Studying the optical properties of various materials requires a thorough comprehension of reflectance and transmittance calculations under both directional and diffuse illumination. Integrating spheres play a vital role in analyzing materials that absorb and scatter light, including milk, corals, and phantoms.

This manuscript aims to present a clear and comprehensive guide to reflectance and transmittance calculations for both types of illumination. By establishing a robust method to assess the optical properties of the medium, we seek to improve the accuracy and versatility of integrating sphere theory.

**Radiation exchange between diffusing surfaces**

This methodology allows for a deeper understanding of the optical properties of a wide range of materials, making integrating sphere theory more adaptable for numerous applications, such as samples with varying surface properties, including optical phantoms.

donde *θe** *es el ángulo entre la dirección de emisión y la normal a la superficie.

Let us now consider a second diffusing surface, *dAr*. The flux (or power) received by this surface takes into account both the radiance emitted by the first surface. The total flux can be written as

where

We can see that,

where

Then,

where we have set

Equation (1) represents a relationship between the fluxes (powers) emitted and received by the surfaces.

Let

then,

Suppose an element of a flat diffusing surface is illuminated by a beam of irradiance *Io* [power/area]. The surface has reflectivity *m* and the angular distribution of the reflected light is Lambertian. The radiance emitted by this element is

From this, we can deduce that

Substituting these quantities into equation (2), we have that within a sphere

where *A* is the area of the sphere.

The last equation highlights an essential property of the integrating sphere.

**Reflectance and transmittance calculations**

In this section, expressions for the detected powers are derived in terms of the apertures of the sphere and the reflectivities of the different elements involved.

*Reflection geometry*

Consider a sphere with a total area

whose interior is coated with a diffusing material of reflectivity *m*. The sphere has three apertures. The aperture where we place the sample of area *As*, the detector aperture with area 𝐴𝛿, and the beam entrance aperture with area *Ah*. We assume that the sample has diffuse reflectance *Rd* and that the detector has reflectance *r*. The fraction of the sphere's area covered by the diffusing material (i.e., the interior area of the sphere, excluding the apertures) is

*Incident light on the wall*

Following the development of Ref. Pickering et al., let us first consider the case where a beam with power *Po* directly strikes the sphere's wall, as illustrated in Figure 1a. The flux, or power reflected by the wall, is

Assuming that the surface acts as a Lambertian diffuser, the flux is reflected uniformly over the entire surface of the sphere, as we saw in the previous section. Then, the flux reaching an area element *dA* is proportional to the flux illuminating the sphere (*mPo*), the fraction of the sphere's area this area represents, and the specular reflectance *Rsp* produce by the sample. Therefore, from this first reflection, a detector with area 𝐴𝛿 will receive a flux

where *A* is the total interior area of the sphere, including the apertures.

The detector reflects a fraction *r* (the detector's reflection coefficient). Similarly, a fraction *m* of the light reaching the walls is reflected, and the sample reflects a fraction *Rd* (the sample's reflection coefficient, which is usually the parameter of interest). Thus, the total light reflected in the second reflection would be

Defining

we can rewrite the light reflected in the second reflection as

We see that *F* represents the fraction of the incident light that is diffusely reflected by all the reflecting components of the sphere.

The light associated with this second reflection is distributed uniformly within the sphere, and the fraction that reaches the detector is

Similarly, from the following (third) reflection, the detector will receive a flux

It is clear, then, that from the n-th reflection, the detector will receive a flux

Summing these terms ad infinitum, we have the total flux received by the detector

The geometric series converges if *F < 1* (which is the case), so the total power received by the detector can be expressed as

Substituting the value of *F* in the previous equation, we have

This expression relates the power received by the detector to the incident power, the sphere parameters, and the sample's reflectivity. From the previous equation, it is possible to find an expression for the medium's diffuse reflectance,

One might assume that using this expression could enable the measurement of a sample's diffuse reflectance; however, it can be demonstrated that this approach is neither practical nor accurate.

*Diffuse diffuse reflectance*

A more effective way to deduce the *Rd* parameter is to express it in terms of detected powers. To do this, let's consider three illumination scenarios, as illustrated in Figure 3. First, without placing any sample on the sphere, we detect a power using equation (3),

Second, by placing a reflectance standard with a rough surface that does not produce specular reflection in the sample port, we obtain a power

Here, 𝑅𝑑^(𝑠𝑡𝑑) represents the reflectance of the standard.

Finally, when we place the sample in the sample port, we detect a power

Now, we divide equation (4) by (5)

or,

Resolving for the diffuse reflectance of the standard,

Similarly, from equations (4) and (6), we can obtain

Finally, by dividing the last two equations, we find a relationship for the diffuse-diffuse reflectance in terms of the detected powers and the reflectance of a standard (see Teran et al., 2019)

We can observe that this expression depends on the quotient of power detected in three different configurations. An analysis of uncertainty propagation demonstrates that this relationship works well with highly reflective samples. However, caution must be exercised when dealing with opaque samples.

*Transmission geometry*

In the case of diffuse incidence on the sample, the secondary source illuminating the sphere has a power given by:

where *Po'* represents the incident power, and *Td* is the transmittance of the medium under diffuse incidence. We use an apostrophe to differentiate quantities in the transmittance configuration from those in reflectance.

Using a similar procedure to the one employed for measuring the flux or power reaching the detector in reflection, we find that for diffuse incident light

The resulting formula is analogous to expressions (3) for the case of diffuse illumination in reflection. We face the same challenge of estimating the value of *Td* without knowing *Rd*. However, we can tackle this using the same approach employed for directional diffuse reflectance. By dividing last equation by (3), we obtain the diffuse-diffuse transmittance (see Teran et al., 2019)

It is crucial to note that while this simple expression is derived, it necessitates a more complex configuration involving diffuse illumination on the sample. This is achieved using a double integrating sphere arrangement, which can be quite challenging to analyze.

**Final remarks**

In conclusion, this manuscript presents a comprehensive and clear guide to calculating reflectance and transmittance for diffuse illumination. By establishing a robust methodology to assess the optical properties of various materials, we have improved the accuracy and versatility of integrating sphere theory. This approach allows for a deeper understanding of the optical properties of a wide range of materials and makes integrating sphere theory more adaptable for numerous applications, including samples with varying surface properties, such as optical phantoms.

We have also provided a detailed analysis of radiation exchange between diffusing surfaces and derived equations for various scenarios within the integrating sphere, considering the relationship between reflectance, transmittance, and absorption properties. These equations offer a solid foundation for researchers to study and analyze the optical properties of diverse materials, ultimately contributing to advances in numerous fields, such as material science, biomedical optics, and environmental monitoring.

Future work could explore the application of this methodology to more complex scenarios, including different geometries and materials, or extend the analysis to cover wavelength-dependent optical properties. This will further enhance the adaptability and applicability of integrating sphere theory, opening new avenues for research and technological advancements.

**References**

E. Terán, E. R. Méndez, R. Quispe-Siccha, A. Peréz-Pacheco, and F. L. S. Cuppo, "Application of single integrating sphere system to obtain the optical properties of turbid media," OSA Continuum 2, 1791-1806 (2019)

__https://doi.org/10.1364/OSAC.2.001791__John W. Pickering, Christian J. M. Moes, H. J. C. M. Sterenborg, Scott A. Prahl, and Martin J. C. van Gemert. Two integrating spheres with an intervening scattering sample. J. Opt. Soc. Am. A, 9(4):621–631, Apr 1992.

__https://doi.org/10.1364/JOSAA.9.000621__

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