Small Integrating Sphere Light Source with High Radiance Uniformity

Abstract

An integrating sphere that yields uniform and intense radiance is needed for the spectral radiance responsivity calibration of absolute radiation thermometers. However, such an integrating sphere is difficult to realize owing to the common trade-off between spatial uniformity and throughput. Typically, the smaller sphere yields the higher throughput but worse uniformity. This paper reports a considerably uniform integrating sphere despite its small size. First, we introduce a simple approach, based on the perfect Lambertian diffuse reflection theory and the measured bidirectional reflectance distribution function of polytetrafluoroethylene (PTFE), for estimating the spatial uniformity of a 50 mm-diameter, pressed PTFE integrating sphere as a function of the incidence angle of an external light source. Based on the estimated results, we made an integrating sphere with $45^\circ$-incidence angle. This sphere showed the spatial uniformity of ± 0.01 % within 5 mm in diameter and the angular uniformity of ± 0.005 % over an angular range of ± $1.58^\circ$. To the best of our knowledge, the $45^\circ$-incident integrating sphere with a diameter of 50 mm introduced in this study is the best design for obtaining a uniform and intense light source suitable for measuring the spectral radiance responsivity of an absolute radiation thermometer among the integrating spheres reported thus far.

Appendix: Derivation of the Total Radiance Ratio of \({\varvec{L}}\left(\mathbf{x}\right)\) to \({\varvec{L}}\left({\mathbf{x}}_{0}\right)\)

Starting from Eq. 6 of the perfect Lambertian

$$L={L}_{1}+{L}_{\mathrm{R}}=\left(1-\rho \right)L+\rho L.$$

(12)

By approximated generalization of Eq. 6 [See Sec. 2.2], the total radiance at surface elements ${\mathrm{x}}_{0}$ and $\mathrm{x}$ on the target arc can be written as

$$L\left({\mathrm{x}}_{0}\right)={L}_{1}\left({\mathrm{x}}_{0}\right)+{L}_{\mathrm{R}}\left({\mathrm{x}}_{0}\right),$$

(13)

$$L\left(\mathrm{x}\right)={L}_{1}\left(\mathrm{x}\right)+{L}_{\mathrm{R}}\left(\mathrm{x}\right).$$

(14)

Under an approximation of ${L}_{\mathrm{R}}\left({\mathrm{x}}_{0}\right)={L}_{\mathrm{R}}\left(\mathrm{x}\right)\equiv {L}_{\mathrm{R}}$, Eq. 13 and 14 can be written as

$$L\left({\mathrm{x}}_{0}\right)={L}_{\mathrm{R}}\left[1+\frac{{L}_{1}\left({\mathrm{x}}_{0}\right)}{{L}_{\mathrm{R}}}\right],$$

(15)

$$L\left(\mathrm{x}\right)={L}_{\mathrm{R}}\left[1+\frac{{L}_{1}\left(\mathrm{x}\right)}{{L}_{\mathrm{R}}}\right].$$

(16)

Dividing Eq. 16 by Eq. 15, the ratio of the total reflected radiance between the two surface elements is

$$\frac{L\left(\mathrm{x}\right)}{L\left({\mathrm{x}}_{0}\right)}=\left[1+\frac{{L}_{1}\left(\mathrm{x}\right)}{{L}_{\mathrm{R}}}\right]\cdot {\left[1+\frac{{L}_{1}\left({\mathrm{x}}_{0}\right)}{{L}_{\mathrm{R}}}\right]}^{-1}.$$

(17)

Since $\frac{{L}_{1}\left({\mathrm{x}}_{0}\right)}{{L}_{\mathrm{R}}}\mathrm{and} \frac{{L}_{1}\left(\mathrm{x}\right)}{{L}_{\mathrm{R}}}\ll 1$, Eq. 17 can be approximated as

$$\frac{L\left(\mathrm{x}\right)}{L\left({\mathrm{x}}_{0}\right)}\cong 1+\frac{{L}_{1}\left(\mathrm{x}\right)}{{L}_{\mathrm{R}}}-\frac{{L}_{1}\left({\mathrm{x}}_{0}\right)}{{L}_{\mathrm{R}}}-\frac{{L}_{1}\left(\mathrm{x}\right)}{{L}_{\mathrm{R}}}\cdot \frac{{L}_{1}\left({\mathrm{x}}_{0}\right)}{{L}_{\mathrm{R}}}+\dots .$$

(18)

Taking only three terms of the first-order approximation, Eq. 18 can be rewritten as

$$\frac{L\left(\mathrm{x}\right)}{L\left({\mathrm{x}}_{0}\right)}\cong 1+\frac{\left[{L}_{1}\left(\mathrm{x}\right)-{L}_{1}\left({\mathrm{x}}_{0}\right)\right]}{{L}_{\mathrm{R}}}.$$

(19)

To simplify Eq. 19 as a function of ${L}_{1}\left(\mathrm{x}\right)/{L}_{1}\left({\mathrm{x}}_{0}\right)$, we approximate $\frac{{L}_{1}\left({\mathrm{x}}_{0}\right)}{{L}_{\mathrm{R}}}$ to $\frac{1-\rho }{\rho }$ using Eq. 12 of $\frac{{L}_{1}}{{L}_{\mathrm{R}}}=\frac{1-\rho }{\rho }$ for the perfect Lambertian. Applying this approximation to Eq. 19,

$$\frac{L\left(\mathrm{x}\right)}{L\left({\mathrm{x}}_{0}\right)}\cong 1+\frac{\left[{L}_{1}\left(\mathrm{x}\right)-{L}_{1}\left({\mathrm{x}}_{0}\right)\right]}{{L}_{1}\left({\mathrm{x}}_{0}\right)}\cdot \frac{\left(1-\rho \right)}{\rho }.$$

(20)

Re-arranging Eq. 20,

$$\frac{L\left(\mathrm{x}\right)}{L\left({\mathrm{x}}_{0}\right)}\cong 1+\left[\frac{{L}_{1}\left(\mathrm{x}\right)}{{L}_{1}\left({\mathrm{x}}_{0}\right)}-1\right]\cdot \frac{\left(1-\rho \right)}{\rho }.$$

(21)

$$\frac{L\left(\mathrm{x}\right)}{L\left({\mathrm{x}}_{0}\right)}\cong 1+\frac{\left(1-\rho \right)}{\rho }\bullet \frac{{L}_{1}\left(\mathrm{x}\right)}{{L}_{1}\left({\mathrm{x}}_{0}\right)}-\frac{\left(1-\rho \right)}{\rho }.$$

(22)

Approximating $\rho$ to 1 in the denominator of Eq. 22, we finally obtain the total radiance ratio of $L\left(\mathrm{x}\right)$ to $L\left({x}_{0}\right)$ as follows:

$$\frac{L\left(\mathrm{x}\right)}{L\left({\mathrm{x}}_{0}\right)}\cong \left(1-\rho \right)\frac{{L}_{1}\left(\mathrm{x}\right)}{{L}_{1}\left({\mathrm{x}}_{0}\right)}+\rho .$$

(23)