Refractive Optics Chapter 26

Refractive Optics  Refraction  Refractive Image Formation  Optical Aberrations  The Human Eye  Optical Instruments

Refraction  Refractive Index  Snell’s Law  Total Internal Reflection  Polarization  Longitudinal Focus Shift  Dispersion

Refraction: Refractive Index Speed of light in vacuum: c = 3.00×10 8 m/s Speed of light in anything but vacuum: < c Index of refraction: n is a dimensionless ratio ≥ 1

Refraction: Refractive Index Index of refraction: n depends on:  material  wavelength of light

Refraction: Snell’s Law When light passes from one material into another:

Refraction: Snell’s Law When light passes from a less-dense (lower index) medium into a more-dense (higher index) medium, the light bends closer to the surface normal.

Refraction: Snell’s Law

Snell’s Law: Total Internal Reflection Consider light passing from a more-dense medium into a less-dense one (example: from water into air). The angle of refraction is larger than the angle of incidence.

Snell’s Law: Total Internal Reflection If the angle of incidence is large enough, the angle of refraction increases to 90° n = n 2 n = n 1

Snell’s Law: Total Internal Reflection At that point, none of the light is transmitted through the surface. All of the light is reflected (total internal reflection). The angle of incidence for which this happens is called the critical angle. n = n 2 n = n 1

Snell’s Law: Total Internal Reflection We can easily calculate the critical angle by imposing the additional condition on Snell’s Law:

Snell’s Law: Polarization We can calculate the angle of incidence for light entering a more-dense medium from a less-dense medium so that the reflected and refracted rays are perpendicular: n = n 2 n = n 1

Snell’s Law: Polarization By inspection of our drawing, we see that the perpendicularity of the reflected and transmitted rays requires that: n = n 1 n = n 2

Snell’s Law: Polarization Snell’s Law: Substitute for  2 :

Snell’s Law: Polarization Snell’s Law: Substitute for  2 : angle-difference identity:

Snell’s Law: Polarization  B is called Brewster’s angle.

Snell’s Law: Polarization When light is incident on a dielectric at Brewster’s angle:  the reflected light is linearly polarized, perpendicular to the plane of incidence  the transmitted light is partially polarized, parallel to the plane of incidence n = n 1 n = n 2

Snell’s Law: Longitudinal Focus Shift Rays are converging to form an image:

Snell’s Law: Longitudinal Focus Shift Insert a window: the focus is shifted rightward (delayed)

Snell’s Law: Longitudinal Focus Shift The amount of the longitudinal focus shift:

Snell’s Law: Longitudinal Focus Shift If an object is immersed in one material and viewed from another: “apparent depth”

Snell’s Law: Longitudinal Focus Shift The longitudinal focus shift and apparent depth relationships presented: are paraxial approximations. Even flat surfaces exhibit spherical aberration in converging or diverging beams of light.

Snell’s Law: Dispersion As we noted earlier, the index of refraction depends on:  the material  the wavelength of the light The dependence of refractive index on wavelength is called refractive dispersion.

Snell’s Law: Dispersion If each wavelength (color) has a different value of n, applying Snell’s law will give different angles of refraction for a common angle of incidence.

Refractive Image Formation: Lenses Just as we used curved (spherical) mirrors to form images, we can also use windows with curved (spherical) surfaces to form images. Such windows are called lenses. A lens is a piece of a transmissive material having one or both faces curved for image-producing purposes. (A lens can also be a collection of such pieces.)

Refractive Image Formation: Lenses Lens forms (edge views) Positive: center thicker than edge Negative: edge thicker than center

Refractive Image Formation: Lenses Positive: also called “converging” Negative: also called “diverging”

Refractive Image Formation: Lenses Real image formation by a positive lens:

Refractive Image Formation: Lenses Positive lens, d o > 2f:

Refractive Image Formation: Lenses Positive lens, d o = 2f:

Refractive Image Formation: Lenses Positive lens, f < d o < 2f:

Refractive Image Formation: Lenses Positive lens, d o = f:

Refractive Image Formation: Lenses Positive lens, d o < f:

Refractive Image Formation: Lenses Negative lens, d o >> f:

Refractive Image Formation: Lenses Negative lens, d o > f:

Refractive Image Formation: Lenses Negative lens, d o < f:

Refractive Image Formation: Lenses How are the conjugate distances measured? “Thin lens:” a simplifying assumption that all the refraction takes place at a plane in the center of the lens.

Refractive Image Formation: Lenses A better picture: “thick lens:” The conjugate distances are measured from the principal points.

Refractive Image Formation: Lenses A catalog example: Image from catalog of Melles Griot Corporation

Refractive Image Formation: Lenses The lens equation: Magnification: Combinations: one lens’s image is the next lens’s object.

Refractive Image Formation: Lenses Sign conventions  Light travels from left to right  Focal length: positive for a converging lens; negative for diverging  Object distance: positive for object to left of lens (“upstream”); negative for (virtual) object to right of lens  Image distance: positive for real image formed to right of lens from real object; negative for virtual image formed to left of lens from real object  Magnification: positive for image upright relative to object; negative for image inverted relative to object

Aberrations Image imperfections due to surface shapes and material properties. Not (necessarily) caused by manufacturing defects. A perfectly-made lens will still exhibit aberrations, depending on its shape, material, and how it is used.

Aberrations The basic optical aberrations  Spherical aberration: the variation of focal length with ray height  Coma: the variation of magnification with ray height  Astigmatism: the variation of focal length with meridian  Distortion: the variation of magnification with field angle  Chromatic: the variation of focal length and/or magnification with wavelength (color)

Lens Power The reciprocal of the focal length of a lens is called its power. This isn’t power in the work-and-energy sense. It really means the efficacy of the lens in converging rays to focus at an image. It can be positive or negative. If thin lenses are in contact, their powers may be added. Unit: if the focal length is expressed in meters, the power is in diopters (m -1 ).

The Human Eye Horizontal section of right eyeball (as seen from above). Illustration taken from Warren J. Smith, Modern Optical Engineering, McGraw-Hill, 1966)

The Human Eye Characteristics  Field of view (single eye): 130° high by 200° wide  Field of view both eyes simultaneously: 130° diameter  Visual acuity (resolution): 1 arc minute  Vernier acuity: 10 arc seconds accuracy; 5 arc seconds repeatability  Spectral response: peaks at about = 0.55  m (yellow- green). Response curve closely matches solar spectrum.  Pupil diameter: ranges from about 2 mm (very bright conditions) to about 8 mm (darkness).

The Human Eye Function  Image distance is nearly fixed (determined by eyeball shape and dimensions  Viewing objects significantly closer than infinity: accommodation  Far point: the farthest-away location at which the relaxed eye produces a focused image (normally infinity)  Near point: the closest location at which the eye’s ability to accommodate can produce a focused image (“normal” near point is 25 cm for young adults)

The Human Eye Defects and Problems  Myopia (nearsightedness)  Too much power in cornea and lens (or eyeball too long)  Far point is significantly closer than infinity  Corrected with diverging lens (negative power)

The Human Eye Defects and Problems  Hyperopia (farsightedness)  Too little power in cornea and lens (or eyeball too short)  Near point is significantly farther away than 25 cm  Corrected with converging lens (positive power)

The Human Eye Defects and Problems  Astigmatism  Different radii of curvature in horizontal and vertical meridians of the cornea  More power in one meridian than the other  Corrected with oppositely-astigmatic lens (toroidal surface)

The Human Eye Defects and Problems  Presbyopia (“elderly vision”)  Significantly decreased accommodation  Normal effect of aging (lens hardens, becomes difficult to squeeze)  Requires positive-power correction for near vision

Optical Instruments Angular Size of Objects and Images Angular size is the angle between chief rays from opposite sides or ends of the object. Angular sizes of object and image are equal.

Optical Instruments Angular Size of Objects and Images small-angle approximation:

Optical Instruments Angular Size of Objects and Images The larger  is, the more retinal pixels (rod and cone cells) are covered by the image. (Better, more detailed picture.)

Optical Instruments Angular Size of Objects and Images Optical instruments present an image to the eye that has a larger angular size than it would without the instrument.

Optical Instruments: Simple Magnifier (“Magnifying Glass”) Enlarged virtual image of object has larger angular size

Optical Instruments: Simple Magnifier (“Magnifying Glass”) The value of d o depends on d i (how the person uses the magnifier). Image at infinity: Image at near point:

Optical Instruments: Compound Microscope Compound microscope: consists of an objective lens and an eyepiece. Illustration from the online catalog of Melles Griot Corporation.

Optical Instruments: Compound Microscope Magnification (“official” Cutnell & Johnson version): where: f o is the objective focal length f e is the eyepiece focal length L is the distance between objective and eyepiece N is the near point distance

Optical Instruments: Compound Microscope Magnification (useful): where:  M o is the objective magnification  M e is the eyepiece magnification, and  the eyepiece and objective are separated by the mechanical tube length for which they were designed (if not, the image quality will be poor anyway). 160 mm is standard in the U.S.

Optical Instruments: Telescope Objective lens forms real image of distant object (at infinity) Eyepiece acts as simple magnifier: presents enlarged virtual image of real image, located at infinity.

Optical Instruments: Telescope Telescopes are afocal: both object and image are located at infinity.

Optical Instruments: Telescope The magnification is the ratio of the objective to eyepiece focal lengths.

Optical Instruments: Telescope Here is a common reflecting form: Newtonian

Optical Instruments: Telescope Another widely-used reflecting form: Cassegrain

Optical Instruments: Telescope Astronomical refracting telescope: has inverted image

Optical Instruments: Telescope Galilean refracting telescope: has upright image

Optical Instruments: Telescope Erecting relay lens configuration. Has upright image and a place to put a reticle. Rifle scopes, spotting scopes, etc.