Mode conversion efficiency to Laguerre-Gaussian OAM modes using spiral phase optics

Andrew Longman; Robert Fedosejevs

doi:10.1364/OE.25.017382

1. Introduction

Over the past 25 years there has been growing interest in exploiting the orbital angular momentum (OAM) properties of light [1]. Studies into the generation and uses of laser generated OAM beams including cold atom trapping [2,3], communications [4,5], and enhanced imaging techniques [6,7] have provided a rich area of research. Recently, interest has grown with the application of OAM in laser-plasma interactions [8,9] corresponding to particle guiding [10,11] and magnetic field generation [12–15]. Additionally, applications of OAM modes in high harmonic generation have gained much interest in the last few years [16,17]. Laser beams carrying OAM can be described using Laguerre-Gaussian (LG) modes that propagate with a rotating Poynting vector about the beam axis. These LG modes are characterized by two numbers, the azimuthal and radial integers (denoted ℓ and p respectively). An infinite number of orthogonal modes can be generated and propagated in theory, but in practice the mode structure and purity is limited by the generation technique.

Since Allen’s paper [1], many methods for generating OAM modes have been introduced and developed; from mixing and converting various Hermite-Gaussian modes [18], to diffractive optical elements [19]. Spatial light modulators are a popular choice for high purity OAM mode generation due to their flexibility to generate any mode on demand, but are costly and only useful for lower power applications [4]. Generation of OAM modes by spiral phase plate (SPP), introduced by Beijersbergen in 1994 [20], has attracted a lot of interest due to its mode conversion quality, cost, and ease of use. The ability to convert a Gaussian TEM₀₀ mode into a distribution of LG modes strongly centered around a single mode makes the SPP extremely attractive to users of high powered lasers who cannot easily modify the cavity modes of the laser front end or intermediate laser amplifier stages. Such phase plates can be designed to meet any output ℓ and p mode requirements with high purity limited only by the manufacturing processes and conversion efficiency.

The SPP has its own intrinsic set of azimuthal and radial mode numbers L and P respectively, which need not be integers. SPP’s [20] are simple in their construction in that the spatial phase pattern of the LG mode, namely the helical wavefront, is imprinted onto a glass substrate via a spiral increase in substrate thickness with the total step height H as shown in Fig. 1(a). The spiral height is proportional to the laser wavelength and material index given by the following: H = Lλ/ (n − 1) [20–22]. Here, λ is the incident laser wavelength and n is the index of refraction of the SPP. It was reported [20], that the conversion efficiency of an L = 1, P = 0 spiral phase plate was 78.5% from the TEM₀₀ mode to the first LG₁₀ mode. It was later reported, that the conversion efficiency was higher yielding a conversion efficiency of around 93% to the LG₁₀ mode [21]. The difference arises from the fact that the first report assumed an output beam waist equal to the input Gaussian beam waist while the second report allowed the output beam waist to be a free parameter. However, the second report does not give details about the relation of the output LG mode beam waist to the input Gaussian beam waist.

Fig. 1 (a) Continuous spiral phase optic with L = 1 and P = 0 also indicating the total spiral height H. (b) 16 step spiral phase optic with L = 1 and P = 0. Both plates shown with an 800nm step height.

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Spiral phase mirrors (SPM’s) are an extension to this generation method using a reflective optical element [23,24] where the height of each spiral step is one half the input wavelength. Since their first publication, SPP’s and SPM’s have been manufactured using a variety of processes with deposition and erosion techniques. We will call these transmission and reflection optics collectively spiral phase optics (SPO’s).

The 2004 paper by Sueda et al. [25], introduced the idea of using a stepped spiral phase plate (SSPP) [Fig. 1(b)], rather than a continuous spiral [Fig. 1(a)] as first proposed [20]. By depositing steps using electron beam evaporation, fabrication of an SSPP is simpler with a small loss in conversion efficiency. It was reported [25] that the expected conversion efficiency to the LG₁₀ mode dropped from a value of 78.5% to 77.5% when switching from a continuous phase plate to a 16 step L = 1, P = 0 spiral phase plate, again assuming the same output beam waist as input beam waist.

When considering the use of spiral phase optics, both continuous and stepped in ultrafast laser systems, the laser bandwidth can become a significant fraction of the central wavelength. Conversion efficiency calculations of SPO’s in laser beamlines have often assumed that the incident light is monochromatic, neglecting this effect. While a few reports have tried to address this issue [26], little work has been done in calculating the effects of broadband beams and spiral steps on mode generation purity and conversion efficiency. Here we show for the first time, an analytical expression for the conversion efficiency from a Gaussian TEM₀₀ laser mode to an arbitrary Laguerre-Gaussian mode for continuous, stepped, and non-integer charged spiral phase optics for the lowest order radial component (P = 0). This analysis also includes an estimation of the effects from broad laser bandwidth using numerical techniques.

2. Mode decomposition

As a Gaussian TEM₀₀ mode is transmitted through or reflected from a spiral phase optic, the spatial phase of the beam is shifted by the spatial phase pattern of the optic. Initially this mode in the near field remains spatially a phase shifted Gaussian TEM₀₀ mode, but as it propagates to the far field the beam phase begins to reshape its intensity profile that can be described by a distribution of LG modes. The higher order p LG modes diverge more quickly than lower order modes leaving the lowest order p modes dominating in the far field near the axis. Conversion to each of these modes can be calculated from the inner product of the input and output modes with a transmission operator T, representative of the spatially varying phase shift from the optic. We can express the transmission operator for a continuous (infinite number of steps) SPO with P = 0 by the following expression:

T = \exp (- i L ϕ)

Here we define L as the topological charge of the SPO, which dictates the fundamental azimuthal mode number of the output beam ℓ. In this paper we limit ourselves to a transmission operator that has a null radial index, that is, P = 0 as they are the most common in use today. Higher order P mode SPO’s will be discussed separately due to the mathematical complexities they introduce.

The general expression for an arbitrary normalized LG mode amplitude is given by [27]:

\begin{array}{l} u_{ℓ p} & = \sqrt{\frac{2 p!}{π (p + | ℓ |)!}} \frac{1}{w (z)} {[\frac{r \sqrt{2}}{w (z)}]}^{| ℓ |} \exp [\frac{- r^{2}}{w^{2} (z)}] L_{p}^{| ℓ |} (\frac{2 r^{2}}{w^{2} (z)}) \\ \times \exp [- i ℓ ϕ] \exp [\frac{i k_{0} r^{2} z}{2 (z^{2} + z_{R}^{2})}] \exp [- i ψ (z)] \end{array}

Here, $z_{R} = π w_{0}^{2} / λ$ is the Rayleigh range with wavelength λ and wavevector k₀ = 2π/λ, $w (z) = w_{0} {[(z^{2} + z_{R}^{2}) / z_{R}^{2}]}^{1 / 2}$ is the radius of the beam with beam waist w₀ and ψ(z) = [(2p + |ℓ| + 1) arctan (z/z_R)] is the Gouy phase. The radial and azimuthal mode numbers of the LG mode, p and ℓ respectively, describe an array of unique modes where p is zero or any positive integer and ℓ is any real integer [28]. The associated Laguerre polynomials $L_{p}^{ℓ}$ can be given by the expression:

L_{p}^{ℓ} (x) = \sum_{n = 0}^{p} {(- 1)}^{n} \frac{{(p + ℓ)}^{!}}{(p - n)! (ℓ + n)! n!} x^{n}

The conversion efficiency from one orthogonal mode to another by means of a transfer function is generally given by the inner product [18]:

η_{ℓ p} = {| 〈 u_{ℓ p} | T | u_{m n} 〉 |}^{2} = {| \int_{0}^{2 π} \int_{0}^{\infty} u_{ℓ p}^{*} (r, ϕ, z) T (r, ϕ) u_{m n} (r, ϕ, z) r d r d ϕ |}^{2}

The modes are normalized such that a unit transfer function in the inner product will result in 0 or 1, and squaring the magnitude of the inner product gives the normalized conversion efficiency amplitude η_ℓp with transmission function T. Setting the mode numbers p = ℓ = 0 in Eq. (2), we obtain the familiar TEM₀₀ mode common to most laser cavity outputs. Expanding the inner product in Eq. (4) with an input TEM₀₀ mode and using a continuous spiral phase optic yields an integral with 2 free parameters, the beam waists and their axial positions z. In many previous works [18,20,25,29], the beam waists were assumed equal and analyzed with the waist positions both set to z = 0 to yield a unique mode decomposition. There is however, no physical requirement for the beam waist sizes to be equal [21, 22, 28]. Whilst the position of the beam waist may also vary, such a change leads to a modification of the Gouy phase for each beam and the integral becomes non-analytic. We therefore allow the beam waist to vary but maintain that the beam waists be set to z = 0. Using an input TEM₀₀ mode with beam waist w₀(z = 0), and choosing an output mode with mode numbers ℓ and p and beam waist w₁(z = 0), the inner product Eq. (4) is given by:

\begin{array}{l} 〈 u_{ℓ p} | T | u_{00} 〉 & = \frac{2}{π} \sqrt{\frac{p!}{(p + | ℓ |)!}} \frac{1}{w_{0} w_{1}} \int_{0}^{2 π} \int_{0}^{\infty} {[\frac{r \sqrt{2}}{w_{1}}]}^{| ℓ |} \exp [- r^{2} (\frac{1}{w_{0}^{2}} + \frac{1}{w_{1}^{2}})] \\ \times L_{p}^{| ℓ |} (\frac{2 r^{2}}{w_{1}^{2}}) \exp [i (ℓ - L) ϕ] r d r d ϕ = R_{| ℓ | p} Φ_{ℓ L} \end{array}

The double integral may be separated into the product of two integrals, denoted R_ℓp and Φ_ℓL which represent the radial and azimuthal integrals respectively. Substituting the series expansion for the associated Laguerre polynomials, the radial part of the integral including the multiplicative coefficients becomes:

\begin{array}{l} R_{| ℓ | p} & = \frac{2}{π} \sqrt{\frac{p!}{(p + | ℓ |)!}} \frac{2^{\frac{| ℓ | 2}{}}}{w_{0} w_{1}^{| ℓ | + 1}} \\ \times \int_{0}^{\infty} r^{| ℓ | + 1} \exp (- β r^{2}) \sum_{m = 0}^{p} \frac{{(- 1)}^{m} (| ℓ | + p)!}{(p - m)! (| ℓ | + m)! m!} \frac{2^{m} r^{2 m}}{w_{1}^{2 m}} d r \end{array}

With,

β = \frac{1}{w_{0}^{2}} + \frac{1}{w_{1}^{2}}

For the case where the SPO has a P = 0 radial index, one can find an analytic solution to the radial integral utilizing the following identity [30]:

\int_{0}^{\infty} x^{n} \exp (- a x^{b}) d x = \frac{1}{b} a^{- 1 / b (n + 1)} Γ (\frac{n + 1}{b})

where Γ represents the gamma function. For the case of the continuous SPO, we can also evaluate the azimuthal integral below to yield 2π when the values of L and ℓ are equal:

Φ_{ℓ L} = \int_{0}^{2 π} \exp (i (ℓ - L) ϕ) d ϕ = {2 π |}_{ℓ = L}

When the difference between the topological charges of the SPO L and output mode ℓ is a nonzero integer, the integral evaluates to zero. As already stated, the azimuthal charge of the output beam must be an integer, but the topological charge of the SPO can be designed to have any value. Non-trivial solutions exist when the SPO carries a non-integer topological charge and will be discussed in more detail in the next section. Combining and simplifying the results from Eq. (6) and Eq. (9) yields the following result:

η_{ℓ p} = {| R_{| ℓ | p} Φ_{ℓ = L} |}^{2} = 2^{| ℓ | + 2} p! (| ℓ | + p)! γ^{2} {[\sum_{m = 0}^{p} \frac{{(- 2)}^{m} Γ (μ)}{(p - m)! (| ℓ | + m)! m! {(1 + γ^{2})}^{μ}}]}^{2}

Here, we define the beam waist ratio, γ = w₁/w₀ and define μ = m + |ℓ|/2 + 1. This gives the general conversion efficiency of a TEM₀₀ mode into any LG mode through an integer charged, continuous SPO with P = 0 and ℓ = L. The mode conversion efficiency can be thus calculated as a function of the beam waist ratio, the results of which are plotted in Fig. 2 for the case of L = ℓ = 1. The output from the SPO gives a spectrum of LG modes centered around a fundamental azimuthal mode (L = ℓ) with corresponding radial modes p. It was shown [20] that a continuous spiral phase plate can convert a TEM₀₀ mode into an LG₁₀ mode with 78.5% maximum efficiency assuming the input and output beam waists are equal. However, the actual conversion efficiency into an LG₁₀ mode is higher reaching just over 93% when the ratio of the beam waists is set to an optimum value. The maximum conversion efficiency to the fundamental mode can be found by finding the extremum of Eq. (10) where the slope is zero relative to γ while setting p = 0, giving the optimum value of γ as:

γ = \frac{1}{\sqrt{| ℓ | + 1}}

This expression gives the optimal beam waist ratio for conversion into any LG_ℓ0 mode yielding a value of η₁₀ = 0.9308 at $γ = 1 / \sqrt{2}$ for a L = 1, P = 0 SPO. The remaining 7% of the input beam energy in this case is converted into higher order radial modes illustrated in Fig. 2. The conversion efficiency value of these additional modes is summarized in Table 1. An interesting observation made here is the conversion efficiency to odd p modes goes to zero at some value of the beam waist ratio with the p = 1 mode going to zero at the optimal beam waist ratio for all ℓ = L values.

Fig. 2 Conversion efficiency of a continuous spiral phase optic with L = 1 to the first 5 LG_ℓp modes. The peak of the LG₁₀ mode is located at $γ = 1 / \sqrt{2}$ where γ is the ratio of the output beam waist to the input beam waist.

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Table 1. Conversion efficiencies from a TEM₀₀ mode to various LG modes for given SPO step numbers N. The SPO carries a null radial index (P = 0) and results are shown for both L = 1 and L = 2 optics. The N = ∞ corresponds to a continuous phase optic.

View Table

The reason for an optimum output beam waist parameter w₁ which is smaller than the starting input beam waist parameter, w₀, can be understood as follows. For a given beam waist parameter the root mean square (RMS) intensity radius of an LG₁₀ mode is larger than the RMS intensity radius of a TEM₀₀ mode [31]. Thus if one converts a TEM₀₀ with a given RMS intensity radius into an LG₁₀ mode with the same RMS intensity radius, the matching beam radius of the LG₁₀ mode is necessarily smaller. For higher ℓ mode numbers the optimum matching radius becomes even smaller as shown in Eq. (11). An assumption made in this analysis which can experimentally lower this conversion efficiency is the domain of the radial integral, assumed to have an infinite boundary. Realistically, SPO’s are finite in radius, and therefore we assume that in practice the incident Gaussian spot size is small compared to the SPO diameter.

As one increases the topological charge of the SPO, the conversion efficiency to the fundamental ℓ = L mode decreases as shown in Fig. 3. The limit as L approaches infinity of Eq. (10) goes to zero. This implies that there is some limit to the efficient generation of OAM modes through the use of an SPO. If for example we require the conversion efficiency to the fundamental mode to be greater than 50%, we find that the maximum topological charge is L = 11. Likewise for a conversion efficiency of 20%, the charge on the plate can be no more than L = 82. The fabrication of high topological charge spiral phase mirrors (up to L = 100) have been reported previously [24], and have given interesting results. Conversion efficiencies to the fundamental mode were not however explicitly stated, it was stated however that the conversion efficiency was reduced in high charge plates due to scattering from the area between steps. Large topological charge modes have narrow ring-like intensity patterns limiting the amount of coupling from a broad Gaussian beam into such modes.

Fig. 3 Conversion efficiency of a continuous spiral phase optic with charge L to the L = ℓ LG mode assuming the optimum output beam waist given by Eq. (11). Displayed are the results for the first four even modes: p = 0, 2, 4, 6 on a log-log scale.

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3. Stepped spiral phase optics

Extending the analysis to include stepped spiral phase optics requires modification of the transmission operator T, but not the radial integral R_|ℓ|p in the P = 0 case. In essence, stepped spiral phase optics (SSPO’s) are a spiral staircase consisting of N steps per revolution as shown in Fig. 1(b). Assuming all the steps are equally spaced and are of equal size, the transmission operator can be modified to match the SSPO phase in each step n given by:

T = \exp [- i L (\frac{2 π n}{N})]

Again, the L value represents the intrinsic topological charge of the SSPO, additionally denoting n as the individual step number. Integrating each discrete step requires us to sum over all the steps to find the total contribution to the scaling factor. This can be written generally as the expression below.

Φ_{ℓ L N} = \sum_{n = 0}^{N - 1} \int_{2 π n / N}^{2 π (n + 1) / N} \exp [i (ℓ ϕ - L \frac{2 π n}{N})] d ϕ

For the case where the SSPO and output mode azimuthal charges are equal (ℓ = L), the integral can be evaluated and the sum converges to a finite value given by:

Φ_{L N} = \frac{- i N}{L} [\exp (\frac{i L 2 π}{N}) - 1]

As one would expect, taking the limit of this expression when the number of steps approaches infinity yields 2π. This result is summarized in Fig. 4, and shows the conversion scaling factor |Φ_LN|² normalized to 4π² as a function of N for various values of L. Setting N = 16 and γ = 1 in Eq. (14) returns a conversion efficiency of 77.5%, matching that in [25]. As expected, there is a minimum number of 2 steps required to convert to an LG₁₀ mode, increasing as L increases. For the case of L = 1, a 16 step SSPO gives relative reduction in conversion efficiency of 1.28%. Similarly for an L = 2 plate, 32 steps gives the same relative reduction in conversion efficiency of 1.28%. These values are summarized in Table 1. It can be seen that an efficient SSPO (less than 1.3% conversion efficiency loss compared to a continuous plate), requires no more than 16L steps, for a topological charge L SSPO.

Fig. 4 Multiplicative scaling factor |Φ_LN|² as a function of the number of SSPO steps N for the L = ℓ = 1, 2, 3 cases, normalized to 4π². As the number of steps approaches ∞, all cases approach a value of 1.

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Returning to the case where the SSPO has a non-integer topological charge, the transmission operator remains unchanged, but the integral solution is different. Summing over each of the integral terms converges again to a finite value as shown below:

Φ_{ℓ L N} = \frac{- i}{ℓ} [\exp (\frac{i ℓ 2 π}{N}) - 1] \frac{\exp (i 2 π (ℓ - L)) - 1}{\exp (i 2 π (ℓ - L) / N) - 1}

The limit as the total number of steps goes to infinity results in the solution:

lim_{N \to \infty} Φ_{ℓ L N} = \frac{- i}{ℓ - L} [\exp (i 2 π (ℓ - L)) - 1]

This expression in Eq. (15) is quite general in that it includes both variables of the SSPO (N and L) and allows the use of an arbitrary output mode. As previously discussed, the output LG mode must contain an integer value azimuthal charge, however the design of the SPO can be such that L can be chosen to be any value. When L is a non-integer, additional azimuthal output modes ℓ_i are also generated with the fundamental output charge ℓ. The fundamental charge is therefore defined as being the closest integer to L. The additional ℓ_i modes generated produce a mixed state of LG modes whose net sum of angular momentum must equal the charge of the SPO. Conversion efficiency to the fundamental mode for a non-integer SPO is summarized in Fig. 5, where the mode conversion efficiency scaling factor |Φ_ℓLN|², normalized to 4π², is plotted as a function of the intrinsic charge of a continuous SPO converting to an LG₁₀ output mode. The conversion efficiency to the fundamental mode is maximized when the SPO and output mode charges are equal, as expected. One sees that a fractional change in SPO charge decreases the conversion efficiency, dropping off significantly as the change approaches half an integer. However, for small mismatches of intrinsic charge due to step height and laser wavelength, the conversion efficiency remains close to its maximum value. From Fig. 5 it is clear that the efficiency change is less than 10% when the charge of the SPO is changed by a significant amount of 0.15. This is further summarized in Fig. 6 where we have plotted the spectral content of an output beam from a continuous L = 1.2 SPO for the case of $γ = 1 / \sqrt{2}$ . Here we see the ℓ = 1, p = 0 is still the fundamental mode as expected, with a conversion efficiency of 0.815, but adjacent ℓ_i modes are also present, including the unconverted TEM₀₀ mode.

Fig. 5 Multiplicative scaling factor |Φ_LN|² as a function of intrinsic topological charge of the spiral phase optic L to the LG₁₀ mode. Inset into the image is a wider field of view showing the behavior away from (ℓ = L = 1), note the zeros of the function at all integer charges not equal to ℓ.

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4. Lasers with broad bandwidth

With this insight, we can investigate further by looking at the effect of a laser with bandwidth on the conversion efficiency, which in principle is the same as changing the charge of the SPO relative to the laser wavelength. Today, ultrafast lasers can carry a considerable bandwidth, pushing upwards of 200nm [32]. This spectrum can be roughly described by a Gaussian distribution in frequency centered around the laser frequency. To model this, we return to the phase transmission operator T for a SSPO, modified to include the laser central frequency ν₀ and some arbitrary frequency value ν.

T = \exp [- i L (\frac{2 π n ν}{N ν_{0}})]

Again, this has no effect on the radial integral of Eq. (7) when examining the case of P = 0. If we convolve the conversion efficiency scaling factor |Φ_ℓLN|² with a Gaussian distribution of frequencies, centered around ν₀ we obtain the following:

\begin{array}{l} {| Φ_{ℓ L N ν} |}^{2} & = \int_{- \infty}^{\infty} \frac{1}{σ \sqrt{2 π}} \exp [- \frac{1}{2} {(\frac{ν - ν_{0}}{σ})}^{2}] \\ \times {| \sum_{n = 0}^{N - 1} \int_{2 π n / N}^{2 π (n + 1) / N} \exp [i (ℓ ϕ - L \frac{2 π n ν}{N ν_{0}})] d ϕ |}^{2} d ν \end{array}

We introduce the frequency standard deviation σ, which can be related to the FWMH (full width half maximum) of the laser bandwidth by the following identity: $FWHM = 2 \sqrt{2 ln 2} σ$ . Evaluation of the azimuthal integral is still possible and leads to the result:

\begin{array}{l} {| Φ_{ℓ L N ν} |}^{2} & = \int_{- \infty}^{\infty} \frac{1}{σ \sqrt{2 π}} \exp [- \frac{1}{2} {(\frac{ν - ν_{0}}{σ})}^{2}] \\ \times {| \frac{- i}{ℓ} [\exp (\frac{i ℓ 2 π}{N}) - 1] \frac{\exp (i 2 π (ℓ - L \frac{ν}{ν_{0}})) - 1}{\exp (i 2 π (ℓ - L \frac{ν}{ν_{0}}) / N) - 1} |}^{2} d ν \end{array}

This general expression can be computed numerically. It may also be simplified further for specific cases of the SPO; continuous, integer SPO charge etc. The conversion efficiency scaling factor for a continuous SPO with equal charges (L = ℓ) at the central wavelength is given in Fig. 7 for various representative wavelengths. From this figure, it is clear that when the laser bandwidth is equivalent to 10% of the laser wavelength, the conversion efficiency scaling factor is very close to 1. It should also be noted that as various spectral modes propagate to the far field, the different phase shifts of the overlapping spectral components may lead to complex spatial and temporal beam profiles.

Fig. 6 Laguerre-Gauss mode spectrum (LG_{ℓ_ip}) emanating from a continuous L = 1.2, P = 0 SPO. The azimuthal mode number ℓ_i is on the x-axis, the radial mode number p is on the y-axis and the total conversion efficiency η_Lℓp is displayed on the vertical z-axis on a log scale.

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Fig. 7 Conversion efficiency scaling factor Φ_ℓLNν as a function of laser bandwidth in nanometers for the case of ℓ = L and N = ∞. Four central wavelengths λ₀ are plotted: 1064nm, 800nm, 532nm, 400nm.

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We have also neglected until now, effects of group velocity dispersion (GVD) in the SPP substrate. It is well understood that an ultrafast pulse will broaden when propagating in a material. This is not the case for SPM’s since there is no dispersion from an ideal reflecting surface, giving them an advantage in the ultrafast regime.

5. Conclusion

An analytical model for the conversion efficiency of a generic SPO with radial mode P = 0 was derived. A further analytic expression has been derived to include the effects of SSPO’s. Further integral expressions have been derived for the case of non-integer topological charge SPO’s and used to evaluate the effects of broad laser bandwidth. It was found that conversion from a TEM₀₀ mode to the lowest order radial LG mode (p = 0) is most efficient when the ratio of output and input beam waists is chosen to be $1 / \sqrt{| ℓ | + 1}$ . The fraction of light not converted to this primary mode is converted into higher order radial modes which diverge more quickly than the fundamental mode yielding a beam profile dominated by the lowest order p mode in the far field. Additional ℓ_i modes are generated for the non-integer and broadband laser cases. The fractional power converted into these modes is found to be low for small deviations in the SPO charge L as applicable to standard broad laser pulses.

It was also shown that as the topological charge L of the SPO is increased, the conversion efficiency decreases, independently of whether the SPO is smooth or stepped inferring a limit to the purity of the generated fundamental output mode.

In the case of SSPO’s, it was shown that using 16L steps results in less than a 1.3% relative reduction in conversion efficiency. The present results allow optimum design of SSPO’s, either transmissive or reflective for the conversion of a TEM₀₀ mode to any fundamental OAM mode.

Funding

This work was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) research grant number RGPIN-2014-05736.

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SPO, Waist ratio	L = 1, $γ = 1 / \sqrt{2}$			L = 2, $γ = 1 / \sqrt{3}$
η_ℓp	η₁₀	η₁₁	η₁₂	η₂₀	η₂₁	η₂₂
N = 8	0.8840	0	0.0327	0.6839	0	0.0641
N = 16	0.9189	0	0.0340	0.8013	0	0.0751
N = 32	0.9279	0	0.0344	0.8330	0	0.0781
N = ∞	0.9308	0	0.0345	0.8437	0	0.0791

Mode conversion efficiency to Laguerre-Gaussian OAM modes using spiral phase optics

Abstract

1. Introduction

2. Mode decomposition

3. Stepped spiral phase optics

4. Lasers with broad bandwidth

5. Conclusion

Funding

References and links

Cited By

Figures (7)

Tables (1)

Equations (19)

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