This multilayered post can be followed from one PDF document, or by the explicit links given below in the description of the whole study. The post covers:
- Working with students to make a light intensity detector using a photodiode. It measures photon flux density in the visible portion of the electromagnetic spectrum.
- We went on to use it as the detector in a (spectro)photometer for measuring the concentration of methylene blue dye illuminated with light from a high-intensity yellow LED.
- The main point I just completed writing up is the use of radiative transport equations that I developed for estimating scattered light within a uniform canopy of plants. The solutions for the fluxes of a direct beam and diffuse light together are analytic, in term of algebraic and exponential quantities.
- The model also is useful for simulating the propagation of light inside leaves for modeling photosynthetic rates of leaves with different structures and pigmentations. I have a number of publications on this (which I can link later, when I find PDFs of them. One interesting prediction I made is that leaves with half-normal chlorophyll content should allow sharing of light with leaves deeper in a dense canopy, ultimately giving an 8% increase in biomass and yield. John Hesketh’s group at the University of Illinois tested this in the field and got an 8% increase over fully green leaves! (Pettigrew WT, Hesketh JD, Peters DB, et al. (1989) Characterization of canopy photosynthesis of chlorophyll-deficient soybean isolines. Crop Science 29:1025-1029).
- Recently (Nov.-Dec 2017) I extended it to multiple layers of different optical properties. The challenge was testing it rigorously and making a comprehensive explanation with text and equations.
Back toward teaching: I wanted to verify that the photodiode circuit responded linearly to flux density. I made a simple error in using layer scattering media too close to the detector, invalidating Beers’ law for the direct beam alone. However, I then dove into the propagation of direct and diffuse light for its inherent interest.
The lead PDF document here has several sections:
- The most recent inquiry: is the photodiode responding linearly to photon flux density?
- The radiative transport equations
- A few notes about extensions to nonuniform canopies
Within the lead document are links to several others. The links are imbedded within the lead document; the links are also noted directly here:
- A short write-up of the electronic circuit for the photodiode detector
- “Fixing-approximations.pdf:” A set of notes on improving a whole-canopy flux model (light, CO2 uptake and respiration, transpiration) in the representation of:
- The enzymatic model of photosynthetic carbon fixation, bridging the cases of high light and low light
- The equations for radiative transport, with their full derivation and some numerical results
- A discussion of extensions of the model for leaves varying in absorptivity with depth in the canopy, or that are clumped, or that vary in temperature as they transpire water at different rates
- In turn, this PDF references a short Fortran 90 program I wrote to solve the radiative transport equations
- Also, a link to a 2013 publication I had with Zhuping Sheng, modeling all the fluxes of a pecan orchard. The relevance is that I cited in this second PDF the modeling of light in a regular array of tilted, ellipsoidal canopies of individual trees. Sheng did the experimental measurement of fluxes with eddy covariance equipment. I modeled the results, with one surprising finding that pecan trees, unlike every other plant I’ve studied, do not reduce its stomatal conductance and thus its transpiration in very dry conditions. They operate at high transpiration rates and poor water-use efficiency in these conditions.
- A couple of references to publications:
- A model of radiative transport in layered plant canopies represented by finite layers (a finite-element model), as an integral equation that’s readily solved numerically. I cite this publication because within it I discuss the changing angular distribution of diffuse light with depth.
- The clever method of colleague and friend Michael Chelle and his former advisor Bruno Andrieu for radiative transport in an arbitrary assemblage of light-scatters. The method is called nested radiosity, accounting essentially exactly for nearby scatterers affecting light at a given leaf and then via a nice smoothing approximation (mean field) for more distant scatterers.