Consider some issues related with practical evaluation of the speckle-metrics <*J*_{PIB}> and

For a moving target surface, these metrics can be estimated by time averaging the measured power-in-the-bucket signal

*J*_{PIB}(

*t*) (We assume that

*J*_{PIB}(

*t*) is a stationary and ergotic random process.)

where *T* is the averaging (sampling) time. Note that

corresponds to the estimation of the mean value (dc component) of the signal

*J*_{PIB}(

*t*) over the time interval [

*t, t*+

*T*]. In practice, the computation of

and

in Eqs. (5.13) and (5.14) can be performed using either digital or analog signal multiplication and integration.

Another practical approach for speckle-metric evaluation is based on spectral analysis of sufficiently long sections of the signal *J*_{PIB}(*ξ*) (*t*≤*ξ*<*t*+*T*). In this approach, registered sections of the signal *J*_{PIB}(*ξ*) are first used for the computation of random spectrum realizations. These spectrum realizations are then averaged to obtain the power spectrum estimation

In the spectral approach, the PIB metric <*J*_{PIB}> corresponds to the zero spectral component of the power spectrum:

The speckle-metric

is determined by integrating the power spectrum

over the entire frequency band [see Eq. (5.11)].

An important advantage of the speckle-metrics (5.10) and (5.12) over the speckle-size metric (3.6) discussed in Section 3.5 is that the values <*J*_{PIB}> and

can be computed using measurements of the one-dimensional signal

*J*_{PIB}(

*t*) obtained from a single photodetector, while speckle-size metric estimation requires processing of the two-dimensional speckle-intensity patterns

*I*_{sp}(

**r**) (speckle images) registered by a high-resolution photo-array.

The primary disadvantage of the PIB speckle- metrics <*J*_{PIB}> and

is that their estimation requires the processing of sufficiently long sections of the signal

*J*_{PIB}(

*ξ): t*≤

*ξ*<

*t*+

*T*. This processing time is an important issue for TIL systems operated in the presence of dynamically changing (e.g., turbulence-induced) phase distortions, as the total time

*τ*_{J}+

*T*+

*τ*_{proc} required for speckle-metric estimation, including the sampling time

*T* and the signal processing time

*τ*_{proc} must be small in comparison with the characteristic time of phase distortion change:

*τ*_{J}≪

*τ*_{at}. For simplicity, we assume that the signal processing time

*τ*_{proc} is small in comparison with the sampling time and can be ignored, so that

*τ*_{J}≃T.

Consider the speckle-metric estimation error *ε*_{T} resulting from the PIB signal averaging over the finite sampling time* T*. This error depends on the shape of the PIB signal power spectrum *G*_{PIB}(*ω*) and the sampling time *T* [51]. For the error to be small requires that *T*≫1/*ω*_{PIB}, where *ω*_{PIB} is the power spectrum bandwidth (cutoff) frequency. In this case, the error variance

asymptotically approaches

*G*_{PIB}(

*ω*=0)/

*T*, indicating that a significant error contribution originates from the low-frequency spectral components [51].

Examples of typical power spectra

corresponding to different beam sizes on the moving target surface are shown in

Figure 8. The power spectrum for the smaller beam (curve 1) is wider than the spectrum for the larger beam (curve 2). This dependence of the power spectrum on the target hit-spot size supports the physical basis for the described speckle metrics.

Figure 8 Normalized power-in-the-bucket fluctuation power spectra

(

*f*=

*ω*/2

*π*) experimentally obtained with the optical TIL system in

Figure 5. The power spectra correspond to the target hit-spot size

*b*_{s}≃0.2 mm (1) and

*b*_{s}≃0.6 mm (2). The speckle field is produced by scattering the outgoing beam off the rough surface of a rotating metal disc. The linear speed in the vicinity of the target hit spot is

*v*_{s}≃8 m/s. The power spectrum

is computed by averaging a set of three random spectra calculated based on sequential sampling of the PIB signal

*J*_{PIB}(

*nT*+

*m*Δ

*t*) over time

*T*=13.5 ms, where

*n*=0, 1, 2,

*m*=0, …, 2047, and Δ

*t*≃6.6 μs. Bars with central frequencies

*f*_{1} and

*f*_{2} and spectral widths Δ

_{1} and Δ

_{2} illustrate the band-pass filtering technique used for computation of the spectral metric (5.16). The values

*P*(

*f*_{1},Δ

_{1}) and

*P*(

*f*_{2},Δ

_{2}) correspond to the power spectrum integrated over the band-pass filter and are given by heights of the bars.

The use of the relatively short sampling time *T* causes large fluctuations in the low-frequency spectral components as clearly seen in Figure 8 where accuracy in determining spectral components is low. It follows that to decrease the error *ε*_{T} (without increasing the sampling time *T*) requires a decrease in the low-frequency contributions.

This goal can be achieved by increasing the power spectrum width (cutoff frequency *ω*_{PIB}) by fast steering of the outgoing laser beam. To estimate the requirements for steering speed, consider a beam with a Gaussian target-plane intensity distribution of size *b*_{s}. In accordance with Eq. (5.9), the power spectrum cutoff frequency is *ω*_{PIB}=v_{s}/*b*_{s}.

Assume that for accurate speckle-metric estimation the required condition 1/*ω*_{PIB}≪*T*≪*τ*_{at} is satisfied if 1/*ω*_{PIB}=b_{s}/*v*_{s}=10^{-2}*T*=10^{-4}*τ*_{at}. From this equality, we obtain * v*_{s}≃(*b*_{s}/*τ*_{at})‧10^{4}.

Consider as an example beam steering along a circuit of radius *a*_{st} that defines beam steering amplitude. In this case, for the steering frequency *f*_{s} (cycles/s), we obtain *f*_{s}=1/*T*_{s}=*v*_{s}(2*πa*_{st})^{-1}=10^{4}*b*_{s}/(2*πa*_{st}τ_{at}). For estimation, let *τ*_{at}=5‧10^{-3} s, and *a*_{st}=*b*_{s} (minimum distance that provides surface roughness update along the steering beam trajectory). It follows that *f*_{s}≃10 MHz. This beam steering frequency can be achieved using coherent fiber-array beam projection systems with stair-mode wavefront dithering technique as described in Section 3.4 (see Figure 4) [33, 52].

The speckle-metric estimation error *ε*_{T} can be reduced using spectral filtering of the signal *J*_{PIB}(*t*) prior to its processing. This implies that changes in the target hit-spot size can be estimated by integrating the power spectrum

only within one or several spectral regions (spectral bands) where the accuracy in determining the signal spectral components is high, as illustrated in

Figure 8.

Consider the *N* selected power spectrum bands of widths Δ_{j} and central frequencies *ω*_{j}, (*j*=1, …, *N*) and assume that the spectral components inside these spectral bands are integrated. This corresponds to the use of a bank of band-pass spectral filters. Output signal from the *j*th band-pass filter can be represented in the form

This band-pass filtering of the PIB fluctuation signal *J*_{PIB}(*t*) gives rise to a spectral metric of the type [14, 46]

where *β*_{j} is the weighting coefficient. In contrast with the speckle-metric

defined by Eq. (5.12), the power spectrum frequency components below

*ω*_{1}-Δ

_{1}/2 and higher than

*ω*_{N}+Δ

_{N}/2 do not contribute to the spectral metric

*J*_{S}. Control of the parameters upon which spectral metric (5.16) depends (coefficients

*β*_{j}, band-pass filter widths Δ

_{j}, and central frequencies

*ω*_{j}) allows optimization of the dependence of metric

*J*_{S} on the target hit-spot intensity distribution.

In contrast with speckle metric

whose value is directly associated with the target-plane metric

*J*_{2}, a similar type of analytical expression linking metric

*J*_{S} with a physically meaningful target-plane metric is not available. Nevertheless, both experiments and the following discussions support the arguments provided for the derivation of the metric (5.16) and demonstrate that with a correct selection of parameters in (5.16), the metric

*J*_{S} can be used as a speckle metric, with its global maximum corresponding to the undistorted target hit-spot beam intensity distribution [14].

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