A diffraction-limited light field of wavelength, Ao propagates in air as shown in the following figure. It starts with a flat phase front (R, 0) at plane 1 and as such has a minimum waist there of woo. 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 n-1 (air) wavelength, Plane 2 1. Using equations 2 and 3, derive the following equation for the q-parameter, q, at plane 2:1 92-ina/o+d Eq 2: L=1 - Eq 3 q₂ = Aq, +B , = 3 = 0 Sub into eq 2 91 = 1x02 qi FO хог To using the transfer matrive for free propagation M==.1] [8] Ass, Bad, csom From eas 92 = COq+4 Da +1 92=9+0 = い Sub in q = 110² 9/2= 1x02 +d To 2. The q-parameter at plane 2 is also given from 1/4 1/R2-12/o (equation 2). Combine this equation with the one your derived in part 1, so removing qz. Then equate the complex parts of this equation to derive and simplify an equation for org at plane 2 in terms of d, and . 2 - 1% 92 - From Part 1, 2 = += 912 1402 +d Multiply by complexe conjugate dixo² += 9 +1782 76 x Cd - ²) d- 140² = 26 d² + comparing w Real: You = 言 -176 = 22 d² + 12" Imaginary! To To Solve for P₂ d² + 1204 202 402 20 20 012² +120+ 7²² + 1204 22 191 2022- = XX π 2 +0 V 102= σ = To² ² + n² 54 7282 +1) 4. Continuing again from part 2, now equate the real parts of the equation you obtained from combining 1/92 = 1/Â₂ — iλo/πw (eq. 2) with the equation you derived in part 1, so removing 92. From this, derive and simplify an equation for R2 at plane 2 in terms of d, σ, λo and constants only. Then simplify this equation in two limits, to give: R₂ where d → 0 and; R2 where d increases such that d »ñσ²/λ。 (another way to define the far-field). Illustrate both of these limits in a labelled sketch of a propagating laser beam, showing caustics and wavefronts. [2]

A diffraction-limited light field of wavelength, Ao propagates in air as shown in the following figure. It starts with a flat phase front (R, 0) at plane 1 and as such has a minimum waist there of woo. 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 n-1 (air) wavelength, Plane 2 1. Using equations 2 and 3, derive the following equation for the q-parameter, q, at plane 2:1 92-ina/o+d Eq 2: L=1 - Eq 3 q₂ = Aq, +B , = 3 = 0 Sub into eq 2 91 = 1x02 qi FO хог To using the transfer matrive for free propagation M==.1] [8] Ass, Bad, csom From eas 92 = COq+4 Da +1 92=9+0 = い Sub in q = 110² 9/2= 1x02 +d To 2. The q-parameter at plane 2 is also given from 1/4 1/R2-12/o (equation 2). Combine this equation with the one your derived in part 1, so removing qz. Then equate the complex parts of this equation to derive and simplify an equation for org at plane 2 in terms of d, and . 2 - 1% 92 - From Part 1, 2 = += 912 1402 +d Multiply by complexe conjugate dixo² += 9 +1782 76 x Cd - ²) d- 140² = 26 d² + comparing w Real: You = 言 -176 = 22 d² + 12" Imaginary! To To Solve for P₂ d² + 1204 202 402 20 20 012² +120+ 7²² + 1204 22 191 2022- = XX π 2 +0 V 102= σ = To² ² + n² 54 7282 +1) 4. Continuing again from part 2, now equate the real parts of the equation you obtained from combining 1/92 = 1/Â₂ — iλo/πw (eq. 2) with the equation you derived in part 1, so removing 92. From this, derive and simplify an equation for R2 at plane 2 in terms of d, σ, λo and constants only. Then simplify this equation in two limits, to give: R₂ where d → 0 and; R2 where d increases such that d »ñσ²/λ。 (another way to define the far-field). Illustrate both of these limits in a labelled sketch of a propagating laser beam, showing caustics and wavefronts. [2]

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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 4 and illustrate it, can you help me solve it and show the illustration please, so i can see how to approach and sketch it, please dont use AI.

A diffraction-limited light field of wavelength, Ao propagates in air as shown in the following figure. It starts
with a flat phase front (R, 0) at plane 1 and as such has a minimum waist there of woo. 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
n-1 (air)
wavelength,
Plane 2
1. Using equations 2 and 3, derive the following equation for the q-parameter, q, at plane 2:1
92-ina/o+d
Eq 2: L=1
-
Eq 3 q₂ = Aq, +B
,
=
3
= 0
Sub into eq 2
91 = 1x02
qi
FO
хог
To
using the transfer matrive for free propagation
M==.1] [8] Ass, Bad, csom
From eas
92 = COq+4
Da +1
92=9+0
=
い
Sub in q = 110²
9/2= 1x02 +d
To
2. The q-parameter at plane 2 is also given from 1/4 1/R2-12/o (equation 2). Combine this
equation with the one your derived in part 1, so removing qz. Then equate the complex parts of this equation
to derive and simplify an equation for org at plane 2 in terms of d, and .
2
-
1%
92
-
From Part 1, 2 =
+=
912
1402
+d
Multiply by complexe conjugate dixo²
+=
9 +1782
76
x
Cd - ²)
d- 140²
=
26
d²
+
comparing w
Real:
You
=
言
-176
=
22
d² + 12" Imaginary! To
To Solve for P₂
d² + 1204
202
402
20
20
012² +120+
7²² + 1204
22
191
2022-
=
XX
π
2
+0
V
102=
σ
=
To² ² + n² 54
7282
+1)

4. Continuing again from part 2, now equate the real parts of the equation you obtained from combining
1/92 = 1/Â₂ — iλo/πw (eq. 2) with the equation you derived in part 1, so removing 92. From this, derive
and simplify an equation for R2 at plane 2 in terms of d, σ, λo and constants only. Then simplify this equation
in two limits, to give: R₂ where d → 0 and; R2 where d increases such that d »ñσ²/λ。 (another way to
define the far-field). Illustrate both of these limits in a labelled sketch of a propagating laser beam, showing
caustics and wavefronts.
[2]

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