1. INTRODUCTION

A Gaussian ${\mathrm{TEM}}_{00}$ is the fundamental mode emitted by sources such as He–Ne lasers. In numerous laser-based optical systems, propagation of the ${\mathrm{TEM}}_{00}$ Gaussian mode through optical components and modules as well as indoor and outdoor free-space optical communication links is involved. There are several situations where profiling a Gaussian beam is pivotal and necessary. Some of these include

Although there is no single figure-of-merit to judge the performance of any given profiler design, nonetheless, a superior beam profiler design/method has to be accurate, efficient, repeatable, cost-effective, and possibly fast if measurement accuracy can be compromised.

Various beam profiling methods have been proposed in prior articles and deployed for commercial use with varying levels of commercial success. These include beam profiling with a classical moving knife edge [5–7], digital knife-edge profiling using a spatial light modulator (SLM) [8,9], pinhole beam profiling [10,11], beam profiling using an analog chopping wheel [12], photographing with a two-dimensional photodetector (PD) array [13,14], and various other methods such as the ones demonstrated in [15–17].

In this paper, we present two simple beam profiling techniques that involve the use of circular apertures such as commonly available graduated irises [18] or circular apertures of different other types. The techniques are simple because they are easy to reproduce with a simple apparatus available in most optics laboratories. Moreover, as some basic optical equipment is required to reproduce them, the proposed techniques are also cost-effective. The fundamental principle of each of the proposed techniques revolves around positioning a circular aperture in the path of the beam, and between measurements blocking a different fraction of the beam power from reaching a PD placed after the aperture. This goal can be possibly achieved from two beam profiling schemes, which we propose and demonstrate. These are:

The first method that we propose requires changing the clear aperture diameter of the circular aperture and measuring the optical power transmitted through the aperture for each diameter setting. This technique, in essence, is very similar to the classical knife-edge profiling approach presented in [5], but the beam profile is estimated using the circular aperture of the varying radius instead of a moving knife edge. The resulting setup only requires an iris with a tunable radius such as [18], and no translation stages, resulting in a much simpler, cheaper, and compact profiler design. The measured data are then used to obtain a least squares best-fit estimate of the $1/{\mathrm{e}}^2$ beam radius.

The second method is very similar to pinhole beam profiling with the distinct advantage that the clear aperture of the iris need not be extremely small in order to emulate a “pinhole.” This is possible due to the analytical expressions that we derive for power transmittance of a Gaussian beam through any decentered circular aperture. This method allows for estimating the beam profile in a faster manner with significantly fewer measurements compared to conventional pinhole profiling, which assumes an infinitesimal area of the pinhole, consequently requiring the pinhole to scan the entire beam profile in very small steps and estimate beam waist in a “brute force” manner.

In the first part of the paper, we derive an exact solution of the transmitted Gaussian beam power through a decentered circular aperture. This is an important step for beam profiling as both of the proposed methods rely on fitting measured data to a known mathematical expression. The derivation of an exact analytical expression for power transmission through a laterally decentered circular aperture with respect to the beam center is nontrivial because the peak irradiance location (location of beam center) does not coincide with the center of the circular aperture. This results in solving a radially asymmetric integral of optical irradiance in order to determine transmitted optical power. A numerical approach to roughly estimate power transmission of a Gaussian Beam through a decentered circular aperture was previously discussed in Ref. [19]. In Ref. [19], the authors set an integral with an integrand that is a function of the lateral displacement of the circular aperture center with respect to the beam center, but then the authors do not attempt to solve this integral, declaring that an antiderivative does not exist and very vaguely applying Simpson’s rule to find an estimate of power transmission without discussing the numerical expression obtained. We, on the other hand, show that an exact antiderivative does exist and, despite the fact that it is not a closed form solution due to a summation of infinite terms, the solution is a converging sum, and the accuracy of the power transmission estimate can be set as required by selecting a sufficient number of terms. In other words, unlike in [19], we provide an actual mathematical expression for power transmission and a truncated version of which with “sufficient number of terms” (that is a subset of the exact solution expressed as a complete convergent summation of infinite terms) can be used in a regression-based estimation of beam profile.

In the second part of the paper, we use the mathematical expression, derived in the first part, to estimate the irradiance profile of a Gaussian beam. We demonstrate an agreement between our experimental results and the theoretical predictions from the analytical expressions that we derive. We also compare our experimental results to sliding knife-edge-based estimates obtained through a knife-edge scanning setup for the same Gaussian beams that were profiled with circular apertures.

2. POWER TRANSMISSION OF THE ${\mathsf{TEM}}_{\mathsf{00}}$ MODE DECENTERED WITH RESPECT TO A CIRCULAR APERTURE

To profile a Gaussian beam with a circular aperture, the first step is to determine the optical power that is transmitted through a laterally decentered circular aperture. In other words, it is crucial to determine the effect on transmitted optical power due to a lateral offset between the optical axis of the aperture and the propagation axis of the incoming Gaussian beam. The case of a Gaussian beam, with a $1/{\mathrm{e}}^2$ waist radius of “$w$,” incident on a circular aperture of radius “$a$,” is depicted in Fig. 1, where Fig. 1(a) shows the much-discussed case of an incident beam that is centered with respect to the circular aperture center, whereas Fig. 1(b) illustrates the situation that we discuss in this paper where an incident Gaussian beam is laterally displaced by $d=\sqrt{d_x^2+d_y^2}$ with respect to the aperture center.

 

Fig. 1. Incident Gaussian beam (a) centered and (b) decentered at a circular aperture of radius “$a$.”

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Deriving an analytical expression for a decentered beam is pivotal for the profiling schemes that we present, as both methods involve propagation of the beam through a decentered circular aperture. In the case of the first technique, a variable radius circular aperture with a fixed center transmits a fraction of beam power depending on the aperture radius (while the centers of the beam profile and the aperture would most likely be laterally decentered). On the other hand, profiling by translating a fixed radius aperture as a finite-sized scanning pinhole involves relative decentering of the beam and aperture radii in all measurements. The proposed methods do not (and should not) depend on a zero centering between the beam center and the aperture center and deliver accurate estimates of beam waist regardless.

We begin by stating the irradiance profile $I(x^{\prime },y^{\prime },z^{\prime })$ of a Gaussian beam, propagating in the $z$ direction, at the plane of the aperture located at $z=z^{\prime }$. Irradiance distribution $I(x^{\prime },y^{\prime },z^{\prime })$ is expressed as

3. GAUSSIAN BEAM PROFILING WITH CIRCULAR APERTURES

In this section, we discuss profiling of Gaussian beams with a simple circular aperture. As mentioned in Section 1, two possible schemes described in this paper are

Here, we discuss these techniques in detail. In the previous section, an exact analytical expression for power transmittance through a decentered circular aperture was derived. We show that this expression can be used to profile any Gaussian beam ${\mathrm{TEM}}_{00}$ mode, i.e., determine the radial offset of the beam center with respect to the center of the aperture and, more importantly, estimate the $1/{\mathrm{e}}^2$ beam waist at the plane of the aperture.

A. Beam Profiling with a Circular Iris at a Fixed Location and a Tunable Radius

We first explore the possibility of profiling a Gaussian beam with a circular aperture, the radius of which can be altered. An iris diaphragm is a circular aperture that is commercially available at a low cost and is present in most typical optics laboratories. Moreover, graduated iris diaphragms—where the radius of the iris aperture can be set to known radii—are also commercially available off-the-shelf [18].

For best performance, an iris with a maximum diameter that is comparable or larger than the $1/{\mathrm{e}}^2$ beam diameter is preferred. The beam profiler module consists of an iris followed by any DC-coupled PD, which is designed to operate at the wavelength of the Gaussian beam. A lens is used to collect the transmitted optical beam through the aperture and efficiently focus at the active area of the PD. This profiler setup is shown in Fig. 2.

 

Fig. 2. Beam profiling with a fixed circular aperture location and varying aperture radius.

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The beam profiler module is simply positioned such that the transmitted beam power is made for different settings of the iris radius. The recorded measurements are then simply used to perform a regression-based optimization using the derived analytical expression of Eq. (14) to obtain an optimal best-fit estimate of “$d$” and “$w$” in the least squares sense. In this way, the $1/{\mathrm{e}}^2$ beam waist radius and the distance of the beam center are estimated with respect to the known central location of the circular iris. Using an SLM such as the one used in [8], the entire operation of the proposed beam profiler can be digitized, eliminating the need to physically change the iris radius but instead forming a micromirror-based aperture on the digital micromirror device (DMD).

B. Beam Profiling with Translating a Circular Iris of Fixed Radius

The other beam profiling technique that we propose involves translating a circular iris across the cross-sectional irradiance distribution of the beam and making measurements of the transmitted power at different iris locations. To improve resolution of measurements, the radius “$a$” of the iris can be set to a small value such that $a\ll w$.

In this case, as we have already derived the analytical expressions for power transmission through a decentered circular aperture, estimating the beam profile does not necessarily follow the “brute force” method of profiling with an infinitesimally small pinhole aperture. Instead, measurements of power transmission are recorded at known displacements of a finite-sized circular aperture from a reference position. With a theoretical framework of power transmission from Eq. (14) available to us, we again apply a least-squares-based optimization on the measured transmitted power data to obtain a best-fit estimate of the beam waist—and that too with the aid of far fewer measurements than a pinhole beam profiling procedure. This technique is illustrated in Fig. 3.

 

Fig. 3. Beam profiling by translating a circular aperture of a fixed radius.

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As is shown in Fig. 3, measurements with this technique require translating the fixed radius iris using motion stages. Similar to [8]—where a moving knife edge was emulated using a DMD—a moving circular aperture can also be digitally reproduced with similar SLMs, thus allowing for the possibility of an all-digital bulk motion-free beam profiling with a scanning circular aperture.

Both proposed beam profiling techniques have been experimentally verified. In the next section, we present experimental data and results to demonstrate the working of the proposed beam profiler methods.

These experimental results validate the analytical expressions that we derived for power transmission and determine the waist of the Gaussian beam to a good accuracy. For the sake of comparison, we also profiled each test beam with a standard knife-edge procedure. We then compare beam profile estimates from the two proposed circular-aperture-based methods with the knife-edge-based irradiance profile estimates of each test beam, respectively.

4. EXPERIMENTS AND RESULTS

For our experiments, we profile the irradiance distribution of a Gaussian beam from the same laser source at two different locations using the two proposed methods. As a diverging beam from a laser source was profiled at two different locations, we—in effect—profile two different Gaussian waists using each method. As mentioned earlier, in addition to these measurements—for comparison—we also profiled each of the two beam waists with the standard moving knife-edge method. Measurement data for all three techniques are presented in Table 1 for the first beam location and Table 2 for the second beam location.

Table 1. Measurement Data of First Beam with a) Varying Circular Aperture Radius, b) Changing Circular Aperture Location, and c) Moving Knife Edge

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Table 2. Measurement Data of Second Beam with a) Varying Circular Aperture Radius, b) Changing Circular Aperture Location, and c) Moving Knife-Edge

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A freely diverging beam from a standard He–Ne laser (Thorlabs HNL008R-EC 633 nm) was profiled at 20 cm and 30 cm from the laser exit aperture. This meant that we profiled two different waists from the same laser source, which we refer to in the text as the first and second beam, respectively. Instead of deploying a mechanical iris, we instead used a DMD with a two-dimensional array of two-state micromirrors for emulating a digital circular iris and a moving digital knife edge for the two proposed methods and the moving knife-edge experiments, respectively. A DMD-based beam profiling was performed instead of profiling with an actual circular iris because in doing so we not only validate the proposed methods of beam profiling but also demonstrate the possibility of digitally automating the profiling procedure.

For the experiments, a DLP3000 DMD with a micromirror pixel pitch of 7.637 μm was used. The PD used for the experiments was a Thorlabs S120C detector [22]. For the moving knife-edge measurements with a DMD, the method in [8] was replicated. To profile a beam with a DMD, black-and-white images, with black area surrounding a white circle at different locations/circle radii, were displayed on the DMD with the white circular region on the DMD pixel grid denoting the equivalent of the clear aperture of a circular iris. Our simple experimental setup is shown in Fig. 4, where a PD, when used in conjunction with the DMD (as was also previously shown in [8] for knife-edge profiling), records only the power that is incident on white micromirrors (mirrors set to a $+\theta$ state) and not the micromirrors set to the $-\theta$ state (black micromirrors). We used a spherical focusing lens with a 2.54-cm diameter and a focal length of 30 mm for our experiments. The focusing lens was placed at roughly 5 cm from the DMD for each of the measurements. The PD was placed a further 5 cm behind the lens, as shown in Fig. 4. As our beam profiling planes were chosen at 20 cm and 30 cm from the exit aperture of the laser source, the DMD was placed at these two locations for each of the profiling experiments, respectively.

 

Fig. 4. Experimental setup for beam profiling with circular apertures.

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Next, experimental measurements were made for the proposed methods. For the first method, the projected white circular pattern on the DMD was displayed at a reference (0,0) DMD location. This radius of the displayed circular pattern was changed without altering its center location.

For each new value of the “white” radius displayed on the DMD, the corresponding power was recorded at the PD. This measurement was performed for the first beam and second beam, and the respective measurement data are presented in Table 1 and Table 2, respectively.

Similarly, for the second proposed method, an image of a circle of radius $a=0.67\mathrm{mm}$ was projected onto the DMD. The circle location was shifted, and the optical power received at each unique location of the circle center was measured. Again, measurements for the first and second beams are also presented in Tables 1 and 2, respectively. Samples of images mapped onto the DMD for both profiling schemes are presented in Fig. 5.

 

Fig. 5. Selected displayed images on DMD for expanding and scanning aperture schemes for beam location 2.

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Subsequently, we plot the power transmission measurements for different aperture radius/aperture location/knife-edge position settings for each profiling method, respectively. Figure 6 plots the recorded data for the first beam from Table 1, and Fig. 7 plots Table 2 data for the second beam.

 

Fig. 6. Data points and estimated best-fit curve for the first beam location with (a) expanding circular aperture, (b) scanning circular aperture, and (c) knife-edge technique.

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Fig. 7. Data points and estimated best-fit curve for the second beam location with (a) expanding circular aperture, (b) scanning circular aperture, and (c) knife-edge technique.

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The computed values of the $1/{\mathrm{e}}^2$ beam radius “$w$” from the expanding aperture technique, the translating circular aperture technique, and the knife-edge technique are denoted as $w_{\text{Expand}}$, $w_{\text{Shift}}$, and $w_{\text{Knife}}$, respectively. Beam waist estimates $w_{\text{Expand}}$ and $w_{\text{Shift}}$ are computed using a standard regression-based least squares curve fit of the measurement data to Eq. (14), while $w_{\text{Knife}}$ is obtained by a least squares curve fit of the measurement data to the knife-edge function,

As is observed from the computed values of the waist radii for each beam, the proposed techniques deliver results that are comparable to knife-edge estimates. For the first beam, $w_{\text{Expand}}$ and $w_{\text{Shift}}$ are estimated to be 593 μm and 584 μm, respectively, while a knife-edge estimate $w_{\text{Knife}}$ of the same beam is obtained to be 622 μm. Measurements for the second beam yield similar results with $w_{\text{Expand}}$ and $w_{\text{Shift}}$ estimated to be 785 μm and 838 μm, respectively, compared to a knife-edge estimate $w_{\text{Knife}}$ of 846 μm.

The accuracy of results from each method depends on whether the plane of beam profiling (which, in our case, is the DMD plane) is orthogonal to the beam propagation or not. Projected circular apertures or the knife edge as seen by the incident beam results in estimation inaccuracies as the fitting model assumes perfectly normal incidence. This is an inherent drawback with any beam profiling method. Another source of error in profiling with a DMD is the discrete spatial resolution that a DMD offers due to finite-sized micromirrors, which reduce measurement resolution of the spatial power distribution and cause estimation errors using a model that assumes spatial continuity. Hence, when a circular pattern of an intended radius is projected onto the DMD, a circular pattern with rough edges and a slightly different effective or average radius compared to the intended radius is actually obtained on the DMD—resulting in minor errors in each measurement—possibly resulting in an overall estimation error as the model assumes that intended radii were displayed on the DMD for each circular pattern projected.

To quantify how much the measured data vary in relation to theoretical prediction from Eq. (14), we evaluated the coefficient of determination ($R^2$ value) for the experimental results plotted in Figs. 6 and 7. The coefficient of determination (or $R^2$ value) is typically a measure of the quality of measurement data points compared to the theoretically expected values, and it is stated as

For our experiments, we calculated the $R^2$ values for each measurement dataset in Figs. 6(a)–6(c) and Figs. 7(a)–7(c), respectively. We summarize these $R^2$ values in Table 3. The $R^2$ values obtained for each experiment are very close to unity. For the four beam profiling datasets involving circular apertures (two each for expanding aperture and scanning aperture techniques), the near-unity $R^2$ value signifies an excellent agreement between the theoretical prediction from Eq. (14) and experimental results for the two proposed beam profiling methods.

Table 3. Measured ${\mathsf{R}}^{\mathsf{2}}$ Values of Each Experimental Dataset Verifying the Validity of the Proposed Theoretical Expressions

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5. CONCLUSION

In this paper, the authors present two novel schemes for profiling the fundamental ${\mathrm{TEM}}_{00}$ mode. The first scheme requires a circular aperture with a variable radius positioned at a fixed central location, while the second proposed scheme involves a scanning circular aperture of a fixed radius. We show that profiling Gaussian beams with the proposed methods can be easily implemented either using a simple circular iris with a variable radius or any fixed iris placed on a translation stage. This results in a simple profiler design at a low cost. We also show that profiling with these proposed methods requires an exact mathematical expression for power transmission of a Gaussian ${\mathrm{TEM}}_{00}$ mode through a decentered circular aperture, which we also derive in this paper. The working principle of the proposed methods was validated through experimental results, which were also been compared to standard knife-edge-based profiling results. Experimental results are shown to be in strong agreement with the presented theory as well as comparable to beam waist estimates from a similar knife-edge profiling setup. Actual experiments were performed using a DMD-based setup instead of using mechanical irises. This was done to demonstrate a bulk motion-free, automated, digitally-controlled realization of the proposed beam profiler schemes.