Algorithms
Some algorithms of this thesis in Wolfram language.
On-axis stigmatic collector singlet lens
1 (∗ parameters ∗)
2 za[ra ]:=−raˆ2/(4∗50);
3
4 f [ ra ]:=−ta+n t+tb+Sign[ta] Sqrt[raˆ2+(ta−za[ra]) ˆ2];
5
6 \[Tau][ ra ]:= za[ra]−t−tb;
7
8 \[CurlyRho]z[ra ]:=(( zaˆ\[Prime])[ra] (ra+(−ta+za[ra]) (zaˆ\[Prime])[ra ]) )/(−Sign[ta]n Sqrt[raˆ2+(
ta−za[ra])ˆ2] (1+((zaˆ\[Prime])[ra ]) ˆ2) )+Sqrt[1−(ra+(−ta+za[ra]) (zaˆ\[Prime])[ra]) ˆ2/( nˆ2 (ra ˆ2+(ta−za[ra])ˆ2) (1+((zaˆ\[Prime])[ra ]) ˆ2) ) ]/ Sqrt[1+((zaˆ\[Prime])[ra ]) ˆ2];
9
10 \[CurlyRho]r[ra ]:=( ra+(−ta+za[ra]) (zaˆ\[Prime])[ra ]) /(−Sign[ta]n Sqrt[raˆ2+(ta−za[ra])ˆ2] (1+((
zaˆ\[Prime])[ra ]) ˆ2) )−((zaˆ\[Prime])[ra] Sqrt[1−(ra+(−ta+za[ra]) (zaˆ\[Prime])[ra]) ˆ2/( nˆ2 (ra ˆ2+(ta−za[ra])ˆ2) (1+((zaˆ\[Prime])[ra ]) ˆ2) ) ]) /Sqrt[1+((zaˆ\[Prime])[ra ]) ˆ2];
11
12 \[CurlyTheta][ra ]:=(−n f [ ra]−ra
13 \[CurlyRho]r[ra]−\[CurlyRho]z[ra] \[Tau][ra]+Sign[n]Sign[tb]Sqrt[(n f [ ra]+ra \[CurlyRho]r[ra]+\[
CurlyRho]z[ra] \[Tau][ra ]) ˆ2−(1−nˆ2) (raˆ2−f[ra]ˆ2+\[Tau][ra ]ˆ2) ]) /(1−nˆ2)
14
15 zb[ra ]:= za[ra]+\[CurlyTheta][ra ]\[ CurlyRho]z[ra];
16 rb[ ra ]:= ra+\[CurlyTheta][ra ]\[ CurlyRho]r[ra ];
17
18 (∗ Define system configuration’s ∗)
19 n = 1.5;
20 t = 5;
21 ta = −60;
22 tb = 60;
23 Rmax = 12.5;
24 zb[0]
25
26 (∗ System plot ∗)
27 g1 =Plot[{za[ra ]}, {ra, −Rmax, Rmax}, PlotStyle −> {Black},
28 AspectRatio −> Automatic, AxesLabel −> {r, z}];
29 g2 =ParametricPlot[{rb[ra], zb[ra ]}, {ra, −Rmax, Rmax},
30 AspectRatio −> 1/1, AxesOrigin −> {0, 0}, PlotStyle −> {Black},
90
31 Mesh −> None];
32 Show[g1, g2, PlotRange −> All]
On-axis stigmatic collimator singlet lens
1 (∗ parameters ∗)
2 za[ra ]:=−raˆ2/(4∗50);
3
4 \[CurlyRho]z[ra ]:=(( zaˆ\[Prime])[ra] (ra+(−ta+za[ra]) (zaˆ\[Prime])[ra ]) )/(−Sign[ta]n Sqrt[raˆ2+(
ta−za[ra])ˆ2] (1+((zaˆ\[Prime])[ra ]) ˆ2) )+Sqrt[1−(ra+(−ta+za[ra]) (zaˆ\[Prime])[ra]) ˆ2/( nˆ2 (ra ˆ2+(ta−za[ra])ˆ2) (1+((zaˆ\[Prime])[ra ]) ˆ2) ) ]/ Sqrt[1+((zaˆ\[Prime])[ra ]) ˆ2];
5
6 \[CurlyRho]r[ra ]:=( ra+(−ta+za[ra]) (zaˆ\[Prime])[ra ]) /(−Sign[ta]n Sqrt[raˆ2+(ta−za[ra])ˆ2] (1+((
zaˆ\[Prime])[ra ]) ˆ2) )−((zaˆ\[Prime])[ra] Sqrt[1−(ra+(−ta+za[ra]) (zaˆ\[Prime])[ra]) ˆ2/( nˆ2 (ra ˆ2+(ta−za[ra])ˆ2) (1+((zaˆ\[Prime])[ra ]) ˆ2) ) ]) /Sqrt[1+((zaˆ\[Prime])[ra ]) ˆ2];
7
8 \[CurlyTheta][ra ]:=((−1+n)t−ta+za[ra]+Sign[ta]Sqrt[raˆ2+ (za[ra]−ta) ˆ2]) /( n−\[CurlyRho]z[ra]);
9 zb[ra ]:= za[ra]+\[CurlyTheta][ra ]\[ CurlyRho]z[ra];
10 rb[ ra ]:= ra+\[CurlyTheta][ra ]\[ CurlyRho]r[ra ];
11
12 (∗ Define system configuration’s ∗)
13 n = 1.5;
14 t = 5;
15 ta = −60;
16 Rmax = 15;
17 zb[0]
18
19 (∗ System plot ∗)
20 g1 =Plot[{za[ra ]}, {ra, −Rmax, Rmax}, PlotStyle −> {Black},
21 AspectRatio −> Automatic, AxesLabel −> {r, z}];
22 g2 =ParametricPlot[{rb[ra], zb[ra ]}, {ra, −Rmax, Rmax},
23 AspectRatio −> 1/1, AxesOrigin −> {0, 0}, PlotStyle −> {Black},
24 Mesh −> None];
25 Show[g1, g2, PlotRange −> All]
On-axis stigmatic singlet lens infinite object finite image
1 (∗ parameters ∗)
2 za[ra ]:=Module[{h},h=−raˆ2/(4∗50)];
3
4 f [ ra ]:=Module[{h},h=n t+tb−za[ra]];
5
6 \[Tau][ ra ]:= Module[{h},h=za[ra]−t−tb];
7
8 \[CurlyRho]z[ra ]:=Module[{h},h=(zaˆ\[Prime])[ra]ˆ2/(n+n (zaˆ\[Prime])[ra]ˆ2)+Sqrt[1+(−1+1/(1+(za ˆ\[Prime])[ra]ˆ2))/nˆ2]/Sqrt[1+(zaˆ\[Prime])[ra ]ˆ2]];
9
10 \[CurlyRho]r[ra ]:= Module[{h},h=(zaˆ\[Prime])[ra] (1/(n+n (zaˆ\[Prime])[ra]ˆ2)−Sqrt[1+(−1+1/(1+(
zaˆ\[Prime])[ra]ˆ2))/nˆ2]/Sqrt[1+(zaˆ\[Prime])[ra ]ˆ2]) ];
11
12 \[CurlyTheta][ra ]:= Module[{h},h=(−n f[ra]−ra \[CurlyRho]r[ra]−\[CurlyRho]z[ra] \[Tau][ra]+Sign[n]
Sign[tb]Sqrt[(n f[ra]+ra \[CurlyRho]r[ra]+\[CurlyRho]z[ra] \[Tau][ra ]) ˆ2−(1−nˆ2) (raˆ2−f[ra ]ˆ2+\[Tau][ra ]ˆ2) ]) /(1−nˆ2)]
13
14 zb[ra ]:=Module[{h},h=za[ra]+\[CurlyTheta][ra]\[CurlyRho]z[ra]];
15
16 rb[ ra ]:=Module[{h},h=ra+\[CurlyTheta][ra]\[CurlyRho]r[ra]];
17
18 (∗ Define system configuration’s ∗)
19 n = 1.5;
20 t = 5;
21 tb = 60;
22 Rmax = 18.5;
23 zb[0]
24
25 (∗ System plot ∗)
26 g1 =Plot[{za[ra ]}, {ra, −Rmax, Rmax}, PlotStyle −> {Black},
27 AspectRatio −> Automatic, AxesLabel −> {r, z}];
28 g2 =ParametricPlot[{rb[ra], zb[ra ]}, {ra, −Rmax, Rmax},
29 AspectRatio −> 1/1, AxesOrigin −> {0, 0}, PlotStyle −> {Black},
30 Mesh −> None];
31 Show[g1, g2, PlotRange −> All]
Single-lens telescope
1 (∗ parameters ∗)
2 za[ra ] := raˆ2/80;
3 4
5 \[CurlyRho]z[ra ] :=
6 Derivative[1][za ][ ra ]ˆ2/( n + n Derivative[1][za ][ ra ]ˆ2) + Sqrt[(
7 nˆ2 + (−1 + nˆ2)Derivative[1][za ][ ra ]ˆ2) /(
8 nˆ2 (1 +Derivative[1][za ][ ra ]ˆ2) ) ]/ Sqrt[
9 1 +Derivative[1][za ][ ra ]ˆ2];
10 11
12 \[CurlyRho]r[ra ] := −((
13 Derivative[1][za ][
14 ra ] (−1 +
15 n Sqrt[1 + Derivative[1][za ][ ra ]ˆ2] Sqrt[(
16 nˆ2 + (−1 + nˆ2)Derivative[1][za ][ ra ]ˆ2) /(
17 nˆ2 (1 +Derivative[1][za ][ ra ]ˆ2) ) ]) ) /(
18 n (1 +Derivative[1][za ][ ra ]ˆ2) ) ) ;
19
20 \[CurlyTheta][ra ] := ((−1 + n) t ) /( n − \[CurlyRho]z[ra])
21
22 zb[ra ] := za[ra ] + \[CurlyTheta][ra] \[CurlyRho]z[ra];
23
24 rb[ ra ] := ra + \[CurlyTheta][ra] \[CurlyRho]r[ra ];
25
26 (∗ Define system configuration’s ∗)
27 n = 1.5;
28 t = 15;
29 Rmax = 20;
30 zb[0]
31
32 (∗ System plot ∗)
33 g1 =Plot[{za[ra ]}, {ra, −Rmax, Rmax}, PlotStyle −> {Black},
34 AspectRatio −> Automatic, AxesLabel −> {r, z}];
35 g2 =ParametricPlot[{rb[ra], zb[ra ]}, {ra, −Rmax, Rmax},
36 AspectRatio −> 1/1, AxesOrigin −> {0, 0}, PlotStyle −> {Black},
37 Mesh −> None];
38 Show[g1, g2, PlotRange −> All]
On-axis stigmatic triplet lens
1 <<MaTeX‘
2 3 n0=1;
4 n1=1.5;
5 n2=1;
6 n3=1.5;
7 n4=1;
8 9 f1=60;
10 f2=80;
11 f3=−70;
12
13 t0=−50;
14 t1=8;
15 t2=11;
16 t3=9;
17 t4=90;
18 t=t1+t2+t3+t4;
19
20 step=0.05; (∗0.05∗)
21 rmax=17.5;
22 density=1;
23 d=22;
24
25 Z1[R1 ]:=R1ˆ2/(4f1);
26
27 na[ra ]:=Sqrt[Z1’[ra ]ˆ2+1];
28
29 Zz2[R2 ]:=t1−R2ˆ2/(4f2);
30
31 Zz3[R3 ]:=t1+t2−(R3)ˆ2/(4f3 ) ;
32
33 z1=Table[Z1[R1],{R1,step,Floor[rmax],step}];
34
35 dz1=Table[Z1’[R1],{R1,step,Floor[rmax],step}];
36
37 r1=Table[R1,{R1,step,Floor[rmax],step}];
38
39 zz2=Table[Zz2[R2],{R2,step,Floor[rmax],step}];
40
41 zz3=Table[Zz3[R3],{R3,step,Floor[rmax],step}];
42 43 44
45 L1=Join[Reverse[Table[{z1[[i]],r1[[ i ]]},{ i ,1, Length[z1]}]],List [{0,0}], Table[{z1[[ i ]],− r1 [[ i ]]},{ i ,1,Length[z1]}]];
46
47 \[CurlyRho]z1=(n0(dz1(r1+(−t0+z1) dz1)))/(n1 Sqrt[r1ˆ2+(t0−z1)ˆ2] (1+(dz1)ˆ2))+Sqrt[1−(n0ˆ2 (r1 +(−t0+z1) dz1)ˆ2)/(n1ˆ2 (r1ˆ2+(t0−z1)ˆ2) ((1+(dz1)ˆ2)))]/Sqrt[(1+(dz1)ˆ2)];
48
49 \[CurlyRho]r1=(n0(r1+(−t0+z1) dz1))/(n1 Sqrt[r1ˆ2+(t0−z1)ˆ2] (1+(dz1)ˆ2))−(dz1 Sqrt[1−(n0ˆ2 (r1 +(−t0+z1) dz1)ˆ2)/(n1ˆ2 (r1ˆ2+(t0−z1)ˆ2) ((1+(dz1)ˆ2)))])/Sqrt[(1+(dz1)ˆ2)];
50
51 \[CurlyTheta]1=1/\[CurlyRho]r1ˆ2 (−r1 \[CurlyRho]r1−2 f2 \[CurlyRho]z1+2 Sqrt[f2] Sqrt[\[
CurlyRho]r1ˆ2 (t1−z1)+r1 \[CurlyRho]r1 \[CurlyRho]z1+f2 \[CurlyRho]z1ˆ2]);
52
53 r2=r1+\[CurlyTheta]1 \[CurlyRho]r1;
54
55 z2=z1+\[CurlyTheta]1 \[CurlyRho]z1;
56
57 L2=Join[Reverse[Table[{Re[z2[[i]]],Re[r2[[i ]]]},{ i ,1, Length[z2]}]],List[{ t1 ,0}], Table[{Re[z2[[i ]]],−Re[r2[[ i ]]]},{ i ,1, Length[z2]}]];
58
59 dr21=ListConvolve[{1,−1},r2]/ListConvolve[{1,−1},r1];
60
61 dr2=Join[dr21,{Last[dr21]}];
62
63 dz21=ListConvolve[{1,−1},z2]/ListConvolve[{1,−1},r1];
64
65 dz2=Join[dz21,{Last[dz21]}];
66
67 \[CurlyRho]r2=−((n1 dr2((r1−r2) dr2+(z1−z2)dz2))/(n2 Sqrt[(r1−r2)ˆ2+(z1−z2)ˆ2] (dr2ˆ2+dz2ˆ2)))
−(dz2 Sqrt[1−(n1ˆ2 ((r1−r2) dr2+(z1−z2)dz2)ˆ2)/(n2ˆ2 ((r1−r2)ˆ2+(z1−z2)ˆ2) (dr2ˆ2+dz2ˆ2))])/
Sqrt[(dr2ˆ2+dz2ˆ2)];
68
69 \[CurlyRho]z2=−((n1 dz2 ((r1−r2) dr2+(z1−z2)dz2))/(n2 Sqrt[(r1−r2)ˆ2+(z1−z2)ˆ2] (dr2ˆ2+dz2ˆ2))) +(dr2Sqrt[1−(n1ˆ2 ((r1−r2) dr2+(z1−z2)dz2)ˆ2)/(n2ˆ2 ((r1−r2)ˆ2+(z1−z2)ˆ2) (dr2ˆ2+dz2ˆ2))])/
Sqrt[(dr2ˆ2+dz2ˆ2)];
70
71 \[CurlyTheta]2=1/\[CurlyRho]r2ˆ2 (−r2 \[CurlyRho]r2−2 f3 \[CurlyRho]z2+2 Sqrt[f3] \[Sqrt]((t1+t2−
z2) \[CurlyRho]r2ˆ2+r2 \[CurlyRho]r2 \[CurlyRho]z2+f3 \[CurlyRho]z2ˆ2));
72
73 r3=r2+\[CurlyTheta]2 \[CurlyRho]r2;
74
75 z3=z2+\[CurlyTheta]2 \[CurlyRho]z2;
76
77 L3=Join[Reverse[Table[{Re[z3[[i]]],Re[r3[[i ]]]},{ i ,1, Length[z3]}]],List[{ t1+t2 ,0}], Table[{Re[z3[[i ]]],−Re[r3[[ i ]]]},{ i ,1, Length[z3]}]];
78
79 dr31=ListConvolve[{1,−1},r3]/ListConvolve[{1,−1},r1];
80
81 dr3=Join[dr31,{Last[dr31]}];
82
83 dz31=ListConvolve[{1,−1},z3]/ListConvolve[{1,−1},r1];
84
85 dz3=Join[dz31,{Last[dz31]}];
86
87 \[CurlyRho]r3=−((n2 dr3((r2−r3) dr3+(z2−z3) dz3))/(n3 Sqrt[(r2−r3)ˆ2+(z2−z3)ˆ2] ((dr3)ˆ2+(dz3)ˆ2) ))−(dz3Sqrt[1−(n2ˆ2 ((r2−r3) dr3+(z2−z3) dz3)ˆ2)/(n3ˆ2 ((r2−r3)ˆ2+(z2−z3)ˆ2)((dr3)ˆ2+(dz3) ˆ2))])/Sqrt[((dr3)ˆ2+(dz3)ˆ2)];
88
89 \[CurlyRho]z3=−((n2 dz3 ((r2−r3) dr3+(z2−z3) dz3))/(n3 Sqrt[(r2−r3)ˆ2+(z2−z3)ˆ2] ((dr3)ˆ2+(dz3) ˆ2)))+(dr3Sqrt[1−(n2ˆ2 ((r2−r3) dr3+(z2−z3) dz3)ˆ2)/(n3ˆ2 ((r2−r3)ˆ2+(z2−z3)ˆ2)((dr3)ˆ2+(dz3) ˆ2))])/Sqrt[((dr3)ˆ2+(dz3)ˆ2)];
90
91 F=−n0 t0+n1 t1+n2 t2+n3 t3+n4 t4−(n0Sqrt[r1ˆ2+(−t0+z1)ˆ2]+n1 Sqrt[(−r1+r2)ˆ2+(−z1+z2)ˆ2]+n2 Sqrt[(−r2+r3)ˆ2+(−z2+z3)ˆ2]);
92
93 \[CurlyTheta]3=1/(n3ˆ2−n4ˆ2) (F n3+n4ˆ2 (r3 \[CurlyRho]r3+(−t+z3) \[CurlyRho]z3)−1/2 \[Sqrt](−4 (−n3ˆ2+n4ˆ2) (−Fˆ2+n4ˆ2 (r3ˆ2+(t−z3)ˆ2))+4 (F n3+n4ˆ2 (r3 \[CurlyRho]r3+(−t+z3) \[CurlyRho ]z3))ˆ2));
94
95 r4=r3+\[CurlyTheta]3 \[CurlyRho]r3;
96
97 z4=z3+\[CurlyTheta]3 \[CurlyRho]z3;
98
99 L4=Join[Reverse[Table[{Re[z4[[i]]],Re[r4[[i ]]]},{ i ,1, Length[z4]}]],List[{ t1+t2+t3 ,0}], Table[{Re[z4 [[i ]]],−Re[r4[[ i ]]]},{ i ,1, Length[z4]}]];
100
101 dr41=ListConvolve[{1,−1},r4]/ListConvolve[{1,−1},r1];
102 dr4=Join[dr41,{Last[dr41]}];
103
104 dz41=ListConvolve[{1,−1},z4]/ListConvolve[{1,−1},r1];
105 dz4=Join[dz41,{Last[dz41]}];
106
107 \[CurlyRho]r4=−((n3 dr4((r3−r4) dr4+(z3−z4) dz4))/(n4 Sqrt[(r3−r4)ˆ2+(z3−z4)ˆ2] ((dr4)ˆ2+(dz4)ˆ2) ))−(dz4Sqrt[1−(n3ˆ2 ((r3−r4) dr4+(z3−z4) dz4)ˆ2)/(n4ˆ2 ((r3−r4)ˆ2+(z3−z4)ˆ2)((dr4)ˆ2+(dz4) ˆ2))])/Sqrt[((dr4)ˆ2+(dz4)ˆ2)];
108
109 \[CurlyRho]z4=−((n3 dz4 ((r3−r4) dr4+(z3−z4) dz4))/(n4 Sqrt[(r3−r4)ˆ2+(z3−z4)ˆ2] ((dr4)ˆ2+(dz4) ˆ2)))+(dr4Sqrt[1−(n3ˆ2 ((r3−r4) dr4+(z3−z4) dz4)ˆ2)/(n4ˆ2 ((r3−r4)ˆ2+(z3−z4)ˆ2)((dr4)ˆ2+(dz4) ˆ2))])/Sqrt[((dr4)ˆ2+(dz4)ˆ2)];
110
111 Surface := {ParametricPlot[{raˆ2/(4 f1), ra}, {ra, −rmax, rmax},
112 PlotStyle −> {Black}, Axes −> False,
113 PlotRange −> {{−60, 130}, {−1.5 rmax, 1.5 rmax}}],
114 ParametricPlot[{t1 − raˆ2/(4 f2) , ra}, {ra, L2 [[1, 2]],
115 L2[[Length[L2], 2]]}, PlotStyle −> {Black}],
116 ParametricPlot[{t1 + t2 − raˆ2/(4 f3) , ra}, {ra, L3 [[1, 2]],
117 L3[[Length[L3], 2]]}, PlotStyle −> {Black}], ListLinePlot[{L4}],
118 Graphics[{Black,
119 Line[{{rmaxˆ2/(4 f1), rmax}, {rmaxˆ2/(4 f2), rmax}}]}]}
120
121 Show[Surface, ImageSize −> {1280, Automatic}]
Off-axis stigmatic singlet lens
1 (∗ parameters ∗)
2 za[ra ]:=Module[{h},h=raˆ2/100];
3
4 \[CurlyRho]r0[ra ]:=Module[{h},h=−ha/(n Sqrt[(ha)ˆ2+(ta)ˆ2])]
5
6 \[CurlyRho]z0[ra ]:=Module[{h},h=Sqrt[1−(−ha)ˆ2/(nˆ2 ((ha)ˆ2+(ta)ˆ2))]]
7
8 f [ ra ]:=Module[{h},h=Sqrt[(ta)ˆ2+(ha)ˆ2]+n t+Sqrt[(\[CurlyRho]r0[ra]t−hb)ˆ2+(tb)ˆ2]−Sqrt[(ra−ha) ˆ2+(za[ra]−ta)ˆ2]];
9
10 \[Tau][ ra ]:= Module[{h},h=za[ra]−t \[CurlyRho]z0[ra]−tb];
11
12 \[CurlyRho]r[ra ]:= Module[{h},h=(−ha+ra+(−ta+za[ra]) (zaˆ\[Prime])[ra])/(n Sqrt[(ha−ra)ˆ2+(ta−za [ra])ˆ2] (1+((zaˆ\[Prime])[ra])ˆ2) )−((zaˆ\[Prime])[ra] Sqrt[1−(−ha+ra+(−ta+za[ra]) (zaˆ\[Prime ])[ra])ˆ2/(nˆ2 ((ha−ra)ˆ2+(ta−za[ra])ˆ2) (1+((zaˆ\[Prime])[ra ]) ˆ2) ) ]) /Sqrt[1+((zaˆ\[Prime])[ra ])
ˆ2]];
13
14 \[CurlyRho]z[ra ]:=Module[{h},h=((zaˆ\[Prime]) [ra](−ha+ra+(−ta+za[ra]) (zaˆ\[Prime])[ra]))/(n Sqrt[(ha−ra)ˆ2+(ta−za[ra])ˆ2] (1+((zaˆ\[Prime])[ra ]) ˆ2) )+ Sqrt[1−(−ha+ra+(−ta+za[ra]) (zaˆ\[
Prime])[ra])ˆ2/(nˆ2 ((ha−ra)ˆ2+(ta−za[ra])ˆ2) (1+((zaˆ\[Prime])[ra ]) ˆ2) ) ]/ Sqrt[1+((zaˆ\[Prime])[
ra ]) ˆ2]];
15
16 \[CurlyTheta][ra ]:= Module[{h},h=(−n f[ra]−(ra−hb) \[CurlyRho]r[ra]−\[CurlyRho]z[ra] \[Tau][ra]+
s1Sqrt[(n f[ra]+(ra−hb)\[CurlyRho]r[ra]+\[CurlyRho]z[ra] \[Tau][ra ]) ˆ2−(1−nˆ2) ((ra−hb)ˆ2−f[
ra]ˆ2+\[Tau][ra]ˆ2) ]) /(1−nˆ2)]
17
18 zb[ra ]:=Module[{h},h=za[ra]+\[CurlyTheta][ra]\[CurlyRho]z[ra]];
19
20 rb[ ra ]:=Module[{h},h=ra+\[CurlyTheta][ra]\[CurlyRho]r[ra]];
21
22 (∗ Define system configuration’s ∗)
23 s1=1;
24 n=1.5;
25 t=10;
26 ta=−20;
27 tb=25;
28 ha=5;
29 hb=−3;
30 Rmax=14;
31 Sqrt[(zb[0]−za[0])ˆ2+(rb[0]−0)ˆ2]
32
33 (∗ System plot ∗)
34 g1=Plot[{za[ra]},{ra,−Rmax,Rmax},PlotStyle−>{Black},AspectRatio−>Automatic,AxesLabel
−>{r,z}];
35
36 g2=ParametricPlot[{rb[ra],zb[ra]},{ra,−Rmax,Rmax},AspectRatio−>1/1, AxesOrigin−>{0,0}, PlotStyle−>{Black}, Mesh −>None];
37 Show[g1,g2,PlotRange−>All]
Conclusions
In conclusion, this doctoral work found not only the general equation of stigmatic lenses but also a new methodology for optical design. This methodology is purely analytical free of approximations and iterations, it is entirely based on Fermat’s principle. The myth of the nonexistence of a stigmatic lens collapsed and at the same time. The uniqueness of stigmatic solutions was demonstrated. This study opens a new paradigm in optical design, which nat-urally generated a lot of work and implications. We study rotationally symmetrical stigmatic lenses, free-form lenses, free-form axicons, stigmatic telescopes, stigmatic systems with an arbitrary number of refractive surfaces, off-axis aberration-free systems, lenses that meet the Abbe sine condition for each ray. There are other implications in the conclusions of this work, but only the time will tell who they are.
The following list shows the works carried out and published during my period as a doctoral student.
Books:
1. Rafael G. Gonz´alez-Acu ˜na, H´ector Chaparro Romo and Julio Guiterrez-Vega, Analyt-ical Lens Design, Institute of Physics Publishing, April 2020.
2. Rafael G. Gonz´alez-Acu ˜na and H´ector Chaparro Romo, Stigmatic Optics, Institute of Physics Publishing, Oct 2020.
3. Rafael G. Gonz´alez-Acu ˜na, H´ector Chaparro Romo and Israel Mel´endez Optics and Artificial Vision, Institute of Physics Publishing, Dec 2020.
Papers:
1. Rafael G. Gonz´alez-Acu ˜na, Julio Guiterrez-Vega, Generalization of axicons shape:
the gaxicon, Journal of the Optical Society of America A, 2018.
2. Rafael G. Gonz´alez-Acu ˜na, H´ector Chaparro Romo, General formula for bi-aspheric singlet lens design free of spherical aberration, Applied Optics, 2018. Editor’s picks.
3. Rafael G. Gonz´alez-Acu ˜na, Abundio Davila, Julio Guiterrez-Vega, Optical flow of non-integer order in particle image velocimetry techniques, Singal Processing, 2019.
97
4. Rafael G. Gonz´alez-Acu ˜na, H´ector Chaparro Romo, Julio Guiterrez-Vega, General formula for freeform lens free of spherical aberrations and astigmatisim, Applied Op-tics, 2019.
5. Rafael G. Gonz´alez-Acu ˜na, Maximino Avenda˜no-Alejo, Julio Guiterrez-Vega, Singlet lens for generating aberration-free patterns on deformed surfaces, Journal of the Opti-cal Society of America A, 2019.
6. Rafael G. Gonz´alez-Acu ˜na, Julio C. Guitierrez-Vega, Analytic formulation of a refractive-reflective telescope free of spherical aberration, Optical Engineering, 2019.
7. Rafael G. Gonz´alez-Acu ˜na, Julio C. Guitierrez-Vega, General formula to eliminate spherical aberration produced by an arbitrary number of lenses, Optical Engineering, 2019.
8. Rafael G. Gonz´alez-Acu ˜na, Julio C. Guitierrez-Vega, A transition integral transform obtained by the generalization of the Fourier transform, Ain Shams Engineering Jour-nal, 2019.
9. Rafael G. Gonz´alez-Acu ˜na, Julio C. Guitierrez-Vega, Analytic design of a spherochro-matic singlet, Journal of the Optical Society of America A, 2020.
10. Rafael G. Gonz´alez-Acu ˜na, H´ector Chaparro Romo, Julio C. Guitierrez-Vega, Ana-lytic aplanatic singlet lens: setting and design for three point objects and images in the meridional plane, Optical Engineering, 2020.
11. Rafael G. Gonz´alez-Acu ˜na, H´ector Chaparro Romo, Julio C. Guitierrez-Vega, Ana-lytic solution of the eikonal for a stigmatic singlet lens, Physica Scripta, 2020.
12. Rafael G. Gonz´alez-Acu ˜na, Michelle C. Rocha, Julio C. Guitierrez-Vega, Freeform axicon with azimuthal variation, Journal of Modern Optics, 2020.
13. Rafael G. Gonz´alez-Acu ˜na, H´ector Chaparro Romo, Julio C. Guitierrez-Vega, Exact equations for stigmatic singlet design meeting the Abbe sine condition, Optics Commu-nications, 2020.
14. Rafael G. Gonz´alez-Acu ˜na, H´ector Chaparro Romo, Julio C. Guitierrez-Vega, Unique-ness of stigmatic solutions, Journal of the Optical Society of America A, 2020.
Proceedings:
1. Rafael G. Gonz´alez-Acu ˜na, Julio Guiterrez-Vega; General formula for aspheric col-limator lens design free of spherical aberrationCurrent Developments in Lens Design and Optical Engineering XX, July, 2019 San Diego, USA.
2. Rafael G. Gonz´alez-Acu ˜na, Julio Guiterrez-Vega; General formula of the refractive telescope design free spherical aberration Novel Optical Systems, Methods, and Ap-plications XXII, July, 2019 San Diego, USA.
3. Rafael G. Gonz´alez-Acu ˜na, Julio Guiterrez-Vega; General formula for freeform colli-mator lens design free of spherical aberration and astigmatisimNovel Optical Systems, Methods, and Applications XXII, July, 2019 San Diego, USA.
4. Rafael G. Gonz´alez-Acu ˜na, Julio Guiterrez-Vega; Analytical solution to the Wasserman-Wolf problemX Iberoamerican Optics Meeting, September, 2019 Cancun, Mexico.
5. Rafael G. Gonz´alez-Acu ˜na, Julio Guiterrez-Vega; Analytical solution to Levi-Civita problemX Iberoamerican Optics Meeting, September, 2019 Cancun, Mexico.
6. Rafael G. Gonz´alez-Acu ˜na, Julio Guiterrez-Vega; The entrance pupil of an on-axis stigmatic singlet lens Current Developments in Lens Design and Optical Engineering XXI, September, 2020 San Diego, USA.
7. Rafael G. Gonz´alez-Acu ˜na, Julio Guiterrez-Vega; Analytic formulation of a sphe-rochromatic collimator lensCurrent Developments in Lens Design and Optical Engi-neering XXI, September, 2020 San Diego, USA.
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