Notes on Optics

Refraction

Index refraction is related to the actual speed of light in a medium. We know the speed of light in a vacuum (total absence matter) is a constant c = 300,000 km/s.

If v_l is the speed of a particular wavelength of light (l) in a medium, then the medium has an index of refraction:

n_l = c / v_l

The subscript denotes the wavelength dependence. This leads to different optical properties at different wavelengths for refracting telescopes, and the problem of chromatic aberration.

Refraction is caused when light travels between different media. The incident angle (q₁) and refracted angle of light (q₂) are related by the indices of refraction of both media. Note: The angles are measured from perpendicular of the interface. This equation is called Snell’s Law:

n_1l sin q₁ = n_2l sin q₂

The construction of lenses is done by shaping both sides of the lens and choosing transparent material with a particular index of refraction.

The key property of a lens is the focal length f_l which may depend on wavelength. There are two types of lenses. A converging lens takes parallel light rays from a distant source and brings them all to a focal point a distance f_l in front of the lens. A diverging lens takes parallel light rays spreads them out so they appear to all emerge from a single point a distance - f_l behind the lens. Note f_l is positive for a converging lens and negative for a diverging lens.  Note:  Because indices of refraction depend on wavelength, so do focal lengths.  The choice of material or use of special coatings is needed to minimize chromatic abberation.

Reflection

Light reflects in a simple wavelength-independent manner.  A reflective surface can also be shaped to create a focus much like a lens.  The law of reflection is simply:

q₁ =  q₂

where both angles are measured from perpendicular of the interface.  A converging mirror has a positive focal length, and a diverging mirror has a negative focal length.  The greates advantage of mirrors is the lack of chromatic aberration.

Telescopes, Focal Length and Plate Scale

A Telescope if constructed with one or more lenses or mirrors.  A telescope brings light entering the front aperture to a focus (focal plane).  One can either place a camera or spectrograph at the focus, or send light through an eyepiece.  Eyepieces are necessary for light to enter eyes as parallel rays.  They can be changed in order to adjust magnification and field of view.   The largest optical element (lens or mirror) is called a primary.  Some telescopes are made of multiple elements with diffent focal lengths.   These elements are placed at the right distances to achieve and overal effective focal length (equations not shown here).  From this point, focal length refers to the effective focal length of the series of optical elements (excludes eyepieces).

Let q be the angle between a source and the optical axis and y be the distance from the focal point along the axis to the image of the source on the focal plane.  These are related by the equation:

y =  f tan q

We can approximate tan q = q  because the angle is small:

y =  f q

A differential relation gives

dq/dy  = 1/f

This relation is known as the plate scale and allows one to translate a small distance in the focal plane (dy) to a small angular distance on the sky (dq).    Larger focal length means that distances in the focal plane translate into smaller angles on the sky.  Thus, focal length is related to the maximum field of view for a telescope.

Diffraction and Telescope Resolution

A single slit aperture creates and interference pattern due to diffraction.  The pattern is alternating bright and dark lines where along the optical axis there is a bright line.  Dark lines off axis are created by destructive interference.  The condition for destructive interference is that the difference in path length of light from the slit center and the slit edge equals half of a wavelength.

(D/2) sin q = l / 2

This condition reduces to

sin q = l / D

A more complete analysis finds that dark fringes occur if

sin q =  m l / D   where m = 1, 2, 3, 4, .....

An analysis for a circular aperture gives a diffraction pattern with nearly all the light in a bright central circle.  The remaining light appear as concentric cicular rings that grow fainter going outwards.  One finds that the angle of full-width at half the maximum (FWHM) intensity is related to the diameter of a circular and the wavelength of light.  The diameter of the primary optical element acts as the aperture size (D) for diffraction patterns in the telescope.  Diffraction limits the amount of detail that is observed.  Angular resolution on the sky is measured by the FWHM caused by diffraction:

q _FWHM =  1.22 l / D  in radians  = 2.5 X 10⁵ l / D  in arcseconds

Focal Ratio and Sensitivity

Focal ratio is defined as the ratio of focal length to aperture and is computed with the equation:

F = f / D

Focal often quoted along with aperture as specifications of a telescope.  Somtimes focal ratio is expressed as f/# where # equals the focal ratio F.
Example:  A telescope with f / 10 has a focal ratio of 10.  If the aperture D = 0.1 meters, then the focal length f = 1.0 meters.

The sensitivity of a telescope is can be computed for two extreme cases.
(1)  If the source of light has large anugular size and is easily resolved, the illumination J is used to measure the sensitivity of a telescope.

J proportional to 1 / F²
(2)  If the source of light is an unresolved point source of light, the LGP due to the aperture area determines sensitivity.
LGP proportional to D²

Most circumstances are intermediate between cases (1) and (2), or have objects in both extremes.

Magnification

Ray optics allow one to determine the effect of eyepieces on magnification as seen by the human eye.  The magnification is given by:

m = f_obj / f_eye

where f_obj is the effective focal length of the telescope and f_eye is the focal length of the eyepiece.