We investigate the optical properties of hydrogen as it undergoes a transition from the insulating molecular to the metallic atomic phase, when heated by a pulsed laser at megabar pressures in a diamond anvil cell. Most current experiments attempt to observe this transition by detecting a change in the optical reflectance and/or transmittance. Theoretical models for this change are based on the dielectric function calculated for bulk, homogeneous slabs of material. Experimentally, one expects a hydrogen plasma density that varies on a length scale not substantially smaller than the wave length of the probing light. We show that taking this inhomogeneity into account can lead to significant corrections in the reflectance and transmittance. We present a technique to calculate the optical properties of systems with a smoothly varying density of charge carriers, determine the optical response for metallic hydrogen in the diamond anvil cell experiment and contrast this with the standard results. Analyzing recent experimental results we obtain $sigma^{Drude}_{DC}=(2.1 pm 1.3) times 10^3$ ($Omega$ cm)$^{-1}$ for the conductivity of metallic hydrogen at 170 GPa and 1250 K.
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