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I am currently studying Laser Systems Engineering by Keith Kasunic. Chapter 1.2.1 Temporal Coherence says the following:

The coherence time $\tau_c$ over which the emitted wavelengths are considered to be in phase – that is, are temporally coherent – thus depends inversely on the absolute value of the wavelength difference $\vert \Delta \lambda \vert$ or frequency difference $\Delta \nu = c \vert \Delta \lambda \vert / \lambda^2$ [from differentiating Eq. (1.1)]: $$\tau_c \approx \dfrac{1}{2 \vert \Delta \nu \vert} = \dfrac{1}{2c} \dfrac{\lambda^2_o}{\vert \Delta \lambda \vert} \ \ \ \ \ \text{[sec]} \tag{1.3}$$ clearly showing that a narrow-spectrum laser with small $\Delta \nu$ has a longer time over which different frequencies propagate before they are no longer considered to be in phase. Typical numbers for a HeNe laser are used for interferometric optical testing are a coherence time $\tau_c = 0.33$ nsec and a coherence length $d_c = c \tau_c = 100$ mm for a linewidth $\Delta \lambda = 2$ pm (see Table 1.3). Note that the results shown in Fig. 1.11(b) – where the different wavelengths are out of phase at a time $t \approx 10^{-8}$ sec – are not consistent with the estimates from Eq. (1.3), as the equation assumes that $\Delta \lambda$ is small in comparison with the center wavelength $\lambda_o$.

Eq. (1.1) is given as follows:

A laser is a source of both light and heat. Light is an electromagnetic wave with a wavelength $\lambda$ and frequency $\nu$ with energy propagating at the speed of light $c$ in a vacuum: $$\nu = \dfrac{c}{\lambda} \ \ \ \ \ \text{[Hz]} \tag{1.1}$$

How does differentiating $\nu = \dfrac{c}{\lambda}$ get us (1.3)? I don't even understand how this is a function in the first place, since $c$ is just the speed of light in vacuum and $\lambda$ is the wavelength.

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I don't know much about the topic but this can be done as follows :

$$\nu=\frac{c}{\lambda}$$

$$\frac{d\nu}{d\lambda}=-\frac{c}{\lambda^2}$$ $$\frac{\Delta \nu}{\Delta \lambda}=-\frac{c}{\lambda^2}$$ $$|\Delta \nu|=\frac{c}{\lambda^2}|\Delta\lambda|$$

That's the exact thing that is used.

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