2. The q-parameter at plane 2 is also given from 1/92 = 1/R2 - iλo/w (equation 2). Combine this equation with the one your derived in part 1, so removing 92. Then equate the complex parts of this equation to derive and simplify an equation for w₂ at plane 2 in terms of d, λo and σ. = - 92 Rz 102 = 1502 То +d From Part 1, 92 के भगवद्ध Multiply by complexe conjugate d- ixσ² = Cd-ing 92 d+1782 x d-200² (d-21 = 20 % d² + 1204 782 = - iTo Comparing withE Real 22 d q. d²+1204, Imaginary! To To Solve for vo₂ x²²² = d² + x204 πως 2 То 72 402 Пог 2 = 30 012 +120 2 x² √² + 12 04 22 % изг = 102= то 2 d² + π +0 = ใ ૧ = +. 1204 7282 +1) 1282 of. 1+70343 π24 3. Simplify the equation you derived in part 2 in two limits, to give: w₂ where d→ 0 and; w₂ where d increases such that w₂ »σ (the far-field). Relate this to an illustration by the clearly marking your variables in your illustration. Illustrate how a laser beam propagates through it's minimum waist, σ by sketching caustics and also show the behaviour of those caustics in the far-field (d→ ∞o). A diffraction-limited light field of wavelength, Ao propagates in air as shown in the following figure. It starts with a flat phase front (R₁co) at plane 1 and as such has a minimum waist there of w₁ = σ. It diffracts as it propagates to plane 2 over a distance, d. Its q-parameters at plane 1 and plane 2 are q₁ and q2 respectively. Plane 1 W₁ = σ R₁ → ∞o d n = 1 (air) wavelength, 20 Plane 2 W₂ 1. Using equations 2 and 3, derive the following equation for the q-parameter, q2 at plane 2: 92 = inσ²/10 + d Eq 2: L - то Eq 3 q₂ = Aq, +B Ri πω R₁ → ∞ = о 101=0 Sub into eq 2 R₂ qi пор =-170 пог ノ 91 = 1402 To using the transfer matripe for free propagation AB [Tod/n] = ["0"] = [A B] M = From ea 92= (Dq1+d 92=9, +1 Sub in gri 92=1403 = g+d +1 1593 = + d A=1, B=d, C=0, D=

2. The q-parameter at plane 2 is also given from 1/92 = 1/R2 - iλo/w (equation 2). Combine this equation with the one your derived in part 1, so removing 92. Then equate the complex parts of this equation to derive and simplify an equation for w₂ at plane 2 in terms of d, λo and σ. = - 92 Rz 102 = 1502 То +d From Part 1, 92 के भगवद्ध Multiply by complexe conjugate d- ixσ² = Cd-ing 92 d+1782 x d-200² (d-21 = 20 % d² + 1204 782 = - iTo Comparing withE Real 22 d q. d²+1204, Imaginary! To To Solve for vo₂ x²²² = d² + x204 πως 2 То 72 402 Пог 2 = 30 012 +120 2 x² √² + 12 04 22 % изг = 102= то 2 d² + π +0 = ใ ૧ = +. 1204 7282 +1) 1282 of. 1+70343 π24 3. Simplify the equation you derived in part 2 in two limits, to give: w₂ where d→ 0 and; w₂ where d increases such that w₂ »σ (the far-field). Relate this to an illustration by the clearly marking your variables in your illustration. Illustrate how a laser beam propagates through it's minimum waist, σ by sketching caustics and also show the behaviour of those caustics in the far-field (d→ ∞o). A diffraction-limited light field of wavelength, Ao propagates in air as shown in the following figure. It starts with a flat phase front (R₁co) at plane 1 and as such has a minimum waist there of w₁ = σ. It diffracts as it propagates to plane 2 over a distance, d. Its q-parameters at plane 1 and plane 2 are q₁ and q2 respectively. Plane 1 W₁ = σ R₁ → ∞o d n = 1 (air) wavelength, 20 Plane 2 W₂ 1. Using equations 2 and 3, derive the following equation for the q-parameter, q2 at plane 2: 92 = inσ²/10 + d Eq 2: L - то Eq 3 q₂ = Aq, +B Ri πω R₁ → ∞ = о 101=0 Sub into eq 2 R₂ qi пор =-170 пог ノ 91 = 1402 To using the transfer matripe for free propagation AB [Tod/n] = ["0"] = [A B] M = From ea 92= (Dq1+d 92=9, +1 Sub in gri 92=1403 = g+d +1 1593 = + d A=1, B=d, C=0, D=

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Hi, i have done questions 1 and 2 of these pratice questions and i have attached the pictures of my workings, but i am struggling on how to do question 3 and illustrate it, can you help me solve it and show the illustration please, so i can see how to do it, please dont use AI

2. The q-parameter at plane 2 is also given from 1/92 = 1/R2 - iλo/w (equation 2). Combine this
equation with the one your derived in part 1, so removing 92. Then equate the complex parts of this equation
to derive and simplify an equation for w₂ at plane 2 in terms of d, λo and σ.
=
-
92
Rz
102
= 1502
То
+d
From Part 1, 92
के भगवद्ध
Multiply by complexe conjugate d- ixσ²
=
Cd-ing
92 d+1782
x
d-200²
(d-21
=
20
%
d² +
1204
782
=
-
iTo
Comparing withE
Real
22
d
q.
d²+1204, Imaginary! To
To Solve for vo₂
x²²² = d² + x204
πως
2
То
72
402
Пог
2
=
30
012 +120
2
x² √² + 12 04
22
%
изг
=
102=
то
2
d²
+
π
+0
=
ใ
૧
=
+.
1204
7282
+1)
1282
of.
1+70343
π24
3. Simplify the equation you derived in part 2 in two limits, to give: w₂ where d→ 0 and; w₂ where d
increases such that w₂ »σ (the far-field). Relate this to an illustration by the clearly marking your variables
in your illustration. Illustrate how a laser beam propagates through it's minimum waist, σ by sketching caustics
and also show the behaviour of those caustics in the far-field (d→ ∞o).

A diffraction-limited light field of wavelength, Ao propagates in air as shown in the following figure. It starts
with a flat phase front (R₁co) at plane 1 and as such has a minimum waist there of w₁ = σ. It diffracts as
it propagates to plane 2 over a distance, d. Its q-parameters at plane 1 and plane 2 are q₁ and q2 respectively.
Plane 1
W₁ = σ
R₁ → ∞o
d
n = 1 (air)
wavelength, 20
Plane 2
W₂
1. Using equations 2 and 3, derive the following equation for the q-parameter, q2 at plane 2:
92 = inσ²/10 + d
Eq 2: L
-
то
Eq 3 q₂ = Aq, +B
Ri
πω
R₁ → ∞
=
о
101=0
Sub into eq 2
R₂
qi
пор
=-170
пог
ノ
91 = 1402
To
using the transfer matripe for free propagation
AB
[Tod/n] = ["0"] = [A B]
M =
From ea
92= (Dq1+d
92=9, +1
Sub in gri
92=1403
=
g+d
+1
1593
=
+
d
A=1, B=d, C=0, D=

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