n1/do + n2/di = n2-n1/R
Image formation in the eyes is described by the thin-lens equations: P = 1/do + 1/di = 1/f and hi/ho = -di/do = m. Accommodation is the process where the eye produces a real image on the retina by adjusting its focal length and power.
Nearsightedness (myopia) is the inability to see distant far objects and is corrected with a diverging lens to reduce the optical power of the eye. Farsightedness (hyperopia) is the inability to see near close objects and is corrected with a converging lens to increase the optical power of the eye. In myopia and hyperopia, the corrective lenses produce images at distances that are between the person's near and far points so that images can be seen clearly.
For close vision, the eye is fully accommodated and has its greatest power, whereas for distance vision it is totally relaxed and has its smallest power. The loss of ability to accommodate with age is called presbyopia, which is corrected by the use of a converging lens to add power for close vision.
Color vision is possible in the eye because it has four types of light receptors (rods) and three types of color-sensitive cones. The rods are good for night vision, peripheral vision, and motion changes, while the cones are responsible for central vision and color. The eyes can perceive many hues from light having mixtures of wavelengths. In a simple theory of color vision, there are three primary colors, which correspond to the three types of cones, and that various combinations of the primary colors produce all the hues. The true color of an object is related to its relative absorption of various wavelengths of light. The color of a light source is related to the wavelengths it produces. Color constancy is the ability of the eye-brain system to determine the true color of an object illuminated by various light sources. The retinex theory of color vision explains color constancy by suggesting the existence of three retinexes or image systems, associated with three types of cones that are compared to obtain sophisticated information.
Cameras are devices that use combinations of lenses to create an image for recording. Digital photography is based on charge-coupled devices (CCDs) that break an image into tiny pixels that can be converted into electronic signals.
A simple magnifier is a converging lens and produces a magnified virtual image of an object located within the focal length of the lens. Angular magnification accounts for magnification of an image created by a magnifier and it is equal to the ratio of the angle subtended by the image to that subtended by the object when the object is observed by the unaided eye. Angular magnification is greater for magnifying lenses with smaller focal lengths. Simple magnifiers can produce as much as ten times the magnification. The angular magnification M of a simple magnifier is M = θimage/θobject.
The angular magnification of an object a distance L from the eye for a convex lens of focal length f held a distance l from the eye is M = (25cm/L)(1+L-l/f) The range of angular magnification for a given lens for a person with a near point of 25 cm is 25 cm/f <= M <= 1 + 25 cm/f.
Many optical devices have more than one lens or mirror and these devices are analyzed by considering each element sequentially. The image formed by the first is the object for the second, and so on. The same ray-tracing and thin-lens techniques apply to each lens element.
The overall magnification of a multiple-element system is the product of the linear magnifications of its individual elements times the angular magnification of the eyepiece. For a two-element system with an objective and an eyepiece, M = m(obj) M(eye), where m(obj) is the linear magnification of the objective and M(eye) is the angular magnification of the eyepiece.
The microscope is a multiple-element-component system that contains more than a single lens or mirror that enables humans to see details the unaided eye cannot see. Both the eyepiece and objective contribute to the magnification. The numerical aperature (NA) of an objective is NA = n sin α where n is the refractive index and α is the angle of acceptance.
Immersion techniques are often used to improve the light gathering ability of microscopes, where the specimen is illuminated by transmitted, scattered, or reflected light through a condenser. The f/# describes the light gathering ability of a lens: f/# = f/D ≈ 1/2NA.
The magnification of a compound microscope with the image at infinity is Mnet = -(16cm)(25cm)/(f(obj)f(eye), where 16 cm is the standardized distance between the image-side focal point of the objective lens and the object-side focal point of the eyepiece. 25 cm is the normal near point distance, f(obj) and f(eye) are the focal distances for the objective lens and the eyepiece, respectively.
Simple telescopes can be made with two lenses and they are used for viewing objects at large distances and use the entire range of the electromagnetic spectrum. The angular magnification M for a telescope is M = θ'/θ = -f(obj)/f(eye), where θ is the angle subtended by an object viewed by the unaided eye, θ' is the angle subtended by a magnified image, and f(obj) and f(eye) are the focal lengths of the objective lens and the eyepiece, respectively.
The lens maker's equation: 1/f = (n2/n1 -1)(1/R1 - 1/R2) relates focal length (f) to refractive index (n) and the radii of curvature of its surfaces (R1, R2). This equation helps opticians and engineers design lenses for specific focal points, with positive focal lengths for converging (convex) lenses and negative for diverging (concave) lenses