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Abstract
A new method for the characterization of coherent laser beams is proposed. It is based on the non-iterative solution of the transport-of-intensity-equation. The phase to recover is decomposed into paraxial properties of laser beams and a set of lateral shifted radial basis functions, which allows for the derivation of a direct solution of the phase by a least-squares fit without the need of an initial guess. The method is tested with synthetic data to deduce an accuracy metric. Additionally, two real laser beams are characterized. Including the real light source in terms of the reconstructed field allows for a more holistic simulation of optical systems.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The characterization of laser beams is an important task for a multitude of applications. Typically, the beam quality is quantified by the $M^2$ parameter, exploiting the similarity to an ideal Gaussian beam. The most important deviations from the ideal value of $M^2=1$ are phase aberrations, amplitude variations or a certain degree of partial coherence. A separation of these three reasons cannot be performed by calculating the scalar measure of $M^2$ [1]. Within this work, partial coherence is not further considered. Especially in applications where the beam is entering beam shaping optics, the measure of the $M^2>1$ is not helpful, since it is not a direct measure of the amplitude and phase. To extend the quality-metric beyond the $M^2$ of a coherent beam, its amplitude and phase must be analyzed. While the intensity can be simply measured by a camera, the phase cannot. To measure the phase, interferometric techniques or wavefront sensors can be used.
To overcome the challenge of simultaneous measurement of amplitude and phase, the method of phase-retrieval can be used. It is of great interest for a lot of applications from the past up to recent time. There are several methods to tackle this inverse problem. Many of them are based on the inverse-Fourier-transform-algorithm (IFTA) principle [2,3]. They are often applied in the field of beam-shaping with arbitrary phase-functions. For the characterization of optical systems, there are specialized algorithms involving the optimization of a parametrized physical model or the extended Nijboer-Zernike theory [4,5]. Another field of algorithms, which tackle the phase-retrieval problem, are based on solving the transport-of-intensity-equation (TIE) [6]. Their applications range from the field of surface testing to phase-imaging techniques in microscopy [7–9].
The TIE is a partial-differential equation, that connects the intensity distribution at a certain position with the related phase distribution at the same position. Therefore, it allows for a simple experimental setup for quantitative phase measurements, that consists of a camera to acquire the intensity, mounted on a movable stage. There are many distinct methods published dealing with the solution of the TIE. In the original paper of Teague, the solution via a Green’s function and direct numerical integration was proposed [6]. This corresponds to a solution of the TIE for each individual pixel of the measured intensity points. Often, a solution for the TIE is found by transforming it into a Poisson equation and solving it in the frequency domain [7,10]. Gureyev et al. solved the TIE via certain sets of basis functions like Zernike-polynomials or Fourier-harmonics [11,12].
In this work, we describe the phase of coherent laser beams by the paraxial properties, consisting of tilt and curvature, and a set of shifted radial-basis-functions (RBFs). The RBFs are useful because of their local support, which makes them flexible in describing symmetry-free surfaces [13] and their simple differentiability. This approach can be directly inserted into the TIE leading to a system of linear equations.
This article is structured into the following sections. After the introduction, the TIE is derived from the Helmholtz-equation. Afterwards, the proposed method to solve the TIE based on the decomposition of the phase into tilt, curvature and higher orders is presented. Additionally, the critical parameters are examined. It is tested with synthetic data to derive the accuracy of the retrieved wavefront based on the measured and reconstructed stack of intensities. Thereafter, the technique is used to characterize and reconstruct two real measured laser beams to validate the method. In the final section, the work is concluded and an outlook is given.
2. Theory of the TIE
The underlying theory to retrieve the phase $\phi$ of an unknown, coherent field by measurements of the intensity $I$ is based on the TIE [6]. Therefore, the TIE is derived from the time-independent Helmholtz-equation, which reads as
where $U=U(\vec{r})$ is the monochromatic field defined in Cartesian coordinates with the wavenumber $k=2\pi /\lambda =\sqrt {k_x^2+k_y^2+k_z^2}$. $\lambda$ is the wavelength and $k_x, k_y, k_z$ are the components of the wavevector in the corresponding directions. The paraxial wave-equation can be derived by separating the fast oscillating term Additionally, small angles $\left (k_z \approx k\right )$ and the slowly varying envelope are assumed, meaning that Therefore, the terms of $k_x, k_y$ and $\frac {\delta ^2 V}{\delta z^2}$ can be neglected. The result is the paraxial wave-equation with the lateral nabla-operator $\nabla _\perp$. The TIE is obtained by inserting the following ansatz-function into Eq. (4) where $I$ is the intensity and $\phi$ is the phase of the field $V$ and separating between real and imaginary parts of the equation. The real part leads to the transport-of-phase-equation and the imaginary part is known as the TIE [6,14] Equation (6) is the TIE and it is further used to retrieve the phase of a coherent field. The intensity has to be measured at different positions in space to calculate the derivatives at a distinct plane $z_0$. To improve the readability, we define thes components3. TIE for coherent laser beams
In this section, the TIE is solved with particular focus on coherent laser beams and the critical parameters of the reconstruction of the phase are discussed.
3.1 Solution of the TIE
To solve the TIE in the context of laser beams the phase is split into basic terms describing the paraxial properties of the beam and higher order terms describing the phase perturbations. The paraxial properties of the beam are the tilt $\phi _t$ and the curvature of the phase $\phi _c$. They are described by lower order polynomials, whereas the higher order phase terms $\phi _{RBF}$ are described by a set of RBFs
The tilt-term is defined as with the corresponding coefficients $t_x, t_y$. The curvature of the phase is described by the coefficient $c$ and assuming the dominant contribution is circular symmetric. For the description of the higher order phase-terms, the following linear combination of $M$ RBFs is used3.2 Critical parameters of the TIE
The critical parameters for the reconstruction with the described set of equations are the noise within the measurement of the intensity, the corresponding gradient of the intensity and the size of the region, where the phase is reconstructed.
The noise in the data acquisition is assumed to be mainly introduced by white noise and quantization-errors due to the bit-depth of the available cameras. Both problems are tackled by recording the intensity at a single position with different exposure-times. The resulting intensity-stack is then merged to a high-dynamic-range (HDR) image [15,16]. The HDR image provides an overall improved signal-to-noise ratio and even provides signal in regions, where conventionally no or only saturated signals can be measured due to under- or over-exposure, respectively.
The gradient of the intensity is calculated based on the actual intensity. Hence, it is also influenced by noise and its accuracy can be improved by reducing the noise. An additional point for calculating the correct gradient is the sampling in axial and lateral direction. In principle, the distance $\Delta z$ between different measurement planes can be chosen as small as possible. It is only limited by the accuracy of the used stage. Therefore, the accuracy can be arbitrary large along $z$ for a noise-free signal. If noise is considered, $\Delta z$ must be chosen carefully. Small distances lead to a small signal change $I(z_2)-I(z_1)$, which may be disturbed by noise, therefore the signal-change is dominated by noise. If $\Delta z$ is too large, the signal change is larger than the noise, but may exceed the range, where a finite difference approximation for the derivative is valid. Therefore, the optimal distance is dependent on the signal change and the noise [17,18]. Teague investigated this theoretically in [6] and gave expressions for an optical choice of $\Delta z$ for known noise and phase. Waller demonstrated similar results by investigating errors of noisy signal derivatives dependent on the finite difference order, $\Delta z$ and varied noise-levels [10]. The lateral derivatives of the intensity are influenced in the same manner as the axial derivative. The difference is, that the separation between data-points is fixed by the sensors pixel-pitch. The only possibility to modify this, is to enlarge the lateral extend of $I$. This can be done by magnifying the beam by an appropriate telescope or by moving to another axial position of reconstruction $z_0$. For the proposed method in this work, $\nabla I$ is calculated by a Savitzky-Golay-filter [19]. It was already investigated in conjunction with the TIE in [18]. This filter basically convolutes a (typically low order) polynomial to the data points in order for smoothing the data and calculating the derivatives. The controllable parameters are the window-size and the polynomial order. Another practical important point is the position $z_0$, where the phase is retrieved, since this strongly changes $I_z$. In case of an ideal Gaussian beam [20], the axial intensity is symmetrical around the focus. Therefore, $I_z$ is exactly zero at the focus. Reconsidering the impact of the noise on $I_z$, it is clear, that the signal-change is strongly influenced by noise at the focus. To maximize the value along $z$, we compute an optimal position $z_0$. This position is identified by calculating the intensity-weighted root-mean-square (rms$_{\textrm {w}}$) value of $I_z$ along $z$ and selecting the position where this is maximal.
The size of the region where the phase is calculated, is dependent on the intended application. For applications like phase imaging, the phase information might not be of interest in the full area of illumination. In the case of the characterization of Gaussian like laser beams with considerably large perturbations, the practical limit of detectable intensity defines the size of the region of reconstruction, coming from the basic principle, that the phase is undefined, if there is no intensity [21]. For further purposes of propagation of the calculated field, the phase must be retrieved for all points of measured intensity.
4. Wavefront accuracy
The assessment of the accuracy of the retrieved phase is an important aspect in the quantitative characterization of a coherent field. In order to fully substitute wavefront measuring equipment like interferometers or Hartmann-Shack sensors, we prefer a consistency check of the reconstructed field with the measured stack of intensities. In order to do so, the reconstructed field is propagated to the planes of measurement and the intensities are compared. If the phase is identical, the difference in intensities must be zero. Thus, if there is a mismatch between the reconstructed phase and the phase of the measured beam, this becomes evident due to propagation, resulting in intensity differences between the measured and the reconstructed beam.
Our approach is to quantify the intensity differences and find a relation to the corresponding phase mismatch. This is done by simulating different Gaussian-like beams, which are perturbed by various amounts of spherical aberration and astigmatism. Additionally, the far field divergence angle $\Theta$ is varied. The synthetic beams are then propagated into the focal region and the phase is retrieved by the proposed method. The accuracy of the retrieved phase can be controlled by the number, shape and distribution of the used RBFs. A more detailed discussion is given in [13]. It is quantified by the weighted rms difference between ideal and retrieved phase $\textrm {rms}_{\textrm {w}}\left (\Delta \phi \right )$. In order to unify the results, the axial length is scaled in Rayleigh-units in air ($RU$)
The intensities are compared within an interval of $\pm 1RU$ by computing the intensity weighted rms difference $\textrm {rms}_{\textrm {w}}\left (\Delta I\right )$.The results are plotted in Fig. 1. The accuracy of the retrieved phase strongly correlates with the difference of the intensities. The achieved accuracy of the phase increases with increasing similarity of intensities. Consequently, larger values of $\textrm {rms}_{\textrm {w}}\left (\Delta I\right )$ correspond to enlarged values of $\textrm {rms}_{\textrm {w}}\left (\Delta \phi \right )$, but simultaneously its spreading is also increased. The dependency between the accuracy of the phase and the intensity difference is described by a third order polynomial, which can also be seen in Fig. 1. The corresponding coefficients are found in the legend. This description allows to quantify the results of real measured laser beams and is used as accuracy estimation in the following section.
Fig. 1. Accuracy of the reconstructed wavefront in relation to the error of the reconstructed intensity stack within $\pm 1RU$. The data-points contain the results of different strengths of spherical aberration and astigmatism, as well as different far field divergence angles.
5. Demonstration of the approach with experimental data
In this section, the proposed method to reconstruct the field of a coherent laser beam is applied to real measured data. First, the experimental setup to acquire the intensity data is introduced. Then, the measured data of two different light sources are used to reconstruct the beams with the prescribed method. The outcome of both measurements are discussed afterwards.
5.1 Experimental setup
The experimental setup to measure the intensity of a Gaussian-like laser beam consists of a light source, two lenses and a camera mounted on a linear stage. A sketch of the setup is shown in Fig. 2. As a light source, a laser diode with $\lambda =$ 663 nm at 1 mW power is used (Lambda Mini, RGB-Photonics GmbH, Kehlheim, Germany). The spectral width is specified with 0.7 nm at full width at half maximum. Since this is small in comparison to the wavelength, it is neglected. The light emitted by the laser diode is coupled into a single-mode fiber and afterwards transmitted through a collimator (60FC-4-M12-33). The beam with a diameter of 2.16 mm is then focused by an interchangeable focus module in order to achieve different focal lengths. Both, the collimation and focusing module are from Schäfter+Kirchoff GmbH (Hamburg, Germany). In Sec. 5.2, a focal length of 325 mm is used (5M-S325-33-S), resulting in a far field divergence of $\Theta =$ 3 mrad. The second focus module with a focal length of 150 mm (5M-S150-33-S) and a corresponding far field divergence of $\Theta =$ 7 mrad is evaluated in Sec. 5.3. To scan the beam along $z$, a dynamic stage is used (M-ILS300LM-S, Newport Corporation, Irvine, USA) and a beam profiling camera is mounted onto it (SP928, Ophir Spiricon Europe GmbH, Darmstadt, Germany). The camera captures 12 bit images with a resolution of 1928 $\times$ 1448 pixels. The pixel pitch is 3.69 µm.
5.2 Reconstruction of a laser beam with $\Theta =$ 3 mrad
For the retrieval of the phase, the intensities inside a radius of 0.49 mm around the centroid are considered. To describe the phase, 3000 RBFs are distributed inside this radius. The axial derivative term $I_z$ is computed by a sixth-order polynomial based on 51 planes along $z$, separated by 0.2 mm. The lateral terms $I_x,I_y$ are also computed by a sixth-order polynomial, but based on eleven intensity-values. Hence, the smoothing is much stronger in axial direction, than in lateral direction.
The results of the retrieval are shown in Fig. 3. In the upper row, the $y$-$z$ cross-section of the measured intensity stack is shown. The corresponding intensities of the reconstructed field are shown in the center row. In the bottom plot, the on-axis intensities of the measurement and the reconstruction are presented. The intensity weighted rms difference of the intensities within $\pm 1RU$ it is $\textrm {rms}_{\textrm {w}}\left (\Delta I\right )=1.23\%$. Comparing this to the results of the Fig. 1, the accuracy of the retrieved phase is estimated to $\textrm {rms}_{\textrm {w}}\left (\Delta \phi \right )\approx 0.045\lambda$. The $M^2$-value of the reconstructed field is calculated based on the spatial moments from the measured intensity and the angular moments, based on the retrieved phase [22]. For the reconstructed field, a value of $M^2=1.3$ is obtained.
Fig. 3. Two-dimensional cross-sections $I(x=0,y,z)$ of the measured and reconstructed intensities of the beam with $\Theta =$ 3 mrad. The intensity values are normalized to the global maximum. Below, the corresponding on-axis intensities $I(x=0,y=0,z)$ are plotted for comparison.
5.3 Reconstruction of a laser beam with $\Theta =$ 7 mrad
The parameters of the retrieval process for this laser beam are equal to the $\Theta =$ 3 mrad case, beside the radius within the phase is retrieved. It is is adapted to 0.22 mm in order to account for the larger $\Theta$ and thus, the stronger focusing effect.
In Fig. 4, the results of the reconstruction are depicted. The top and center row show the $y$-$z$ cross-sections of the measured and computed intensities, respectively. In the bottom row, the comparison of the on-axis intensities are plotted. The accuracy of the retrieved phase is again estimated based on Fig. 1. Within $\pm 1RU$ the intensity difference is $\textrm {rms}_{\textrm {w}}\left (\Delta I\right )=1.01\%$, which corresponds to an accuracy of the phase to $\textrm {rms}_{\textrm {w}}\left (\Delta \phi \right )\approx 0.034\lambda$. The $M^2$-value of the reconstructed field is $M^2=1.7$.
Fig. 4. Two-dimensional cross-sections $I(x=0,y,z)$ of the measured and reconstructed intensities of the beam with $\Theta =$ 7 mrad. The intensity values are normalized to the global maximum. Below, the corresponding on-axis intensities $I(x=0,y=0,z)$ are plotted for comparison.
5.4 Discussion
The experimental setup of both measured laser beams differs by the used focusing modules. From a correction point of view, the focus module with the larger $\Theta$ was expected to perform slightly worse, since larger $\Theta$ provokes larger angles in the ray-bending, typically leading to an enhancement of spherical aberrations. By comparing the $M^2$-values of both reconstructions, these expectations were met. Both stacks show certain rings. Such rings typically arise due to spherical aberration and are often only seen on one side of the caustic [23], which is also the case here on the right hand side of the caustic for both measurements. These rings can also be seen in Fig. 3 and Fig. 4 as asymmetric intensities around the caustic and the modulations of the on-axis intensity. In Fig. 4, the modulations are more pronounced confirming expectation of the stronger spherical aberrations for a larger $\Theta$. To prove the observations quantitatively, the retrieved field is propagated back to the position of the focus module and the corresponding wavefront is fitted by Zernike-Fringe polynomials. The norm-radius is 2.5 mm, which is the clear aperture radius of the lens. The largest coefficient is seen for fourth order spherical aberration ($c_9$ in Zernike-Fringe notation) in both measurements. For the $\Theta =$ 3 mrad case, it is $c_9=0.17\lambda$ and approximately doubled in the case of $\Theta =$ 7 mrad with $c_9=0.38\lambda$. Thus, the previously made interpretations are confirmed.
The rms values do not offer any insights on particular problems in the reconstruction as edge diffraction due to an imperfect radius of reconstruction or issues with dead pixels of the camera. Therefore, the visual comparison of the reconstructed beam with measurement is useful. The sensitivity of the visual agreement can also be understood in the comparison of results of the two presented beams. The intensities of the retrieved field in the $\Theta =$ 7 mrad case match the measurement better in the $\Theta =$ 3 mrad case. In particular this can be seen in the agreement of the oscillations of the on-axis intensities on the right hand side of the caustics. The intensities of the larger $\Theta$ matches in amplitude and frequency, while for the small $\Theta$ a mismatch can be seen. These obvious performance differences based only on visual interpretation of the data are also supported by the corresponding quantitative results based on the weighted rms differences of the intensities. Here, the retrieved phase of the second measurement performs better. To understand this, we investigated the intensities of both measurements. A clear difference is the number of dead pixels on the camera. Only for the first measurement, we could identify a large amount of dead pixels. The intensity values of the $\Theta =$ 3 mrad were filtered by a median-filter to achieve the presented results. This was unnecessary for the other measurement.
The influence of $M^2$ degradation by pure quartic spherical aberration on an ideal Gaussian beam was investigated in [24]. Here, the amplitude is an ideal Gaussian, which is typically not fulfilled in real measurements. However, the $M^2$ is influenced by variations in phase and amplitude which are covered by the presented method. This indicates the precarious situation of the scalar $M^2$ as a measure of beam quality.
In general, the performance of the reconstruction and propagation of both measured laser beams shows good visual agreement with the measured data. The weighted rms differences of the intensities within $\pm 1RU$ are less than $1.3\%$. Comparing this to the investigations made in Sec. 4, the estimated residual weighted rms wavefront error of the proposed method to retrieve the phase is less than $0.05\lambda$ for the experimental data.
6. Conclusion
In this paper, a new method based on non-iterative phase retrieval method based on the TIE is proposed to characterize and reconstruct coherent laser beams. The TIE is solved by decomposing the phase into paraxial beam properties and a set of shifted RBFs leading to a linear system of equations, which is solved by a least-squares-fit. The reconstructed field can be used to characterize the laser beam or to further propagate the laser beam through complex optical systems in order to enhance the simulation complexity by including the real light source. To assess the quality of the retrieved phase, a metric based on synthetic data is introduced. Finally, the method is applied to retrieve the phase of two laser beams based on real measurements. The corresponding weighted rms phase errors are estimated to be less than $0.05\lambda$.
Funding
Bundesministerium für Bildung und Forschung (0.13039/501100002347, 03WKCX1C).); Carl-Zeiss-Stiftung.
Disclosures
The authors declare no conflicts of interest.
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