amsikking: SOLS transmission

Description

In a single-objective light-sheet (SOLS) microscope there are more optics on the emission path compared to a regular microscope. So how does the transmission of a SOLS microscope compare to a standard microscope? Or similar microscopes like OPM, eSPIM or Lattice?

Here's a simple analytical estimate for the transmission of a SOLS (or similar) microscope:

• To quickly see the results for a particular setup simply edit and run this Python script: SOLS_transmission.py

Results

The original SOLS design from 2019 transmits about ~50% of the light of the equivalent widefield epifluorescence microscope (the 'Benchmark'). The 'SOLS re-optimised' configuration with ~65% transmission can be achieved by swapping out the original lens-galvo-lens scanner for a 'lensless' scanner (like a pair of larger galvos), and optimizing the tilt of microscope 3 from 30deg to 25deg. The larger scanning optics in this configuration are typically slower, but should still be faster than sample scanning (e.g. the Lattice).

Calculation

The analysis is limited to microscopes that respect a planar boundary for compatibility with standard slides, dishes and multiwell plates (i.e. no additional optics above the coverslip, or below the front face of the objective for dipping systems). In this regime the maximum available light from a spherical emitter is a hemisphere of rays. The estimate for transmission can then be split into:

1. Ray transmission: what fraction of the hemisphere of rays are transmitted by the system \(T_{rays}\).
2. Optic transmission: what fraction of the rays are transmitted after optical losses (for example by reflection and absorption) \(T_{optics}\).

The total transmission of the system \(T\) is then the product of the ray and optic transmissions: \[ T = T_{rays} \; T_{optics} \tag{1}\]

Ray transmission

Ideally behavior is assumed from the objectives. The combined ray transmission is found with: \[ T_{rays} = T_1 \; T_2 \; T_3 \tag{2}\]

Objective 1

If we have a primary objective with a collection half angle \(\theta_1\), then the fraction of rays we can transmit from a hemisphere \(T_1\) is: \[ T_1 = 1 - \cos\theta_1 \tag{3}\] where, \[ \theta_1 = \arcsin{\left ( \frac{NA_1}{n_1} \right )} \tag{4}\] and \(NA_1\) and \(n_1\) are the numerical aperture and refractive index of the immersion of objective 1. Equation (3) is found by taking the ratio of the area of a spherical cap to that of a hemisphere:

Objective 2

We will impose the 'perfect 3D imaging' condition from the original remote refocus paper (Botcherby 2007) and apply the preservation of angles between the object and the remote space (i.e. \(\theta_{1ray} = \theta_{2ray}\)). In this regime if \(\theta_2\) is the collection half angle of objective 2 then we have two possible results for the transmission \(T_2\):

1. If \(\theta_2 \geq \theta_1\) then all the rays are transmitted: \[T_2 = 1 \tag{5}\]
2. If \(\theta_2 \lt \theta_1\) then we must take the ratio of the fraction of light that objective 2 can transmit compared to objective 1 (i.e. a ratio of spherical caps): \[T_2 = \frac{1 - \cos\theta_2}{1 - \cos\theta_1} \tag{6}\]

Objective 3

In contrast to a 'straight through' remote refocus, in a SOLS style microscope (Millett-Sikking 2019) objective 3 (with collection half angle \(\theta_3\)) is tilted away from the primary optic axis by an angle \(\theta_t\). The tilting can be thought of as tilting the spherical transmission cap of objective 3 away from the spherical transmission cap of objective 2, which gives rise to four possible conditions:

1. If \(\theta_t \geq \theta_2 + \theta_3\) then the spherical cap of objective 3 is completely outside the spherical cap of objective 2 and no rays are transmitted: \[T_3 = 0 \tag{7}\]
2. If \(\theta_t + \theta_2 \leq \theta_3\) then the spherical cap of objective 2 is completely inside the spherical cap of objective 3, and all the rays are transmitted: \[T_3 = 1 \tag{8}\]
3. If \(\theta_t + \theta_3 \leq \theta_2\) then the spherical cap of objective 3 is completely inside the spherical cap of objective 2, and (like equation (6)) we must take the ratio of the ratio of spherical caps: \[T_3 = \frac{1 - \cos\theta_3}{1 - \cos\theta_2} \tag{9}\]
4. For any other combination of \(\theta_2\), \(\theta_3\) and \(\theta_t\) the fraction of transmitted rays is given by the ratio of the intersection of the spherical caps with the spherical cap of objective 2: \[T_3 = \frac{(\pi - A\cos b - B \cos a - C)}{\pi (1 - \cos a)} \tag{10}\] See here for the origin of equation (10) and how to covert \(A\), \(B\), \(C\), \(a\) and \(b\) back to the variables \(\theta_2\), \(\theta_3\) and \(\theta_t\).

Optic transmission

The combined optic transmission is simply found with: \[ T_{optics} = (T_{ob} \; T_{tl})^{n_{ob}} \; T_{sl}^{n_{sl}} \; T_{mi}^{n_{mi}} \; T_{di}^{n_{di}} \; T_{fi}^{n_{fi}} \tag{11}\] where \(T_{ob}\), \(T_{tl}\), \(T_{sl}\), \(T_{mi}\), \(T_{di}\) and \(T_{fi}\) are the nominal transmission efficiencies for objectives (87.5%), tube lenses (97.5%), scan lenses (90%), mirrors (99%), dichroics (95%) and filters (95%) respectively, and the exponents are the number of each element in the emission path (for example \(n_{mi} = 3\) would be 3 mirrors). The numbers in brackets are the typical transmission values at ~532nm that were used in the provided script (SOLS_transmission.py).

Benchmark

With a 'planar boundary' constraint the ideal microscope would collect hemispheric rays at 100% efficiency. In practice a standard epifluorescence microscope has minimal emission path optics (i.e. maximum transmission) which provides a sensible 'benchmark' for comparison:

Here the Nikon 100x1.35 Silicone oil objective, together with a single tube lens, dichroic and emission filter was used to represent the 'best in class' microscope. The 100x1.35 Sil has one of the highest angular collections currently available and is optimally refractive index matched for in vivo 3D imaging.