# Design of a High-Performance Digital Slit-Lamp Microscope with Five-Switched Zoom
## 2 Design of the DSLM Optical System
### Design Specifications and Initial Structure  
As mentioned earlier, the optical system of the DSLM is divided into three subsystems: the front objective, Galilean telescope, and photographic lens. In this section, we analyze and calculate the design specifications and initial structures of these three parts. Their derivation is described as follows. The total visual magnification (\(M_{TOT}\)) of the system can be expressed as [22]  
\[M_{TOT}=\left(-{f^{\prime}}_{T}/{f^{\prime}}_{F\O}\right)\times\Gamma_{TEL} \times M_{E}, \tag{1}\]  
where \(\Gamma_{TEL}\) is the magnification of the Galilean telescope. The quantity \(M_{E}\) is the magnification of the eyepiece, which is set as 12.5\(\times\). The quantities \({f^{\prime}}_{F\O}\) and \({f^{\prime}}_{T}\) are the focal lengths of the front objective and tube lens, respectively. As described previously, when \(M_{TOT}=16\times\), \(\Gamma_{TEL}=1\times\) (the Galilean telescopes are empty), so \(\left(-{f^{\prime}}_{T}/{f^{\prime}}_{F\O}\right)=1.28\). It can be deduced that the corresponding magnifications of the Galilean telescope (\(\Gamma_{TEL}\)) are 0.4\(\times\), 0.625\(\times\), 1.6\(\times\), and 2.5\(\times\), while the total visual magnifications (\(M_{TOT}\)) are 6\(\times\), 10\(\times\), 25\(\times\), and 40\(\times\), respectively. Note that in the front objective, the field of view (FOV) and f-number decrease as the magnification increases [23]. An FOV of 6\(\times\) and f-number of 40\(\times\) should be taken as the initial specifications of the front objective. Generally, the aperture diameter (\(D_{FO}\)) of the front objective is no less than 40 mm at 40\(\times\) magnification (set to 40 mm for \(D_{FO}\)). For convenient detection and treatment, a large working distance of the optical system is necessary, so we set the focal length of the front objective (\({f^{\prime}}_{FO}\)) to 110 mm. Hence, the f-number is \({f^{\prime}}_{FO}/D_{FO}=2.75\) for the front objective. The FOV can be obtained by [22]  
\[\beta=\left(-{f^{\prime}}_{T}/{f^{\prime}}_{FO}\right)\times\Gamma_{TEL}={y^{ \prime}}/{y}, \tag{2}\]  
\[\tan\omega_{FO}={y}/{f^{\prime}}_{FO}. \tag{3}\]  
where \(\beta\) is the vertical magnification of the combined microscope objective. \(\Gamma_{TEL}\) of 0.4\(\times\) was selected for Equation (2). In this design, the field number (F. N.) of view, defined as the diameter of the bright circle which can be seen in the eyepiece, was set to 20 mm. The corresponding image height (\(y^{\prime}\)) of the microscope objective is 10 mm in practical applications. Therefore, the FOV, 2\(\omega_{FO}=20.13^{\circ}\), can be calculated for the front objective. Two Galilean telescopes are installed in the rotating drum and used to provide four magnification rates in the system. The distance between the objective and eyepiece of the Galilean telescope was taken to be 30 mm. Therefore,  
\[{f^{\prime}}_{1}-f_{2}=30, \tag{4}\]  
where \({f^{\prime}}_{1}\) (\(f_{1}\)) and \({f^{\prime}}_{2}\) (\(f_{2}\)) are the focal lengths of the objective and eyepiece of the Galilean telescope. Also,  
\[\Gamma_{TEL}=-{f^{\prime}}_{1}/{f^{\prime}}_{2}. \tag{5}\]  
We can calculate the focal lengths, \({f^{\prime}}_{1}\) and \({f^{\prime}}_{2}\), according to Equations (4) and (5) when \(\Gamma_{TEL}\) = 1.6\(\times\) and 2.5\(\times\). The optical path from the front objective is shared by two identical Galilean telescopes, so the maximum aperture diameter of the Galilean telescope objective was set to \(D_{TEL}=D_{FO}/2=20\) mm. The f-number of the objective of the Galilean telescope can be obtained by calculating the ratio of \({f^{\prime}}_{1}\) to \(D_{TEL}\).