Detection of laser beams based on intensity interferometry

Abstract

We describe an approach to detecting off-axis radiation of laser beams propagating in scattering media, especially in the atmosphere, in the presence of background (solar) radiation. The method relies on a generalization of the conventional intensity interferometry (II) theory to scenarios involving coexisting sources of relatively long (laser radiation) and much shorter (background) coherence times. In such circumstances, the high coherence of the laser light allows its discrimination against even much stronger, but low-coherence, background. We propose a simple detection system consisting of a small array of photodetectors (e.g. photodiodes) and estimate the ratio of the background-to-laser irradiances at which the laser radiation is expected to be detectable.

Keywords:

• Laser radiation
• scattering
• coherence
• solar background
• intensity interferometry

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Outweighed, at the time, by its insensitivity to detrimental effects of atmospheric turbulence, as discussed following Equation (3(3) $C({\mathit{\varrho }}_0):=\frac{⟨\Delta J({\mathit{\varrho }}_0/2)\Delta J(-{\mathit{\varrho }}_0/2)⟩}{\sqrt{\phantom{g}\phantom{(1)}⟨(\Delta J({\mathit{\varrho }}_0/2){)}^2⟩⟨(\Delta J(-{\mathit{\varrho }}_0/2){)}^2⟩}}\equiv \frac{⟨\Delta J_{+}\Delta J_{-}⟩}{\sqrt{\phantom{g}\phantom{(1)}⟨(\Delta J_{+}{)}^2⟩⟨(\Delta J_{-}{)}^2⟩}},$(3) ).

2 In the definition (2(2) $\delta =\frac{1}{2}\Delta \tau \eta \mathcal{E}A,$(2) ) it is assumed that the aperture area is much smaller than the radiation's coherence area (as defined below, Equations (55a(55a) $\Delta A{\phantom{j}}_{\mathrm{c}}=\int {\mathrm{d}}^2\varrho F{\phantom{j}}_{\mathrm{c}}(\mathit{\varrho })=\sqrt{\frac{\pi }{2}}\Delta \varrho {\phantom{j}}_{\mathrm{c}\parallel }\Delta \varrho {\phantom{j}}_{\mathrm{c}\perp }=\frac{{\lambda }^{2}R}{\sqrt{\pi }\varphi w{\phantom{j}}_{\mathrm{B}}}\approx 1.27{\mathrm{mm}}^2$(55a) )). In general, A has to be replaced with the aperture autocorrelation area, discussed in Section 3.

3 We will, alternatively, express irradiance in units of watts per meter squared, by using the conversion factor $1\mathrm{W}/{\mathrm{m}}^2=(\lambda /hc)\mathrm{photons}/\mathrm{s}\approx 7.5\cdot {10}^{18}\mathrm{photons}/(\mathrm{s}{\mathrm{m}}^2)$, the last numerical value corresponding to the SWIR wavelength $\lambda =1.5\text{μ}\mathrm{m}$.

4 An exhaustive overview of the literature is given in the recent dissertation [20].

5 By a photosensor we mean a ‘single-pixel’ photodetector, such as a PIN photodiode, an avalanche photodiode, etc.

6 Equation (5(5) $I_{\pm }(t):={\int }_{{\mathcal{P}}_{\pm }}{\mathrm{d}}^2{\varrho }^{\prime }|u_{\mathcal{P}}(t,{\mathit{\varrho }}^{\prime }){|}^2\approx {\int }_{\mathcal{A}{\phantom{j}}_{\pm }}{\mathrm{d}}^2\varrho |u(t,\mathit{\varrho }){|}^2,$(5) ) implies that the ensemble average of the intensity is given by the surface integral of the irradiance.

7 In fact, in SII one can also go beyond the approximations valid for very small apertures and introduce a ‘partial coherence factor’ Δ associated with larger apertures of the telescopes ([27], Appendix A and [2, p. 60]). Our Equations (16b(16b) $\begin{array}{rl}⟨\Delta I_{\pm }\left(\frac{1}{2}\tau \right)\Delta I_{\pm }(-\frac{1}{2}\tau )⟩& \approx \frac{1}{2}|\gamma (\tau ){|}^2{\mathcal{E}}^2{\int }_{\mathcal{A}{\phantom{j}}_{\pm }}{\mathrm{d}}^2{\varrho }_1{\int }_{{\mathcal{A}}_{\pm }}{\mathrm{d}}^2{\varrho }_{2}F({\mathit{\varrho }}_1-{\mathit{\varrho }}_2)\\ & \equiv \frac{1}{2}|\gamma (\tau ){|}^2{\mathcal{E}}^2{A}_{\pm }{\overline{A}}_{\pm }\end{array}$(16b) ) and (16c(16c) $\begin{array}{rl}⟨\Delta I_{+}\left(\frac{1}{2}\tau \right)\Delta I_{-}(-\frac{1}{2}\tau )⟩& \approx \frac{1}{2}|\gamma (\tau ){|}^2{\mathcal{E}}^2{\int }_{\mathcal{A}{\phantom{j}}_{+}}{\mathrm{d}}^2{\varrho }_1{\int }_{{\mathcal{A}}_{-}}{\mathrm{d}}^2{\varrho }_{2}F({\mathit{\varrho }}_1-{\mathit{\varrho }}_2)\\ & \equiv \frac{1}{2}|\gamma (\tau ){|}^2{\mathcal{E}}^2\sqrt{\phantom{g}\phantom{(1)}{A}_{+}{A}_{-}}\overline{A}{\phantom{j}}_{+-},\end{array}$(16c) ) provide a generalization of these results.

8 In the following, we will use either the frequency or the wavelength, always assuming $\omega =2\pi c/\lambda$.

9 We add here the superscript ϕ to emphasize the irradiance's dependence on the detector FOV.

10 For reasons mentioned below, we neglect here L compared to R.

11 We use here the symbol $\mathrm{Jinc}$ to differentiate this function from the conventional $\mathrm{jinc}(x)=2{\mathrm{J}}_1(\pi x)/(\pi x)$.

12 Any effects of possible optical filters have also to be taken into account in the integral (49(49) $\int \mathrm{d}\omega \eta (\omega )\Gamma {\phantom{j}}_{\mathrm{b}\omega }(0;\mathbf{0},\mathbf{0})=\int \frac{\mathrm{d}\omega }{2\pi }\eta (\omega )\frac{\mathrm{d}\mathcal{E}{}_{\mathrm{b}}^{\varphi }(\omega )}{\mathrm{d}\omega /2\pi }\equiv [\eta \mathcal{E}{}_{\mathrm{b}}^{\varphi }{]}_{\mathrm{tot}}=\Omega {\phantom{j}}_{\varphi }\int \mathrm{d}\omega \eta (\omega )\mathcal{I}{\phantom{j}}_{\mathrm{b}}(\omega ),$(49) ).

13 Derivation of the expression for the function $V(t)$ in Equation (52(52) $V(t)=\frac{2}{\pi }({\cos }^{-1}t-t\sqrt{1-t^2})\mathrm{for}t\le 1,V(t)=0\mathrm{otherwise},$(52) ) can be found, e.g. in [32], Figure 6, where a similar aperture geometry problem is analyzed.

14 Although (by Equation (23(23) $\overline{\delta }{\phantom{j}}_{\pm }:=\frac{1}{2}\Delta \tau \eta \mathcal{E}{\overline{A}}_{\pm }\mathrm{and}\overline{\delta }{\phantom{j}}_{+-}:=\frac{1}{2}\Delta \tau \eta \mathcal{E}\overline{A}{\phantom{j}}_{+-},\mathrm{satisfying}\overline{\delta }{\phantom{j}}_{+-}\le (\overline{\delta }{\phantom{j}}_{+}\overline{\delta }{\phantom{j}}_{-}{)}^{1/2}.$(23) )) $\overline{\delta }{}_{\pm }^{\mathrm{c}}$ in the denominator are, generally, larger than $\overline{\delta }{}_{+-}^{\mathrm{c}}$ appearing in the numerator, estimates of Section 5.1 suggest that the difference is not significant.

15 We recall that our elementary detector is a lens focusing light on a single-pixel photosensor.

16 The laser beam expected to be relatively stable, as it has to track the target. Although the target may be moving, the beam direction would be unlikely to change appreciably.

Additional information

Funding

This material is based upon work supported by the U.S. Air Force Office of Scientific Research under award number FA 9550-19-1-0144.