The analytical approach used in this paper is designed for application to uniaxial birefringent cylinders in a double-cylinder configuration, with the birefringent axis lying parallel to the silk axis. The double-cylinder configuration arises naturally from the silk-spinning process, where two spinnerets each yield a single cylinder, which then adhere together. The principal refractive indices of the double cylinder are labeled;

$${n}_{p}={n}_{p}-i{\kappa}_{p}\mathrm{,}$$(1)

$${n}_{s}={n}_{s}-i{\kappa}_{s}\mathrm{,}$$(2)

where **n**_{p} and **n**_{s} are the complex refractive indices for light polarized parallel and perpendicular to the birefringent axis, respectively. If the double cylinders are homogeneous, full characterization of the refractive indices requires evaluation of four quantities: *n*_{p}, *n*_{s}, *κ*_{p}, and *κ*_{s}. As silks generally do not exhibit any obvious linear dichroism, the optical absorption is assumed to be isotropic, hence,

$${\kappa}_{p}={\kappa}_{s}=\kappa \mathrm{.}$$(3)

As with all immersion-based analysis techniques, double cylinders are immersed in a liquid with refractive index **n**_{L}
=*n*_{L}−*iκ*_{L} such that *n*_{s}<*n*_{L}<*n*_{p}. As the double cylinders’ refractive index at some polarization orientation, *θ*, is given by

$$n\mathrm{(}\theta \mathrm{)}=\frac{{\text{cos}}^{2}\theta}{{n}_{p}^{2}}+\frac{{\text{sin}}^{2}\theta}{{n}_{s}^{2}}\mathrm{,}$$(4)

there exists a polarization orientation, *θ*_{0}, such that *n*(*θ*_{0})=*n*_{L}.

The double cylinder was positioned a distance *z*_{0} behind the focal (object) plane of the imaging objective, where *z*_{0} was defined as the position that yielded the highest irradiance contrast when imaging the scattered field with a parallel-polarized incident field at normal incidence. Modeled scattered fields on the X−Y plane intersecting the *z*-position where the on-axis irradiance was highest, could then be directly compared with experimental images and circumvent the need to accurately measure the position of the double cylinder with respect to the focal plane.

Silks were then imaged at both *p*- and *s*-polarizations. The visibility of the double cylinder under illumination was quantified by three contrast parameters, *C*_{p}, *C*_{s}, and *C*_{0}, defined as

$${C}_{p}={I}_{\text{max}}\mathrm{/}{I}_{0}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{(}\theta =0\mathrm{)}\mathrm{,}$$(5)

$${C}_{s}={I}_{\text{min}}\mathrm{/}{I}_{0}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{(}\theta =\pi /2\mathrm{)}\mathrm{,}$$(6)

$${C}_{0}={I}_{\text{min}}\mathrm{/}{I}_{0}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{(}\theta ={\theta}_{0}\mathrm{)},$$(7)

where *I*_{max} and *I*_{min} denote the maximum and minimum irradiances, respectively, in the measured irradiance profiles, and *I*_{0} is the incident (or background) irradiance. Uncertainties in *C*_{p}, *C*_{s}, and *C*_{0} (labeled *δC*_{p}, *δC*_{s}, and *δC*_{0}) were calculated as twice the standard deviation of the normalized irradiance.

As the polarizations are separable in the case of a cylindrical scatterer, the theoretical relationship between normalized irradiance in the object volume (*I*_{p}, *I*_{s}) and the optical properties of the double cylinder can be summarized as

$${I}_{i}={\displaystyle \int {f}_{i}\mathrm{(}{n}_{i}\mathrm{,}\text{\hspace{0.17em}}{n}_{L}\mathrm{,}\text{\hspace{0.17em}}k\mathrm{,}\text{\hspace{0.17em}}a\mathrm{)}dk}\mathrm{,}$$(8)

where *i*=*p*, *s*; *a* is the radii of each cylinder in the double-cylinder structure, and **k** is the incident wave vector. The functions *f*_{p} and *f*_{s} are derived from electromagnetic scattering theory [10, 11]. It was assumed, based on previous TEM studies of spider dragline silk, that the double cylinders were in contact, with the same radius and refractive index, and with both cylinder axes in a plane parallel to the focal plane of the imaging objective.

To obtain a similar relation for the irradiance in the image volume, the functions *f*_{p} and *f*_{s} were convolved with the point-spread function of the imaging objective, *p*.

$${{I}^{\prime}}_{i}={\displaystyle \int p\ast {f}_{i}\mathrm{(}{n}_{i}\mathrm{,}\text{\hspace{0.17em}}{n}_{L}\mathrm{,}\text{\hspace{0.17em}}k\mathrm{,}\text{\hspace{0.17em}}a\mathrm{)}dk}={F}_{i}\mathrm{(}{n}_{i}\mathrm{,}\text{\hspace{0.17em}}{n}_{L}\mathrm{,}\text{\hspace{0.17em}}{k}^{\prime}\mathrm{,}\text{\hspace{0.17em}}a\mathrm{)}$$(9)

where the prime denotes the image volume, ∗ denotes a 2-D convolution, and **k**′ denotes the central illumination wave vector. *F*_{i} represents the same thing as *f*_{i}, except with the numerical aperture of the incident field and imaging objective accounted for. As the double cylinder is closely refractive index matched with the surrounding medium, and **k**′ is usually fixed normal to the focal plane with magnitude defined by the measurement wavelength, Eq. (9) can be approximated as

$${{I}^{\prime}}_{i}\approx {F}_{i}\mathrm{(}\Delta {n}_{i}\mathrm{,}\text{\hspace{0.17em}}\Delta \kappa \mathrm{,}\text{\hspace{0.17em}}a\mathrm{}\mathrm{)}\mathrm{,}$$(10)

where Δ*n*_{i}=*n*_{i}−*n*_{L} and Δ*κ*=*κ*−*κ*_{L} (*i*=*p*, *s*).

Optical properties of double cylinders were inferred by determining the optical constants that minimized the following error terms:

$$\Delta {C}_{p}=\text{\hspace{0.17em}}\mathrm{|}\text{max}\mathrm{(}{{I}^{\prime}}_{p}\mathrm{)}-{C}_{p}\mathrm{|}$$(11)

$$\Delta {C}_{s}=\text{\hspace{0.17em}}\mathrm{|}\text{min}\mathrm{(}{{I}^{\prime}}_{s}\mathrm{)}-{C}_{s}\mathrm{|}$$(12)

$$\Delta {C}_{0}=\text{\hspace{0.17em}}\mathrm{|}\text{min}\mathrm{(}{{I}^{\prime}}_{p}{\text{cos}}^{2}{\theta}_{0}+{{I}^{\prime}}_{s}{\text{sin}}^{2}{\theta}_{0}\mathrm{)}-{C}_{0}|\mathrm{,}$$(13)

where max and min denote the maximum and minimum values in the specified quantities. While the irradiance is imaged over a 2-D plane, the symmetry of the scatterer means that the profile can be reduced to a 1-D cross-section.

To minimize the error terms, Δ*C*_{p} and Δ*C*_{s} were optimized for several estimates of *κ*. As the remaining value of Δ*C*_{0} was approximately linear with *κ*, new estimates for *κ* that simultaneously minimized all three error terms could be computed reasonably quickly with a Newton’s method-type approach.

Uncertainties in *n*_{p}, *n*_{s}, and *κ* were derived by linearizing *f*_{p} and *f*_{s} about the measured values for these quantities. For brevity, perfect positive correlation was assumed (yielding the largest uncertainty estimates).

$$\delta \Delta \kappa \left|\frac{\partial {C}_{0}}{\partial \Delta \kappa}\right|=\delta {C}_{0}+\delta a\left|\frac{\partial {C}_{0}}{\partial a}\right|$$(14)

$$\delta \Delta {n}_{p}\left|\frac{\partial {C}_{p}}{\partial {n}_{p}}\right|=\delta {C}_{p}+\delta \Delta \kappa \left|\frac{\partial {C}_{p}}{\partial \Delta \kappa}\right|+\delta a\left|\frac{\partial {C}_{p}}{\partial a}\right|$$(15)

$$\delta \Delta {n}_{s}\left|\frac{\partial {C}_{s}}{\partial {n}_{s}}\right|=\delta {C}_{s}+\delta \Delta \kappa \left|\frac{\partial {C}_{s}}{\partial \Delta \kappa}\right|+\delta a\left|\frac{\partial {C}_{s}}{\partial a}\right|\mathrm{,}$$(16)

where *δ*Δ*n*_{p}, *δ*Δ*n*_{s}, and *δ*Δ*κ* represent uncertainties in Δ*n*_{p}, Δ*n*_{s}, and Δ*κ*, respectively. Partial derivatives in Eq. (14) were computed numerically in the process of optimizing the error terms using the procedure outlined above. Given the optical properties of the double cylinder and liquid are uncorrelated, the final uncertainties for *n*_{p}, *n*_{s}, and *κ* were calculated as

$${\mathrm{(}\delta {n}_{p}\mathrm{)}}^{2}={\mathrm{(}\delta {n}_{L}\mathrm{)}}^{2}+{\mathrm{(}\delta \Delta {n}_{p}\mathrm{)}}^{2}$$(17)

$${\mathrm{(}\delta {n}_{s}\mathrm{)}}^{2}={\mathrm{(}\delta {n}_{L}\mathrm{)}}^{2}+{\mathrm{(}\delta \Delta {n}_{s}\mathrm{)}}^{2}$$(18)

$${\mathrm{(}\delta \kappa \mathrm{)}}^{2}={\mathrm{(}\delta {\kappa}_{L}\mathrm{)}}^{2}+{\mathrm{(}\delta \Delta \kappa \mathrm{)}}^{2}\mathrm{,}$$(19)

where *δn*_{L} and *δκ*_{L} represent uncertainties in *n*_{L} and *κ*_{L}, respectively, and were determined using specifications provided by the manufacturer of the refractive index liquid.

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