Depth profiles in confocal optical microscopy: a simulation approach based on the second Rayleigh-Sommerfeld diffraction integral

Rosario Esposito; Giuseppe Scherillo; Marianna Pannico; Pellegrino Musto; Sergio De Nicola; Giuseppe Mensitieri

doi:10.1364/OE.24.012565

1. Introduction

Confocal spectroscopy is a well-known, non-destructive analytical technique which provides high spatial resolution coupled with high sensitivity to the molecular and structural features of a sample. In the last decade this technique has been established as a powerful tool for the analysis of polymeric materials, especially when structural and/or compositional heterogeneities are present [1–7], as is the case of semi-crystalline polymers and of immiscible, phase-separated polymer blends. Confocal spectroscopy finds extensive application also for the inspection of thick biological specimens stained with fluorescent labels where it has contributed to the current renaissance in light microscopy [8,9].

The depth mode measurement in confocal Raman spectroscopy consists in focusing the laser beam at a certain point of the sample, acquiring the emitted Raman radiation and, then, by moving the sample stage along the direction of the microscope optical axis (z axis). Repeated measurements in several points along this axis provides a signal intensity as a function of depth within the sample. Numerous contributions have appeared in the literature in which the depth-scanning mode was employed for investigating the distribution of molecular species in a wide range of matrices; this techniques was found particularly suitable for optically transparent samples [5,10].

However, focusing the laser beam onto a series of aligned small spot does provide neither a faithful reproduction of actual concentration profiles nor a good spatial resolution. In fact, laser refraction perturbs the collection efficiency of the confocal system causing intensity reduction and broadening of the detected Raman signal with focusing depth [11]. Theoretical approaches have been developed in confocal imaging for reliable data analyses in presence of refractive-index-mismatched media. The refractive index mismatch between the surroundings and the specimen has been accounted for in the framework of the scalar wave optics to predict the axial intensity behavior for a plane wave incident upon a high numerical aperture objective lens. A phase aberration function was introduced to design beam-shaping elements for reduction of spherical aberration in the focusing properties of the system [12–16].

Everall was the first to point out the misinterpretation of data obtained by confocal Raman microscopy when laser refraction through a planar dielectric interface is ignored [17, 18]. Indeed, a simple analysis based on the principles of geometrical optics clearly evidences that the spatial resolution decreases when focusing deeper and deeper within the sample and shows that the spatial distribution of the illuminated volume depends both on the pinhole size and the penetration depth [11, 19–21]. The pioneering work of Everall stimulated further contributions. Baldwin at al. [22] developed a simple model which accounted only for the refraction and neglected the diffraction. The model described the decrease in the collected Raman intensity with focusing depth according to depth-profiling experiments on transparent samples. Then, a simple modification to this model was proposed by Bruneel et al. [23] which considered the introduction of an additional term to describe spherical aberration contributions caused by off-axis laser intensity distribution. A rigorous treatment was developed by Sourisseau et al. who modeled the axial and lateral diffraction as well as the transmission of the aperture [24], predicting the depth profiles. However, this procedure was computationally expensive and more tractable approaches have been proposed incorporating the effects of both diffraction and confocal aperture in semi-empirical models [25–28]. Macdonald et al. described a novel photon scattering approach which neglected the refraction at the sample surface and allowed to predict the depth profile response when confocal Raman spectroscopy is applied both to silicon and to a number of polymeric materials of varying optical clarity [29]. A general approach based on regularized deconvolution has been reported by Tomba et al. [30]. The approach is designed to recover realistic Raman intensity depth profiles from distorted measurements carried out with dry objectives by using a strategy that does not assume any specific shape for the recovered profile and adopts a simple predictive model to describe the depth response.

Recently, Maruyama et al. [31] have introduced a modeling approach that takes into account the wave distortion caused by the transmission of a Gaussian laser beam through the sample-air interface and the diffraction effects caused by the objective lens. The refractive index mismatch effects have been described by using the Huygens-Fresnel principle expressed as a superposition of spherical waves with an inclination factor. Then the propagation of the field amplitude through the optical system to the image screen has been described in the framework of scalar wave optics theory. The power image of a point emitter (input response function) has been calculated by using a two-dimensional (2-D) Fourier transform of a kernel which is expressed as a 2-D integral. Finally, the intensity depth profile is obtained by calculating numerically a three-dimensional (3-D) integral of the input response function and the Gaussian distribution of the exciting beam. The comparison with experimental results showed that this model is able to describe quite accurately depth profiles up to a penetration depth of about 30μm.

In this contribution we present a theoretical approach capable of recovering the actual spatial distribution of intensity in Raman confocal microscopy with an extended depth of field. The effects of the refractive index mismatch is taken into account by treating the surface of separation between the sample and the medium surrounding the objective microscope as a diffraction plane. The method is applicable also to the case in which the emitters are close to sample surface, a situation in which the scalar wave optics based on the Fresnel-Kirchhoff diffraction integral is known to lead to mathematical inconsistency in the calculation of the field [32,33]. The adopted method uses the second Rayleigh-Sommerfeld diffraction integral [34] within the framework of the scalar wave theory, to describe the diffraction effects and the propagation of the distorted field through a system of two converging lenses. (sec. 2.1). The illumination volume within the sample is determined by the spatial distribution of the intensity of the focused transmitted Gaussian laser beam in the aberration free case (sec. 2.2). We show that it is possible to obtain at a simple expression of the point spread function through a circular pinhole in terms of a 1D integral. The advantage of this method compared to approach of [31], is that it allows for a better understanding of the role of the physical parameters involved in the theory and that it also presents a substantial simplification of the formulation of the forward and inverse problem.

The theoretical expression for the Raman signal as a function of the nominal depth of the focused beam is calculated from the point spread function and the spatial distribution of the Raman emitters assumed to emit incoherently. A comparison with the experimental data requires accurate estimates of the position of the sample surface and of the amplitude factor that accounts for the unknown power of the focused laser beam. To this regard, a step-by-step procedure is usually employed for estimating these parameters until a reduction in the final chi-squared is obtained, but the efficacy of such a procedure depends critically on the expertise of the operator and on a proper guess of the beam waist w₀ which is known only indirectly from the spatial resolution of an objective microscope (see [31]). To circumvent problems related to the limited reliability of this procedure, we have adopted here a new iterative multi-curve fitting scheme based on the Brent Method [35] to determine the minimum of the chi-squared value between the experimental data and the model predictions as a function of beam waist w₀, which is now used as a fitting parameter. In turn, for each value of w₀, the fitting procedure uses the conjugate gradient method to estimate the values of the position of the sample surface and of the amplitude factor (sec. 3) [36], which are used as well as fitting parameters. The accuracy of the predictions of the proposed theoretical model has been investigated through comparison with experimental measurements of the Raman depth profiles for different pinhole diameters, performed by us on polystyrene sheets and with depth profile data available in the literature [31].

2. Theory

2.1. The system response function of the confocal Raman microscope: the Rayleigh-Sommerfeld approach

In this section we illustrate the calculation procedure of the response function of a confocal Raman microscope by using the second Rayleigh-Sommerfeld diffraction integral [34] for determining the spatial distribution of the field focused on the image plane. This approach accounts for the boundary conditions in the presence of refractive index mismatch across an interface which causes a distortion of the spherical wave emitted by a source P located at a depth Δ on the optical axis z inside the sample [32,33].

Figure 1 shows the configuration of a confocal Raman microscope from the point of view of the point-like emitter P. An infinity-corrected objective lens L₀ is coupled confocally to a tube lens L_t to generate the image on the plane where a pinhole intercept the field near the optical axis. The pinhole is located at a distance f_t (the tube focal length); Σ is the surface of separation between the sample with refractive index n₂ and the medium with refractive index n₁ which surrounds the lens L₀. The field at a point P₁of the interface Σ is given by

U_{P} (P_{1}) = U_{0} \frac{\exp (- 1 n_{2} k_{e m} s)}{s} k_{e m} = \frac{2 π}{λ_{e m}}

where U₀ is the amplitude factor, λ_em is the emitted wavelength and s is the distance between the source at P and the point P₁ where the field is evaluated (see Fig. 2).

Fig. 1 Schematic of a confocal Raman microscope from the point of view of a point-like emitter P located at a depth Δ inside a sample with refractive index n₂. Σ is the surface of separation between the sample and the surrounding medium of refractive index n₁.

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Fig. 2 Schematic representation of the transmitted field across the interface Σ. An optical ray (red dotted line) emitted by P moves towards the sample surface. The ray intercepts the interface at P₁ and changes its direction (red solid line) due to the refractive index mismatch between the sample and the surrounding medium. The refracted ray hits the lens surface L₀ at P′ that is distant d from the interface Σ and is distant r′ from P₁. The unit vector n is the surface normal.

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According to the second Rayleigh-Sommerfeld diffraction integral, the distribution of the diffracted light at P′ (x′,y′) on the lens surface L₀ can be obtained from the normal derivative of the field at the interface Σ with an integral representation given by:

\begin{array}{l} U (x^{'}, y^{'}, d | 0, 0, Δ) = - \frac{1}{2 π} \int_{\sum} \frac{\partial U_{P} (P_{1})}{\partial n} \frac{\exp (- 1 n_{1} k_{e m} r)}{r^{'}} d \sum \\ w i t h r^{'} = \sqrt{{(x - x^{'})}^{2} + {(y - y^{'})}^{2} + d^{2} .} \end{array}

From Eqs. (1) and (2) we obtain:

\begin{array}{l} U (x^{'}, y^{'}, d | 0, 0, Δ) = Δ U_{0} \int_{0}^{\infty} \frac{e^{(- 1 n_{2} k_{e m} s)}}{s} \frac{\exp (- 1 n_{1} k_{e m} r^{'})}{r^{'}} (\frac{1 n_{2} k_{e m}}{s} + \frac{1}{s^{2}}) r d r \\ w i t h s = \sqrt{r^{2} + Δ^{2}} \end{array}

where Δ is the distance of the source P from the interface Σ.

The kernel in Eq. (3) can be simplified within the framework of Fresnel approximation. The validity limit of the Fresnel propagation through the space between the sample surface Σ and the objective lens L₀ is given in Goodman’s and Saleh textbooks [32, 37] by the condition $a^{2} θ_{m}^{2} / 4 λ_{e m} f_{m} ≪ 1$ , where the quantity a = f_m × NA is the maximum radial distance and θ_m = a/f_m is the largest angle taken by the beam. This condition is overly pessimistic as explained by Goodman [32]. Indeed, Papoulis shows in [38] that the Fresnel approximation can also be used for a point emitter on the surface Σ provided that its distance d from the lens aperture satisfies the condition d > 400 ×λ_em. This condition is largely satisfied for wavelength in the visible spectral range and distance d comparable with the typical focal length f_m of a microscope objective. By applying the Fresnel approximation to the spherical wave exp(−n₁k_emr′)/r′ in the Eq. (3), the complex amplitude of the diffracted field U(x′,y′,d|0,0,Δ) can be written in the following simplified form:

\begin{array}{l} U (x^{'}, y^{'}, d | 0, 0, Δ) = \\ = \frac{Δ U_{0}}{d} e^{(- 1 n_{1} k_{e m} d)} \int_{0}^{\infty} \frac{e^{(- 1 n_{2} k_{e m} s)}}{s} (\frac{1 n_{2} k_{e m}}{s} + \frac{1}{s^{2}}) \exp [- 1 n_{1} k_{e m} \frac{{(x - x^{'})}^{2} + {(y - y^{'})}^{2}}{2 d}] r d r . \end{array}

Within the framework of the Fresnel approximation and taking into account the quadratic phase factor introduced by the objective of focal length f_m, the complex amplitude of the field at distance l on the tube lens L_t reads

U (x^{'}, y^{'}, d + 1 | Δ) = \frac{e^{- i k_{e m} l}}{1} \iint U (x^{″}, y^{″}, d | Δ) \times e^{1 n_{1} k_{e m} \frac{{x^{″}}^{2} + {y^{″}}^{2}}{2 f_{m}}} e^{- i 1 n_{1} k_{e m} \frac{{(x^{″} - x^{'})}^{2} + {(y^{″} - y^{'})}^{2}}{21}} d x^{″} d y^{″} .

Finally, the propagation of the field over a distance f_t from the tube lens gives the field at the image plane S

U (x^{'}, y^{'}, L | Δ) = \frac{e^{- 1 k_{e m} f_{t}}}{f_{t}} \iint U (x^{‴}, y^{‴}, d + 1 | Δ) \times e^{1 k_{e m} \frac{{x^{‴}}^{2} + {y^{‴}}^{2}}{2 f_{t}}} e^{- 1 k_{e m} \frac{{(x^{‴} - x^{″})}^{2} + {(y^{‴} - y^{″})}^{2}}{2 f_{t}}} d x^{‴} d y^{‴}

where L = d + l + f_t (see Fig. 1) is the distance of the interface Σ and the image plane S.

The Fresnel diffraction approximation used for obtaining Eqs. (5) and (6) is ensured if the condition $a^{2} θ_{m}^{2} / 4 λ_{e m} f_{t} ≪ 1$ is fulfilled for the beam path between the lens L₀ and the image plane S [37], with the maximum radial distance a = f_m × NA and largest angle θ_m = a/f_t. In the case at hand, for an objective microscope with numerical aperture NA = 0.9, focal distance f_m = 1.8 × 10³ μm, a tube lens focal f_t = 252 × 10³ μm and a wavelength λ_em ~ 0.5 μm, the condition is fulfilled since we have $a^{2} θ_{m}^{2} / 4 λ_{e m} f_{t} \underline{\approx} 2.15 \times 10^{- 4}$ .

Equation 6 is the expression of the system response function of the confocal Raman microscope derived by using the second Rayleigh-Sommerfeld diffraction integral. After somewhat lengthy but straightforward algebra and passing to cylindrical coordinates in the image plane this expression simplifies and reduces to the evaluation of 1-D integral, namely:

U (r^{'}, L | Δ) \propto Δ \int_{0}^{\infty} \frac{e^{(- 1 n_{2} k_{e m} s)}}{s} (\frac{1 n_{2} k_{e m}}{s} + \frac{1}{s^{2}}) \times e^{1 f r^{2}} J_{0} (w r) r d r

with

z = f_{m} - d f = \frac{n_{1} k_{e m}}{2 z} w = \frac{n_{1} k_{e m} f_{m}}{f_{t} z} r^{'}

where J₀ is the Bessel function of the first kind of order zero. The spatial distribution of the normalized intensity on the image plane S (Fig. 1) is given by

I_{p} (r^{'}, L | Δ) = \frac{{| U (r^{'}, L | Δ) |}^{2}}{2 π \int_{0}^{\infty} r^{'} {| U (r^{'}, L | Δ) |}^{2} d r^{'}} .

This result can be easily extended to case in which the emitter P is located off axis. For a unit point emitter with cylindrical coordinates (ρ_p,θ_p,Δ) we have [38]:

I (r^{'}, θ^{'} | ρ_{p}, θ_{p}, Δ) = \frac{1}{{| M |}^{2}} I_{p} (\sqrt{{r^{'}}^{2} + M^{2} ρ_{p}^{2} - 2 M r^{'} ρ_{p} \cos (θ^{'} - θ_{p}), L | Δ}),

where M = f_t/f_m is the magnification of the confocal system. Equation 9 reduces to the on-axis system response function given by Eq. (8) when the radial distance ρ_p of the source from the optical axis is zero.

2.2. Depth profiles: theoretical analysis

We assume that the incident field is a fundamental Gaussian beam focused on the object plane. Figure 3 depicts a collimated Gaussian laser beam with excitation wavelength λ_ex transmitted through the notch filter towards the objective lens L₀. The propagation direction coincides with the optical axis and the beam is focused through the interface Σ to provide a spatially confined excitation source. The actual focus position depends on the refractive index mismatch at Σ and it is given by p = (n₂/n₁)z where z = f_m − d is the nominal focal position below Σ [39].

Fig. 3 Schematic of a confocal Raman microscope from the point of view of the input source. The focal length of the lens L₀ is f_m. The nominal focal position is z = f_m −d and the actual focus position compared to nominal focal position is p = (n₂/n₁)z.

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The spatial distribution of the focused intensity in the excitation volume inside the sample is given by:

I_{i l l} (ρ_{p}, Δ, z) = I_{0} {[\frac{w_{0}}{w (Δ, z)}]}^{2} e^{- \frac{2 ρ_{p}^{2}}{w^{2} (Δ, z)}},

where ρ_p and Δ are the radial distance from the optical axis and the depth respectively of an illuminated point. The quantity w(Δ,z) denotes the beam-waist radius at the axial position and is given by the following formula:

w (Δ, z) = w_{0} \sqrt{1 + {(\frac{Δ - (n_{2} / n_{1}) z}{n_{2} z_{0}})}^{2}},

where

z_{0} = π w_{0}^{2} / λ_{e x}

is the Rayleigh range. In particular the beam-waist radius reaches the minimum value w₀ at the actual focus position p = (n₂/n₁)z where the intensity I₀ corresponds to the maximum of the excitation field. The intensity distribution at different position Δ of the point emitters along the optical axis has a Lorentzian shape (see Fig. 4):

I_{i l l} (0, Δ, z) = \frac{I_{0}}{1 + {[\frac{Δ - (n_{2} / n_{1}) z}{z_{0}}]}^{2}} .

Fig. 4 The normalized on-axis gaussian laser beam intensity I_ill/I₀ vs the depth Δ−(n₂/n₁)z, that is the distance of the point emitter from the actual focus position.

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A molecule located inside the excitation volume at depth Δ and radial distance ρ_p is a point-like Raman emitter which radiates with intensity proportional to the excitation intensity I_ill(ρ_p,Δ,z). According to the results of the previous section, the spherical wave emitted by the molecule propagates through the Raman optical system to form an image on the plane S with intensity distribution I_ill(ρ_p,Δ,z)×I(r′,θ′|ρ_p,θ_p,Δ) where I(r′,θ′|ρ_p,θ_p,Δ) is the system response function represented by the Eq. (9) in the previous section.

In the case of Raman emission determined by a set of randomly vibrating molecules, the total integrated intensity distribution on the image plane is given by:

P (r^{'}, θ^{'}, w_{0}, z) = ∭ ρ_{p} I_{i l l} (ρ_{p}, Δ, w_{0}, z) \times I_{p} (r^{'}, θ^{'} | ρ_{p}, θ_{p}, Δ) d ρ_{p} d θ_{p} d Δ,

where we have made explicit the dependence on the waist radius w₀. The resulting signal is the power detected through the pinhole of radius D/2 centered on the optical axis and it reads as:

P_{t o t} (w_{0}, z) = 2 π \int_{0}^{D / 2} \int_{0}^{2 π} r^{'} P (r^{'}, θ^{'}, w_{0}, z) d r^{'} d θ^{'} .

From Eqs. (13) and (14) we have:

P_{t o t} (w_{0}, z) = 2 π \iint ρ_{p} I_{i l l} (ρ_{p}, Δ, w_{0}, z) P_{p} (ρ_{p}, Δ) d ρ_{p} d Δ .

The quantity P_p(ρ_p,Δ) in Eq. (15) is the power carried by a point emitter through the pinhole surface and it is given by:

P_{p} (ρ_{p}, Δ) = \int_{0}^{D / 2} \int_{0}^{2 π} I_{p} (r^{'}, θ^{'} | ρ_{p}, θ_{p}, Δ) r^{'} d r^{'} d θ^{'} .

As it can be seen this quantity does not depend on the angular position θ_p of the emitter because of the cylinder symmetry.

3. Comparison with experimental results

Experimental measurements have been performed by using a commercial confocal laser Raman micro-spectroscope (Horiba-Jobin Yvon Mod. Labspec Aramis) equipped with a motorized x-y-z stage, a continuously size-adjustable square-shaped confocal pinhole, and a laser source with wavelength λ_ex = 0.532μm. The width of the slit immediately in front of the spectrometer is always fully opened so that all the light passing through the confocal pinhole travels the spectrometer to the CCD detector. The optical configuration adopted for measurements consists of an infinity-corrected Olympus MPlan achromatic 100× objective lens L₀ with f_m = 1.8 × 10³ μm and numerical aperture NA = 0.9 that is coupled to the tube lens L_t with distance f_t = 252 × 10³ μm from the image plane S which gives a magnification M = f_t/f_m = 140.

Figure 5 shows the normalized intensity of the 997 cm⁻¹ peak for the case of a polystyrene thick sheet as a function of the nominal penetration depth z of the focused laser beam (black points). The measurements have been carried out by using two pinholes sizes D = 200 μm and D = 400 μm. In the LabRam instrument, the pinhole is squared and the value one sets in the theoretical model Eq. (15) corresponds to the half diagonal D/2 of the square since it has been very clearly shown that the results are sufficiently independent of the geometry, at least for circles and squares, with differences that vary in a range of some 2% [40]. Despite the fact that the expected dependence of peak intensity on the nominal depth should result in a constant value, a decreasing trend is instead experimentally detected, due to the refractive index mismatch. As will be further detailed in the following, the model provides a very satisfactory qualitative and quantitative interpretation of these depth profile data.

Fig. 5 (a) Normalized confocal Raman depth profiles of a polystyrene thick sheet for pinholes sizes, D = 200 μm and D = 400 μm. Black points refer to the experimental data (intensity of the 997cm⁻¹ Raman peak); (b) Normalized confocal Raman depth profiles of of a transparent homogeneous plastic plate made of allyl-diglycol-carbonate (NT43-927, Edmund Optics, Inc.) for three pinholes size, D = 400μm, 700μm, 1000μm. Black points refer to the experimental data reported in [31] (intensity of the 2955cm⁻¹ Raman peak). Solid red line represent the theoretical model predictions.

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Fitting theoretical predictions to the experimental results for the confocal Raman depth profile allows the estimation of the focused beam waist w₀, the amplitude factor A, that accounts for the unknown power of the laser beam, and the sample surface position z₀, according to the following model equation:

P (A, w_{0}, z_{0}, z) = A \times P_{t o t} (w_{0}, z - z_{0}) .

In the following we describe a multi-curve-fitting procedure to fit several depth profile (DP) curves simultaneously. The method works by assigning first the value of the waist radius w₀ as a common parameter for the DP curves and then determining the parameters A and z₀ for each experimental DP curve. According to this method the sum of the individual reduced chi-squared values $χ_{i = 1 . . N}^{2} (w_{0})$ is minimized to fit N curves to N model Eq. (17) simultaneously:

χ_{t o t}^{2} (w_{0}) = \frac{1}{N} \sum_{i = 1}^{N} χ_{i}^{2} (w_{0})

χ_{i}^{2} (w_{0}) = \frac{1}{M_{i}} \sum_{m = 1}^{M} \frac{{(E_{m, i} - P (A_{i}, w_{0}, z_{0}, z_{m, i}))}^{2}}{σ_{m, i}^{2}} .

In Eq. (19)E_m,_i is the measurement of the i^th DP curve taken with error σ_m,_i at the nominal depth z_m,_i and M_i is the corresponding number of measurements. For a given value of w₀ the minimization of the reduced chi-squared value $χ_{t o t}^{2} (w_{0})$ is achieved through an iterative scheme based on the conjugate gradient method that allows to retrieve the best fitted values of the parameters A_i and z₀,_i. These values depend, in turn, on the choice of the waist radius w₀. The value of w₀ is then retrieved by minimizing the reduced chi-squared $χ_{t o t}^{2} (w_{0})$ by using a minimization procedure based on the Brent Method [35]. This approach relies on the fact that the chi-squared $χ_{t o t}^{2} (w_{0})$ is a function of the waist w₀ and it exhibits a minimum $χ_{t o t}^{2} (w_{0, \min})$ when w₀ = w₀,_min. Deviations from the minimum can be fitted to a good approximation by a quadratic law and the value of the waist radius w₀ can be recovered by the following iterative scheme:

w_{0, m i n} = w_{0} - \frac{1}{2} \frac{{(w_{0} - w_{a})}^{2} [χ_{t o t}^{2} (w_{0}) - χ_{t o t}^{2} (w_{b})] - {(w_{0} - w_{b})}^{2} [χ_{t o t}^{2} (w_{0}) - χ_{t o t}^{2} (w_{a})]}{(w_{0} - w_{a}) [χ_{t o t}^{2} (w_{0}) - χ_{t o t}^{2} (w_{b})] - (w_{0} - w_{b}) [χ_{t o t}^{2} (w_{0}) - χ_{t o t}^{2} (w_{a})]},

where w₀ is the previous estimate of the waist radius and

χ_{t o t}^{2} (w_{0})

is the corresponding value of the chi-squared. According to the Brent method, the minimum w₀ is always bracketed with the triplet of points w_a < w₀ < w_b such that the chi-squared value

χ_{t o t}^{2} (w_{0})

is less than both

χ_{t o t}^{2} (w_{a})

and

χ_{t o t}^{2} (w_{b})

. A degenerate case arises when the three points are collinear since Eq. (20) cannot be used but, in this case, the Golden Section Search technique can be applied [35]. The best fitted value w₀,_min = 0.354 μm for the focused waist radius is obtained when the proposed iterative multi-curve-fitting procedure is applied to the experimental data reported in Fig. 5(a). This value agrees well with the resolution limit of the microscope given by the radius of the Airy disk, r_Airy = 0.61λ/NA = 0.36μm, with a discrepancy lower than 2%. The retrieved values of the amplitude factor and the surface position are A₁ = 0.53, A₂ = 0.54 and z_0,1 = −0.61μm, z_0,2 = −0.63μm, which are equal within 2% maximum discrepancy.

As evident in Fig. 5(a), the overall agreement between the theoretical model and the experimental data is excellent. From the analysis of residuals it is found an agreement with dependencies within 2% over the extended spatial range from the raising edge of the DP curves up to a nominal depth z = 150μm.

In order to test the performance of the method in different experimental conditions we have analyzed also the capability of the model in interpreting the experimental data reported in [31]. The experimental data refer to the Raman depth profiling of a transparent homogeneous plastic plate made of allyl-digly-col-carbonate with refractive index n₂ = 1.501 (NT43-927, Edmund Optics, Inc.) and were recorded by using a LabRam HR-800, Horiba-Jobin-Yvon micro-Raman system and an Argon ion laser with excitation wavelength λ_ex = 0.514μm. In this case, the authors report a measured value for the focused laser beam waist w₀ = 0.346μm. By fitting our theoretical model to the experimental data we have instead determined a value w₀ = 0.332μm, with an agreement that is within 4%. As for the amplitude factors, A_i, i = 1,2,3, the maximum deviations between the three retrieved values are within 4%, in good agreement with the assumption that each one represents an estimate of the same amplitude factor, that accounts for the unknown power laser. The red solid lines shown in Fig. 5(b) confirm the excellent agreement between the experimental data and theoretical predictions with discrepancies lower than 3%.

4. Discussion and conclusions

We have developed a theoretical approach for depth profiling of transparent and amorphous samples with low scattering in confocal Raman microscopy. The sources are assumed to be independent point-like dipole with a not well-defined orientation that emit spherical waves as is the case for amorphous polymer samples. The simplifying assumption of the scalar wave optics propagation has been adopted to calculate the system point spread function from the second Rayleigh-Sommerfeld diffraction integral. Particularly, the Fresnel approximation for the light propagation of a point emitter from the sample surface to the objective lens has led to a significant reduction of the computational effort and the point spread function has been formulated in terms of a simple expression involving a 1-D integral.

An iterative multi-fitting scheme, based on the combination of the conjugate gradient method and the Brent algorithm, has been adopted to fit the model to several depth profile curves simultaneously. The fitting procedure allows to retrieve, with a high accuracy, the beam waist, the signal amplitude and the position of the sample surface.

Comparison of experimental results with theoretical predictions shows that the theoretical model describes accurately the depth intensity profiles upto a depth z = 150μm with discrepancies that are lower than 3%. Experimental measurements of the Raman depth profiles have been analyzed for different pinholes with size much greater than the emitter wavelength to neglect the diffraction effects of the light passing through the pinhole, a case not addressed by the model. The minimum pinhole diameter used was 200μm (corresponding to about 2 Airy Units). Moreover, pinholes that are too narrow considerably reduce the intensity and the collecting efficiency of the system by increasing the acquisition time.

The good agreement between the theoretical model and experimental data confirms that the scalar wave optics and Fresnel approximation can be adopted as basic assumptions for describing the depth profiles of samples with spherical symmetric Raman emitters provided that the distance between the sample surface and the objective lens satisfies the condition d > 400λ_em. Otherwise, a rigorous vector-wave treatment is required with a full vector analysis that is computationally much more intensive than a scalar analysis because the various components of the electric field in the image space of the confocal microscope must be considered.

The proposed theoretical model and computational inversion method constitutes a first step towards the development of a manageable, but physically consistent, theoretical framework, amenable to a relatively simple numerical implementation for the solution of the more complex ’inverse’ problem of finding the correct reconstruction of unknown profiles of chemical species and moieties starting from experimental information gathered by micro-Raman depth profiling.

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Depth profiles in confocal optical microscopy: a simulation approach based on the second Rayleigh-Sommerfeld diffraction integral

Abstract

1. Introduction

2. Theory

2.1. The system response function of the confocal Raman microscope: the Rayleigh-Sommerfeld approach

2.2. Depth profiles: theoretical analysis

3. Comparison with experimental results

4. Discussion and conclusions

References and links

Cited By

Figures (5)

Equations (20)

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