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GEOMETRIC OPTICS

I. What is GEOMTERIC OPTICS

In geometric optics, LIGHT is treated as imaginary rays. How these rays interact withat 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 surfacesand 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 pathChange wavelength which means…Change velocity (speed of light)

The velocity DECREASES and the wavelength SHORTENS when light passesfrom a “faster” to a “slower” media.The velocity INCREASES and the wavelength LENGTHENS when light passesfrom a “slower” to a “faster” media.In either case, the FREQUENCY remains the same.

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refraction, continued

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.

When light hits the interface of twomedia at an angle, the lower partof the ray interacts first, thus slowingit down before the rest of the raymeets the interface. This rotatesthe ray toward the normal.

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

nmedium c

vmedium

The medium will commonlybe air, water, glass, plastic n is the refractive index

c is the speed of light in avacuum

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refraction, continued

substance refractive index, n

vacuum 1air 1.000277water 1.333glass 1.50

The table of refractive index values shows you that light slows down onlya 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 highrefractive index (glass), light refracts significantly.

SNELL’S LAW TELLS US HOW MUCH IT REFRACTS.

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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 themedia the light ray passes.

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

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

sin1sin2

v1v2

n2n1

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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 passesthrough 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 whichthe 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 itscolors (ROYGBIV)

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IV. PRISMS

violet redshort 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.

Prisms are used to separate light into its component wavelengths. Prisms demonstratethe optical phenomenon of DISPERSION.

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

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prisms, continued

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

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

nprismnair

sin( )2

sin2

incidentlight

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

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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 throughthe lens FOCAL POINT on the other side of the lens.

C C

horizontal axis orprinciple axis

F is the lens FOCAL POINTC = 2F. C is the lens RADIUS OF CURVATURE

rays that are parallel to the axis arerefracted by the lens into F

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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

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.

eye is over here

REAL IMAGEINVERTED

object

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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 refractedby the lens AWAY from the horizontal axis. Rays extended BACKWARD from therefracted rays will intersect at the FOCAL POINT of the lens.

rays parallel to the axis refracted away from

the axis

refracted rays extendthrough the focalpoint

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D. How is an image produced through a diverging lens

IMAGES WILL ALWAYS BE VIRTUAL, UPRIGHT, AND REDUCED.

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 asthe 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.

eye is over here

VIRTUAL IMAGEUPRIGHT

object

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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 distanceand magnification of an object viewed through a lens.

There are equations for that!

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

1

do1

d i1

f

do is the distance of theobject from the lens

di is the distance of the imagefrom the lens

f is the focal lengthMagnification with a converging lens:

m dido

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

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VI. REFLECTION

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

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

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VII. MIRRORS

A. PLANE or FLAT MIRROR

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

To find the image, therays of reflected lightare extended forward. Thepoint at which two (or more)extended lines cross show where the imageis located.

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

For mirroranalysis,your eye willbe on the same side asthe object(of course!)

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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 POINTA ray THROUGH the center of curvature goes THROUGH the IMAGEA ray THROUGH the focal point reflects back PARALLEL to the axis.THESE THREE RAYS INTERSECT AT THE IMAGE LOCATION

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concave mirrors, continued

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

virtual imageobject

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B. CONVEX MIRROR

CONVEX MIRRORS ALWAYS FORM VIRTUAL IMAGES BEHIND THE MIRROR.

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

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

Draw 3 rays.One parallel to the axisOne through the center of curvatureOne through the focal point.

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“OBJECTS IN MIRROR ARE CLOSERTHAN THEY APPEAR”