## 3 Synthesis of Optical Coherence Function
### Synthesis of Arbitrary-Shaped Optical Coherence Function with SOCF  
An arbitrary-shaped optical coherence function can be synthesized by additionally using intensity modulation and phase modulation to the lightwave, besides using the frequency modulation. Figure 6 shows an experimental setup that has all three modulations [22]. In this setup, a specific LD that has a superior frequency modulation function is used, which is a superstructured grating distributed Bragg reflector laser diode (SSG-DBR-LD) [23]. A multiple-line spectrum is generated, in a time-averaging manner, through modulating the LD frequency by a multiple-step function, as shown in Figure 6i. Then, a periodical delta function-like coherence function is synthesized, as shown in Figure 7a. Since the envelope shape of the synthesized multiple-line spectrum is rectangular, the delta function-like coherence function has relatively large side lobes, as shown in Figure 7a. Figure 6: SOCF setup to synthesize arbitrary-shaped optical coherence functions [22]. Waveform for FM, PM, and IM are shown in Figure 6i,ii,iii, respectively. Figure 7: Parallel shift of the synthesized optical coherence function by using phase modulation, which has a similar waveform to the FM waveform [22]. Figure 7a shows a synthesized delta-function-like optical coherence function without the shift along the position axes, and Figure 7b shows that shifted by the phase modulation, respectively. When using additionally the phase modulation, which must be the same waveform as the FM waveform (see Figure 60(i)), we can shift the whole coherence function along the lightwave propagation axis [24], for example, as shown in Figure 7(b) [22]. The phase modulation (PM) waveform is basically the same shape as the FM. In practice, the PM waveform is folded within \(+\pi\) and \(-\pi\), as shown in Figure 60(ii). When we change the PM amplitude, the synthesized optical coherence function can be shifted. Theoretically, the function to shift the synthesized optical coherence function corresponds to a formula in the Fourier transform theory. In Figure 7(b), it is shown that the synthesized optical coherence function can realize the reduced or zero degree of coherence, even in the case of the zero optical path-length difference [22]. This situation cannot take place in the natural interferences. By additionally using the intensity modulation, which is done by a semiconductor optical amplifier (SOA), as shown in Figure 6, the power of each line spectrum can arbitrarily be set by using a waveform, as shown in Figure 60(iii) [22]. With the intensity modulation, the envelope of the multi-line spectrum can be manipulated. This technique is also familiar in the Fourier transform theory, which is called the \"window\" or the \"apodization\". By the intensity modulation (IM), the side lobes can be reduced, which is shown in Figure 7(a) as an experimental result [22]. By totally using FM, IM, and PM, an almost arbitrary-shaped optical coherence function can be synthesized [22]. Examples are shown in Figure 7(b),c. A periodical triangle-shaped coherence function is synthesized in Figure 7(b). A trapezoid-shaped one is shown in Figure 7(c). The multi-line-shaped spectrum, which was shown in Figures 4 and 6, is useful to synthesize an arbitrary-shaped coherence function. However, it is not easy to make a fine multi-stairs-shaped frequency modulation, because of the non-ideal response of the LD and the driving circuit. For synthesizing the periodical delta function-like optical coherence function, a sinusoidal FM is simple but effective [14; 25]. In Figure 9, the time-averaged light source power spectrum and the synthesized coherence function are shown [14]. Since both edges of the power spectrum have a high energy, the side lobes are relatively large. However, this problem can be solved by an analogue intensity modulation synchronized to the FM waveform.