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The pulse energy shape factor is the dimensionless ratiowhere $U$ is laser pulse energy, $P_{\mathrm{pk}}$ is background-subtracted peak power and $\tau$ is a stated pulse duration. It belongs to the chosen profile and width convention. Here $\tau$ denotes the power full width at half maximum, the convention explicitly used by the primary source documentation.^[ref]

An original counterexample compares a rectangular pulse with a bilateral exponential pulse. Give both the same $P_{\mathrm{pk}}$ and power FWHM $\tau$. The rectangle is nonzero over a width $\tau$, so $U_{\mathrm{rect}}=P_{\mathrm{pk}}\tau$ and ${\kappa }_{\mathrm{rect}}=1$. Forthe half-maximum points are again $\pm \tau /2$, but integration of the two tails gives $U_{\exp }=P_{\mathrm{pk}}\tau /\ln 2$ and ${\kappa }_{\exp }=1/\ln 2\approx 1.4427$.

With $P_{\mathrm{pk}}=2\mathrm{W}$ and $\tau =5\mathrm{ns}$, the energies are $10$ and $14.43\mathrm{nJ}$. The same measured peak and FWHM therefore leave a 44.3 percent difference in this example. This nonuniqueness of pulse energy at fixed peak power and FWHM is a profile ambiguity, not an instrument-unit conversion.

The Gaussian pulse energy and peak power article gives the Gaussian-family relation. Reconstructing an arbitrary pulse's energy requires the full calibrated profile or an additional justified shape model; a width and peak alone cannot supply it.

Unbounded pulse energy at fixed peak and FWHM means there is no universal finite energy upper bound from those two quantities alone. Each pulse can still have finite duration and finite energy. The nonuniqueness of pulse energy at fixed peak power and FWHM article constructs a rectangular core of width $\tau$ and a below-half-maximum pedestal of height $\alpha P_{\mathrm{pk}}$ extending to total duration $D$.

For any desired energy above the core energy, chooseThis is finite for every finite desired $U$ while keeping the peak and power FWHM unchanged. For an original numerical case, take $P_{\mathrm{pk}}=3\mathrm{W}$, $\tau =4\mathrm{ns}$ and $\alpha =0.20$. An energy of $1\mu \mathrm{J}$ requires $D\approx 1.651\mu \mathrm{s}$ in this ideal profile. The core contributes only $12\mathrm{nJ}$.

Additional information changes the problem. If the signal support is bounded by $D_{\max }=20\mathrm{ns}$ and power outside the FWHM interval is at most $0.20P_{\mathrm{pk}}$, thenThese pulse-energy bounds from tail duration and pedestal height require independently justified tail constraints. A FWHM definition alone fixes a threshold width, not the below-threshold integral.^[ref] The construction concerns isolated ideal pulse profiles, not a claim that an actual laser or finite-period pulse train can have arbitrary tails.