1. INTRODUCTION

There are numerous methods to measure the diameter of a Gaussian laser beam, most of which are based on either direct imaging of the beam intensity profile or a spatial scan of the intensity using apertures of various shapes. Cameras have been used to acquire an image of the beam, and from this derive the properties of various beams including Gaussian and Bessel beams [1,2]. The knife-edge (razor-blade) method has been commonly used to measure the beam diameter down to micrometers [3–5]. Combined with the knife-edge technique, piezoelectric detection schemes have been implemented where the laser beam is chopped at a certain frequency and the photo-acoustic signal is analyzed with a reported accuracy of $\sim 10\%$ [6]. The knife-edge method is somewhat cumbersome, as the aperture has to be moved across the beam in a highly controlled fashion, requiring translation stages or motorized slits. Theory for the knife-edge and aperture method is well presented for Gaussian, elliptical, and rectangular beams [7]. For high-power laser beams, thermal effects have been utilized where the spot temperature is monitored with time and compared with the equilibrium temperature to estimate the beam diameter [8]. Also, beam diameters in the micrometer range have been measured with an accuracy of about 7% by scanning the beam through a straight edge and detecting the diffracted intensity only [9]. For relatively smaller beams, quadrant photodiodes have been used to measure the beam width, by using a photodiode mounted on a micrometer resolution translation stage and scanned along the two transverse beam axes [10]. The basic idea of using a moving obstacle to partially block the beam and thus retrieve the Gaussian beam diameter was demonstrated by Yoshida and Asakura [11], where two beams were measured using an aluminum ribbon with a statistical accuracy of 10% without comparison of the measured beam diameters with any other method.

In this paper, we revisit this approach and extend it to elliptical Gaussian beams, Airy spots, and the central peak of Bessel beams. Using only a cylindrical object (e.g., a standard round wire) of appropriate diameter, a photodiode, and an oscilloscope, even an untrained person can easily measure the diameter of a beam with a precision down to 1% and an accuracy of better than 3%. In order to do so, the experimenter simply manually moves the wire across the laser beam and records the minimum and maximum beam power with the photodiode and oscilloscope. By comparing this simple technique to a commercial beam profiler and the knife-edge technique [3,12], we show that the measurement accuracy can easily be better than 3% for diameters of Gaussian beams ranging from hundreds of micrometers to a few centimeters.

2. THEORY

In this section, we present analytical expressions for the estimation of the fraction of the total power of a beam that is transmitted when it is partially blocked using a cylindrical object. We derive a relation between beam diameter, the diameter of the obstruction, and the fraction of the power transmitted.

Let us consider the case in which a cylindrical object of diameter $D$ is moved through an optical beam of an intensity distribution $I(x,y)$ (see Fig. 1). For beams, where the intensity decreases monotonically with the distance from the center, the fraction of the light reaching the detector is minimal when the wire blocks the central part of the beam. The minimum transmission is then given by

 

Fig. 1. Basic principle of the method: a wire moves across a Gaussian laser beam partially blocking it, where $D$ is the diameter of the wire and $\theta$ is the angle between the wire’s normal to the $x$ axis when the wire crosses the center of the beam. Note that the coordinate system is chosen here so that it coincides with the minor and major axes of the beam.

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Many beam profiles are characterized by a parameter related to the width of the beam. In such cases, it is often possible to evaluate Eq. (1) numerically and plot this result as a function for this beam width in convenient units. Figure 2 shows the normalized beam width $w_{50}$ for Gaussian beams (red line), Airy spots (dashed blue), and Bessel beams (dot-dashed green). In order to compare the dependence of the transmission of the different beams, we normalized the curves to one at the point where the minimum intensity drops to 50%. In the following sections we describe the three intensity distributions in detail and show how their beam parameters can be determined from $w_{50}$.

 

Fig. 2. Beam diameter of Gaussian beams (solid red line), Airy spots (blue dashed line), and central peaks of Bessel beams (green dash-dotted line) as a function of the minimum transmissivity $(T_{\min })$. The parameter $w_{50}$ of the left vertical axis can be used together with Eqs. (4), (6), and (8) to accurately determine the beam parameters from $T_{\min }$. The right vertical axis directly shows the normalized Gaussian diameter $(2w_{\mathrm{G}}/D)$ calculated according to Eq. (4). The black circles depict the beam diameters as measured using the slit-based beam profiler divided by the wire thickness, as depicted in Table 1. The error bars are from the beam profiler only. The red dot denotes the value for the smallest beam diameter 630 μm, for which the beam profiler gives the largest relative error. The diameter solid black dot was measured using the knife-edge method.

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A. Gaussian Beams

We start by considering an elliptically shaped Gaussian beam. Without loss of generality, we assume that the major and minor axes of the ellipse coincide with the axis of the coordinate system. The intensity distribution of a Gaussian beam in a transverse $\mathrm{xy}$ plane is then given by

B. Airy Spots

The intensity distribution of the spot resulting from focusing an evenly illuminated circular aperture of radius $w_{\mathrm{t}}$ using an ideal lens of focal length $f$ is called an Airy spot. In the focus it has the intensity distribution $T_{\mathrm{A}}$:

C. Bessel Beams

Another interesting case is the Bessel beam, which has the following intensity distribution:

3. EXPERIMENTAL IMPLEMENTATION

The experimental implementation of this method requires a cylindrical object (e.g., a standard metal wire), a photodetector, and an oscilloscope, as can be seen in Fig. 3. For the measurement of a circular beam, it suffices to manually move the object through the beam and to record the trace on the oscilloscope using an appropriate trigger voltage. The $T_{\min }$ can then be determined from the min–max readings on the oscilloscope, taking into account any offsets that might occur. Care must be taken to ensure the linearity of the photodetector, e.g., by avoiding saturation of the photodiode using a neutral density filter after the wire. The motion of the wire has to be slow enough that the signal is fully contained in the frequency bandwidth of the detection system. In practice, a few kilohertz of bandwidth is easily achieved and fully sufficient. One large practical advantage is that the measurement depends only on the point where the minimum transmission is reached. It does not depend on the path followed to reach that point. For a collimated beam, it also does not depend on the angle of the wire along the optical axis ($z$).

 

Fig. 3. Schematic of the experimental setup. It consists of a laser source, a neutral density filter, mirrors, a beam expander, a wire, a photodiode, and an oscilloscope. Please note that the mirrors and lenses were only used to achieve variable diameter collimated beams in the same setup.

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For an elliptical beam, one passes repeatedly through the beam whilst slowly changing the angle $\theta$. One then optimizes the angle to find the minimum and maximum values of $T_{\min }$, which then correspond to the minor and major axes of the Gaussian beam. The measurement of the two axes can easily be done in one or two minutes. The beam diameters are then calculated again using Eq. (4). Note that the wire method is independent of the angle of the wire with respect to the transverse plane. This stands in stark contrast to the CCD or the moving aperture method, where deviation from the transverse plane creates an apparent ellipticity.

The discussion above assumes that the shape of the beams under investigation is well known. It is interesting to note that for minimum transmissions between 30% and 70%, the curves for the Airy spot, the Bessel beam, and the Gaussian beam are virtually indistinguishable. The difference between them becomes appreciable only when the wire covers most of the central part of the beam and the wings of the intensity distribution become more important. This indicates that the error due to small deviations from the assumed beam shape is very small for $30\%<T_{\min }<70\%$.

In the theoretical analysis we had assumed that all transmitted light is detected. In some circumstances, however, diffraction can lead to some of the transmitted light being scattered into angles high enough that the light misses the collection lens. The lower value of $T_{\min }$ results in an underestimation of the beam waist [see Eq. (4) and Fig. 2]. In order to avoid this, care must be taken that the lens has a sufficiently large numerical aperture ${\mathrm{NA}}_{\mathrm{d}}=\sin (\sigma )$ to collect all the light onto the detector. We can quantify this effect by taking into account [13] that 98% of the incident light power is diffracted from a thin wire over a half-angle $\tan (\sigma )=5\lambda /D$, where $\lambda$ is the wavelength. Assuming for simplicity that the collection optics consist of a lens of diameter $D_{\mathrm{L}}$ and focal length $f$, and that in order to collect 98% of the diffracted power ${\mathrm{NA}}_{\mathrm{c}}\ge {\mathrm{NA}}_{\mathrm{d}}$, we reach the condition $D\ge 10\lambda (\#F)$, where $(\#F)\equiv f/D_{\mathrm{L}}$ is the F-number of the collection optics. For example, using a typical lens of $f=100\mathrm{mm}$, $D_{\mathrm{L}}=25.4\mathrm{mm}$, the numerical aperture of the collection optics is ${\mathrm{NA}}_{\mathrm{c}}\approx 0.126$, and thus the wire diameter should be $D\ge 39\lambda$. This analysis can be expanded to higher or lower collection efficiencies; for example, to collect 99% of the input power, the wire diameter should be $D_{99\%}\ge 20\lambda (\#F)$, while to collect 90%, it is sufficient to use $D_{90\%}\ge 2\lambda f/D_{\mathrm{L}}\sim 6\mathrm{\mu m}$, which is much smaller than the smallest wire diameter used here (220 μm).

4. RESULTS

In order to verify our method, we compared the manual measurements with a commercial beam profiler and the knife-edge method for collimated Gaussian laser beams with beam diameters ranging from $2w=630\mathrm{\mu m}$ to 3 cm.

A sketch of the optical setup is shown in Fig. 3. We couple 0.4 mW of optical power from a 780 nm CW diode laser into a FC-APC single-mode optical fiber and collimate the light using an achromatic lens doublet. The resulting beam waist is $2w=1.36\mathrm{mm}$ with an ellipticity of $w_{\mathrm{x}}/w_{\mathrm{y}}\cong 1$ measured by a commercial moving-slit beam profiler (Ophir-Nanoscan 2 Ge/9/5). We expanded and reduced the beam waist using two different $4f$ lens systems resulting in five different beam diameters ($2w=0.63,1.36,2.80,4.22$, and 30 mm).

Figure 4 shows a typical oscilloscope trace of a wire of 1 mm diameter being manually moved through the beam. As described in the previous section, we can then calculate the Gaussian beam diameter using Eq. (4), where $T_{\min }$ is the ratio of the minimum to the maximum signal in Fig. 4. The diameter of the wire is expected to have a considerable impact on the ease and precision of the measurement. Too small of a wire diameter $(D\ll 2w)$ will result in too large of a transmissivity, which can be difficult to determine with good precision. Too large of a wire $(D\gtrsim 2w)$, on the other hand, emphasizes the wings of the beams, which tend to deviate more readily from the ideal Gaussian. In order to assess the effect of the wire diameter $D$ on the accuracy of the measurement, we measured one and the same beam using different wires of various diameters. We find that the mean standard deviation of the wire measurements is 1%. Furthermore, using the slit-based beam profiler, we determined the diameter of the beam to be $2w=2.8\mathrm{mm}$ in agreement to within 1%–13% with our wire-based measurements. The results for the comparison of this measurement to the wire-based method are shown in the upper part of Table 1. For $(2\lesssim 2w/D\lesssim 5)$, the wire method fully agrees with the measurements of the commercial device and also knife-edge-based measurements. We find that the method has a larger discrepancy (13%) with the beam profiler for $(D\gtrsim 2w)$, which might be due to larger wire diameters emphasizing existing deviations from the Gaussian profile in the wings of the beam.

 

Fig. 4. Oscilloscope trace of the transmitted power of a Gaussian beam of $2w=2.8\mathrm{mm}$ diameter with a 1 mm diameter wire being scanned across it. The solid blue lines correspond to the minimum and maximum voltages with $T_{\min }=U_{\min }/U_{\max }$. The dashed red line stands for the trigger voltage ($U_{\mathrm{trig}}=56.7\mathrm{mV}$) used.

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Table 1. Comparison of Beam Diameter Measurements Using the Wire Method (column 2) with a Commercial Slit-Based Beam Profiler (column 1, $\mathsf{0.6}\mathsf{mm}<\mathsf{2}\mathsf{w}<\mathsf{4.3}\mathsf{mm}$) and—for the 30 mm Beam Only (marked by †)—Using the Knife-Edge Method [3]^a

View Table

In order to test the wire method for a large range of beam parameters, we measured beam diameters ranging from $2w=630\mathrm{\mu m}$ to 30 mm with the wire method and compared them to measurements using a commercial scanning slit beam profiler for $630\mathrm{\mu m}\le 2w\le 4.22\mathrm{mm}$ and the knife-edge method using a fit to the error function ($2w=30\mathrm{mm}$). The results are summarized in the lower part of Table 1. The numbers in brackets are the standard deviations of the experimental points in units of the last digit, except for the fourth column, where the wire diameter was measured with digital calipers and the brackets indicate the stated accuracy of the device. The average standard deviation of the wire measurements was only 1%, demonstrating excellent reproducibility and excellent agreement with the measurements using the commercial device and knife-edge method.

5. CONCLUSIONS

We presented a simple and precise method to accurately measure the width of Gaussian beams, Airy spots, and Bessel beams to very high accuracy as verified experimentally for Gaussian beams. The method works for a very large range of beam widths from a few micrometers with basically no upper limit on the beam size. The lower limit on the beam size is set in practice by the availability of an appropriate size thin wire. The method has an excellent repeatability of about 1%, making it ideal for beam diameter comparisons and beam focusing. Our measurements are in good agreement with the comparative measurements performed with a commercial slit-based beam profiler and the knife-edge technique. Finally, the proposed technique is fully scalable and can be used in confined spaces where a beam profiler cannot be placed or for cases in which the beam width is larger than the beam-profiler aperture.

The simplicity of the proposed technique, which requires only instruments readily available in any optics laboratory, combined with its accuracy and repeatability make it a very interesting, low-cost alternative to the standard beam-profiling techniques.