Irradiance formations of on-axis Lambertian pointlike sources in polygonal total-internal-reflection straight light pipes

Yi-Kai Cheng; Meng-Hua Wang; Jyh-Long Chern

doi:10.1364/JOSAA.24.002748

Journal of the Optical Society of America A
Vol. 24,
Issue 9,
pp. 2748-2757
(2007)
•https://doi.org/10.1364/JOSAA.24.002748

Irradiance formations of on-axis Lambertian pointlike sources in polygonal total-internal-reflection straight light pipes

Yi-Kai Cheng, Meng-Hua Wang, and Jyh-Long Chern

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Yi-Kai Cheng, Meng-Hua Wang, and Jyh-Long Chern, "Irradiance formations of on-axis Lambertian pointlike sources in polygonal total-internal-reflection straight light pipes," J. Opt. Soc. Am. A 24, 2748-2757 (2007)

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Abstract

Different geometric shapes of light pipes cause different irradiance distributions. We analytically explore the irradiance distributions of on-axis Lambertian pointlike sources in polygonal total-internal-reflection (TIR) straight light pipes. It is analytically shown that the irradiance of pentagonal light pipes concentrates on the center of the exit plane. Numerical verifications are also provided, and experimental explorations with different shapes of acrylic light pipes are carried out for comparison. We also analyze the influence of light-pipe length on distribution uniformity and deduce the smallest uniform-mixing length/circumradius ratios for polygonal light pipes.

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Condition	Range/(No. of Reflections)	Vector Form
1:	$0 < d ⩽ d_{1} (0)$	$\overset{⃑}{d} = (d \sin ϕ, - d \cos ϕ)$
$0 ° < ϕ < 5.5 °,$	$d_{1} < d ⩽ d_{1} + d_{2} (1)$	$\overset{⃑}{d} = (d \sin ϕ, (d - 2 d_{1}) \cos ϕ)$
$S (1) \to S (3) \to$	$d_{1} + d_{2} < d ⩽ d_{1} + d_{2} + d_{3} (2)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - (d - d_{1} - d_{2}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - (d - d_{1} - d_{2}) \sin (18 ° + ϕ) \end{matrix})$
$S (4) \to S (5)$	$d_{1} + d_{2} + d_{3} < d ⩽ d_{1} + d_{2} + d_{3} + d_{4} (3)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - d_{3} \cos (18 ° + ϕ) - d_{4} \cos (54 ° - ϕ) + (d - d_{1} - d_{2} - d_{3} - d_{4}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - d_{3} \sin (18 ° + ϕ) - d_{4} \sin (54 ° - ϕ) - (d - d_{1} - d_{2} - d_{3} - d_{4}) \sin (18 ° + ϕ) \end{matrix})$
	$d_{1} + d_{2} + d_{3} + d_{4} < d (4)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - d_{3} \cos (18 ° + ϕ) - d_{4} \cos (54 ° - ϕ) + (d - d_{1} - d_{2} - d_{3} - d_{4}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - d_{3} \sin (18 ° + ϕ) - d_{4} \sin (54 ° - ϕ) - (d - d_{1} - d_{2} - d_{3} - d_{4}) \sin (18 ° + ϕ) \end{matrix})$
$where d_{1} = \frac{R \cos 36 °}{\cos ϕ}, d_{2} = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}, d_{3} = \sqrt{{(x_{3} - x_{2})}^{2} + {(y_{3} - y_{2})}^{2}}, d_{4} = \sqrt{{(x_{4} - x_{3})}^{2} + {(y_{4} - y_{3})}^{2}}$
$(x_{1}, y_{1}) = (R \cos 36 ° \tan ϕ, - R \cos 36 °), (x_{2}, y_{2}) = (\frac{R (1 + 2 \cos 36 °)}{\cot ϕ + \tan 36 °}, R - \tan 36 ° ∙ x_{2}),$
$(x_{3}, y_{3}) = (\frac{y_{2} - \tan (18 ° + ϕ) x_{2} - R}{\tan 36 ° - \tan (18 ° + ϕ)}, R + \tan 36 ° x_{3}), (x_{4}, y_{4}) = (\begin{matrix} \frac{\tan (54 ° - ϕ) x_{3} - y_{3} - R (\cos 36 ° + \sin 36 ° \tan 72 °)}{\tan 72 ° + \tan (54 ° - ϕ)}, \\ - \tan 72 ° x_{4} - R (\cos 36 ° + \sin 36 ° \tan 72 °) \end{matrix})$
2:	$0 < d ⩽ d_{1} (0)$	$\overset{⃑}{d} = (d \sin ϕ, - d \cos ϕ)$
$5.55 ° < Φ <$	$d_{1} < d ⩽ d_{1} + d_{2} (1)$	$\overset{⃑}{d} = (d \sin ϕ, (d - 2 d_{1}) \cos ϕ)$
$22.386 °, S (1)$	$d_{1} + d_{2} < d ⩽ d_{1} + d_{2} + d_{3} (2)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - (d - d_{1} - d_{2}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - (d - d_{1} - d_{2}) \sin (18 ° + ϕ) \end{matrix})$
$\to S (3) \to S (5)$	$d_{1} + d_{2} + d_{3} < d (3)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - d_{3} \cos (18 ° + ϕ) + (d - d_{1} - d_{2} - d_{3}) \cos (ϕ - 18 °), \\ (d_{2} - d_{1}) \cos ϕ - d_{3} \sin (18 ° + ϕ) + (d - d_{1} - d_{2} - d_{3}) \sin (ϕ - 18 °) \end{matrix})$
$where d_{1} = \frac{R \cos 36 °}{\cos ϕ}, d_{2} = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}, d_{3} = \sqrt{{(x_{3} - x_{2})}^{2} + {(y_{3} - y_{2})}^{2}}$
$(x_{1}, y_{1}) = (R \cos 36 ° \tan ϕ, - R \cos 36 °), (x_{2}, y_{2}) = (\frac{R (1 + 2 \cos 36 °)}{\cot ϕ + \tan 36 °}, R - \tan 36 ° ∙ x_{2})$
$(x_{3}, y_{3}) = (\begin{matrix} \frac{\tan (18 ° + ϕ) x_{2} - y_{2} - R (\cos 36 ° + \sin 36 ° \tan 72 °)}{\tan 72 ° + \tan (18 ° + ϕ)}, \\ - \tan 72 ° x_{3} - R (\cos 36 ° + \sin 36 ° \tan 72 °) \end{matrix})$
3:	$0 < d ⩽ d_{1} (0)$	$\overset{⃑}{d} = (d \sin ϕ, - d \cos ϕ)$
$22.386 ° < Φ <$	$d_{1} < d ⩽ d_{1} + d_{2} (1)$	$\overset{⃑}{d} = (d \sin ϕ, (d - 2 d_{1}) \cos ϕ)$
$26.267 °, S (1)$	$d_{1} + d_{2} < d ⩽ d_{1} + d_{2} + d_{3} (2)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - (d - d_{1} - d_{2}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - (d - d_{1} - d_{2}) \sin (18 ° + ϕ) \end{matrix})$
$\to S (3) \to S (1)$	$d_{1} + d_{2} + d_{3} < d ⩽ (3)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - d_{3} \cos (18 ° + ϕ) - (d - d_{1} - d_{2} - d_{3}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - d_{3} \sin (18 ° + ϕ) + (d - d_{1} - d_{2} - d_{3}) \sin (18 ° + ϕ) \end{matrix})$
$where d_{1} = \frac{R \cos 36 °}{\cos ϕ}, d_{2} = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}, d_{3} = \sqrt{{(x_{3} - x_{2})}^{2} + {(y_{3} - y_{2})}^{2}} .$
$(x_{1}, y_{1}) = (R \cos 36 ° \tan ϕ, - R \cos 36 °), (x_{2}, y_{2}) = (\frac{R (1 + 2 \cos 36 °)}{\cot ϕ + \tan 36 °}, R - \tan 36 ° ∙ x_{2}),$
$(x_{3}, y_{3}) = (\frac{y_{2} - \tan (18 ° + ϕ) x_{2} - R}{\tan 36 ° - \tan (18 ° + ϕ)}, y_{3} = R \cos 36 °)$
4:	$0 < d ⩽ d_{1} (0)$	$\overset{⃑}{d} = (d \sin ϕ, - d \cos ϕ)$
$26.267 ° < Φ$	$d_{1} < d ⩽ d_{1} + d_{2} (1)$	$\overset{⃑}{d} = (d \sin ϕ, (d - 2 d_{1}) \cos ϕ)$
$36 °, S (1)$	$d_{1} + d_{2} < d ⩽ d_{1} + d_{2} + d_{3} (2$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ + (d - d_{1} - d_{2}) \cos (54 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ + (d - d_{1} - d_{2}) \sin (54 ° + ϕ) \end{matrix})$
$\to S (2) \to S (3)$	$d_{1} + d_{2} + d_{3} < d ⩽$ $d_{1} + d_{2} + d_{3} + d_{4} (3)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ + d_{3} \cos (54 ° + ϕ) - (d - d_{1} - d_{2} - d_{3}) \cos (54 ° - ϕ) \\ (d_{2} - d_{1}) \cos ϕ + d_{3} \sin (54 ° + ϕ) - (d - d_{1} - d_{2} - d_{3}) \sin (54 ° - ϕ) \end{matrix}$
$\to S (5)$	$d_{1} + d_{2} + d_{3} + d_{4} < d (4)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ + d_{3} \cos (54 ° + ϕ) - d_{4} \cos (54 ° - ϕ) + (d - d_{1} - d_{2} - d_{3} - d_{4}) \cos (ϕ - 18 °), \\ (d_{2} - d_{1}) \cos ϕ + d_{3} \sin (54 ° + ϕ) - d_{4} \sin (54 ° - ϕ) + (d - d_{1} - d_{2} - d_{3} - d_{4}) \sin (ϕ - 18 °) \end{matrix})$
$where d_{1} = \frac{R \cos 36 °}{\cos ϕ}, d_{2} = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}, d_{3} = \sqrt{{(x_{3} - x_{2})}^{2} + {(y_{3} - y_{2})}^{2}}, d_{4} = \sqrt{{(x_{4} - x_{3})}^{2} + {(y_{4} - y_{3})}^{2}}$
$(x_{1}, y_{1}) = (R \cos 36 ° \tan ϕ, - R \cos 36 °), (x_{2}, y_{2}) = (\frac{R (1 + 2 \cos 36 °)}{\cot ϕ + \tan 36 °}, \cot ϕ ∙ x_{2} - 2 R \cos 36 °),$
$(x_{3}, y_{3}) = (\frac{R + \tan (54 ° + ϕ) x_{2} - y_{2}}{\tan 36 ° + \tan (54 ° + ϕ)}, - \tan 36 ° x_{3} + R) . (x_{4}, y_{4}) = (\begin{matrix} \frac{\tan (54 ° - ϕ) x_{3} - y_{3} - R (\cos 36 ° + \sin 36 ° \tan 72 °)}{\tan 72 ° + \tan (54 ° - ϕ)}, \\ - \tan 72 ° x_{4} - R (\cos 36 ° + \sin 36 ° \tan 72 °) \end{matrix})$

Condition	Range/(No. of Reflections)	Vector Form
1: $0 ° < Φ < 11.31 °$ , Square A $\to B \to C \to D \to G$	$0 < d ⩽ d_{1}$ (0)	$\overset{⃑}{d} = \vec{P A}$
	$d_{1} < d ⩽ d_{2}$ (1)	$\overset{⃑}{d} = \vec{P B} ∙ (1, - 1)$
	$d_{2} < d ⩽ d_{3}$ (2)	$\overset{⃑}{d} = \vec{P C}$
	$d_{3} < d ⩽ d_{4}$ (3)	$\overset{⃑}{d} = \vec{P D} ∙ (1, - 1)$
	$d > d_{4}$ (4)	$\overset{⃑}{d} = \vec{P G} ∙ (- 1, - 1)$
$where d_{1} = \frac{w}{\cos ϕ}, d_{2} = 3 w \sqrt{(1 + \tan^{2} ϕ)}, d_{3} = 5 w \sqrt{(1 + \tan^{2} ϕ)}, d_{4} = w \sqrt{(1 + \cot^{2} ϕ)}$
2: $11.31 ° < Φ < 18.43 °$ , Square A $\to B \to C \to F \to G$	$0 < d ⩽ d_{1}$ (0)	$\overset{⃑}{d} = \vec{P A}$
	$d_{1} < d ⩽ d_{2}$ (1)	$\overset{⃑}{d} = \vec{P B} ∙ (1, - 1)$
	$d_{2} < d ⩽ d_{3}$ (2)	$\overset{⃑}{d} = \vec{P C}$
	$d_{3} < d ⩽ d_{4}$ (3)	$\overset{⃑}{d} = \vec{P F} ∙ (- 1, 1)$
	$d > d_{4}$ (4)	$\overset{⃑}{d} = \vec{P G} ∙ (- 1, - 1)$
$where d_{1} = \frac{w}{\cos ϕ}, d_{2} = 3 w \sqrt{(1 + \tan^{2} ϕ)}, d_{3} = w \sqrt{(1 + \cot^{2} ϕ)}, d_{4} = 5 w \sqrt{(1 + \tan^{2} ϕ)}$
3: $18.43 ° < Φ < 31.63 °$ , Square A $\to B \to E \to F \to G$	$0 < d ⩽ d_{1}$ (0)	$\overset{⃑}{d} = \vec{P A}$
	$d_{1} < d ⩽ d_{2}$ (1)	$\overset{⃑}{d} = \vec{P B}$
	$d_{2} < d ⩽ d_{3}$ (2)	$\overset{⃑}{d} = \vec{P E} ∙ (- 1, - 1)$
	$d_{3} < d ⩽ d_{4}$ (3)	$\overset{⃑}{d} = \vec{P F} ∙ (- 1, 1)$
	$d > d_{4} (4)$	$\overset{⃑}{d} = \vec{P G} ∙ (- 1, - 1)$
$where d_{1} = \frac{w}{\cos ϕ}, d_{2} = w \sqrt{(1 + \cot^{2} ϕ)}, d_{3} = 3 w \sqrt{(1 + \tan^{2} ϕ)}, d_{4} = 5 w \sqrt{(1 + \tan^{2} ϕ)}$
4: $31.63 ° < Φ < 45 °$ , Square A $\to B \to E \to F \to H$	$0 < d ⩽ d_{1}$ (0)	$\overset{⃑}{d} = \vec{P A}$
	$d_{1} < d ⩽ d_{2}$ (1)	$\overset{⃑}{d} = \vec{P B} ∙ (1, - 1)$
	$d_{2} < d ⩽ d_{3}$ (2)	$\overset{⃑}{d} = \vec{P E} ∙ (- 1, - 1)$
	$d_{3} < d ⩽ d_{4}$ (3)	$\overset{⃑}{d} = \vec{P F} ∙ (- 1, 1)$
	$d > d_{4}$ (4)	$\overset{⃑}{d} = \vec{P H}$
$where d_{1} = \frac{w}{\cos ϕ}, d_{2} = w \sqrt{(1 + \cot^{2} ϕ)}, d_{3} = 3 w \sqrt{(1 + \tan^{2} ϕ)}, d_{4} = 3 w \sqrt{(1 + \cot^{2} ϕ)}$

No. of Light- Pipe Edges	Smallest Uniform-Mixing $L ∕ R$ Ratio	Uniformity Deviation δ (%)
Circular	N/A	$> 66$
3	3	4.62
4	3	4.59
5	19	4.31
6	3	4.81

Condition	Range/(No. of Reflections)	Vector Form
1:	$0 < d ⩽ d_{1} (0)$	$\overset{⃑}{d} = (d \sin ϕ, - d \cos ϕ)$
$0 ° < ϕ < 5.5 °,$	$d_{1} < d ⩽ d_{1} + d_{2} (1)$	$\overset{⃑}{d} = (d \sin ϕ, (d - 2 d_{1}) \cos ϕ)$
$S (1) \to S (3) \to$	$d_{1} + d_{2} < d ⩽ d_{1} + d_{2} + d_{3} (2)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - (d - d_{1} - d_{2}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - (d - d_{1} - d_{2}) \sin (18 ° + ϕ) \end{matrix})$
$S (4) \to S (5)$	$d_{1} + d_{2} + d_{3} < d ⩽ d_{1} + d_{2} + d_{3} + d_{4} (3)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - d_{3} \cos (18 ° + ϕ) - d_{4} \cos (54 ° - ϕ) + (d - d_{1} - d_{2} - d_{3} - d_{4}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - d_{3} \sin (18 ° + ϕ) - d_{4} \sin (54 ° - ϕ) - (d - d_{1} - d_{2} - d_{3} - d_{4}) \sin (18 ° + ϕ) \end{matrix})$
	$d_{1} + d_{2} + d_{3} + d_{4} < d (4)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - d_{3} \cos (18 ° + ϕ) - d_{4} \cos (54 ° - ϕ) + (d - d_{1} - d_{2} - d_{3} - d_{4}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - d_{3} \sin (18 ° + ϕ) - d_{4} \sin (54 ° - ϕ) - (d - d_{1} - d_{2} - d_{3} - d_{4}) \sin (18 ° + ϕ) \end{matrix})$
$where d_{1} = \frac{R \cos 36 °}{\cos ϕ}, d_{2} = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}, d_{3} = \sqrt{{(x_{3} - x_{2})}^{2} + {(y_{3} - y_{2})}^{2}}, d_{4} = \sqrt{{(x_{4} - x_{3})}^{2} + {(y_{4} - y_{3})}^{2}}$
$(x_{1}, y_{1}) = (R \cos 36 ° \tan ϕ, - R \cos 36 °), (x_{2}, y_{2}) = (\frac{R (1 + 2 \cos 36 °)}{\cot ϕ + \tan 36 °}, R - \tan 36 ° ∙ x_{2}),$
$(x_{3}, y_{3}) = (\frac{y_{2} - \tan (18 ° + ϕ) x_{2} - R}{\tan 36 ° - \tan (18 ° + ϕ)}, R + \tan 36 ° x_{3}), (x_{4}, y_{4}) = (\begin{matrix} \frac{\tan (54 ° - ϕ) x_{3} - y_{3} - R (\cos 36 ° + \sin 36 ° \tan 72 °)}{\tan 72 ° + \tan (54 ° - ϕ)}, \\ - \tan 72 ° x_{4} - R (\cos 36 ° + \sin 36 ° \tan 72 °) \end{matrix})$
2:	$0 < d ⩽ d_{1} (0)$	$\overset{⃑}{d} = (d \sin ϕ, - d \cos ϕ)$
$5.55 ° < Φ <$	$d_{1} < d ⩽ d_{1} + d_{2} (1)$	$\overset{⃑}{d} = (d \sin ϕ, (d - 2 d_{1}) \cos ϕ)$
$22.386 °, S (1)$	$d_{1} + d_{2} < d ⩽ d_{1} + d_{2} + d_{3} (2)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - (d - d_{1} - d_{2}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - (d - d_{1} - d_{2}) \sin (18 ° + ϕ) \end{matrix})$
$\to S (3) \to S (5)$	$d_{1} + d_{2} + d_{3} < d (3)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - d_{3} \cos (18 ° + ϕ) + (d - d_{1} - d_{2} - d_{3}) \cos (ϕ - 18 °), \\ (d_{2} - d_{1}) \cos ϕ - d_{3} \sin (18 ° + ϕ) + (d - d_{1} - d_{2} - d_{3}) \sin (ϕ - 18 °) \end{matrix})$
$where d_{1} = \frac{R \cos 36 °}{\cos ϕ}, d_{2} = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}, d_{3} = \sqrt{{(x_{3} - x_{2})}^{2} + {(y_{3} - y_{2})}^{2}}$
$(x_{1}, y_{1}) = (R \cos 36 ° \tan ϕ, - R \cos 36 °), (x_{2}, y_{2}) = (\frac{R (1 + 2 \cos 36 °)}{\cot ϕ + \tan 36 °}, R - \tan 36 ° ∙ x_{2})$
$(x_{3}, y_{3}) = (\begin{matrix} \frac{\tan (18 ° + ϕ) x_{2} - y_{2} - R (\cos 36 ° + \sin 36 ° \tan 72 °)}{\tan 72 ° + \tan (18 ° + ϕ)}, \\ - \tan 72 ° x_{3} - R (\cos 36 ° + \sin 36 ° \tan 72 °) \end{matrix})$
3:	$0 < d ⩽ d_{1} (0)$	$\overset{⃑}{d} = (d \sin ϕ, - d \cos ϕ)$
$22.386 ° < Φ <$	$d_{1} < d ⩽ d_{1} + d_{2} (1)$	$\overset{⃑}{d} = (d \sin ϕ, (d - 2 d_{1}) \cos ϕ)$
$26.267 °, S (1)$	$d_{1} + d_{2} < d ⩽ d_{1} + d_{2} + d_{3} (2)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - (d - d_{1} - d_{2}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - (d - d_{1} - d_{2}) \sin (18 ° + ϕ) \end{matrix})$
$\to S (3) \to S (1)$	$d_{1} + d_{2} + d_{3} < d ⩽ (3)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ - d_{3} \cos (18 ° + ϕ) - (d - d_{1} - d_{2} - d_{3}) \cos (18 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ - d_{3} \sin (18 ° + ϕ) + (d - d_{1} - d_{2} - d_{3}) \sin (18 ° + ϕ) \end{matrix})$
$where d_{1} = \frac{R \cos 36 °}{\cos ϕ}, d_{2} = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}, d_{3} = \sqrt{{(x_{3} - x_{2})}^{2} + {(y_{3} - y_{2})}^{2}} .$
$(x_{1}, y_{1}) = (R \cos 36 ° \tan ϕ, - R \cos 36 °), (x_{2}, y_{2}) = (\frac{R (1 + 2 \cos 36 °)}{\cot ϕ + \tan 36 °}, R - \tan 36 ° ∙ x_{2}),$
$(x_{3}, y_{3}) = (\frac{y_{2} - \tan (18 ° + ϕ) x_{2} - R}{\tan 36 ° - \tan (18 ° + ϕ)}, y_{3} = R \cos 36 °)$
4:	$0 < d ⩽ d_{1} (0)$	$\overset{⃑}{d} = (d \sin ϕ, - d \cos ϕ)$
$26.267 ° < Φ$	$d_{1} < d ⩽ d_{1} + d_{2} (1)$	$\overset{⃑}{d} = (d \sin ϕ, (d - 2 d_{1}) \cos ϕ)$
$36 °, S (1)$	$d_{1} + d_{2} < d ⩽ d_{1} + d_{2} + d_{3} (2$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ + (d - d_{1} - d_{2}) \cos (54 ° + ϕ), \\ (d_{2} - d_{1}) \cos ϕ + (d - d_{1} - d_{2}) \sin (54 ° + ϕ) \end{matrix})$
$\to S (2) \to S (3)$	$d_{1} + d_{2} + d_{3} < d ⩽$ $d_{1} + d_{2} + d_{3} + d_{4} (3)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ + d_{3} \cos (54 ° + ϕ) - (d - d_{1} - d_{2} - d_{3}) \cos (54 ° - ϕ) \\ (d_{2} - d_{1}) \cos ϕ + d_{3} \sin (54 ° + ϕ) - (d - d_{1} - d_{2} - d_{3}) \sin (54 ° - ϕ) \end{matrix}$
$\to S (5)$	$d_{1} + d_{2} + d_{3} + d_{4} < d (4)$	$\overset{⃑}{d} = (\begin{matrix} (d_{1} + d_{2}) \sin ϕ + d_{3} \cos (54 ° + ϕ) - d_{4} \cos (54 ° - ϕ) + (d - d_{1} - d_{2} - d_{3} - d_{4}) \cos (ϕ - 18 °), \\ (d_{2} - d_{1}) \cos ϕ + d_{3} \sin (54 ° + ϕ) - d_{4} \sin (54 ° - ϕ) + (d - d_{1} - d_{2} - d_{3} - d_{4}) \sin (ϕ - 18 °) \end{matrix})$
$where d_{1} = \frac{R \cos 36 °}{\cos ϕ}, d_{2} = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}, d_{3} = \sqrt{{(x_{3} - x_{2})}^{2} + {(y_{3} - y_{2})}^{2}}, d_{4} = \sqrt{{(x_{4} - x_{3})}^{2} + {(y_{4} - y_{3})}^{2}}$
$(x_{1}, y_{1}) = (R \cos 36 ° \tan ϕ, - R \cos 36 °), (x_{2}, y_{2}) = (\frac{R (1 + 2 \cos 36 °)}{\cot ϕ + \tan 36 °}, \cot ϕ ∙ x_{2} - 2 R \cos 36 °),$
$(x_{3}, y_{3}) = (\frac{R + \tan (54 ° + ϕ) x_{2} - y_{2}}{\tan 36 ° + \tan (54 ° + ϕ)}, - \tan 36 ° x_{3} + R) . (x_{4}, y_{4}) = (\begin{matrix} \frac{\tan (54 ° - ϕ) x_{3} - y_{3} - R (\cos 36 ° + \sin 36 ° \tan 72 °)}{\tan 72 ° + \tan (54 ° - ϕ)}, \\ - \tan 72 ° x_{4} - R (\cos 36 ° + \sin 36 ° \tan 72 °) \end{matrix})$

Condition	Range/(No. of Reflections)	Vector Form
1: $0 ° < Φ < 11.31 °$ , Square A $\to B \to C \to D \to G$	$0 < d ⩽ d_{1}$ (0)	$\overset{⃑}{d} = \vec{P A}$
	$d_{1} < d ⩽ d_{2}$ (1)	$\overset{⃑}{d} = \vec{P B} ∙ (1, - 1)$
	$d_{2} < d ⩽ d_{3}$ (2)	$\overset{⃑}{d} = \vec{P C}$
	$d_{3} < d ⩽ d_{4}$ (3)	$\overset{⃑}{d} = \vec{P D} ∙ (1, - 1)$
	$d > d_{4}$ (4)	$\overset{⃑}{d} = \vec{P G} ∙ (- 1, - 1)$
$where d_{1} = \frac{w}{\cos ϕ}, d_{2} = 3 w \sqrt{(1 + \tan^{2} ϕ)}, d_{3} = 5 w \sqrt{(1 + \tan^{2} ϕ)}, d_{4} = w \sqrt{(1 + \cot^{2} ϕ)}$
2: $11.31 ° < Φ < 18.43 °$ , Square A $\to B \to C \to F \to G$	$0 < d ⩽ d_{1}$ (0)	$\overset{⃑}{d} = \vec{P A}$
	$d_{1} < d ⩽ d_{2}$ (1)	$\overset{⃑}{d} = \vec{P B} ∙ (1, - 1)$
	$d_{2} < d ⩽ d_{3}$ (2)	$\overset{⃑}{d} = \vec{P C}$
	$d_{3} < d ⩽ d_{4}$ (3)	$\overset{⃑}{d} = \vec{P F} ∙ (- 1, 1)$
	$d > d_{4}$ (4)	$\overset{⃑}{d} = \vec{P G} ∙ (- 1, - 1)$
$where d_{1} = \frac{w}{\cos ϕ}, d_{2} = 3 w \sqrt{(1 + \tan^{2} ϕ)}, d_{3} = w \sqrt{(1 + \cot^{2} ϕ)}, d_{4} = 5 w \sqrt{(1 + \tan^{2} ϕ)}$
3: $18.43 ° < Φ < 31.63 °$ , Square A $\to B \to E \to F \to G$	$0 < d ⩽ d_{1}$ (0)	$\overset{⃑}{d} = \vec{P A}$
	$d_{1} < d ⩽ d_{2}$ (1)	$\overset{⃑}{d} = \vec{P B}$
	$d_{2} < d ⩽ d_{3}$ (2)	$\overset{⃑}{d} = \vec{P E} ∙ (- 1, - 1)$
	$d_{3} < d ⩽ d_{4}$ (3)	$\overset{⃑}{d} = \vec{P F} ∙ (- 1, 1)$
	$d > d_{4} (4)$	$\overset{⃑}{d} = \vec{P G} ∙ (- 1, - 1)$
$where d_{1} = \frac{w}{\cos ϕ}, d_{2} = w \sqrt{(1 + \cot^{2} ϕ)}, d_{3} = 3 w \sqrt{(1 + \tan^{2} ϕ)}, d_{4} = 5 w \sqrt{(1 + \tan^{2} ϕ)}$
4: $31.63 ° < Φ < 45 °$ , Square A $\to B \to E \to F \to H$	$0 < d ⩽ d_{1}$ (0)	$\overset{⃑}{d} = \vec{P A}$
	$d_{1} < d ⩽ d_{2}$ (1)	$\overset{⃑}{d} = \vec{P B} ∙ (1, - 1)$
	$d_{2} < d ⩽ d_{3}$ (2)	$\overset{⃑}{d} = \vec{P E} ∙ (- 1, - 1)$
	$d_{3} < d ⩽ d_{4}$ (3)	$\overset{⃑}{d} = \vec{P F} ∙ (- 1, 1)$
	$d > d_{4}$ (4)	$\overset{⃑}{d} = \vec{P H}$
$where d_{1} = \frac{w}{\cos ϕ}, d_{2} = w \sqrt{(1 + \cot^{2} ϕ)}, d_{3} = 3 w \sqrt{(1 + \tan^{2} ϕ)}, d_{4} = 3 w \sqrt{(1 + \cot^{2} ϕ)}$

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Journal of the Optical Society of America A