We next take into account the surface roughness of DF and estimate how such roughness leads to a randomness of arrival times. To proceed, we consider a simple case where the roughness is represented by a step function with a height discrepancy Δh on the diffuser surface [Fig. 2]. The time delay of a pulse p1 induced by Δh is simply Δh/C_{Glass} − Δh/C_{Air} = (n_{Glass} − n_{Air}) · (Δh/C_{0}) ≈ 0.5 Δh / C_{0}. Further, the time delay caused by the roughness in a region A_{r} of radius r on the diffuser surface is projected into a region
\documentclass[12pt]{minimal}\begin{document}$A_r^{\prime }$\end{document}
$Ar\u2032$ of radius
r′ =
f_{O}/
f_{D} r on the image plane [Fig.
2]. The maximal time delay Δ
t_{RD} within
\documentclass[12pt]{minimal}\begin{document}$A_{r^{\prime }}$\end{document}
$Ar\u2032$ may thus be approximated as
Display Formula2\documentclass[12pt]{minimal}\begin{document}
\begin{equation}
\Delta t_{RD} = 0.5\: \frac{\Delta h}{C_0},
\end{equation}
\end{document}
$\Delta tRD=0.5\Delta hC0,$ where Δ
h is the maximal surface height discrepancy within
A_{r}. In general, the roughness of a ground-glass diffuser is generated by grinding a flat surface of glass with particles of size less than a certain length
D. Thus, we expect Δ
h → 0 when
r → 0, and Δ
h ≈
D if
r ≫
D, as shown in Fig.
2. To take into account these asymptotic estimations, we used a simple approximation here: Δ
h ≈ α · 2
r if α · 2
r <
D and Δ
h ≈
D if α · 2
r ⩾
D, where α is a dimensionless parameter describing the roughness of a diffuser. Using this approximation, we obtain a simple estimation of the difference of arrival times Δt
_{RD} within
\documentclass[12pt]{minimal}\begin{document}$A_{r^{\prime }}$\end{document}
$Ar\u2032$,
Display Formula3\documentclass[12pt]{minimal}\begin{document}
\begin{eqnarray}
\Delta t_{RD}= \left\lbrace
\begin{array}{l l}
\displaystyle\frac{\alpha \:f_D}{C_0\cdot f_O}\cdot r^{\prime }&\quad \mbox{if }\displaystyle\frac{\alpha \:f_D}{f_O}\cdot r^{\prime }<0.5\, D\\[12pt]
\displaystyle\frac{0.5\, D}{C_0}&\quad \mbox{if }\displaystyle\frac{\alpha\: f_D}{f_O}\cdot r^{\prime }\ge 0.5\, D\\
\end{array} \right.
=\frac{1}{C_0}\cdot \mbox{Min}\left[\frac{\alpha \:f_D}{f_O}\: r^{\prime },\:0.5\:D\right].
\end{eqnarray}
\end{document}
$\Delta tRD=\alpha fDC0\xb7fO\xb7r\u2032if\alpha fDfO\xb7r\u2032<0.5D0.5DC0if\alpha fDfO\xb7r\u2032\u22650.5D=1C0\xb7Min\alpha fDfOr\u2032,0.5D.$ For the out-of-focus point
q^{′} shown in Fig.
1 (inset),
\documentclass[12pt]{minimal}\begin{document}$A_{r^{\prime }}$\end{document}
$Ar\u2032$ corresponds to the area covered by the cone angle θ, and so we have
\documentclass[12pt]{minimal}\begin{document}$r^{\prime }\approx z\cdot \theta \approx \frac{\rm NA}{n}z$\end{document}
$r\u2032\u2248z\xb7\theta \u2248 NA nz$ and
Display Formula4\documentclass[12pt]{minimal}\begin{document}
\begin{equation}
\Delta t_{RD}(z) =\frac{1}{C_0}\cdot \mbox{Min} \left[\:\frac{\alpha \:f_D}{f_O}\cdot \frac{\rm NA}{n}\cdot z\:, \: 0.5\,D\:\right].
\end{equation}
\end{document}
$\Delta tRD(z)=1C0\xb7Min\alpha fDfO\xb7 NA n\xb7z,0.5D.$ Combining Eqs.
1,
4, we finally obtain the effective pulse duration at an out-of-focus point
q^{′} at distance
z from the IP, namely
Display Formula5\documentclass[12pt]{minimal}\begin{document}
\begin{eqnarray}
\tau _{\rm eff}(z) &=& \tau _0 +\Delta t_{RD}+\Delta t_{G}
\end{eqnarray}
\end{document}
$\tau eff (z)=\tau 0+\Delta tRD+\Delta tG$ Display Formula6\documentclass[12pt]{minimal}\begin{document}
\begin{eqnarray}
&=&\tau _0 + \frac{\mbox{Min} \left[\:\frac{\alpha \:f_D}{f_O}\frac{\rm NA}{n} z\:, \: 0.5\,D\:\right]}{C_0} + \frac{(f_D+f_O-d) {\rm NA}^2}{2\:C_0\:n\:f_O^2} z^2\nonumber\\
&&+\,n\:\frac{n-\sqrt{n^2-{\rm NA}^2}}{C_0\:\sqrt{n^2-{\rm NA}^2}}z,
\end{eqnarray}
\end{document}
$=\tau 0+Min\alpha fDfO NA nz,0.5DC0+(fD+fO\u2212d) NA 22C0nfO2z2+nn\u2212n2\u2212 NA 2C0n2\u2212 NA 2z,$ where τ
_{0} is the pulse width of the laser source.