Reflectance matrix approach to absolute photoluminescence measurements with integrating spheres

Luke J. Sandilands; Joanne C. Zwinkels

doi:10.1364/OE.27.000423

1. Introduction

The accurate absolute measurement of photoluminescent materials is important across a range of applications, including quality control in color-intensive industries, biomedical/chemical analysis, and material science [1]. For example, the quantum yield, defined as the number of emitted photons per number absorbed photons, is a key performance metric in the evaluation of phosphorescent modules for LED-based luminaires and other optoelectronic materials [2]. Similarly, the commercial value of optically-brightened paper products is directly tied to their perceived brightness and whiteness. Accurate measurement of these colorimetric quantities is therefore an essential component of quality control and standardization in the pulp and paper industry [3].

Absolute photoluminescence (PL) measurements are often accomplished using an integrating sphere (IS) setup, as this allows the collection of all emitted photons regardless of the spatial distribution of the emission [4]. However, such IS-based measurements are complicated by multiple reflection effects that make the sample illumination and sphere throughput sample dependent. A typical experimental configuration is depicted in Fig. 1, where a photoluminescent sample mounted on an IS is excited by optical radiation at normal incidence. The relevant multiple reflection effects may be separated into two components: secondary excitation and PL reabsorption/emission. In the former, the reflected portion of the excitation flux undergoes multiple reflections in the sphere and so provides secondary diffuse illumination of the sample. In the latter, for materials with finite overlap between their photoluminescence excitation and emission profiles (i.e., small Stokes shift), emitted radiation may be reabsorbed and potentially reemitted by the sample, thereby modifying the sphere throughput and distorting the apparent amplitude and spectral profile of the PL. These multiple reflection effects are known to lead to significant error in quantities characterizing the optical properties of photoluminescent materials such as quantum yield or colorimetric coordinates [5–7].

Several schemes have been proposed in the literature for performing absolute quantum yield measurements using an IS [8–14]. In particular, the method due to de Mello $e t a l .$ [4] is widely used and has been extended by Ahn $e t a l .$ to account for PL reabsorption/emission [5]. However, these schemes suffer from a number of drawbacks. First, knowledge of the ‘true’ emission spectrum (i.e., undisturbed by multiple reflection effects) is required, which may not be feasible in cases where the PL displays an angular dependence, e.g., in LED phosphor modules [15]. Second, the quantum yield is assumed to be independent of wavelength and the sphere throughput is assumed to be unperturbed by the sample in the long wavelength limit, neither of which is necessarily true in the general case [16]. Finally, existing protocols apply only to the quantum yield. The correction of other quantities, such as the colorimetric coordinates, has not been addressed.

Motivated by these issues, we have developed a theory describing an IS with photoluminescent surfaces. The theory is based on the Donaldson reflectance factor matrix formalism for describing reflection from a photoluminescent surface [17, 18]. In conjunction with a bispectral luminescence data set (i.e., obtained by spectrally scanning over both the excitation and emission profiles of the photoluminescent sample), this theory allows for the multiplereflection effects occurring within the IS to be fully corrected for. We tested our matrix theory experimentally by performing absolute PL measurements of a photoluminescent standard using two differently-sized spheres. Multiple reflection effects are expected to become more pronounced in smaller diameter spheres, as the likelihood of reflected or emitted radiation being reabsorbed by the sample is increased. The two data sets are in excellent agreement and therefore strongly support the validity of the proposed matrix method.

Fig. 1 Multiple reflection effects occurring in an integrating sphere with photoluminescent sample.

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2. Matrix theory of integrating spheres with photoluminescent surfaces

The spectral flux density $ϕ_{r} (λ)$ (W/nm) radiating from a photoluminescent surface may, in the steady state and at low excitation densities, be written as

ϕ_{r} (λ) = ρ (λ) ϕ_{e} (λ) + \int_{0}^{\infty} β (λ, μ) ϕ_{e} (μ) d μ .

Here $ρ (λ)$ is the spectral reflectance, $ϕ_{e} (μ)$ (W/nm) is the incident flux density, $β (λ, μ)$ (1/nm) is the bispectral luminescent reflectance factor, and μ and λ (nm) are the excitation and emission wavelengths, respectively. Note that $ρ (λ)$ and $β (λ, μ)$ depend implicitly on the illumination and viewing conditions and that $β (λ, μ) = 0$ when $λ \leq μ$ . It should be stressed that $ρ (λ)$ and $β (λ, μ)$ fully encode the optical properties of a photoluminescent surface: absolute quantities such as the quantum yield and the colorimetric coordinates may be computed provided $ρ (λ)$ and $β (λ, μ)$ are known [1].

For the reasons discussed earlier, Eq. (1) is not straightforward to solve in an IS, as $ϕ_{e} (μ)$ depends self-consistently on $ϕ_{r} (λ)$ [Fig. 1]. As a first step in addressing this, it is convenient to recast Eq. (1) in matrix form. For a discrete set of m wavelengths equally spaced by $Δ μ$ , Eq. (1) may be expressed as

ϕ_{r} (λ_{i}) = ρ (λ_{i}) ϕ_{e} (λ_{i}) + Δ μ \sum_{j = 0}^{m} β (λ_{i}, μ_{j}) ϕ_{e} (μ_{j}) .

In matrix form this becomes

ϕ_{r} = ρ ϕ_{e} + Δ μ β ϕ_{e} = D ϕ_{e},

where ϕ_r and ϕ_e are

m \times 1

vectors specifying the flux at different wavelengths, ρ and β are

ρ (λ_{i})

and

β (λ_{i}, μ_{j})

in

m \times m

matrix form, and D is an

m \times m

Donaldson reflectance factor (DRF) matrix [17, 18]. For a small enough experimental bandwith, the diagonal elements of D are simply the values of

ρ (λ)

at each wavelength. For a non-photoluminescent surface, D is purely diagonal.

The advantage of the DRF matrix formalism is that it allows the usual, multiple reflection theory of the IS to be readily generalized to accommodate photoluminescent surfaces [18]. Consider an IS of radius R, as depicted in Fig. 1. Flux ϕ_o from an external source of optical radiation enters through an open port and illuminates a first strike region at normal incidence whose reflective properties are described by D_f . The reflected flux from the first strike region then undergoes multiple reflections within the sphere. Assuming a perfect spherical geometry and that all surfaces within the sphere are ideally Lambertian (diffusing), the flux from each of these reflections is evenly distributed across the entire sphere surface [18]. The total flux ϕ_t incident on the sphere surface outside the first strike region is therefore given by the sum of these multiple reflections:

\begin{array}{l} ϕ_{t} = ϕ_{1} + ϕ_{2} + ϕ_{3} + \dots \\ = D_{f} ϕ_{o} + \bar{D} D_{f} ϕ_{o} + {\bar{D}}^{2} D_{f} ϕ_{o} + \dots \\ = [I + \bar{D} + {\bar{D}}^{2} + \dots] D_{f} ϕ_{o} \\ = M D_{f} ϕ_{o} . \end{array}

Here $\bar{D}$ is the area-averaged DRF matrix for the sphere interior surface and is given by $\bar{D} = \sum_{p} a_{p} D_{p}$ where the sum runs over all distinct regions p of the sphere surface (sphere wall, sample, ports,...) characterized by DRF matrix D_p and fractional area a_p. The fractional area a_p of a given region p is simply $A_{p} / A_{S}$ where A_p is the area of p and $A_{S} = 4 π R^{2}$ is the total area of the sphere. The sphere response matrix M is closely related to the usual wavelength-dependent sphere multiplier and is purely diagonal when no section of the sphere surface is photoluminescent.

Fig. 2 Two-monochromator configuration with integrating sphere for measuring the D_s matrix of a photoluminescent sample.

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3. Application of the matrix theory to absolute photoluminescence measurements using the two-monochromator method

Fully characterizing the optical properties of a photoluminescent surface requires a bispectral measurement (i.e., spectrally scanning over both the excitation and emission profiles of the material) [17]. Given the matrix form of Eq. (4), this is also true of the throughput of an IS with photoluminescent surfaces. The appropriate bispectral measurements may be performed using a two-monochromator method, such as in the experimental configuration illustrated schematically in Fig. 2. Here an excitation monochromator in combination with a broadband light source functions as a tunable, quasimonochromatic source of optical radiation. This optical radiation is directed onto the sample under study at normal incidence. The sample, as well as a non-photoluminescent reference, are mounted on different ports of an IS. An emission monochromator, combined with a suitable photodetector, views a section of the IS wall and provides spectrally-resolved information on the optical flux in the IS. As discussed in detail in the Appendix, properly treating the two-monochromator configuration requires extending Eq. (4) in two ways in order to account for the spectral characteristics of the excitation and emission units.

The first extension concerns the effective bandwidth $Δ μ$ (nm) which fixes the wavelength sampling interval and is required in order to compute β from D [Eq. (3)]. This quantity is determined by an integral over the normalized convolution of the spectral line spread functions of the excitation and emission units, which we label $x (μ - μ_{j})$ and $e (λ - λ_{j})$ , respectively. Specifically,

Δ μ = \frac{\int_{- \infty}^{\infty} x e (τ) d τ}{x e (0)},

where

x * e (τ)

is the convolution of

x (μ - μ_{j})

and

e (λ - λ_{j})

for wavelength offset τ. In other words,

Δ μ

corresponds to the effective spectral width of the instrumental slit scattering function and may be measured experimentally by fixing the excitation unit at an arbitrary reference wavelength and scanning the emission unit about this reference wavelength in small intervals (i.e., varying τ) [19, 20]. The second extension requires the addition of a further matrix K to account for the spectral responsivity of the emission unit as a whole (diffraction grating, photodetector, amplifier, etc.). The relevant measurement expression for the two-monochromator configuration is therefore

v_{j} = K M D_{f} ϕ_{o, j},

where v_j (V) is a vector made up of the emission unit readings at different wavelengths and

ϕ_{o, j}

is a flux vector representing monochromatic excitation at μ_j.

Combined with bispectral data obtained using the two-monochromator configuration, the DRF matrix method allows for an absolute measurement of D_s that is free of multiple reflection effects. A suitable experimental protocol is as follows. First, bispectral data should be collected over a square $m \times m$ grid of excitation and emission wavelengths, so that the relevant matrices may be inverted. The spacing of this grid is fixed by $Δ μ$ . Bispectral data should be collected in two sample illumination configurations: direct and diffuse. In the direct configuration, the sample is positioned directly in the excitation beam as in Fig. 2. In the diffuse configuration, the positions of the sample and non-photoluminescent reference are interchanged, meaning the sample is diffusely illuminated. For an ideal integrating sphere, the sphere spectral response is unchanged by this permutation. The diffuse illumination data set therefore amounts to characterizing the bispectral responseof the sphere with the sample present. Following Eq. (6), the resulting data sets may be written

V_{s} = K M D_{s} Φ_{o},

V_{r} = K M D_{r} Φ_{o} .

Here subscripts s and r refer to the direct and diffuse configurations while D_s and D_r are the DRF matrices for the sample and reference, respectively. The matrices V_s and V_r are the bispectral data sets arranged in matrix form. Lastly, Φ_o is a diagonal matrix representing the relative excitation spectral flux density at different excitation wavelengths. In principle, stray light in the excitation unit monochromator would introduce off-diagonal elements in Φ_o but is ignored here. The sphere response matrix M

is assumed to be unchanged by the sample/reference interchange. Combining Eqs. (7) and (8) to eliminate KM yields

D_{s} = D_{r} Φ_{o} V_{r}^{- 1} V_{s} Φ_{o}^{- 1} .

For the non-photoluminescent reference, D_r is simply the spectral reflectance of the reference and may be determined with other techniques [21]. Eq. (9) therefore provides a measurement of D_s that fully accounts for multiple reflection effects within the IS.

An additional benefit of the DRF matrix method outlined above is that it also accounts for stray light in the emission unit monochromator and for other sources of PL, such as contamination of the IS interior surface[22–24]. Since PL is often spectrally broad and of weak intensity, it can be a challenge to unambiguously discriminate the true sample PL from these other effects [25]. It is well known that instrumentalstray light in a grating monochromator may be modelled in terms of matrices [26]. In the terminology introduced here, stray light effects are included via the off-diagonal components of K. Similarly, contamination of the IS by a photoluminescent substance simply leads to an additional off-diagonal contribution to $\bar{D}$ and therefore to M. Since KM, the combined IS-emission unit response, is eliminated in Eq. (9), stray light and spurious contaminant PL are both accounted for by the matrix method.

4. Experimental validation

Validation experiments were performed with the National Research Council Reference Goniospectrofluorimeter, a well-characterized two-monochromator instrument that is described in detail elsewhere [27, 28]. The instrument employs a 450 W Xe arc lamp as an excitation source, a pair of grating monochromators (Model 207, McPherson, USA) for the excitation and emission units, and a thermoelectrically-cooled photomultiplier tube for photodetection (R928, Hamamatsu, Japan). The nominal bandpass of both the excitation and emission monochromators is 5 nm. A Si photodiode (THE-2100, EG&G, USA) is used to monitor the excitation unit output and to compensate for time-dependent fluctuations of the source intensity. As a test sample, we used a doped sintered polytetrafluoroethylene (PTFE) photoluminescent standard (USFS-200, Labsphere, USA). Importantly, the excitation and emission profiles of this photoluminescent standard reveal a significant overlap, meaning PL reabsorption/emission is likely to be relevant. As a non-photoluminescent reference, we used either a commerical sintered PTFE standard (Spectralon, USRS-99-020, Labsphere, USA) or a pressed PTFE standard fabricated in house. The spectral diffuse reflectance of both references were measured using a spectrophotometer with a sphere accessory (Lambda 19, Perkin-Elmer, USA) traceable to NRC’s absolute diffuse reflectance scale. Illumination was at near-normal angle of incidence ( $\sim 8^{\circ}$ ) with the specular component included.

Fig. 3 (a)-(c) Matrices V_s, V_r, and D_s measured with the 20 cm sphere. (d)-(f) Matrices V_s, V_r, and D_s measured with the 30 cm sphere. V_s and V_r have been normalized to their maximum values. (g) Normalized slit scattering function NSSF(λ) measured at 410 nm. (h) Diagonal elements $ϕ_{o} (μ)$ of the spectral excitation matrix Φ_o. (i)Sample spectral reflectance $ρ_{s} (μ)$ , corresponding to diagonal elements of matrix D_s measured in the 20 cm sphere.

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Table 1. Dimensions of the two integrating spheres.

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To validate the proposed matrix method, we measured the photoluminescent standard using two ISs of different diameters (20 cm and 30 cm). The dimensions of the two spheres are summarized in Table 1. Both sphere interiors were coated with Spectraflect (Labsphere, USA). Multiple reflection effects should be more pronounced in the 20 cm sphere with larger fractional port area, as the fraction of reflected or emitted radiation reabsorbed by the sample is increased. If the matrix method properly corrects for these effects, the results obtained using the two spheres should agree. The bandwidth parameter $Δ μ$ was evaluated using Eq. (5) at 410 nm, an integration range of $\pm 10$ nm, a wavelength interval of 1 nm, and with non-photoluminescent standards at the sample and reference ports. Bispectral data for the direct and diffuse illumination cases were collected at 10 nm intervals between 300 nm and 700 nm and interpolated to match the experimental $Δ μ = 5.58$ nm. The excitation spectral flux distribution Φ_o was measured using a calibrated Si photodiode (S1337-1010BQ, Hamamatsu, Japan). The sample reflectance factor matrix D_s was then determined using Eq. (9).

Fig. 4 (a) Emission profile $D_{s} (λ)$ under μ = 350 nm excitation for the 20 and 30 cm data sets. (b) Spectral quantum yield QY(μ) for the 20 and 30 cm data sets. (c) Difference between the spectral QY derived from the 20 and 30 cm data sets compared with the experimental reproducibility. Inset to (b): Sample absorption $1 - ρ_{s} (μ)$ measured in the 30 cm sphere.

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The bispectral data and D_s matrices for the two spheres are shown in Fig. 3. The direct illumination data (V_s) for both spheres reveals a broad feature for $λ > μ$ due to sample PL [Figs. 3(a) and 3(d)]. The two-peaked structure of the PL indicates that two different species of fluorophore may be present in the sample. Similar features are evident in the diffuse illumination/20 cm data [Fig. 3(b)], although the magnitude of the sample PL is reduced in this case as the sample only receives diffuse secondary illumination. In contrast, the diffuse illumination/30 cm data [Fig. 3(e)] reveals qualitatively different behaviour. Instead of the two-peaked sample PL, a single broad feature is visible near ( $λ, μ$ ) = (550 nm, 450 nm). Since this feature is only present in the 30 cm sphere data, it may be assigned primarily to unwanted PL from the 30 cm sphere surface [25]. The weak feature visible just off the diagonal in all four illumination/sphere data sets may be assigned to stray light in the emission unit monochromator. The sample D_s

matrices derived from the bispectral data are shown in Figs. 3(c) and 3(f). As expected, the two-peaked sample PL is clearly visible in both cases, while the emission unit stray light and sphere wall PL are suppressed. Below 400 nm in the emission channel, the experimental noise becomes more evident, especially in the 30 cm case, due to the poor sensitivity of the emission unit in this spectral range. Figs. 3(g)–3(i) summarize the normalized slit scattering function data used to determine $Δ μ$ (NSSF), the diagonal elements of the source spectral distribution matrix (ϕ_o), and the diagonal elements (ρ_s) of D_s, respectively.

The sample D_s matrices obtained from the 20 cm and 30 cm data sets are compared in Fig. 4. Representative emission profiles $D_{s} (λ)$ for μ = 350 nm are shown in Fig. 4(a). Excellent agreement is found both in terms of absolute scale and spectral profile. A similar level of agreement is found for the spectral quantum yield (QY) shown in Fig. 4(b), which is computed via

Q Y (μ) = \frac{\int_{350}^{700} λ β_{s} (λ, μ) d λ / μ}{1 - ρ_{s} (μ)},

where the integral runs over the emission spectral range of the PL [29]. Starting at μ = 500 nm, the QY curves rapidly increase, reflecting the onset of fluorophore absorption, before reaching a relatively constant QY

\sim 72 %

near 400 nm. A sharp decrease in QY then occurs at 340 nm, which may reflect the onset of a distinct photoexcited state with lower effective QY. As with the emission profiles, the spectral quantum yields from the two spheres are in excellent agreement, both in terms of absolute scale and spectral line shape. In fact, the two data sets are indistinguishable to within the reproducibility of our measurements. From repeated runs with the 30 cm sphere, we estimate that the standard deviation is about

0.3 %

for the integrated photoluminescent signal and

0.45 %

for ρ_s. The total expanded (k = 2) uncertainty due to experimental reproducibility is estimated by adding these two components in quadrature and multiplying by a coverage factor of 2. In Fig. 4(c), we compare this total reproducibility with the difference in spectral quantum yields ΔQY determined from the 20 cm and 30 cm data. Except for a single point at μ = 500 nm, where both the PL and the absorption

1 - ρ_{s}

become very small, ΔQY is well within the estimated experimental reproducibility. The excellent agreement between the D_s matrices obtained for different diameter spheres indicates that any multiple reflection effects are fully accounted for and so provides strong support for the validity of the matrix method.

Fig. 5 (a) Emission profiles $D_{s} (λ)$ under μ = 350 nm excitation and (b) spectral quantum yields QY(μ) from the full matrix method for the 20 cm sphere compared with the results of the partial matrix method for the 20 and 30 cm spheres.

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The DRF matrix approach allows us to quantitatively assess the magnitude of the sample-dependent changes to the illumination and sphere throughput. We do this by comparing the results of the full matrix treatment [Figs. 3 and 4] with apartial matrix method. In the partial method, the diffuse illumination bispectral scan is performed with a second non-photoluminescent reference, rather than the photoluminescent sample, present at the IS reference port. In other words, the sphere multiplier matrix M is determined without the sample present on the sphere. The partial method D_s is then computed via Eq. (9) using this alternate data set in place of V_r. The partial method therefore mimics conventional calibration schemes [13, 30], while also accounting to some extent for unwanted PL from the IS surface and stray light in the emission unit. Differences between the partial and full matrix approach may therefore be largely attributed to sample-dependent multiple reflection effects.

The results of the partial and full matrix methods are compared in Fig. 5. In Fig. 5(a), the emission profile $D_{s} (λ)$ under μ = 350 nm illumination is shown for the 20 cm/partial, 30 cm/partial, and 20 cm/full cases. The absolute value of $D_{s} (λ)$ in the 20 cm/partial case is significantly overestimated compared with the full matrix method, while the spectral profile is also distorted (note the relative height of the two emission peaks). The former may be primarily attributed to secondary diffuse illumination, while the latter is due to PL reabsorption/emission, as this tends to skew the spectrum towards longer wavelengths. The difference between the partial and full treatments is less significant for the 30 cm sphere: the absolute value of the emission is in good agreement, while only a small spectral distortion is evident towards shorter wavelengths. Similar results are obtained for the spectral QY profiles [Fig. 5(b)]. The 20 cm/partial case overestimates the QY by as much as 13% ( $\sim 7 %$ on average) compared to the full result, while for the 30 cm/partial case the difference is at most 7% (1% on average) and is almost negligible at shorter wavelengths. Overall, the results of Fig. 5 support the expectation that multiple reflection effects are more pronounced in the smaller diameter sphere with larger fractional port area, as the fraction of reflected or emitted radiation returning to the sample surface for reabsorption is increased.

5. Summary and outlook

In conclusion, a matrix theory describing an IS with photoluminescent surfaces has been developed. In combination with a bispectral data set, the matrix theory allows for the sample-dependent multiple reflection effects that normally plague IS-based absolute PL measurements to be fully accounted for. The bispectral data set is collected using a two-monochromator configuration and by mounting both the sample and a non-luminescent reference on the sphere and permuting their positions in order to realize both direct and diffuse sample illumination conditions. The matrix method has been validated experimentally, with data for a photoluminescent standard collected using two spheres of different diameter showing excellent agreement. The advantages of the matrix method presented here are most pronounced in cases with significant PL reabsorption/emission (i.e., samples with small Stokes shifts) and/or luminescence from the IS walls. When these problematic multiple reflection effects are negligible, we expect that the standard methodology for measuring QY should produce reliable results [4].

It should be emphasized that while we have focused here on the measurement of surface PL from a diffuse reflector, the matrix method readily generalizes to other common experimental geometries (transmittance, liquid sample in cuvette, thin film, etc.) provided both direct and diffuse illumination bispectral data can be acquired. We wish to make two comments on the broader applicability of the matrix method. First, the meaning of the sample matrix depends on geometry. For a cuvette placed inside of the sphere, for example, the sample matrix should be interpreted as a directional-spherical scattering matrix, rather than a reflectance matrix. In this case, the diagonal elements of the sample matrix correspond to the fraction of the excitation flux transmitted or scattered over the full 4π solid angle (in other words, 1 - A where A is the sample absorptance), while the off-diagonal elements correspond to the total emitted PL. Second, a possible complication may arise when the spatial structure of the sample emission differs strongly from that of the reference. Consider, for example, an optically-flat thin film sample and a diffuse reflecting reference. In this case, the first highly-specular reflection from the sample surface may well fall on a region of the sphere interior whose local reflectance differs from the area-averaged response of the entire sphere interior, leading to an error in the measured sample reflectance. Similar considerations apply to any spatial structure in the PL emission, which is also referenced to the radiation reflected from the reference. A full uncertainty analysis would have to account for the contribution of such variations.

Appendix A Details of the matrix method applied to the two-monochromator configuration

The two-monochromator method, depicted schematically in Fig. 2 of the main text, allows for the excitation and emission characteristics of a photoluminescent surface to be fully characterized. In order to properly apply the DRF matrix method to this scheme, it is necessary to account for the spectral characteristics of the excitation and emission units. The spectral characteristics of the excitation and emission units are defined as follows. We focus on discrete sets of m excitation and emission wavelengths μ_i and λ_j, as in a real experiment. These sets are taken to be equal, i.e., μ_i = λ_i. Due to the finite bandpass of the excitation unit, the optical flux $X_{j} (μ)$ (W/nm) delivered to the sample by the excitation unit maybe expressed as

X_{j} (μ) = ϕ_{o} (μ_{j}) x (μ - μ_{j}),

where μ_j is the nominal excitation wavelength,

x (μ - μ_{j})

is the linespread function of the excitation monochromator, and

ϕ_{o} (μ_{j})

(W/nm) is the spectral flux density at

μ = μ_{j}

. Similarly, the spectral response (V/W) of the emission unit is

E_{i} (λ) = A (λ_{i}) e (λ - λ_{i}),

where λ_i is the nominal excitation wavelength,

e (λ - λ_{i})

is the linespread function of the emission monochromator, and

A (λ_{i})

(V/W) represents the spectral response of the emission unit plus IS wall response at

λ = λ_{i}

. The linespread functions are assumed to be independent of λ_i and μ_j, all dependence on the nominal wavelengths being subsumed into

ϕ_{o} (μ_{i})

and

A (λ_{j})

, and are defined such that

e (0) = x (0) = 1

.

At a given λ_i and μ_j, the signal $v_{i j}$ measured by the emission unit is given by a sum of contributions $v_{i j} = v_{i j}^{1} + v_{i j}^{2} + \dots$ , corresponding to the multiple reflections $ϕ_{j} (λ) = ϕ_{j}^{1} (λ) + ϕ_{j}^{2} (λ) + \dots$ occuring within the integrating sphere under excitation at μ_j. Following similar reasoning to section 2 of the main text, the first reflection spectral flux $ϕ_{j}^{1} (λ)$ for excitation at μ_j is

ϕ_{j}^{1} (λ) = \int_{0}^{\infty} [ρ_{f} (μ) δ (μ - λ) + β_{f} (λ, μ)] X_{j} (μ) d μ,

where

ρ_{f} (μ)

and

β_{f} (λ, μ)

describe the optical properties of the first strike region. The corresponding signal measured by the emission unit is therefore

\begin{array}{l} v_{i j}^{1} = \int_{0}^{\infty} E_{i} (λ) ϕ_{j}^{1} (λ) d λ \\ = \int_{0}^{\infty} E_{i} (λ) \int_{0}^{\infty} X_{j} (μ) [ρ_{f} (μ) δ (μ - λ) + β_{f} (λ, μ)] d μ d λ \\ \approx A (λ_{i}) ϕ_{o} (μ_{j}) ρ_{f} (μ_{j}) δ_{i j} \int_{0}^{\infty} e (λ - λ_{i}) x (λ - λ_{i}) d λ \\ + A (λ_{i}) ϕ_{o} (μ_{j}) β_{f} (λ_{i}, μ_{j}) \int_{0}^{\infty} e (λ - λ_{i}) d λ \int_{0}^{\infty} x (μ - μ_{j}) d μ \\ \approx K_{i} [ρ_{f} (μ_{j}) δ_{i j} + Δ μ β_{f} (λ_{i}, μ_{j})] ϕ_{o} (μ_{j}) \\ \approx K_{i} D_{f, i j} ϕ_{o} (μ_{j}) . \end{array}

Here we have assumed that the spectral overlap between $e (λ - λ_{j})$ and $(λ - λ_{j})$ is negligible for $i \neq j$ . The instrumental response K_i is defined as

K_{i} = A (λ_{i}) \int_{0}^{\infty} d λ e (λ - λ_{i}) x (λ - λ_{i}),

and the effective bandwidth

Δ μ

as

Δ μ = \frac{\int_{0}^{\infty} x (μ - μ_{j}) d μ \int_{0}^{\infty} e (λ - λ_{j}) d λ}{\int_{0}^{\infty} x (λ - λ_{j}) e (λ - λ_{j}) d λ} = \frac{\int_{- \infty}^{\infty} x e (τ) d τ}{x e (0)},

where the second equality in Eq. (16) follows from the convolution theorem [31]. The effective bandwith parameter

Δ μ

is the appropriate wavelength interval for the bispectral data and is also needed to relate

s y m b o l D

to β [Eq. (3) in the main text]. In matrix form, Eq. (14) reads

v_{j}^{1} = K D_{f} ϕ_{o, j} .

The contributions from further reflections are derived in a similar if tedious manner. For example, the second reflection spectral flux $ϕ_{j}^{2} (λ)$ is

ϕ_{j}^{2} (λ) = \int_{0}^{\infty} [\bar{ρ} (γ) δ (γ - λ) + \bar{β} (λ, γ)] ϕ_{j}^{1} (γ) d γ

where

\bar{ρ} (γ)

and

\bar{β} (λ, γ)

are the area-averaged spectral reflectance and bispectral luminescence reflectance factor of the sphere interior, as defined in the main text. The second reflection makes a contribution

v_{i j}^{2}

to the total signal that is given by

\begin{array}{l} v_{i j}^{2} = \int_{0}^{\infty} E_{i} (λ) ϕ_{j}^{2} (λ) d λ \\ = A (λ_{i}) \int_{0}^{\infty} e (λ - λ_{i}) \bar{ρ} (λ) ϕ_{1}^{j} (λ) d λ + A (λ_{i}) \int_{0}^{\infty} e (λ - λ_{i}) \bar{β} (λ, γ) ϕ_{1}^{j} (γ) d γ d λ \\ \approx K_{i} \bar{ρ} (λ_{i}) D_{f, i j} ϕ_{o} (μ_{j}) + A (λ_{i}) ϕ_{o} (μ_{j}) \int_{0}^{\infty} \int_{0}^{\infty} e (λ - λ_{i}) \bar{β} (λ, γ) ρ_{f} (γ) x (γ - μ_{j}) d γ d λ \\ + A (λ_{i}) ϕ_{o} (μ_{j}) \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} e (λ - λ_{i}) \bar{β} (λ, γ) β_{f} (γ, μ) x (μ - μ_{j}) d γ d λ d μ \\ \approx K_{i} \bar{ρ} (λ_{i}) D_{f, i j} ϕ_{o} (μ_{j}) + A (λ_{i}) ϕ_{o} (μ_{j}) ρ_{f} (μ_{j}) \bar{β} (λ_{i}, μ_{j}) \int_{0}^{\infty} e (λ - λ_{i}) d λ \int_{0}^{\infty} x (γ - μ_{j}) d γ \\ + A (λ_{i}) ϕ_{o} (μ_{j}) \int_{0}^{\infty} \bar{β} (λ_{i}, γ) β_{f} (γ, μ_{j}) d γ \int_{0}^{\infty} e (λ - λ_{i}) d λ \int_{0}^{\infty} x (μ - μ_{j}) d μ \\ \approx K_{i} \bar{ρ} (λ_{i}) D_{f, i j} ϕ_{o} (μ_{j}) + K_{i} Δ μ ρ_{f} (μ_{j}) \bar{β} (λ_{i}, μ_{j}) ϕ_{o} (μ_{j}) \\ + K_{i} Δ μ ϕ_{o} (μ_{j}) \int_{0}^{\infty} \bar{β} (λ_{i}, γ) β_{f} (γ, μ_{j}) d γ \\ \approx K_{i} \bar{ρ} (λ_{i}) D_{f, i j} ϕ_{o} (μ_{j}) + K_{i} Δ μ ρ_{f} (μ_{j}) \bar{β} (λ_{i}, μ_{j}) ϕ_{o} (μ_{j}) \\ + K_{i} {(Δ μ)}^{2} ϕ_{o} (μ_{j}) \sum_{l = 0}^{m} \bar{β} (λ_{i}, γ_{l}) β_{f} (γ_{l}, μ_{j}) . \end{array}

In matrix form, the second reflection contribution becomes

\begin{array}{l} v_{j}^{2} \approx K \bar{ρ} D_{f} ϕ_{o, j} + Δ μ K \bar{β} ρ_{f} ϕ_{o, j} + {(Δ μ)}^{2} K \bar{β} β_{f} ϕ_{o, j} \\ \approx K [\bar{ρ} + Δ μ \bar{β}] D_{f} ϕ_{o, j} \\ \approx K \bar{D} D_{f} ϕ_{o, j} \end{array}

The contributions to the measured signal may be summed up, leading to Eq. (6) in the main text:

v_{j} = K [I + \bar{D} + {\bar{D}}^{2} + \dots] D_{f} ϕ_{o, j} = K M D_{f} ϕ_{o, j} .

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Sphere diameter (cm)	Port diameter (cm)	Fractional port area $a_{p o r t}$
20.3	3.8	$8.8 \times 10^{- 3}$
30.0	3.8	$4.0 \times 10^{- 3}$

Reflectance matrix approach to absolute photoluminescence measurements with integrating spheres

Abstract

1. Introduction

2. Matrix theory of integrating spheres with photoluminescent surfaces

3. Application of the matrix theory to absolute photoluminescence measurements using the two-monochromator method

4. Experimental validation

5. Summary and outlook

Appendix A Details of the matrix method applied to the two-monochromator configuration

References

Cited By

Figures (5)

Tables (1)

Equations (21)

Optics Express