# Perceived Brightness and Resolution of Holographic Augmented Reality Retinal Scan Glasses
## 3 Results  
### Retinal Scan Luminance and Ambient Contrast Ratio  
To express the brightness of an RS display in terms of luminance, we determine the luminance of a Lambertian display that would cause the same retinal illuminance \(\mathrm{E_{R}}\) as the RS glasses (see Table 1 in Appendix A for an overview of mathematical symbols). \[\mathrm{E_{R_{Lam}}}=\mathrm{E_{R_{Scan}}} \tag{3}\]  
To find the RS retinal illuminance \(\mathrm{E_{R_{Scan}}}\), we first calculate the luminous flux \(\mathrm{\Phi_{V_{Scan}}}\) from the radiant flux \(\mathrm{\Phi_{E_{Scan}}}\), the luminous efficacy constant \(\mathrm{K_{m}}=683\,\frac{\mathrm{\mu}m}{W}\), and the luminous efficiency function \(\mathrm{V(\lambda)}\). The luminous efficacy constant \(\mathrm{K_{m}}\) tells us how many lumens result from 1 watt of optical power. It acts as the conversion factor from radiometric to photometric quantities. The luminous efficiency function \(\mathrm{V(\lambda)}\) denotes how bright light with different wavelengths appears to the human eye. In practice, the resulting integral is approximated by a finite sum of measurements. \[\mathrm{\Phi_{V_{Scan}}}=\mathrm{K_{m}}\int\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}( \lambda)}\ \mathrm{d}\lambda\approx\mathrm{K_{m}}\sum_{\lambda}\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}(\lambda)} \tag{4}\]  
Next, we calculate the illuminated retinal area \(\mathrm{A_{R_{Scan}}}\) geometrically from the RS field of view angles \(\alpha\) and \(\beta\) and the eye's focal distance \(\mathrm{f_{e}}\). \[\mathrm{A_{R_{Scan}}}=\mathrm{f_{e}}\ 2\tan\Bigl{(}\frac{\alpha}{2}\Bigr{)}\ \mathrm{f_{e}}\ 2\tan \Bigl{(}\frac{\beta}{2}\Bigr{)}=4\ \mathrm{f_{e}}^{2}\tan\Bigl{(}\frac{\alpha}{2}\Bigr{)}\ \tan \Bigl{(}\frac{\beta}{2}\Bigr{)} \tag{5}\]  
Plugging Equations (4) and (5) into Equation (1) gives us the retinal illuminance for an RS display. \[\mathrm{E_{R_{Scan}}}=\frac{\mathrm{\Phi_{V_{Scan}}}}{\mathrm{A_{R_{Scan}}}}= \frac{\mathrm{K_{m}}\sum_{\lambda}\mathrm{V(\lambda)}\ \mathrm{\Phi_{E_{Scan}}(\lambda)}}{4\ \mathrm{f_{e}}^{2}\tan \bigl{(}\frac{\alpha}{2}\bigr{)}\ \tan\Bigl{(}\frac{\beta}{2}\Bigr{)}} \tag{6}\]  
Next, we express the Lambertian display retinal illuminance \(\mathrm{E_{R_{Lam}}}\) in terms of the Lambertian display luminance \(\mathrm{L}\)[13]. For an observer with pupil area \(\mathrm{A_{P}}\) at distance \(\mathrm{R}\) to the display, the solid angle \(\Omega\) will be given by the ratio of \(\mathrm{A_{P}}\) and \(\mathrm{R}^{2}\). \[\Omega=\frac{\mathrm{A_{P}}}{\mathrm{R}^{2}} \tag{7}\]  
Plugging Equation (7) into Equation (2) and rearranging terms yields the luminous flux \(\mathrm{\Phi_{V_{Lam}}}\) entering the pupil as a function of the Lambertian display luminance \(\mathrm{L_{Lam}}\), pupil area \(\mathrm{A_{P}}\), display area \(\mathrm{A_{S}}\) and display distance \(\mathrm{R}\). \[\mathrm{L_{Lam}}=\frac{\mathrm{\Phi_{V_{Lam}}}}{\mathrm{\frac{A_{P}}{\mathrm{ R}^{2}}\ A_{S}}}\ \Leftrightarrow\ \mathrm{\Phi_{V_{Lam}}}=\frac{\mathrm{L_{Lam}\ A_{P}\ A_{S}}}{\mathrm{R}^{2}} \tag{8}\]To find the retinal image area \(\rm A_{R_{Lam}}\), we use the lens formula from geometrical optics. A display with area \(\rm A_{S}\) at a distance \(\rm R\) to the eye creates an image with area \(\rm A_{R}\) at focal distance \(\rm f_{e}\) on the retina. The magnification between \(\rm A_{S}\) and \(\rm A_{R}\) is given by the square of the ratio between image distance \(\rm f_{e}\) and object distance \(\rm R\). \[\rm A_{R_{Lam}}=\rm A_{S}\left(\frac{\rm f_{e}}{\rm R}\right)^{2} \tag{9}\]  
Plugging Equations (8) and (9) into Equation (1) gives us the retinal illuminance for a Lambertian display:  
\[\rm E_{R_{Lam}}=\frac{\rm\Phi_{V_{Lam}}}{\rm A_{R_{Lam}}}=\frac{\rm L_{Lam}\;A _{P}\;A_{S}}{\rm R^{2}\;A_{S}\left(\frac{\rm f_{e}}{\rm R}\right)^{2}}=\frac{ \rm L_{Lam}\;A_{P}}{\rm f_{e}{}^{2}} \tag{10}\]  
Finally, we can set both retinal illuminances \(\rm E_{R_{Lam}}\) and \(\rm E_{R_{Son}}\) as equal.