The purpose of this note is to consider the geometric optics design analytically. By applying some basic equations, we now seek to design a proof-of-concept transmission grating-based visible range hyperspectral imager (Build I). This procedure will be transferred to the design of FINCH EYE after developing an understanding of the GRISM and the SNR in subsequent notes.

Our motivation is to select components for a visible/NIR transmission grating-based hyperspectral imager - Build I - that the team will use as a proof-of-concept testing platform. The aim is to use cheap and readily available optical components to learn (and make mistakes) before advancing to the significantly more challenging and expensive SWIR imager. A transmission grating dispersive element is adopted for simplicity of design and evaluation. While the opportunities for meaningful spectroscopy work are limited, the hope is that Build I can be used to distinguish between different outdoor materials.

Consider the ray trace diagram for the spectral and spatial components of the transmission grating-based push-broom imager to guide the following discussion. Most of these components are sourced from Thorlabs.com, which has a large selection of optical components available with fast deliveries.

Camera

The camera is chosen based on its sensor properties. We require a monochrome CMOS camera to gather visible range light, the RGB filter matrix is not desired. Thorlabs offers several models (Zelux, Kiralux, Quantalux) with varying number of pixels and noise levels. Passive cooling should be sufficient for high signal-level visible light applications. For the time being, considerations on form factor are secondary, and the model that offers NIR-enhanced detection efficiency is selected (Thorlabs CS135MUN). The boost in efficiency over the 750nm-900nm range is substantial, and may enable some work in the near infra-red. The efficiency is about 10% lower in he peak visible range wavelengths, but there would be many more visible photons than NIR photons from our broadband illumination source to compensate the difference (e.g. sunlight) .

Some key camera specifications are as follows:

Several important specifications for camera sensors are omitted from the discussion, particularly those related to noise, frame rate, and temperature control. These will be discussed in a future note.

Objective Lens

The choice of lens follows from the camera sensor. The chosen Kiralux camera (1.3MP) has a 1/2” format. Historically, objective lenses were designed for specific photographic film size standards, which encode some aspect ratio and diagonal length, and the convention carried over to camera sensors. We need to choose an objective lens with a format that’s at least larger than the sensor, else we will not utilize all the available pixels. Likewise, we do not need to get an objective lens with a much bigger format (as it’s more expensive), because we’ll be limited by the size of the sensor.

Other key parameters of objective lenses are the effective focal length, the aperture or f/# (the ratio of focal length to the effective aperture diameter), and the working distance. Smaller f/# means the diameter of the lens is relatively larger, hence the objective lens can gather more light onto the sensor. The working distance refers to the minimum distance an object can be placed away from the objective lens for it to be able to produce a sharp image. The objective lenses have knobs to slide the position of some lenses to adjust the focus, and another knob controlling an iris which restricts how much light enters the system.

While lenses with longer focal lengths enable greater spatial resolution on a distant object, it would be difficult to emulate earth-observation with test targets in a laboratory setting. Therefore, a short focal length lens is also ordered in order to facilitate bench-top evaluation. For example, the 50mm focal length lens MVL50M23 could be a suitable candidate. The shorter working distance of 20cm is much easier to accommodate than the 2-meter working distance of the MVL100M23 (f=100mm) lens, in an indoor setting at least.

Among the 50-mm focal length lenses, there is a choice of different f/#’s. The lower the f/#, the “faster” the lens (more light throughput), but also more expensive the lens because its hard to compensate aberrations at large angles of incidence. For example, the f/2.8 lens, f/1.4 lens, and f/0.98 lens cost $250, $500, and $1000, respectively. Since we anticipate having sufficient brightness of the target source in laboratory settings, we don’t need to pursue the fast f/# lenses, especially as it will require all subsequent optics (and their alignment) to be as high of a quality as the objective lens. We proceed with the f/2.8 variant, whose specifications and MTF chart are given below.

At this point, we can determine the obtainable spatial resolution if the lens was mounted directly to the camera. From imaging theory, the minimum resolvable angle is

$$ \theta_{px}=p/f=4.8\mu m / 50 mm \approx 0.1mrad $$

so at the 200mm working distance the sensor should resolve down to 20um, but not without loss in contrast. It is well known that as the spatial frequency of an image increases, it is harder to maintain the same level of contrast. As the spacing between adjacent black (0 intensity) and white (1 intensity) lines decreases, the lines blur out on the sensor and the contrast is reduced (to, e.g., 0.1 and 0.9 intensity). Compare the left and right images below (Edmund Optics Tutorial). The “density” of alternating lines is typically given in line-pairs per millimeter on the camera sensor, where a line pair is one black-white cycle. The contrast is given as:

$$ \frac{(I_{max}- I_{min}){image}}{(I{max}- I_{min})_{object}} $$