05-733-009 Solution Set #3. Assume that the esolution limit of the eye is acminute. At what distance can the eye see a black cicle of diamete 6" on a white backgound? One acminute is, so conside a tiangle with shot side of 3inand angle of 60 0. By (VERY BASIC) tigonomety, we have: tan = 3in = z 0 z 75 mm = tan 0 tan = tan 0 0 π h 80 adians π i =tan =.454 0 4 adians 0 80 so half of the cicle subtends about 0.45 milliadian= 45 μadian. This is the adius of the esolution spot, do the angula diamete would be: z = =.9 0 4 adians 50 mm = 53.9m = 0, 66 in = 79 ft =.908 0 km 4
. Calculate the linea sepaation of two objects on the suface of the moon that may be baely esolved by the 00" telescope on Paloma Mountain; neglect atmospheic effects. Assume that the obsevation is in visible light so that λ 0 = 0.5 μm. Accoding to the Rayleigh limit, two Aiy disk pattens with equal iadiances ae just baely esolved if the cental peak of one coincides with the fist minimum of the othe. ( θ) min '. λ 0 The distance to the moon is appoximately 40,000 miles (the distance I dove my last ca befoe its demise): L = 40, 000 mi =.4 0 5 mi 580 ft in mm 5.4 mi ft in = 3.864 0 5 km mm = 00 in 5.4 in =5.08 m 0.5 μm ( θ) min '. 5.08 m =.0 0 7 adian adian = 60 acseconds acminute 60acminutes degee acseconds 0665 360 degee π adian = =.0 0 5 acseconds adian adian = acsecond = 5 0 6 adians =5μadians useful numbe to emembe: acsecond =5 μadian ( θ) min '.0 0 7 adian = 0. μadian 0. = 5 = 0.04 At the distance to the moon, the angula subtense of the esolvable spot has linea dimension: =.0 0 7 3.864 0 5 km = 46.4m
3. Detemine the diamete of a telescope that would be equied to esolve the two equally bight components of a double stat whose linea sepaation is 0 8 km at a distance of 0 light yeas. The distance z in the figue in # is now 0 light yeas. Light tavels at velocity The numbe of seconds in a yea: v = 3 0 8 m s 60 s min 60 min h 4 h d 365.5 d y = 3.56 0 7 s y = 3, 000, 000 s y (anothe useful numbe to emembe) distance taveled by light in a yea: The tangent of the angle is z = 3 0 8 m s 3.56 s 07 y = 9. 468 0 km y = 0 light yeas = 9. 468 0 3 km / 0 4 km tan θ = z = 0 8 km 9. 468 0 3 km =.056 0 6 adians =.06 μadian = 0. acsecond The equied diamete of the telescope would satisfy:. λ 0.056 0 6 adians λ 0 =..056 0 6 adians In visible light with λ 0 =0.5 μm, the diamete is: 0.5 μm..056 0 6 adians = 0.578 m = 578 mm =.76 in = ft many advanced amateu astonomes have telescopes of this size. 3
4. The neoimpessionist painte Geoge Seuat poduced paintings composed of a lage numbe of closely spaced dots of unmixed pigments. The colos ae mixed in the eye of the obseve to blend into the desied colos. Assume that the diametes and cente-to-cente spacings of the dots ae both in. How fa must the obseve stand 0 fom the painting to obtain the necessay mixing of colo? This is vey simila to #: we want the distance z to be sufficiently lage so that we cannot esolve adjacent spots. One way would be to again assume that the eye can esolve about 0 of ac and that we want the linea dimension of the blu to subtend at least two spots, we can use the same discussion in # to find: z =.54 mm.908 0 4 = 7.5m = 690 in = 60 ft which is huge. Anothe way would be to assume that a diffaction-limited eye with a pupil diamete of = mm, which is an appopiate numbe fo bight light (a lage diamete might be appopiate since you don t exhibit paintings unde bight lights; this would make the diffaction spot smalle and equie you to move back fathe). The angula diamete of the full Aiy disk fo this estimate is:.44 λ 0 = 0.5 μm.44 mm = 6. 0 4 adians = 60 μadians = 0665 6. 0 4 acsec = 6 acsec = acmin So we e in the same ballpak. Now assume that the Aiy diffaction disk subtends at least two cicles: 6. 0 4 z 0.in = 5mm = z = 5mm = 8m = 6 ft = 35 in 6. 0 4 which is smalle because the diamete of the diffaction spot is lage. As long as you explain you logic, I accept values in the ange 7.5m z 8m 4
5. A glass plate is spayed with cicula dots of pigment of the same size that ae andomly distibuted ove the plate. If a distant point souce of light is obseved by eye though the plate, a diffuse halo is seen whose angula width is. Estimate the diamete of the paticles. The fist point to make is that the diffaction patten fom an opaque spot on a clea backgound and fom a clea apetue on an opaque backgound ae vey simila except at the oigin: Clea apetue: f [x, y] = CY L = F [ξ,η] = πd 0 4 SOMB [ρ] x y = g [x, y] F, SOMB = I [x, y] g [x, y] SOMB ³ ³ Opaque spot: f [x, y] = CY L = F [ξ,η] =δ [ξ,η] πd 0 4 SOMB [ρ] x y x y = g [x, y] δ, F, =( ) δ [x, y] πd 0 4 SOMB ³ = I [x, y] (λ 0z ) δ [x, y] πd 0 4 SOMB ³ ( ) 4 δ [x, y] πd 0 4 (λ 0z ) δ [x, y] SOMB ³ + πd 0 4 SOMB ³ SOMB µ = δ [x, y] πd 0 πd δ [x, y] SOMB [0, 0] + 0 ( ) 4( ) SOMB = µ µ πd 0 πd ( ) δ [x, y]+ 0 4( ) ³ ³ So the iadiance away fom the oigin is popotional to the squae of the besinc (sombeo) function in both cases. The fact that the diffaction pattens ae simila away fom the oigin fo complementay objects aises fom Babinet s pinciple. 5
If the object consists of a single paint dot, the image is as just shown, with spatial vaiation away fom the oigin in the fom of: I [x 6= 0,y 6= 0] SOMB ³ sotheanguladiameteoftheaiydiskis: =. θ = z =.44 λ 0 If we have a lot of paint dots on a egulaly spaced gid (descibed by a COMB function, the object function is: x f [x, y] =CY L COMB x, y y whee x, y ae usually lage than (and often significantly lage). I [x, y] π d 0 4 SOMB ³ SOMB ³ COMB ( x y) COMB x λ0 z x, ³ y y x λ0 z x, ³ y y If ( x) +( y) >d 0, as is usual, then many elements of the COMB function will lie within the cental disk of the Sombeo, so again the cental disk is detemined by the diamete of the paint spot via λ0 z =.44 If the spots ae andomly placed, we still get an iadiance of the same fom as: I [x, y] SOMB ³ R x y λ 0 z λ0 z, ³ d x 0 y whee R is the spectum of the ensemble of spot locations; the cental disk again is detemined by the diamete of the paint spot via λ0 z =.44 6
The halo may be intepeted as the diamete of the Aiy patten. Since the fist zeo is at. λ 0, we can estimate that the angula diamete of the halo is halfway between: D halo λ 0 =.44 If the angula diamete of the halo is π = adian =3. 49 80 0 adian = 30 adian, then the diamete of the paint spot may be estimated: θ = 3.49 0 adian λ0 z =.44 =.44 λ 0 3.49 0 =:69.9 λ 0 If λ 0 =0.5 μm fo visible light, then: z 0 =.44 =.44 0.5 μm 3.49 0 = 34.95 μm = 0.035 mm Since small scales in the object tansfom to lage scales in the diffaction patten, we can use the Faunhofe diffaction to measue small objects accuately! µ λ0 7