GEOMETRIC OPTICS

I. What is GEOMTERIC OPTICS In geometric optics, LIGHT is treated as imaginary rays. How these rays interact with at the interface of different media, including lenses and mirrors, is analyzed.

LENSES refract light, so we need to know how light bends when entering and exiting a lens and how that interaction forms an image.

MIRRORS reflect light, so we need to know how light bounces off of surfaces and how that interaction forms an image.

II. Refraction We already learned that waves passing from one media to another cause light to do two things: Change path Change wavelength which means…Change velocity (speed of light) The velocity DECREASES and the wavelength SHORTENS when light passes from a “faster” to a “slower” media. The velocity INCREASES and the wavelength LENGTHENS when light passes from a “slower” to a “faster” media. In either case, the FREQUENCY remains the same.

1 refraction, continued

When light hits the interface of two media at an angle, the lower part of the ray interacts first, thus slowing it down before the rest of the ray meets the interface. This rotates the ray toward the normal.

The NORMAL LINE is an imaginary line PERPENDICULAR to the interface of two media.

The REFRACTIVE INDEX of a substance tells you how much light will change speed (or bend) when it passes through the substance. It is the ratio of the speed of light in the medium to the speed of light in a vacuum.

The medium will commonly n is the refractive index be air, water, glass, plastic c is the speed of light in a vacuum 2 refraction, continued substance refractive index, n

vacuum 1 air 1.000277 water 1.333 glass 1.50

The table of refractive index values shows you that light slows down only a little in air, but its speed is reduced about 33% in glass.

The higher the refractive index, the slower the speed of light.

If light passes from a medium with low refractive index (air) to one of high refractive index (glass), light refracts significantly.

SNELL’S LAW TELLS US HOW MUCH IT REFRACTS.

3 refraction, continued

SNELL’S LAW: Relates the ratio of the sines of the angle of incidence and angle of refraction of a light ray to the ratio of refractive indices of the substances the light passes.

the 1 and 2 subscript are the media the light ray passes.

For example, substances 1 and 2 might be air and water.

Notice how the angle of incidence and refraction are defined with respect to the NORMAL

4 III. OPTICS DEFINITIONS (LENSES AND MIRRORS) focal point-the point on the axis of a lens or mirror to which parallel rays of light converge or from which they appear to diverge after refraction or reflection radius of curvature-a point beyond the focal point that indicates how curved a lens or mirror is virtual image-an optical image from which light rays appear to diverge, although they actually do not pass through the image real image-An optical image such that all the light from a point on an object that passes through an optical system actually passes close to or through a point on the image. upright image-an optical image that is in the same orientation as the object from which the image comes inverted image- an optical image that is upside-down with respect to the object from which the image comes magnification-A measure of the effectiveness of an optical system in enlarging or reducing an image. dispersion-separation of light of several frequencies, such as white light, into its component. In other words, dispersion is the name given to the separation of white light into its colors (ROYGBIV) 5 IV. PRISMS Prisms are used to separate light into its component wavelengths. Prisms demonstrate the optical phenomenon of DISPERSION.

Prisms are used in a number of REAL LIFE optical applications where light needs to be selectively refracted or reflected.

violet red short λ long λ

REFRACTION DEPENDS ON LIGHT WAVELENGTH OR FREQUENCY

The shorter the wavelength (higher the frequency), the more the light is refracted. Hence, blue light is bent at a greater angle than red light.

6 prisms, continued

incident light

The ANGLE OF MINIMUM DEVIATION, δ, is a parameter used to characterize prisms. The refractive index of the prism is then related to the apex angle, σ of the prism and δ as in the equation above.

δ can be found by adjusting the angle of the incident light so that the light passes through the prism parallel to the base of the prism.

This may seem complicated but it is easy to show in the lab with a laser pointer and a prism (and a sheet of paper that you can draw angles and stuff on).

7 V. LENSES A. CONVERGING Converging (convex) lenses have one or both faces that bulge OUT. It is thicker in the center than at the edges.

CONVERGING lenses FOCUS light rays PARALLEL to the horizontal axis through the lens FOCAL POINT on the other side of the lens.

horizontal axis or principle axis

C C

F is the lens FOCAL POINT C = 2F. C is the lens RADIUS OF CURVATURE rays that are parallel to the axis are refracted by the lens into F

8 B. How is an image produced through a converging lens

IF THE OBJECT IS OUTSIDE THE FOCAL POINT, THE IMAGE WILL BE REAL AND INVERTED.

IF THE OBJECT IS INSIDE THE FOCAL POINT. THE IMAGE WILL BE VIRTUAL AND UPRIGHT

object REAL IMAGE INVERTED

eye is over here

This is how a ray diagram is drawn for a converging lens. It really only takes two rays to tell where the image will be. Two will cross where the image is located. Third ray makes sure you don’t make a mistake!

The principal ray connects the object with the lens and is then refracted THROUGH the FOCAL POINT

The central ray goes STRAIGHT THROUGH the CENTER of the lens to the image without refracting. There is NO refraction at the center of the lens

The focal ray passes through the FOCAL POINT on the object side of the lens. It is refracted such that the ray becomes PARALLEL to the horizontal axis. It crosses the other two rays at the image. 9 C. DIVERGING LENSES

Diverging lenses have one or both faces CONCAVE. IT “cups” in and is thinner at the center.

Light rays that strike a diverging lens parallel to the horizontal axis are refracted by the lens AWAY from the horizontal axis. Rays extended BACKWARD from the refracted rays will intersect at the FOCAL POINT of the lens.

refracted rays extend through the focal point

rays parallel to the axis refracted away from the axis 10 D. How is an image produced through a diverging lens

IMAGES WILL ALWAYS BE VIRTUAL, UPRIGHT, AND REDUCED.

object

eye is over here

VIRTUAL IMAGE UPRIGHT

This is how a ray diagram is drawn for a diverging lens.

The principal ray (1) connects the object with the lens and is then refracted AWAY from the FOCAL POINT. A ray can be extended backward from the refracted ray THROUGH the FOCAL POINT on the same side as the object.

The central ray (3) goes STRAIGHT THROUGH the CENTER of the lens and the image on the same side as the object. There is NO refraction at the center of the lens

The focal ray (2) is refracted PARALLEL to the axis but a forward extension of the ray passes through the FOCAL POINT on the eye side of the lens. A backward extension of the refracted ray is parallel to the axis and goes through the image on the object side of the lens. It crosses the other two extended rays at the image. 11 E. THE LENS EQUATION

Ray diagrams are nice for analyzing the geometry of how light interacts with lenses, but it would be a hassle to draw a scaled ray diagram to determine the distance and magnification of an object viewed through a lens.

There are equations for that!

Locating the distance of the object, image, or focal point:

do is the distance of the object from the lens

di is the distance of the image from the lens

f is the focal length Magnification with a converging lens:

a – magnification means the image is inverted a + magnification means the image is upright

12 VI. REFLECTION

For a REFLECTED RAY, the angle of incidence = angle of reflection. This is sometimes called the “law of reflection”

Just like for refracted rays, reflected ray angles are measured with respect to the normal of the reflecting surface

13 VII. MIRRORS

A. PLANE or FLAT MIRROR

A plane mirror will provide a reflected VIRTUAL image behind the plane of the mirror and the image will be upright and the same size as the object.

For mirror To find the image, the analysis, rays of reflected light your eye will are extended forward. The be on the point at which two (or more) same side as extended lines cross the object show where the image (of course!) is located.

notice the little dotted lines normal to the mirror surface. Those help you draw the reflected ray angle properly.

14 B. CONCAVE MIRRORS

IF THE OBJECT IS OUTSIDE THE FOCAL POINT, THE IMAGE WILL BE REAL AND INVERTED.

IF THE OBJECT IS INSIDE THE FOCAL POINT. THE IMAGE WILL BE VIRTUAL AND UPRIGHT

THE FOCAL POINT AND THE CENTER OF CURVATURE ARE IN FRONT OF THE MIRROR.

Use 3 rays for the ray diagram. A ray parallel to the principle axis reflect THROUGH the FOCAL POINT A ray THROUGH the center of curvature goes THROUGH the IMAGE A ray THROUGH the focal point reflects back PARALLEL to the axis. THESE THREE RAYS INTERSECT AT THE IMAGE LOCATION 15 concave mirrors, continued

object virtual image

in the case of the object being inside of the focal point, the image is located by extending the incident rays through the mirror surface.

16 B. CONVEX MIRROR CONVEX MIRRORS ALWAYS FORM VIRTUAL IMAGES BEHIND THE MIRROR.

THE FOCAL POINT AND THE CENTER OF CURVATURE ARE BEHIND THE MIRROR.

Draw 3 rays. One parallel to the axis One through the center of curvature One through the focal point.

C

“OBJECTS IN MIRROR ARE CLOSER THAN THEY APPEAR” 

For convex mirrors, the image is located by extending the reflected rays through the mirror surface. The extended lines cross at the image location. 17