Also, assume that the lens has a thickness of $t$. Then the phase change when light is transmitted through the lens at a distance $r$ from the axis of symmetry is given by $$\phi(r) = k_0 n(r) t = k_0 t(n_0 - (n_0-n_r)\frac{r^2}{r'^2}).$$

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If we compare this with the phase chage of a general lens with focal length $f$, which is given by $\phi(r) = -k_0 r^2/(2f)$ (with some arbitrary phase constant), comparing both expressions yields us:

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Germanium windows are ideal for IR applications at a broad range of wavelengths, including use in lower power CO2 lasers. Germanium also has low dispersion, ideal for imaging applications. The windows feature an AR coating providing high transmission from 3 to 12 µm.

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The lens has a radius of $r'$. Assume that the index of refraction at $r = 0$ is $n_0$ and at $r = r'$ is $n_r$. We know that the refractive index varies parabolically, meaning that $$n(r) = n_0 - (n_0-n_r)\frac{r^2}{r'^2}.$$

For which we can rearrange for all constants and set them to $0$ since they're arbitrary to finally get the focal length:

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I want to find a closed expression of the focal length of a graded index since I don't manage to find any on the internet. I already checked this out: Determining the focal length of a gradient index lens

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Germanium Optical Windows have good transmittance from 2 to 16 µm uncoated, and very low peak absorption, making the material an excellent candidate for use in C02 laser cavities. Germanium also has high opacity across the entire visible spectrum, making the windows ideal for applications where transmission of only IR wavelengths is desired.

Germanium windows are ideal for IR applications at a broad range of wavelengths, including use in lower power CO2 lasers. Germanium also has low dispersion, ideal for imaging applications. The windows feature an AR coating providing high transmission from 3 to 12 µm.

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Blender allows you to work this way, by specifying the focal length in the Lens panel, and the sensor size in the Camera panel. It even offers a Camera ...

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Am I in the wrong for comparing it with the phase given by a general lens of focal length of $f$, and is my assumption about setting the constants to $0$ right as well? In that case, why / why not?

Germanium Optical Windows have good transmittance from 2 to 16 µm uncoated, and very low peak absorption, making the material an excellent candidate for use in C02 laser cavities. Germanium also has high opacity across the entire visible spectrum, making the windows ideal for applications where transmission of only IR wavelengths is desired.

Use Fermat's prinicple. Let the thickness of the lens be denoted by $d$, then the optical path length from axial point $\mathcal P$ to another axial point $\mathcal Q$ on the other side of the lens is $$OPL[{\mathcal P \mathcal Q}]=\sqrt{p^2+r^2}+n(r)d+\sqrt{q^2+r^2}\\ \approx p+\frac{r^2}{2p} + n(r)d + q+\frac{r^2}{2q}$$ For imaging this must be independent of $r$, that is, $$ \frac{r^2}{2}\big(\frac{1}{q}+\frac{1}{p}\big) + n(r)d =\frac{r^2}{2f}+n(r)d=n_0$$ where $\frac{1}{q}-\frac{1}{p}= \frac{1}{f}$ with $$n(r)=n_0-\frac{r^2}{2f}$$

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Imaging Systems. Optical Coherence Tomography (OCT) is a non-invasive, non-contact imaging modality used to visualize and monitor changes to the morphology of ...

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