I think jello normally has about the refractive index of water, but it’s possible to change it from about 1.3 to about 1.7 by adding sugar, or lower it as low as 0.7 by adding alcohol; both will diffuse through the jello once it is set. To avoid chromatic diffraction patterns, the difference in optical path length should be over 10λ, or about 7μm for red light. This in turn suggests that a variation in refractive index from 1.3 to 1.4 would require layers of some 70μm or more in thickness. But of course that is easy to achieve; what is difficult to achieve is layers of less than about 100μm.
The proposal, in short, is to deposit jello with a 3-D printer in thin layers with continuously variable sugar or alcohol content, in order to approximate a desired optical transformation function. Other food gels such as agar, carrageenan, or konjac are also likely to work well, and have the advantage of avoiding ethical issues with cruelty to animals. As the jello is deposited, it will cool and gel, slowing but not eliminating the diffusion of the dopants.
Discontinuities in refractive index that scatter light in conventional optical systems, producing stray light that limits optical system performance. Dopant diffusion eliminates these discontinuities, replacing them with gradients, permitting order-of-magnitude improvements in optical system complexity.
Active closed-loop control of refractive index of deposited material permits the use of very small differences in refractive index to improve control of optical phase delay. Given a layer thickness of 200μm, a minimally acceptable phase delay error of 100nm would seem to require .05% error in the refractive index of the deposited material, for example, controlling whether the refractive index is 1.4000 or 1.4007. This turns out to be feasible with closed-loop control, as follows.
According to the International Scale of Refractive Indices, water’s refractive index at 40% sugar is 1.3997; at 41% it is 1.4016, a difference of 0.0019 units of IOR over 1%. Ordinary lab refractometers provide a readout to 5 digits of precision. So this level of control amounts to controlling the sugar content of water to a precision of about 0.3%, or 0.3°Bx, which I think is feasible. Indeed, it should be feasible to control it to within the precision of the refractometer, 0.0001 units. This corresponds to a phase error of some 20nm, which is acceptable for even the most demanding optics. Some refractometers have 5× smaller errors than this, which requires wavelength calibration of their light source and temperature compensation of the sample.
Such closed-loop process control using process refractometers has long been common practice in industries that process sugar solutions.
It isn’t necessary to control the IOR of the finished product to this level of precision in an absolute sense, because what matters for refraction is the gradient of IOR; it’s just necessary to avoid random errors between passes and layers. This is fortunate, because the refractive index of the sugar-water system varies with temperature by about 0.0001 per kelvin.
It should be possible to achieve fine control of sugar concentration by mixing two or more homogeneous solutions in varying proportions; for example, a mixture of 36%-sugar jello A and 42%-sugar jello B that is 50-50 would be 39% sugar, while if it’s 55-45, it would be 38.7% sugar. In this way a 5% error in the mixing process amounts to only an 0.3% error in the final concentration.
Some other related solute systems have nonmonotonic refractive-index curves; the ethanol-water system, for example, reaches its maximum refractive index of about 1.3658 at about 80% ethanol, thereafter declining to 1.361. In the neighborhood of this local maximum, fairly large changes in the concentration of ethanol are needed to obtain small changes in refractive index; for example, IOR is 1.3654 at 72% ethanol, but 1.3657 at 76% ethanol. This improves the control over IOR that can be achieved despite errors in mixing.
Ethanol has the major disadvantage that it is a very small molecule, so diffusion and consequent degradation of the instrument is likely to be very fast. Instead, larger molecules than sucrose would be advantageous in order to delay diffusion further.
A perfectly flat “Fresnel” lens of jello with a thickness of about 200μm should be achievable with this approach. A simple waveplate pattern would provide an aspheric focusing lens, for example, although at the cost of dispersion and some stray light from the boundaries between the rings. A thicker lens, probably thicker than normal glass lenses, could reduce the number of rings (perhaps to 1) and thus eliminate the stray light. It should be feasible to fabricate an entire compound microscope or telescope in a single solid block in this way.
To prevent decay of the product (before it is destroyed naturally by diffusion), some kind of antibacterial and antifungal agent should be included. Ethanol at levels above about 15% is one possibility; sodium chloride is another; acids such as acetic acid or citric acid are yet another. Finally, a sufficiently high level of sugar would also prevent decay, but that level might be high enough to prevent gel formation.
It’s probably desirable to encapsulate the final result in some kind of airtight container (sandwiched between glass plates, filling a transparent polyvinylidene-chloride balloon, etc.) to prevent evaporation from converting the result to a xerogel, which would likely impair its optical properties. Alternatively, if the degradation is acceptable, xerogel conversion would protect against decay, dramatically slow solute diffusion after drying, and reduce layer thickness considerably, thus improving phase delay control.
Diffusion could also be controlled by chemically bonding the index-altering solutes to the solid matrix of the gel, perhaps by making the gel itself from a mixture of different substances with different refractive indices, or perhaps just using glucomannan or a similar dense gel former at different concentrations. (What’s the name of that hydrophilic polymer they use for contact lenses and ceramic gelcasting?)
Correcting chromatic aberration probably requires multiple solutes that induce different dispersions, thus varying IOR independently at different wavelengths.
Avoiding air bubbles is of the highest importance. The simplest way to achieve this would be to do the entire printing process under vacuum, but unfortunately that is not an option with a water-based gel at room temperature, because water boils rather violently at room temperature in vacuum. Alternatively perhaps the product could be degassed under vacuum before the gel forms, the way we do with polyester resin casting, if there aren’t too many bubbles. It would be better to use a liquid (like, uh, polyester for resin casting) that has a relatively low vapor pressure and therefore can be printed under vacuum.
These gels are capable of substantial elastic deformation, often with strains exceeding 10%, which provides another axis of variation; they can be designed to provide an optical transformation which depends in a designed way on elastic deformation.
To research: