On the Interference of two Gaussian beams and their ABCD Matrix Representation

Muzammil A. Arain; Guido Mueller

doi:10.1364/OE.17.019181

1. Introduction

Gaussian irradiance beam profiles are the most common irradiance patterns in Laser optics. Most of the lasers emit optical radiation that has an irradiance distribution which is well represented by a fundamental Gaussian mode. The Gaussian mode can be represented by a q-parameter that describes the beam size as well as the radius of curvature (ROC) as a function of the location along the optical axis. In fact, within the paraxial approximation the q-parameter defines a whole set of ortho-normal Hermite-(HG) or Laguerre-Gaussian (LG) modes which can be used to describe the irradiance distribution of any field. The propagation of Gaussian beams through space or via various optical components like mirrors, lenses, Gaussian ducts, and dielectric interfaces have long been calculated using standard ABCD matrices [1]. Use of these matrices greatly simplifies the calculation of beam irradiance profiles after an optical beam has passed through an optical system. An extension of this concept is the use of complex ray matrices to represent more complex situations like astigmatic resonators, and arbitrary complex wave fields [2–6].

Frequently one encounters interference of two optical beams with spatial irradiance distributions which can be approximated by two different fundamental Gaussian modes. Usually this is carried out in the spatial domain by simply adding the amplitudes of the two beams and thus finding the irradiance distribution after the interference [7]. Another technique is to represent one beam in the HG or LG basis of the other beam carrying energy in higher order modes. For example, if the modal mismatch between the two beams is small, one of the beams can be described as the sum of a TEM₀₀ mode and a TEM₁₀ component in the LG-basis which best describes the other mode [8]. The amplitude of the TEM₁₀ component depends upon the mode mismatch between the two interfering beams. For larger mismatches we would have to include a large number of higher order modes to express this situation.

Yet another technique can be to describe the interference of two Gaussian modes into a third Gaussian mode that more closely represents the Gaussian mode of the resultant beam. For keeping the problem simple, we may want to express the sum of the two Gaussian beams as a third fundamental TEM₀₀ Gaussian mode. If we can derive the best matching mode easily, we can represent the interference of two beams in terms of the ABCD formalism. This would open a whole new branch of ABCD beam propagation which can be used to describe imperfectly mode matched interferometers, and coupled cavities which are now solved by either using fast-Fourier transform (FFT) techniques [9] or by using the first principle expressions of Hygen’s integral [10]. It is understood that the proposed method will not describe such complex coupled cavities and interferometer perfectly, but nevertheless will provide an extremely fast way to provide a basis to start more rigorous treatment using computationally extensive methods like [9–12]. However, for this method to be effective, we need to incorporate the relative amplitudes and the relative phase between the two interfering beams in the q-parameter domain. In the next two sections, we will attempt to derive such a relationship.

2. Interference of two Gaussian beams

First we evaluate the interference of two cylindrical symmetric Gaussian irradiance beams when the beam size and the ROC of the two beams do not match and the two beams have different magnitudes. Assume that:

E_{1} = α_{1} e^{- r^{2} [\frac{1}{w_{1}^{2}} + i \frac{π}{λ R_{1}}]},

and

E_{2} = α_{2} e^{- r^{2} [\frac{1}{w_{2}^{2}} + i \frac{π}{λ R_{2}}]}

are the two optical fields where

α_{1, 2}

,

w_{1, 2}

, and

R_{1, 2}

are the amplitudes, beam sizes and the ROCs of the two beams respectively. The parameters of the two beams are then defined by:

\frac{1}{q_{1,2}} = \frac{1}{R_{1,2}} - i \frac{λ}{π w_{1, 2}^{2}} .

Unless q ₁ = q ₂, the superposition will not longer have a pure Gaussian irradiance distribution. However, it can be approximated by a new Gaussian distribution represented by a new q-parameter. The new q-parameter depends upon the original q-parameters as well as the relative magnitudes and phases of the interfering beams. The formalism described in this paper allows deriving the new q-parameter in a way which is compatible with the standard ABCD formalism. The error due to this approximation will also be quantified in this paper.

A good approximation for the q-parameter of the combined field can be calculated from the initial q-parameters q_1,2 and the complex field amplitudes α_1,2. First we expand the two interfering beams using a Maclaurin series expansion of the complex exponential functions:

E_{1} = α_{1} e^{- r^{2} [\frac{1}{w_{1}^{2}} + i \frac{π}{λ R_{1}}]} = α_{1} [1 - \frac{r^{2}}{w_{1}^{2}} - i \frac{π r^{2}}{λ R_{1}} + ...],

E_{2} = α_{2} e^{- r^{2} [\frac{1}{w_{2}^{2}} + i \frac{π}{λ R_{2}}]} = α_{2} [1 - \frac{r^{2}}{w_{2}^{2}} - i \frac{π r^{2}}{λ R_{2}} + ...] .

The sum of these two beams E can be written as:

E = E_{1} + E_{2} = (α_{1} + α 2) [1 - r^{2} \frac{(\frac{α_{1}}{w_{1}^{2}} + \frac{α_{2}}{w_{2}^{2}}) + \frac{i π}{λ} (\frac{α_{1}}{R_{1}} + \frac{α_{2}}{R_{2}})}{(α_{1} + α_{2})} + ...] .

From Eq. (6) we can intuitively derive an expression for the resultant q-parameter. It is evident that the q-parameter should be a weighted sum of inverse q-parameter values of the individual beams. Owing to the fact that while using standard ABCD matrices, no operation can be performed separately on the real and imaginary parts of the q-parameters, we will adopt absolute values for the weighted sum. This evidently will introduce some error that will be quantified in the next section. Thus, we define the resultant beam waist and ROC parameter as:

\frac{1}{w_{e}^{2}} = \frac{(\frac{| a_{1} |}{w_{1}^{2}} + \frac{| a_{2} |}{w_{2}^{2}})}{(| a_{1} | + | a_{2} |)} = \frac{| a_{1} | / w_{1}^{2}}{(| a_{1} | + | a_{2} |)} + \frac{| a_{2} | / w_{2}^{2}}{(| a_{1} | + | a_{2} |)},

and

\frac{1}{R_{e}} = \frac{(\frac{| a_{1} |}{R_{1}} + \frac{| a_{2} |}{R_{2}})}{(| a_{1} | + | a_{2} |)} = \frac{\frac{| a_{1} |}{R_{1}}}{(| a_{1} | + | a_{2} |)} + \frac{\frac{| a_{2} |}{R_{2}}}{(| a_{1} | + | a_{2} |)} .

We can combine Eq. (7) and Eq. (8):

\frac{1}{R_{e}} - i \frac{λ}{π w_{e}^{2}} = \frac{| a_{1} |}{(| a_{1} | + | a_{2} |)} [\frac{1}{R_{1}} - i \frac{λ}{π w_{1}^{2}}] + \frac{| a_{2} |}{(| a_{1} | + | a_{2} |)} [\frac{1}{R_{2}} - i \frac{λ}{π w_{2}^{2}}] .

to determine the new q-parameter that results from the interference of two Gaussian beams:

\frac{1}{q e} = \frac{| a_{1} |}{(| a_{1} | + | a_{2} |)} \frac{1}{q_{1}} + \frac{| a_{2} |}{(| a_{1} | + | a_{2} |)} \frac{1}{q_{2}} .

Thus we have shown that the q-parameter of the new beam is the inverse of the weighted sum of the inverse individual q-parameters. Based upon, Eq. (10), we can define the ABCD matrix that can be used to represent the sum of two optical beams with different amplitudes.

[\begin{array}{l} A & B \\ C & D \end{array}] [\begin{array}{l} 1 \\ {1/q}_{e} \end{array}] = [\begin{array}{l} | a_{1} | / (| a_{1} | + | a_{2} |) & 0 \\ 0 & | a_{1} | / (| a_{1} | + | a_{2} |) \end{array}] [\begin{array}{l} 1 \\ {1/q}_{1} \end{array}] + [\begin{array}{l} | a_{2} | / (| a_{1} | + | a_{2} |) & 0 \\ 0 & | a_{2} | / (| a_{1} | + | a_{2} |) \end{array}] [\begin{array}{l} 1 \\ {1/q}_{2} \end{array}] .

This expression in Eq. (11) can be used to represent the ABCD matrix of the superposition of two Gaussian beams with different amplitudes, phases, and q-parameters in ABCD matrix calculations.

3. Figure of merit for the q-parameter formulation

Obviously, the above derived matrix representation of the interference is only an approximation of the true field. To validate this approximation, we evaluate the overlap integral (FOM from here on) between the beam with different q-parameters represented by Eq. (10) and the numerical sum of the two optical beams with miss-matched parameters. The FOM is given by:

F O M = \frac{{| \int_{- \infty}^{\infty} (E_{1} (r) + E_{2} (r)) \cdot E^{*} (r) d A |}^{2}}{\int_{- \infty}^{\infty} {| (E_{1} (r) + E_{2} (r)) |}^{2} d x \int_{- \infty}^{\infty} {| E (r) |}^{2} d A} .

Here ‘*’ represents conjugate operation and dA represents integral over the area. While evaluating the FOM, the exact expression for the sum of two fields will be used, i.e., considering the amplitude as well as the phase of the two beams. Replacing the values of the electric fields in the above expression and using radial coordinates, we get:

F O M = \frac{{| \int_{0}^{2 π} d φ \int_{0}^{r} [α_{1} e^{- r^{2} [\frac{1}{w_{1}^{2}} + i \frac{π}{λ R_{1}}]} + α_{2} e^{- r^{2} [\frac{1}{w_{2}^{2}} + i \frac{π}{λ R_{2}}]}] \cdot [(α_{1} + α_{2}) e^{- r^{2} [\frac{1}{w_{e}^{2}} - i \frac{π}{λ R_{e}}]}] r d r |}^{2}}{\int_{0}^{2 π} d φ \int_{0}^{\infty} {| (α_{1} e^{- r^{2} [\frac{1}{w_{1}^{2}} + i \frac{π}{λ R_{1}}]} + α_{2} e^{- r^{2} [\frac{1}{w_{2}^{2}} + i \frac{π}{λ R_{2}}]}) |}^{2} r d r \int_{0}^{2 π} d φ \int_{0}^{\infty} {| (α_{1} + α_{2}) e^{- r^{2} [\frac{1}{w_{e}^{2}} + i \frac{π}{λ R_{e}}]} |}^{2} r d r} .

Using some algebraic simplifications and using the fact that:

\int_{0}^{\infty} x e^{- a^{2} x^{2}} d x = \frac{1}{2 a^{2}} .

We can evaluate the above integral as Eq. (15) below. Note that Eq. (15) is an exact expression where the amplitudes and the phases of the two beams have been taken into account. Equation (15) is the FOM for 2-D power mismatch. Equation (15) calculates the mode mismatch between the sum of two Gaussian irradiance beams and their approximate representation given by Eq. (10). The novelty of Eq. (10) is that it can be used in an ABCD matrix format. The loss predicted by Eq. (15) for a given set of parameters, would be the amount of higher order losses in this representation.

F O M = \frac{4 {| {\frac{α_{1}}{\frac{1}{w_{1}^{2}} + \frac{1}{w_{e}^{2}} + i \frac{π}{λ} (\frac{1}{R_{1}} - \frac{1}{R_{e}})}} + {\frac{α_{2}}{\frac{1}{w_{2}^{2}} + \frac{1}{w_{e}^{2}} + i \frac{π}{λ} (\frac{1}{R_{2}} - \frac{1}{R_{e}})}} |}^{2}}{(w_{e}^{2}) | {| a_{1} |}^{2} w_{_{1}}^{2} + {| a_{2} |}^{2} w_{_{2}}^{2} + 2 [{\frac{α_{1} \times α_{2}^{*}}{\frac{1}{w_{1}^{2}} + \frac{1}{w_{2}^{2}} + i \frac{π}{λ} (\frac{1}{R_{1}} - \frac{1}{R_{2}})}} + {\frac{α_{2} \times α_{_{1}}^{*}}{\frac{1}{w_{1}^{2}} + \frac{1}{w_{2}^{2}} + i \frac{π}{λ} (\frac{1}{R_{2}} - \frac{1}{R_{1}})}}] |} .

4. Numerical validation of addition of two beams using q-parameters

To validate the result, we use the FOM described by Eq. (15) and evaluate it for a range of values of beam waist, ROC, amplitude, and phase mismatches. We start with few simple cases and then gradually move to some complicated cases. In all the plots, we will use normalized units for beam sizes and ROCs to make the figures more meaningful. However, we are using 5.2 cm beam size and 1000 m ROC as the reference values. These values are typical of large scale Fabry-Perot cavity enhanced gravitational wave interferometers where the proposed solution has immediate application [13].

4.1 Beam size mismatch and phase difference

Figure 1 shows FOM plotted as a function of phase difference θ between the two beams for various beam size mismatches as noted in the legend. Here we have assumed that $| α_{2} | = 2 | α_{1} |$ and the phase difference varies from zero to 180° between the two angles. As the beam size mismatch increases, the FOM decreases. Also, the FOM has minima when the two beams are out of phase (destructive interference). For the example in Fig. 1, we have used 5.2 cm beam size but the plots are essentially independent of actual beam size and ROCs used. We have tested extensively the proposed FOM as a function of various beam sizes and ROC values and have observed that the mode mismatch depends upon the normalized differences.

Fig. 1 FOM evaluated for various beam size mismatch as a function of phase angle between two Gaussian irradiance beams. Here w ₁ = 5.2 cm, $| α_{2} | = 2 | α 1 | (\cos θ + i \sin θ)$ and θ varies from zero to 360°. Although the plots shown are for 5.2 cm beam, the functional form remains same for widely different beam sizes.

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4.2 Beam size mismatch and relative magnitude

As evident from Fig. 1, out of phase beams represent the worst case scenario for two beams with different beam sizes while in-phase beams are well described by this formalism even for substantial beam size mismatches. Next we evaluate the effect of magnitude ratio on the FOM. We start with two beams with varying beam magnitude ratios but having either 0° or 180° phase difference. The model breaks down when the two beams have the same amplitude but are 180° out of phase as shown in Fig. 2 . However, as the magnitude difference also increases between the two beams, the FOM increases again fairly fast.. This behavior can be explained with the help of Fig. 3 where we have plotted two beams with beam sizes different by a factor of 2 and the amplitude of the beam with smaller beam size is −1.4 times larger than the larger beam. The proposed approximation deteriorates when the sum of the amplitudes changes sign within a distance $r < w$ from the propagation axis. Note that a low FOM value does not mean that the proposed fundamental Gaussian beam parameters are wrong. It shows that now there is more power in the higher order beams and thus more modes need to be included in the analysis to represent the sum of two Gaussian beams.

Fig. 2 FOM evaluated for various beam magnitude ratios. Negative values indicate that the two beams are 180° out of phase with each other. Various lines indicate a different beam size mismatch. Again $w_{1} = 5
.2 cm$ .

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Fig. 3 Plot of two Gaussian irradiance beams with different magnitude and beam size ratio as indicated in the legend. Shown in red is the numerical sum of two beams while the approximation represented by Eq. (10) is shown via cyan color.

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4.3 ROC mismatch

Mismatches in the ROC of the two interfering fields can also be treated with our approach. Figure 4 shows the FOM for various ROC mismatches as a function of the magnitude ratio between the fields; again, negative values correspond to phase differences of 180° or destructive interference. As both modes can always be propagated forward or backwards to the waist of the first mode without changing the FOM, we simplify the discussion by assuming that the first beam is at its waist which implies R₁ is infinite. In our numeric examples, we further assume a beam size of again 5.2cm for both beams. As the natural scale for ROCs is the Rayleigh range Z_r, we expressed R₂ in Fig. 4 as a fraction of the Rayleigh range of the first beam where Z_r = 8 km for λ = 1.064μm. As can be seen from the figure, our approximation will be a very good representation of the final field distribution as long as R₂ > Z_r and the interference is constructive. When the ROC becomes smaller than the Rayleigh range, the sagitta across the beam size approaches λ. For equal amplitudes, the resulting beam will then also include some amounts of higher order modes in addition to our approximated fundamental Gaussian mode. Consequently, the FOM of our approximated solution is reduced. For magnitude ratios near −1, the two spatial profiles interfere destructively and the resulting spatial distribution no longer resembles anything close to a fundamental Gaussian mode. Consequently, our approximation is no longer good. For larger magnitude ratios where one of the beams dominates, our approximation will again provide a good description for the resulting field.

Fig. 4 FOM evaluated for various ROC mismatches normalized to Rayleigh range Z_r of about 8 km. The first beam is considered at the beam waist thus ΔR represents the ROC of the second beam.

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4.4 A general example

Finally, we present the effect of both beam size and ROC mismatch for the beams used in the above examples. Here we use the first beam as reference with a magnitude of 1, i.e., $α = 1, w_{1} = 5.2 cm, R_{1} = 1000 m$ . We vary the beam size and ROC of the second beam with equal ratio. The x-axis shows normalized units. Therefore, x-axis value of −50 and + 50 means that the beam size and the ROC of the second beam are 50% and 150% of the first beam respectively. The result is shown in Fig. 5 . The noticeable features are that the approximation works very good for the case of in-phase interfering beams with even 50% beam parameter mismatch. Approximation introduces error for beams that are close in magnitude but are 180° out of phase with each other.

Fig. 5 FOM evaluated for same normalized beam size and ROC mismatch for various magnitude rations. The parameters of the first beam are kept constant where $α_{1} = 1$ , R₁ = 1000 m, $w_{1} = 5
.2 cm$ . The % change in the beam parameters of the second beam is plotted on x-axis.

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7. Validation of ABCD propagation in optical systems

As an attempt to provide confidence in the use of the proposed ABCD formulism, we use the example of Sec. 4.4 in an optical system shown in Fig. 6 .

Fig. 6 Optical system to test the proposed ABCD formulism using example of Sec. 4.4. Here two beams (Beam 1 (red) and Beam 2 (blue)) are combined using a beam combiner and the resultant beam is calculated numerically as the sum of beam 1 and Beam 2. Beam 3 (green) is the beam obtained via the ABCD formulism proposed in this paper and propagated via the same optical system as Beam 1 and 2.

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We propagate the two beams (Beam 1 and 2) individually through the optical system and calculate the interference of the two beams at various distances from the lens. We also calculate Beam 3 using our proposed method at the beam combiner. Then we propagate this beam through the lens system just like Beam 1 and 2. We evaluate the FOM at various distances from the lens for the numerical sum of Beam 1 and Beam 2 and Beam 3 propagated via the ABCD matrix system. The resultant FOM is invariant for the modal space (or mode mismatch parameters) shown in Fig. 5. Thus we conclude that if the FOM calculated at the point of interference is acceptable, the representation of interference between two Gaussian beams through the ABCD matrix preserves the FOM while propagating through an optical system. This is the key find of this paper that enables us to represent mismatched coupled cavities in a simple way and find their Eigen values.

6. Summary

In conclusion, we have demonstrated that the interference of two beams can be described by the sum of two ABCD matrices. This matrix can accommodate scenarios where the two interfering beams have complex q-parameters. The proposed ABCD matrix also takes into account the amplitudes of the two beams and the phase angle between them and the effect of amplitude and phase angle on the approximation is calculated. However, it should be noted that if the two beams undergoing interference are poorly mode matched to start with, the generated fundamental Gaussian mode will not be a good approximation to the resultant solution. Thus it will be prudent to check the FOM for a specific problem to determine the suitability of this approach.

The proposed ABCD matrix is very helpful in determining beam propagation in interferometers, coupled cavities, and other complex optical systems without using complex FFT models or higher order mode expansions. It is especially useful when following optical components and detectors only transmit or are sensitive to fundamental Gaussian modes as is the case in all large scale gravitational wave detectors.

Acknowledgments

The authors want to acknowledge the support of the LIGO Science Collaboration. Especially the discussions with Hiro Yamamoto, Bill Kells, David H. Reitze, and David B. Tanner were very helpful. This work was supported by the National Science Foundation under grant PHY-0653582.

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On the Interference of two Gaussian beams and their ABCD Matrix Representation

Abstract

1. Introduction

2. Interference of two Gaussian beams

3. Figure of merit for the q-parameter formulation

4. Numerical validation of addition of two beams using q-parameters

4.1 Beam size mismatch and phase difference

4.2 Beam size mismatch and relative magnitude

4.3 ROC mismatch

4.4 A general example

7. Validation of ABCD propagation in optical systems

6. Summary

Acknowledgments

References and links

Cited By

Figures (6)

Equations (15)

Optics Express