## 2 Experimental Setup
Based on [22], \(D_{\rm T}\) = \(-\) 0.0016 ps/nm\(\cdot\)km/\({}^{\circ}\)C, therefore pulses of longer wavelengths travel with higher speed \(\
u\) than those of shorter wavelengths (in our case \(\
u_{\lambda 4}<\
u_{\lambda 2}<\
u_{\lambda 1}<\
u_{\lambda 3}\)). This is contrary to CD where the chromatic dispersion coefficient \(D_{\rm CD}>0\). The impact of \(D_{\rm T}\) is twofold:  
1. It causes the rise of the so-called time-skewing effect illustrated in Figure 2c. We can see that because the code carriers having a longer wavelength travel faster than those with a shorter wavelength, instead of being placed on top of each other at the fibre receiving end after the OCDMA decoder (Figure 1), as is shown in Figure 2b, the code carriers are instead time shifted (Figure 2c). The resulting OCDMA auto-correlation envelope has thus become skewed. 2. Because of finite code carriers' linewidth and \(D_{\rm T}\) being negative, they will experience a temporal narrowing (squeezing) [19] (see Figure 2c). This result is opposite to the CD effect where the chromatic dispersion coefficient \(D_{\rm CD}>0\), and it would therefore cause temporal broadening of the code carriers. Our next step is to formally describe the above. Analyses of Chirp and Temperature Induced Fibre Dispersion on Optical Code Division Multiple Access Auto-Correlation  
We will now develop an equation describing the OCDMA auto-correlation envelope that was recovered by the OCDMA decoder on the receiver site being affected by the initial code carriers' chirp and fibre temperature changes. A temporal Gaussian shape of an optical pulse affected by chirp \(C\) can be expressed as [23]:  
\[P_{\rm L}(t)\ =\ P_{0}\exp\left[-2.77(1+iC)\left(\frac{t}{\tau}\right)^{2} \right], \tag{1}\]  
where \(t\) is time, \(P_{0}\) is a peak power and the coefficient 2.77 indicates that its width \(\tau\) is measured at full width at half maximum (FWHM). Figure 2: Illustration of the effect of temperature induced dispersion in fibre transmission link on incoherent optical code division multiple access (OCDMA). **(a)** Two-dimensional wavelength-hopping time-spreading (2D-WH/TS) transmitted OCDMA code; **(b)** Recovered OCDMA auto-correlation—ideal case; and, **(c)** Recovered OCDMA auto-correlation time-skewed and compressed due to the temperature induced dispersion effects (\(D_{\rm T}<0\)). In our investigation we will use a 2D-WH/TS OCDMA system based on four multi-wavelength picosecond pulses as code carriers, each pulse carrier \(j\) has a line-width of \(\Delta\lambda j\). Carriers are spectrally separated by \(\Delta\lambda j\) where \(j=1\), 2, 3, 4, respectively. Using Equation (1) and following [24] for the OCDMA auto-correlation temporal envelope that was recovered by the OCDMA receiver \(R_{\times}\), we can write the following expression:  
\[S_{\text{L}(\lambda 1-\lambda 4)}(t)\ =\ \sum_{k\ =\ 0}^{w-1}P\exp\left[-2.77(1+iC) \left(\frac{t-k\Delta\tau_{0}}{\tau-\Delta\tau_{0}}\right)^{2}\right], \tag{2}\]  
In the above expression, \(w\) = 4 is the number of code carriers (wavelengths) and it is also known as the code weight. The numerator (\(t-k\Delta\tau_{0}\)) represents time skewing among wavelength code carriers and the denominator (\(\tau-\Delta\tau_{0}\)) reflects the carrier width change due to temperature induced dispersion effects in the fibre link. By following [20], we now develop a mathematical model for describing the effect of the code carriers' chirp and temperature induced dispersion changes in the fibre transmission link on the OCDMA auto-correlation envelope. Because we have used the same OCDMA hardware, we used in [20], the chirp value \(C\) = 2.36 at the OCDMA encoder output (point A, Figure 1) will be considered to be the same for the purpose of this investigation. As has already been mentioned, the 19.5 km fibre transmission link was CD compensated at a room temperature of \(T\) = 20 \({}^{\circ}\)C. In order to CD compensate the 19.5 km SMF-28 fibre, we applied a commercially available DCF module.