**4. Estimation of multi-photon absorption coefficients for biomaterials**

Theoretically, the multi-photon absorption is determined by the following equation,

$$\frac{dI}{dx} = a\_1 I + a\_2 I^2 + a\_3 I^3 + a\_4 I^4 + a\_5 I^5 \tag{3}$$

where I is the intensity of incident light beam propagating along the z-axis. The coefficients α1, α2, α3, α4, α<sup>5</sup> are one-, two-, three-, four- and five-photon absorption coefficients for a given medium, respectively. For a biomaterial, such as spider silk fiber of diameter D0, we define the term absorption, A = (dI/I) (1/D0), which, using Eq. (3) becomes a polynomial of fourth-order, *<sup>A</sup>* <sup>¼</sup> *<sup>α</sup>***<sup>1</sup>** <sup>þ</sup> *<sup>α</sup>***2***<sup>I</sup>* <sup>þ</sup> *<sup>α</sup>***3***I***<sup>2</sup>** <sup>þ</sup> *<sup>α</sup>***4***I***<sup>3</sup>** <sup>þ</sup> *<sup>α</sup>***5***I***<sup>4</sup>**. This mixed fit was used to fit the experimental data. A comparison of mixed fit with cases of pure 2-, 3-, 4-, and 5-photon absorption processes are also shown for comparison. Our fit analysis suggests that as the intensity of the pulses increases, the absorption is dominated by progressively higher-order multi-photon processes. The existence of mixed multi-photon processes for some intensity is also likely [20].

Another technique, "Z-Scan" can also be used to determine multi-photon absorption. The single silk fiber was illuminated with sub-10 fs laser beam at 85 MHz rep rate. The Rayleigh range was approximately, ZR = 25 μm and the silk sample was typically D0 = 1–4 μm in diameter. Thus, we satisfy the thin-sample (ZR> > D0) condition for the Z-scan measurements. The total transmission was measured by collecting the maximum light and focusing it on a photodiode, while the sample was scanned at low power and high power. The sample was translated along the beam path using a computer-controlled stage that allowed us to collect the data at about 2 μm spacing. The normalized data for high power was obtained by subtracting the low power data to minimize any variation due to diffraction effects from silk fibers as shown in **Figure 5**. We fitted our data with the nonlinear transmission function described in Ref. [20], and reference therein. The fit function for 2-photon was, <sup>2</sup>ð Þ¼ *<sup>x</sup> Log* <sup>1</sup> <sup>þ</sup> *<sup>C</sup>*2*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> ð Þ ð Þ *<sup>=</sup>* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> ð Þ, with a fitting parameter C2. The fit function for the 3-photon absorption was, *<sup>T</sup>*3ð Þ¼ *<sup>x</sup> Sinh*�<sup>1</sup> *<sup>C</sup>*<sup>3</sup> 1þ*x*<sup>2</sup> h i*<sup>=</sup>* <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> ½ �. The fit function for 4 photon absorption was, T4 x½ �¼ H2F1 1*=*3,1*=*3,4*=*3, � <sup>1</sup> <sup>∗</sup>*C*<sup>4</sup> 1þ*x*<sup>2</sup> � �<sup>3</sup> � , with C4 was a fit

parameter. The hypergeometric function H2F1 was computed in Mathematica. Our data fitted better with 4-photon absorption fits. This was consistent with our other measurements of nonlinear transmission [20].
