Lecture 2: Basic Astronomical Optics Prisms, Lenses, and Mirrors
Basic Optical Elements Refraction (Lenses) No longer used for large telescopes Widely used for instrument optics Reflection (mirrors) Widely used for large telescopes Widely used for instrument optics Dispersion (prisms, grisms, gratings) Grisms, prisms, and gratings are all used in modern instrumentation Commonly used for spectrographs Prisms convey basic concept
Snell s Law n i i Snell s Law : n i sin ( i ) = n r sin ( r ) n i = index of refraction in incident medium n r = index of refraction in refractive medium i = incident angle r = refractive angle r Refraction of the Earth s atmosphere on stellar light beam and ray bow effect due to the Snell s law. Refraction is also a key tool in designing astronomical optics. n r
What makes a rainbow? Water droplets act like prisms, refracting and reflecting light. White light from the Sun is refracted into colors and reflected. We see droplets in an arc ~42 o away from the direction opposite the Sun.
A 40-42 degree arc around this shadow Many droplets create an arc of reflected light
Snell s Law n i Snell s Law : n i sin ( i ) = n r sin ( r ) i r n r Values of n for various materials: Vacuum 1 Air 1.00029 (STP) Ice 1.31 Water 1.333 Table salt 1.54 Crown glass 1.52-1.62 Sapphire 1.77 Diamond 2.417 In general these values are wavelength-dependent
Snell s Law n i i r n r Refractive indices for different glasses (wikipedia)
Snell s Law Snell s Law : n i sin ( i ) = n r sin ( r ) n i Does light always get transmitted (refracted)? If not, what is the condition for reflection. i There s a critical angle of incidence such that the refracted angle is 90 degrees -- i.e. the refracted ray goes along the surface. Setting r =90, one sees that this is equivalent to sin( i )= n r / n i r n r For larger incident angles, the light is reflected back into the original medium. This is called total internal reflection. Note that internal reflection can only occur in the denser medium (n r < n i ).
Snell s Law Snell s Law : n i sin ( i ) = n r sin ( r ) n i i r n r http://galileo.phys.virginia.edu/classes/usem/sciimg/home_files/introduction.htm
Fiber Light Transmission Total Internal Reflection: sin i 1 n n 2 1 Light leak n 0 TIR n 2 u 1 90 i 1 90 sin 1 n ( n 2 1 ) u 0 u 1 i 1 n 1 Core Cladding 2 n n1 n 1 Fiber Acceptance Cone sin u0 sin u1 Angle: n0 n0 Only incoming beams with incident angles smaller than u 0 will pass the fiber without leakage! 2 2 The numerical aperture: 2 2 NA n 0 sin u0 sin u 0 n 1 n 2
wikipedia Snell s Law: Application to a Prism Prisms are simple dispersive elements First surface disperses the light Second surface disperses further At this surface different wavelengths also have different angles of incidence Illustrates the basic idea of dispersion for spectroscopy http://btc.montana.edu/messenger/instruments/mascs.htm http://web.utk.edu/~wverplan/kantor/cog.html
Dispersing Element 1: Prisms 1 n 2 Deviation angle: sin 1 2 2 n sin 1 sin cos sin 1 1 Deviation angle,, depends on refractive index, n, wedge angle,, and incident angle, 1
Basic Optical Elements Refractive Lenses (Convex and Concave Glass plates and Optical lenses: applied refraction http://uk.geocities.com/nsc_zambia/chart.htm
Basic Optical Elements Refractive Lenses (Convex and Concave) Convex lenses are converging lenses Concave lenses diverging lenses Flipped (opposite) for mirrors f is defined as the focal length -- the distance from the lens to the focus point.
How do lenses work? Lenses are shaped to bend light to a focus: The shape can be a parabola
Basic Optical Elements Refractive Lenses (Convex and Concave Light travelling on-axis doesn t bend How would you change the focal length?
Basic Optical Elements Refractive Lenses (Convex and Concave Light travelling on-axis doesn t bend Is the focal length the same for all wavelengths of light? http://uk.geocities.com/nsc_zambia/chart.htm
Thick Lenses First order optics = Gaussian optics = paraxial optics = perfect optics Unprimed are in object space Primed are in image space Cardinal Points and Planes: F Front focal plane/point F Rear focal plane/point Optical Axis F FFL P P EFL BFL F P Front principal plane/point P Rear principal plane/point EFL: Effective focal length BFL: Back focal length FFL: Front focal length P P F
Paraxial Image Formation Newtonian equations: Origins at focal plane h F f P h P F z z h f h Magnification, m = h /h = f/z = z /f zz = ff
Gaussian equations: Origins at principal plane h f P P z F z F f h Magnification, m = h /h = f/(z-f) = (z - f )/f f z z f ' ' 1 When f = f 1 z 1 z' 1 f
Basic Optical Elements Refractive Lenses (Convex and Concave Combination of concave and convex lenses can be used to make a more complex (compound) lens. http://uk.geocities.com/nsc_zambia/chart.htm
Lenses Rays parallel to the optical axis will pass through the focus Rays through the center will be undeviated. Focal length (f) is distance from lens to focus Focal ratio (f/ratio or f/number) = f/d, where D is the diameter of the lens Small f/ratio is considered fast, large is slow Power of a lens: P=1/f (units of P are diopters for f in meters) How does one compute the focal length? Define R1 and R2 and the radii of curvature for the two sides of the lens: Lens maker s formula: example: In this R 1 >0 R 2 <0 hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html
Lenses Rays parallel to the optical axis will pass through the focus Rays through the center will be undeviated. Focal length (f) is distance from lens to focus Focal ratio (f/ratio or f/number) = f/d, where D is the diameter of the lens Small f/ratio is considered fast, large is slow Power of a lens: P=1/f (units of P are diopters for f in meters) How does one compute the focal length? Two thin lenses in contact: Total focal length is given by: 1/f= 1/f 1 +1/f 2, or P=P 1 +P 2 hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html
Lenses Of course you can create lenses with various curvatures. In all cases, the lens maker s formula can still be used to give you the focal length. wikipedia
Lenses From the basic lens elements we ve discussed, you can also construct compound lenses for various applications (like cameras). The specific example below shows how two materials with different refractive indices can be coupled to minimize chromatic aberration (unwanted dispersion). wikipedia
Keplerian Refracting Telescope Objective Lens f o f e Eyepiece Lens θ o θ e Focal point Angle seen at objective: θ o Angle seen at eyepiece: θ e Objective Focal Length: f o Eyepiece Focal Length: f e The telescope's magnification: M= θ e / θ o =f o /f e
Keplerian Refracting Telescope Field of View Objective Lens f o f e Eyepiece Lens Focal point Field of View
Galilean Refracting Telescope Objective Lens f o Eyepiece Lens θ O θ e Focal point f e Angle seen at objective: θ O Angle seen at eyepiece: θ e Objective Focal Length: f O Eyepiece Focal Length: f e The telescope's magnification: M= θ e / θ o =f o /f e
Mirrors Same concepts of focal length, f-ratio, etc. still apply Concave mirrors commonly used for telescopes (We ll talk about the curvature of these mirrors later.) van.hep.uiuc.edu/.../ 20010605161128.htm
Mirrors Same concepts of focal length, f-ratio, etc. still apply Flat mirrors Used to redirect light path Folding mirrors are flat mirrors used to fold the light path wikipedia
Mirrors Two mirrors to form a reflective telescope Same concepts of focal length, f-ratio, etc. still apply wikipedia
The Cassegrain Telescope f f 1 Secondary Mirror Primary Mirror Secondary magnification: M f f 1 Two-mirror systems control aberration, change magnification Consists of a concave primary and convex secondary Short tube length Convenient focal surface Free of spherical aberration Off-axis image sharpness depends strongly on the shape of the surfaces System Primary Secondary Two popular combinations: Classical Cassegrain Ritchey- Chrétien Paraboloid Hyperboloid Hyperboloid Hyperboloid