Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre-Gaussian beams

Colin JR Sheppard

doi:10.1364/OE.17.003690

1. Duality of beams

The dual of a paraxial beam is a beam in which the transverse distance and angular behaviors are interchanged. Thus, equivalently, the near-field and far field (neglecting the parabolic wavefront) amplitude distributions are interchanged. A definition can be established in terms of the ABCD matrix formalism,¹ so that a dual can be generated by a 2-f optical system, for which A = D = 0. The principle of duality as applied to optical experiments and theories was introduced many years ago by Lohmann.^{2, 3} A beam whose cross-section is a self-Fourier transform ^4–6 is its own dual, of which the standard Hermite-Gaussian (sHG) and Laguerre-Gaussian (sLG) beams are special cases. As the sHGs and sLGs are complete sets, any amplitude in the waist can be expanded as a series of these basis functions. The sHG and sLG beams have relative phase in the far field that depends on the mode numbers, and can take values 0,7π/2,π,3π/2. Thus in terms of sHG or sLG expansions, the dual can be generated by introducing a factor of a power of i depending on the mode numbers. One example of the dual of a beam that has been described is that of a flattened beam.⁷ In the limiting case of the flattened beam, corresponding to a beam from a circular pupil, the focal plane is an Airy disc (N.B. not an Airy function), and its dual is a beam with an Airy disc pupil, with a uniform circular spot in the focal plane. To give the dual of the flattened beam alternating signs are introduced in the terms of the sLG expansion.

Another example is the Bessel-Gauss (BG) beam, which has an amplitude given by the product of a Gaussian and a Bessel function in the waist, and the convolution of a two-dimensional Gaussian and a ring in the far field, and which has as its dual a beam with a Gaussian ring in the waist. The amplitude of a zero order paraxial BG beam can be written⁸

B G_{m} (r, z) = \frac{1}{μ} J_{m} (\frac{br}{μ w_{0}}) \exp (- \frac{r^{2}}{μ w_{0}^{2}} - \frac{b^{2}}{4} + \frac{b^{2}}{4 μ} + imϕ),

where J_m is a Bessel function of the first kind of integer order m, b is a constant defining the relative width of the Bessel and Gaussian functions, w ₀ and z ₀ = kw ² ₀/2 are the beam waist and confocal parameter for the Gaussian, Z = z/z ₀ and μ = 1+iZ, and a factor exp[i(kz-ωt)] with k = 2π/λ, is suppressed. An expression for the BG beam was given⁸ ten years before the reintroduction of the BG beam by Gori et al.,⁹ and almost ten years even before the Bessel beam was proposed by Durnin et al. ¹⁰ The intensity, defined as the squared modulus of the amplitude, in the focal region is illustrated in Fig.1a and 1b for the value b = 10 and m = 0. We can see the annular structure forming in Fig. 1b. If b is complex we obtain the generalized Bessel-Gauss (gBG) beam.¹¹ The dual of the BG beam (dBG) is obtained by putting b as b = ib′ (equivalent to introducing alternating signs in the LG expansion), giving

dB G_{m} (r, z) = \frac{1}{μ} {I_{m} (\frac{b' r}{μ w_{0}}) \exp (\frac{b' r}{μ w_{0}})} \exp [- \frac{1}{μ} {(\frac{r}{w_{0}} - \frac{b'}{2})}^{2} + \frac{{b'}^{2}}{4} + imϕ] .

where I_m is a modified Bessel function of the first kind. This is also called the modified Bessel Gauss (mBG) beam.¹¹ It has been written in this way so that the quantity in braces is approximately constant for Z = 0 and large argument, so the amplitude is only appreciable near r = b′w ₀/2, representing an annular waist. This is shown in Fig.1c for b′ = 10. We see a ring structure in the waist, and subsidiary peaks on the axis either side of focus that are analogous to the Poisson spot in the diffraction pattern of a circular obstruction.

Fig. 1. The intensity in the focal region for BG beams for b = 10 (a, b), and for the dual of BG, dBG, for b’ = 10 (c).

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2. Hermite-Gaussian beams

sHG beams represent an important solution of the scalar paraxial wave equation.¹² In the waist, the sHG beams exhibit a particular scaling ratio between the widths of the Hermite polynomial and the Gaussian. Wünsche^{13, 14} showed that these beams can be generalized to an arbitrary scaling parameter, with the amplitude expressed in closed form for any point in space, called generalized HG (or gHG) beams, but the behaviour was not described explicitly. For the paraxial case any form can be assumed for the waist, as long its Fourier transform exists, but the amplitude might not be expressible simply in closed form for all space. For non-paraxial beams, if there are no evanescent waves, the amplitude in the waist must be band-limited.¹⁵ Siegman investigated the important particular case of the elegant HG (written eHG) beam, in which the arguments of the Gaussian and the Hermite polynomial have the same form at any propagation distance.¹⁶ Pratesi and Ronchi ¹⁷ investigated beams with an arbitrary complex constant parameter. They chose the values of the complex parameter to satisfy a particular condition that was useful for the study of beam propagation in a quadratic index medium. However, any complex value of the parameter gives a valid solution of the paraxial wave equation. In this paper, we investigate the particular condition with a real scaling ratio b. Then, we can show that the amplitude can be written in the form

gH G_{n} (x, z) = \frac{v^{n / 2}}{μ^{n + 1 / 2}} H_{n} [\frac{x}{w_{0}} {(\frac{2}{ν})}^{1 / 2}] \exp (- \frac{x^{2}}{w_{0}^{2} μ})

where H_n is a Hermite polynomial of integer order n defining the mode number, c = b ² -1, and ν = (1+c) + 2icZ + (1-c)Z ² = μ[(1 + c)-i(1-c)Z] . In the waist Z = 0, the argument of the Hermite polynomial is real if c ≥ -1 (b is real). In general c may be complex. On examining the argument of the Hermite function, we can see that it can be made real at an arbitrary plane by taking c as complex, with a phase - 2 arctan Z. In this plane the scaling factor of the Hermite polynomial relative to the Gaussian is b = 1/(1 + c)^1/2. The modulus of c controls the scaling factor, and its argument the plane in which the argument of the Hermite function is real. Thus for Z → ±𢈞, corresponding to the far field, the phase of c is ∓π , and c is real and negative. These negative values of c correspond to the dual of a beam of positive ∣c∣.

We now concentrate particularly on the case when z ₀ and c are real, so that the amplitude in the waist is real. For c = 0, Eq. (3) reduces to the sHG mode,¹² whereas for c = 1 they reduce to Siegman’s eHG modes.¹⁶ For ∣c∣ large, the amplitude reduces to the form

gH G_{n} (x, z) = \frac{{(- c)}^{n / 2}}{(n / 2)!} \frac{1}{μ^{1 / 2}} \exp (- \frac{x^{2}}{w_{0}^{2} μ}), n even

In the far field, Z → ∞, we have for the general case

gH G_{n} (x, z) = \frac{1}{i^{n + 1 / 2}} {(1 - c)}^{n / 2} \sqrt{\frac{k w_{0}^{2}}{2 z}} H_{n} [\frac{k w_{0} (x / z)}{\sqrt{2} {(1 - c)}^{1 / 2}}] \exp {- {[\frac{k w_{0} (x / z)}{2}]}^{2} + \frac{ik x^{2}}{2 z}} .

For the eHG case (c =1), the Hermite polynomial can be evaluated using the limit

{lim}_{ε \to 0} ε^{n} H_{n} (\frac{y}{ε}) = 2^{n} y^{n},

so that it reduces to a simple power^{14, 19}

eH G_{n} (x, z) = \frac{1}{i^{n + 1 / 2}} \sqrt{\frac{k w_{0}^{2}}{2 z}} {[\sqrt{2} k w_{0} (x / z)]}^{n} \exp {- {[\frac{k w_{0} (x / z)}{2}]}^{2} + \frac{ik x^{2}}{2 z}} .

For values of c > 1, then although the argument of the Hermite polynomial is imaginary, in the far field

{(1 - c)}^{n / 2} H_{n} [\frac{k w_{0} (x / z)}{\sqrt{2} {(1 - c)}^{1 / 2}}]

is purely real. The oscillating sign of the Hermite function of real argument becomes positive in the function of Eq. (8), so that its modulus increases monotonically with argument. In this way it bears the same relationship to the Hermite function of a real quantity that the exponential function has to the trigonometrical functions.

In the waist, z = 0

gH G_{n} (x, 0) = {(1 + c)}^{n / 2} H_{n} [\frac{\sqrt{2 x}}{w_{0} {(1 + c)}^{1 / 2}}] \exp (- \frac{x^{2}}{w_{0}^{2}}) .

For c = -1, again we can use Eq. (6) to give

deH G_{n} (x, 0) = {(\frac{2 \sqrt{2 x}}{w_{0}})}^{n} \exp (- \frac{x^{2}}{w_{0}^{2}}) .

This (deHG) is an important special case and can be regarded as the dual⁷ of Siegman’s elegant beam. We have in general for deHG

deH G_{n} (x, z) = \frac{{(- 2 iz)}^{n / 2}}{μ^{(n + 1) / 2}} H_{n} [\frac{x}{w_{0}} {(\frac{i}{Z μ})}^{1 / 2}] \exp (- \frac{x^{2}}{w_{0}^{2} μ}) .

For any gHG, we can construct a dual beam for the same z ₀ by replacing a real c with - c, i.e. cd = -c. The sHG beam is its own dual. Fig. 2 shows the intensity (modulus squared of the amplitude) in the waist for gHG beams for n = 5 for different values of the parameter c. For sHG, c = 0, as is well known the pattern consists of an array of six peaks, which become slightly stronger with distance from the axis. The behavior near to the axis is close to that of a sin² function. As c is increased, the relative strength of the outer peaks is reduced. For eHG (c = 1) the outer two peaks are very weak. As c is decreased below zero, the relative strength of the central peaks is reduced, until for deHG (c = -1) only two peaks remain. For more negative values of c, the width of the central minimum is reduced. As beams with the same absolute value of c but of opposite sign are duals, Fig. 2 also represents the intensity in the far field if the sign of c is changed. For even values of n, there is a peak on the axis the strength of which depends on the value of c. For deHG (c = -1), the central peak vanishes, leaving only two peaks as before. The strength of the central peak increases as c is increased or decreased from this value.

Fig. 2. The intensity in the waist for HG beams for n = 5 for different real values of the parameter c.

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Fig. 3. The intensity in the focal region for HG beams with n = 5 for different real values of the parameter c.

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Figure 3 shows the intensity in the focal region for n = 5 for different values of the parameter c. We see that for sHG beam, as the number of peaks does not change with z, the ridges along the peaks can be easily followed. For deHG (c = -1), for example, it is seen that the two peaks in the waist are not joined continuously to the peaks in the far field as the strongest peaks are the outer peaks in the waist, but the inner peaks in the far field. This behavior is also seen in Fig. 4, which shows the phase in the focal region, for which we must now choose a particular value of kz ₀, taken as 5, which is large enough for the paraxial treatment to be approximately valid. For sHG (c = 0), the phase is clearly separated by π for the three adjacent peaks, and changes progressively in the direction of propagation along each ridge. The behavior is quite different for eHG (c = 1). Similar effects occur for even n, where the central peak in the waist for c = 1 decays on propagation while two peaks increase in strength.

3. Laguerre-Gauss beams

In a similar way, we can show that for the generalized Laguerre-Gaussian (gLG) beams for cylindrical geometry

gH G_{nm} (r, ϕ, z) = \frac{v^{(2 n + m) / 2}}{μ^{2 n + m + 1}} {(\frac{2 r^{2}}{w_{0}^{2} ν})}^{m / 2} L_{n}^{m} (\frac{2 r^{2}}{w_{0}^{2} ν}) \exp (- \frac{r^{2}}{w_{0}^{2} μ} + imϕ) .

where L^m_n is an associated Laguerre polynomial where m, n are integer mode numbers. In the waist

gH G_{nm} (r, ϕ, 0) = {{(1 + c)}^{n} (\frac{2 r^{2}}{w_{0}^{2}})}^{m / 2} L_{n}^{m} (\frac{2 r^{2}}{w_{0}^{2} (1 + c)}) \exp (- \frac{r^{2}}{w_{0}^{2}} + imϕ) .

The behavior is broadly similar to the HG case, except that for the sLG beams the relative strength of the peaks decay with distance from the axis, before eventually increasing. For deLG (c = -1), an annular beam is generated in the waist, which becomes a thin annulus for large n. For non-zero m, the beams exhibit a phase singularity on the axis. In the far field^{14, 20}

gH G_{nm} (r, ϕ, 0) = {(- i)}^{2 n + m + 1} \sqrt{\frac{k w_{0}^{2}}{2 z}} {(1 - c)}^{n} {(- \frac{k^{2} r^{2} / z^{2}}{2 w_{0}^{2}})}^{m / 2}

Then for eLG (c =1), using the identity18

{lim}_{ε \to 0} ε^{n} L_{n}^{m} (\frac{y}{4 ε}) = {(- 1)}^{n} \frac{y^{2 n}}{2^{2 n} n!}

we obtain

eL G_{nm} (r, ϕ, z) = \frac{{(- i)}^{2 n + m + 1}}{n!} \sqrt{\frac{k w_{0}^{2}}{2 z}} {(- \frac{k^{2} r^{2} / z^{2}}{2 w_{0}^{2}})}^{n + m / 2} \exp {- {[\frac{k w_{0} (r / z)}{2}]}^{2} + \frac{ik r^{2}}{2 z} + imϕ} .

For non-zero values of n or m the far field thus represents an annular beam.

For the particular case of deLG, putting c = -1 in Eq. (14)

deL G_{nm} (r, ϕ, z) = {(\frac{- 2 iZ}{μ})}^{n + m / 2} \frac{1}{μ} {(\frac{i r^{2}}{w_{0}^{2} Zμ})}^{m / 2} L_{n}^{m} (\frac{i r^{2}}{w_{0}^{2} Zμ}) \exp (- \frac{r^{2}}{w_{0}^{2} μ} + imϕ) .

In the waist

deL G_{nm} (r, ϕ, 0) = \frac{1}{n!} {(\frac{2 r^{2}}{w_{0}^{2}})}^{n + m / 2} \exp (- \frac{r^{2}}{w_{0}^{2}} + imϕ) .

Thus if n or m are non-zero the beam has an annular form in its waist.

Fig. 4. The phase in the focal region for HG beams for the case when kz ₀ = 5 and n = 5 . Phase is shown wrapped, between -π (black) and π (white), with zero as mid gray. For c = -1 (deHG) there is a phase singularity at the origin.

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4. Discussion

We have introduced specifically the duals of BG, eHG and eLG beams, and described the propagation of the generalized beams gHG and gLG. An important feature of the generalized beams is that c can take continuous, non-integer values. It is known that for large n, eHG or eLG beams are good approximations to the cosine-Gauss (CG) or BG beams, respectively. For example, for HG we have the limit for n → ∞ and x not too large

H_{2 n} (x) \exp (- \frac{x^{2}}{2}) \approx \frac{{(-)}^{n} 4^{n} n!}{\sqrt{πn}} \cos (2 \sqrt{nx})

H_{2 n + 1} (x) \exp (- \frac{x^{2}}{2}) \approx \frac{{(-)}^{n} 4^{n} n!}{\sqrt{π}} \sin (2 \sqrt{nx})

These expressions are analogous to a similar one for Laguerre polynomials given in Ref.20. For a CG beam, the scaling between the cosine and the Gaussian can be varied continuously, but for HG to reduce to a polynomial n must be an integer. From Eq. (19) we see that gHG can be approximated to CG for any positive value of c. This has been pointed out for the LG case by Bandres and Gutiérerrez-Vega.²¹ Thus an approximation to a CG beam based on a gHG beam can be found for an integer value of n, with a value of c near to unity.

Duality can be considered based on the theoretical framework of confluent hypergeometric functions.^21–24 This interesting and powerful approach can suggest new beam forms and make visible the symmetry properties. But the hypergeometric beams are no more general than gLG with complex n, as the Laguerre function for complex n is defined in terms of the confluent hypergeometric function. However, symmetry relationships can be revealed for particular special cases of the confluent hypergeometric function. The relationships between our parameters and the complex beam parameters used by Bandres are q ₀ = -iz ₀,q̃₀ = -iz ₀̃ = iz ₀(1 + c)/(1- c). Hence c = (q̃₀ - q ₀)/(q̃₀ + q ₀), μ = 1 + z/q ₀ and ν = 2μ(z + q̃₀)/(q ₀ + q̃₀) .

For the case of a 2f optical system, we have A = D = 0, B = f,C = -1/ f. Thus the dual of a beam has q _d0 = -f ²/q ₀ = -q ^* ₀, q̃_d0 = -f ²/q̃₀ = -q ₀ q ^* ₀/q̃₀ if we take f ² = q ₀ q ^* ₀. We find that as we stated earlier without proof the dual of a beam has c_d = -c even for the general case when c is complex, and also z _d0 = z ^* ₀, w _d0 = w ^* ₀ if its real part is taken as positive, μ_d = 1-z/q ^* ₀ = 2 - μ ^* and ν_d = -2μ_d(q̃₀/q ^* ₀)(z + q̃_d0)/(q ₀ + q̃₀) .

For the special case when z ₀ is real, the width of the underlying Gaussian is unchanged in the dual beam, and μ_d = μ,

ν_{d} = \frac{ν + c (ν - 4 μ)}{1 - c},

and Bandres’ parameter χ ² becomes^21,24

\frac{χ_{d}^{2}}{χ^{2}} = 1 - \frac{4 cμ}{(1 + c) ν} .

For the one dimensional case in our terminology

gH G_{nm} (x, z) = \frac{ν^{n / 2}}{μ^{m + 1 / 2}} {\frac{1}{Γ {[(1 - 2) / 2]}^{1}} F_{1} (- \frac{n}{2}, \frac{1}{2}; \frac{k x^{2}}{ν z_{0}})

where ₁ F ₁ is a confluent hypergeometric function, Γ is a Gamma function, and in general n can be complex. Similarly for the two dimensional case

gH G_{nm} (x, z, ϕ,) = \frac{ν^{n + ∣ m ∣ / 2}}{μ^{2 n + ∣ m ∣ + 1}} {(\frac{k r^{2}}{ν z_{0}})}^{∣ m ∣ / 2}_{1} F_{1} (- n, 1 + ∣ m ∣; \frac{k r^{2}}{ν z_{0}}) \exp (- \frac{k r^{2}}{2 μ z_{0}} + imϕ),

where m is an integer but n can be complex. Note that Eqs. 22, 23 are slightly different from those of Bandres^{21, 24}, as they are written so that they converge as ν → 0 for n a positive integer. The fractional frHG and frLG beams follow directly from the definition of the Hermite and Laguerre functions of complex order n.^{25, 26} Formally, Gutiérrez-Vega’s²⁵ P_n(x) is just 2^-n π ^1/2 H_n(-x). (The minus sign is a result of Gutiérrez-Vega’s definition of the creation operator.)

As eHG corresponds to (q ₀,q̃₀ → ∞), so deHG can be taken as (q ₀,0). This can be recognized as the one dimensional equivalent of the MLG (“modified LG”) beam of Karimi et al.^21,23 . In fact Eq. (21) is equivalent to Eq. (1) of Karimi, so that his MLG can be identified as the same as the deLG. Thus the dual of Karimi’s MLG is eLG.

From Eq. (21) we immediately see that the BG and dBG beams of the forms J_m,I_m(k_tr) result when(1+c) = lim_n→-∞±4kn/z ₀ k ² _t. The quadratic Bessel Gauss (QBG) beams²⁷ with a Bessel function of order ∣m∣/2 of a quadratic radial dependence can be generated by putting n=-(1+∣m∣)/2 and c = -4iW ₀ ²/w ₀ ², z ₀=-ik(2i/w ₀ ²-1/2W ₀ ²)^-1.²¹ Here w ₀ and W ₀ are the widths of the Gaussian and Bessel functions. Note that z ₀ is not related to w ₀ for the overall beam in the usual way, because a Gaussian factor is generated from the hypergeometric function. The dual of QBG has a modified Bessel function dependence (dQBG or mQBG) (c_d = 4iW ₀ ² w ₀ ², z _d0 = ik(2i/w ₀ ² + 1/2W ² ₀)^-1). For odd m the Bessel function degenerates to a spherical Bessel function, i.e. it can be expressed in terms of sinusoidal functions (or hyperbolic for the case of the dual) of quadratic argument.

Karimi et al.²³ also investigated beams with Bessel functions with quadratic radial dependence. For beams with n = -∣m∣/2 for odd m, the far field gives a phase singularity without the usual r ^∣m∣ dependence and in the waist a sum of modified Bessel functions of quadratic argument. For even m the waist is given by a sum of hyperbolic functions. Actually, Karimi discussed the duals of these beams, with the modified Bessel functions or exponentials in the far field and a simple Gaussian with exp(imϕ) behaviour in the waist, which can be generated experimentally by applying a phase modulation to a TEM₀₀ Gaussian beam.²³

Note that the definition of the dual proposed is not the only possibility, but there are other cases when the ABCD matrix also leads to the relationship q _d0/q̃_d0 = q̃₀/q ₀. An important alternative is f ² = q ₀ q̃₀;q _a0 = -q̃₀ q̃_a0 = q̃₀ . For real z ₀ this case corresponds to the adjoint beams (hence the subscript a) that form a biorthogonal set with the generalized beams.¹⁶ We find that c_a = -c as for the previous dual, but now z _a0 = z ₀(1+c)/(1 - c), so that z _a0(1 + c_a) = z ₀(1 + c) and the scaling of the hypergeometric function, instead of the Gaussian, is unchanged. The adjoint beams are not so suitable for our present analysis, however, as the adjoint of the elegant beam has z _a0 → ∞ (i.e. q _a0 → ∞), so that there is no Gaussian factor in Eqs. 20, 21.¹⁶

Acknowledgment

C. J. R. Sheppard acknowledges support from the Singapore Ministry of Education Tier 1 funding (grants R397000022112 and R397000033112).

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Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre-Gaussian beams

Abstract

1. Duality of beams

2. Hermite-Gaussian beams

3. Laguerre-Gauss beams

4. Discussion

Acknowledgment

References and Links

Cited By

Figures (4)

Equations (27)

Optics Express