Non-paraxial TM<sub>01</sub> and TE<sub>01</sub> from Laguerre-Gauss angular spectrum

Pierre-André Bélanger; Simon Thibault

doi:10.1364/OE.20.020228

1. Introduction

In several applications, a sharply focused light beam is required [1–3]. It has been demonstrated that a radially polarized and an azimuthally polarized light beam can generate respectively a very sharp axial electrical field and a sharp axial magnetic field [4,5]. Such axial electrical and magnetic fields find application in classical optics as well as in nonlinear optics [6,7]. These two types of polarization require the generation of laser TM₀₁ and TE₀₁ beam. Such beam can be generated after passing a Gaussian laser beam through a polarization converter and the field components are evaluated at the focus of the optical system using the well-known Richards–Wolf [8] three propagation integrals..

In this paper we use first the Maxwell equations to write down each component of the field as a spectrum of plane wave. We optimize the spectrum bandwidth product (_{_{$σ_{0}^{2} {\hat{σ}}_{0}^{2}$}}) defined by their variance to obtain an ideal spectrum distribution. Thereafter, we calculate in closed form the main field components with the optimized Laguerre-Gauss spectrum profiles and comparison of our calculated distribution is made with some experimental result of Dorn et al. [1], Dehez et al. [3], Jia et al [9] and Hao et al. [10].

2. Theoretical

In cylindrical coordinates, the axial components of the field E_z and H_z obey the wave equation and therefore they can be written as an angular spectrum A(τ) of plane waves. Both distributions are given at the Eq. (1) where k is the wave number in image space.

E_{z} o r H_{z} = [\int_{0}^{\infty} A (τ) e^{- i k z \sqrt{1 - τ^{2}}} J_{o} (k r τ) τ d τ]

Using the Maxwell equations and η as the impedance in vacuum, the transverse component for a TM₀ and TE₀ distribution is given respectively by the Eqs. (2) (TM) and (3) (TE).

T M {\begin{cases} E_{r} = i \int_{0}^{\infty} A (τ) \sqrt{1 - τ^{2}} e^{- i k z \sqrt{1 - τ^{2}}} J_{1} (k r τ) d τ \\ H_{φ} = \frac{i}{η} \int_{0}^{\infty} A (τ) e^{- i k z \sqrt{1 - τ^{2}}} J_{1} (k r τ) d τ \end{cases}

T E {\begin{cases} H_{r} = - i \int_{0}^{\infty} A (τ) \sqrt{1 - τ^{2}} e^{- i k z \sqrt{1 - τ^{2}}} J_{1} (k r τ) d τ \\ E_{ϕ} = - i η \int_{0}^{\infty} A (τ) e^{- i k z \sqrt{1 - τ^{2}}} J_{1} (k r τ) d τ \end{cases}

We are now interested to find out a spectrum A(τ) that will optimize the spectrum bandwidth product of the electric fields. We will first consider the transverse field E_ϕ which will ensure a finite energy for all fields.

The variance of the spectrum is given by Eq. (4) and the variance of the beam is given by Eq. (5) with the help of the Dirac delta function for Bessel function (Eq. (6)) [11]. For simplicity, A(τ) is noted only as ‘A’ in the equations.

{\hat{σ}}_{0}^{2} = \frac{k^{2} \int_{0}^{\infty} A^{2} τ^{2} \frac{d τ}{τ}}{\int_{0}^{\infty} A^{2} \frac{d τ}{τ}}

σ_{0}^{2} = \frac{\int_{0}^{\infty} | E_{ϕ} | r^{2} (r d r)}{\int_{0}^{\infty} {| E_{ϕ} |}^{2} (r d r)}

\int_{0}^{\infty} J_{n} (a r) J_{n} (b r) r d r = \frac{1}{a} δ (a - b)

Equation (5) can also be expressed in term of the spectrum distribution and the equation is given by Eq. (7).

k^{2} σ_{0}^{2} = \frac{\int_{0}^{\infty} {(\frac{d A}{d τ})}^{2} \frac{d τ}{τ}}{\int_{0}^{\infty} A^{2} \frac{d τ}{τ}}

Equation (7) is obtained using _{$k r J_{1} (k r τ) = - \frac{\partial J_{0} (k r τ)}{\partial τ}$} and integrating by part Eq. (3) for E_ϕ and with the condition A(τ) = 0 at τ = 0 and τ = ∞. The spectrum distribution is read at z = 0 and supposed to be real. We normalized the spectrum variance such that the space and spectrum bandwidth product for a pure Gaussian beam is unity.

Now our goal is to find the minimum spatial variance for a fixed total energy and a fixed spectrum variance. Using Lagrange multipliers (_{$Λ_{0}$}and_{$Λ_{1}$}) we can implement these two constraints and these conditions can be written as a variational problem expressed by the Eq. (8). The first term is the function to minimize (spatial variance), the second term is the energy and the last is the spectrum variance.

\partial [\int_{0}^{\infty} [{(\frac{d A}{d τ})}^{2} \frac{d τ}{τ} + Λ_{0} \int_{0}^{\infty} A^{2} \frac{d τ}{τ} + Λ_{1} \int_{0}^{\infty} A^{2} τ^{2} \frac{d τ}{τ}]] = 0

The Euler-Lagrange equation [12]. for Eq. (8) leads to the differential Eq. (9) for the spectrum distribution A(τ).

\frac{d}{d τ} (\frac{1}{τ} \frac{d A}{d τ}) - Λ_{0} \frac{A}{τ} - Λ_{1} A \cdot τ = 0

Solution of the differential Eq. (9) that are finite at the origin and at infinity are given in Eq. (10) where L_m¹(x) is the Laguerre polynomial of order (1) and (m). The Lagrange multipliers (9) are also given by the Eq. (10).

\begin{array}{l} A_{m} (τ) = A_{0} \cdot τ^{2} e^{- \frac{τ^{2}}{2 f^{2}}} L_{m}^{(1)} (\frac{τ^{2}}{f^{2}}) ​ w i t h m = 0, 1, 2, ... \\ Λ_{0} f^{2} = - 4 (m + 1), Λ_{1} = \frac{1}{f^{4}} \end{array}

The parameter f ² introduced here specifies the width of the angular spectrum. For example, when m = 0, the variance of the spectrum is given by the Eq. (11).

{\hat{σ}}_{0}^{2} = \frac{k^{2} \int_{0}^{\infty} τ^{2} A^{2} τ d τ}{\int_{0}^{\infty} A^{2} τ d τ} = 2 f^{2} k^{2}

The spectrum distribution forms an orthogonal set of functions with the orthogonally given in Eq. (12).

\begin{array}{l} \int_{0}^{\infty} e^{\frac{- τ^{2}}{f^{2}}} L_{m}^{(1)} (\frac{τ^{2}}{f^{2}}) L_{n}^{(1)} (\frac{τ^{2}}{f^{2}}) τ^{3} d τ = 0 n \neq m \\ = \frac{f^{4}}{2} (n + 1) n=m \end{array}

This Laguerre-Gauss spectrum has already been used for calculating the focal spot of radially polarized laser beams by Kozawa and Sato [13] while using the Richards-Wolf [8] integrals. In the space domain and equivalent Laguerre-Gaussian beams have been derived in the paraxial approximation as self-similar beam polarized propagation [14].

For the TM mode, the transverse field E_r will have the same extremum spectrum bandwidth product as Eq. (10) while A(τ) is replaced by _{$A (τ) \sqrt{1 - τ^{2}}$}. However this introduction of the square root term will yield that the energy in the E_z field is infinite. The spectral distribution A(τ) in Eq. (10) for both TE and TM beams will be used in the section 3.

According to Eq. (10), we obtain not a single angular spectrum that minimise the spatial variance but a family of angular spectra. For E_ϕ and E_z, the spectrum-bandwidth product is _{$4 {(m + 1)}^{2}$} and _{$3 {(m + 1)}^{2}$} respectively. Consequently, the spectrum-bandwidth product is minimal for the fundamental mode (m = 0).

3. Comparison to experimental results

In their experimental set up, Dorn et al. [1] have generated a TM₀₁ and TE₀₁ beam distribution after passing a quasi TEM₀₀ Gaussian beam through a four half-wave plates polarization corrector. The output beam is then close to a TM₀₁ or TE₀₁ beam profile depending on the orientation of the polarizers. After strongly focusing the beam, they measured the E_z distribution of the TM₀₁ and the E_ϕ distribution of the TE₀₁ components in the space domain. The focusing generates a beam showing a smaller variance σ₀² (larger f ²) and we estimated that their observed beam distribution can be close to the optimum beam profile according to Eq. (10). Using the relation Eq. (13) for the Laguerre polynomial [15], we calculate the distribution of the E_z and E_ϕ space component for the spectrum of Eq. (10).

x L_{m}^{(1)} (x) = (m + 1) [L_{m} (x) - L_{m + 1} (x)]

After integration the E_z component of the TM mode and the E_ϕ component of the TE mode are given respectively by Eqs. (14) and (15) where _{$c_{0}, c_{1}$} (normalization constant).

E_{z} = c_{0} {(- 1)}^{m} (m + 1) \exp [\frac{- {(k f r)}^{2}}{2}] [L_{m} ({(k f r)}^{2}) + L_{m + 1} ({(k f r)}^{2})]

E_{ϕ} = c_{1} {(- 1)}^{m} \exp [\frac{- {(k f r)}^{2}}{2}] (k f r) L_{m}^{(1)} ({(k f r)}^{2})

Finally, the normalized distribution of the fundamental mode (m = 0) are given in Eqs. (16) and (17) respectively.

\frac{E_{z} (r, 0)}{E_{0}} = \exp [\frac{- {(k f r)}^{2}}{2}] (1 - \frac{{(k f r)}^{2}}{2})

\frac{E_{ϕ} (r, 0)}{E_{0}} = \sqrt{2} (k f r) \cdot \exp [\frac{1}{2}] \cdot \exp [\frac{- {(k f r)}^{2}}{2}]

For the E_ϕ(r,0) we have already obtained that the variance width is _{${σ_{0}^{2} |}_{ϕ} = \frac{2}{k^{2} f^{2}}$}and for the E_z(r,0) field it is possible to show that the variance width is _{${σ_{0}^{2} |}_{z} = \frac{1}{k^{2} f^{2}}$} . For simplicity, the Eqs. (16) and (17) is written using the E_z(r,0) spatial variance σ₀² to Eqs. (18) and (19).

\frac{E_{z} (r, 0)}{E_{0}} = \exp [\frac{- r^{2}}{2 σ_{0}^{2}}] [1 - \frac{r^{2}}{2 σ_{0}^{2}}]

\frac{E_{ϕ} (r, 0)}{E_{0}} = \sqrt{2} e^{\frac{1}{2}} (\frac{r}{σ_{0}}) \exp [\frac{- r^{2}}{2 σ_{0}^{2}}]

The normalized Eqs. (18) and (19) are plotted in the Fig. 1(a) and 1(b) respectively.

Fig. 1 Normalized E_z from Eq. (18) (a) and normalized E_ϕ fields from Eq. (19) (b).

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We can also calculate the spot size component (Eq. (18)). The spot size is defined as the area (not diameter) limited by the full width half maximum (FWHM) intensity value [1]. The spot size is given by Eq. (20).

S p o t S i z e_{F W H M} = \frac{0.0263 λ^{2}}{f^{2}} \approx 1.038 σ_{0}^{2}

This is smaller by a factor of 2 then the spot size of a pure Gaussian beam.

From the experimental results presented by Dorn [1], the measured spot size was 0.2λ² for the E_z component. From our model, this implies that f² = 0.113. They also observed that the power contained in the longitudinal field is nearly 50% of the total transmitted power. With the preceding analytical results we can show that this ratio is given by the Eq. (21). Using f² = 0.113, we obtain 22% which may indicate that the fundamental mode alone cannot explain the result.

R = 2 f^{2}

Furthermore, Eq. (18) predicts for the fundamental E_z field a width σ₀ = 0.439λ = 0.27um. The first zero of the E_z field should be located at r₀ ≈0.4um and the first maximum at 0.55um. Equation (19) for the fundamental E_ϕ field has the first maximum at r ≈0.28um. Again the experimental results seem to be close to the ones associated with the fundamental mode of the Eqs. (18) and (19). However, a closer look shows that the amplitude of the first maximum of the E_z field predicted by Eq. (18) is too low as compared with the experimental result. Moreover Eq. (19) for the E_ϕ field does not predict a second peak as clearly observed in Fig. 3(b) of Dorn [1]. Consequently, our analysis in section 2 has shown that the optimum angular spectrum can have higher order distribution (Eq. (10). Adding the next order with amplitude ‘a’ to the fundamental solution, the E_ϕ and E_z components become the Eqs. (22) and (23).

\frac{E_{z} (r, 0)}{E_{0}} = \exp [^{\frac{- r^{2}}{2 σ_{0}^{2}}}] [1 + \frac{6 a - 1}{2 - 4 a} {(\frac{r}{σ_{0}})}^{2} - \frac{a}{2 - 4 a} {(\frac{r}{σ_{0}})}^{4}]

\frac{E_{ϕ} (r, 0)}{E_{0}} = (\frac{r}{σ_{0}}) \exp [\frac{- r^{2}}{2 σ_{0}}] [1 + \frac{a}{1 - 2 a} {(\frac{r}{σ_{0}})}^{2}]

As shown on Fig. 2(a) and 2(b), a better match can be obtained with the experimental result of Dorn [1] using “a = −2/3”. For the first two order distributions, we can show that the Eq. (21) is now R = 4.17 f². Consequently, the power contained in the longitudinal field is now about 50% of the total transmitted power which is in accordance with Dorn [1].

Fig. 2 a: Normalized E_z from Eq. (23). b: Normalized E_ϕ fields from Eq. (23). For both graphs, a = −2/3 and f ² = 0.113.

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In another experimental setup, Dehez et al. [3] have improved the resolution of a two-photon microscope using a TM₀₁ laser beam. The measured spot size was 0.15λ² for E_z longitudinal field. The profile of the distribution shown in this paper is also compatible with our predicted E_z field for the fundamental TM₀₁ mode. The profile for the TE₀₁ distribution is also close to our predicted field. However, their experimental results do not clearly show the secondary peak of the field and within the experimental error, we cannot conclude that they generated a fundamental TM₀₁ or TE₀₁ beams.

In recent experimental measurement, Hao and Leger [10] recorded the focal pattern of the radially polarized light in a photoresist material. Their experimental points (see Fig. 5(b) of Ref [10].) seems to follow the two peaks of the Laguerre-Gauss profile (21).

4. Discussion

In recent publications (see for example ref [1], [7], [13], and [16]) the TM₀ and TE₀ distributions have been calculated using three propagation integrals derived by Richards and Wolf [8], and very similar profiles as the ones obtained here has been formed. The Richards and Wolf results have been developed after propagating plane waves (E = ηH) represented by geometrical rays through an aplanatic focusing system. Here we have propagated the electric and magnetic field accordingly to Maxwell equations and we have not specified any particular focusing system. Richards and Wolf [8] made their first analysis for an aplanatic focusing system and they used the sine condition for energy. This condition yields the term (cos^1/2θ) in their integral while here the corresponding term is (cosθ) (in our notation cos²θ = 1-τ²). Richards and Wolf [8] assumed that the magnetic fields to be proportional to the electric one, here as shown on Eqs. (2) and (3) they are not exactly proportional. However for the paraxial limit (τ²<<1) the two developments are identical. Our analysis here was made assuming that we were at the focus of an optical system (real spectrum). Propagation of the spectrum for z>0 can be made numerically or after making analytically integration required by each fields [1–3]. For the Laguerre-Gauss spectrum (10) introduced here we can use the following Hermite H_n(x) generating function (24) [15]_.

\exp [\frac{1}{2 f^{2}} (1 - τ^{2}) - i k z \sqrt{1 - τ^{2}}] = \sum_{n = 0}^{\infty} \frac{{(\frac{- 1}{2 f^{2}})}^{\frac{n}{2}}}{n!} H_{n} (\frac{k f z}{\sqrt{2}}) {(1 - τ^{2})}^{\frac{n}{2}}

We generally neglect the contribution of the evanescent waves after propagation and for cosθ ≤ π/2 (τ²≤1) all the integration can be made with a well-known integral [15] of Eq. (25).

\begin{array}{l} \int_{0}^{π / 2} J_{μ} (x \sin t) \sin^{μ + 1} (t) \cos^{2 υ + 1} d t = \\ \frac{2^{υ} Γ (1 + υ)}{x^{υ + 1}} J_{μ + υ + 1} (x) \end{array}

Finally each components of the TM or the TE field can be written in closed form as a summation of Bessel function _{$J_{n + 2} (k r)$}time a Hermite function _{$H_{n} (\frac{k f z}{\sqrt{2}})$}_.

5. Conclusion

In this paper we have shown that solving directly the Maxwell equation for a TM or a TE distribution in an angular spectrum of plane waves in cylindrical coordinates all the field components can be written in term of these integrals that are similar to the integral derived by Richards-Wolf [8]. And after requiring that the spectrum bandwidth product of the transverse electrical field to be minimum in variance, a Gauss-Laguerre profile for the spectrum is recovered.

We show that the experimental result obtained by Dehez [3] is in accord with the basic fundamental mode for the spot size (FWHM). We also show that the more detailed experimental and calculated distribution found by Dorn [1] can be closely described by the combination of the first two Gauss-Laguerre modal distributions.

In the paraxial approximation it is possible to show that each components of the magnetic and electrical field propagate according to a pure self-similar Gauss-Laguerre beam as first shown by Nesterov and Niziev [17].

Acknowledgments

This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the NSERC Industrial Research Chair in Lens Design.

References and links

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13. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007). [CrossRef] [PubMed]

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17. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). [CrossRef]

Non-paraxial TM₀₁ and TE₀₁ from Laguerre-Gauss angular spectrum

Abstract

1. Introduction

2. Theoretical

3. Comparison to experimental results

4. Discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (2)

Equations (25)

Optics Express