White light conical diffraction

R. T. Darcy; D. McCloskey; K. E. Ballantine; B. D. Jennings; J. G. Lunney; P. R. Eastham; J. F. Donegan

doi:10.1364/OE.21.020394

1. Introduction

Conical refraction was first predicted by William Rowan Hamilton [1] and later observed by Humphrey Lloyd at Trinity College Dublin [2]. It occurs when light travels along an optic axis of a biaxial material. Such materials have three principal refractive indices $n_{1} < n_{2} < n_{3} .$ The optic axes are special directions for which the dispersion surface [3] has no defined normal due to the intersection of the sheets of the dispersion surface at the ‘diabolical’ point [4]. This singularity gives rise to a directional degeneracy for the Poynting vector, which now traces out a skewed cone. The approach formulated by Belsky and Khapalyuk [5] and later reformulated by Berry [6] leads to a model which demonstrates a double-ring profile in the focal image plane for the conical beam. This inclusion of diffraction theory gives an accurate description of the process and results in the switch of nomenclature from ‘conical refraction’ to ‘conical diffraction’. The phenomenon has found numerous applications such as the generation of radially polarised light beams [7] and a novel type of laser based on conical diffraction [8]. Berry and Jeffrey have also formulated a paraxial theory to describe light propagating through a biaxial material in a direction not along, but very close to, the optic axis [9]. This model will be used here as it will be seen that the direction of the optic axis (among other parameters) depends on wavelength. Thus, the paraxial theory can be used to obtain a theory with explicit wavelength dependence. We show that this theory accurately reproduces the complex images generated experimentally by the conical diffraction of white light.

The aim of this paper is to present experimental images of a conically diffracted white light beam and compare them to the theoretical intensity profiles. Also examined are the wavelength dependences of both the radius of the conical diffraction ring and the location of the focal image plane. This provides the foundation for future work on both the fundamental physics and applications of polychromatic conical diffraction, such as communication systems incorporating conically diffracted light of multiple wavelengths [10, 11].

2. Theoretical method

In conical diffraction, the Poynting vector at a diabolical point traces out a skewed cone with semi-angle [4]

A = \frac{1}{2} arc \tan [n_{2}^{2} \sqrt{(n_{1}^{- 2} - n_{2}^{- 2}) (n_{2}^{- 2} - n_{3}^{- 2})}] .

A geometrical optics description of this process may be found in [12] and [13], while Fig. 1(a) demonstrates what occurs graphically. Using the paraxial approximation, the radius of the emerging cylinder of light is thus

R_{0} = A l

where

l

is the length of the biaxial material [14]. In keeping with the convention used by Berry and Jeffrey [4], we define the following dimensionless parameters [15] for a Gaussian beam

\begin{array}{l} ρ_{0} = \frac{A l}{w}, & ρ = \frac{R}{w}, & ζ = \frac{l + (z - l) n_{2}}{n_{2} k_{0} w^{2}} \end{array},

where

R

and

z

are cylindrical coordinates with

R

measured from the centre of the emergent beam and

z

measured along the propagation direction.

k_{0} = 2 π / λ

and

w

is the

1 / e

beam radius at the narrowest point of the beam when the biaxial material is not present. This location corresponds to

z = 0.

When

ζ = 0,

the rings take on their sharpest formation. This location, the focal image plane (FIP), can be found by letting

ζ = 0

in Eq. (2) giving

Fig. 1 (a) Dispersion surface close to a diabolical point showing the angle $θ$ the optic axis makes with the $k_{3}$ direction. When light is incident along the optic axis, the singularity at the diabolical point results in a directional degeneracy of the Poynting vector (S), which is shown tracing a skewed cone of semi-angle $A$ with respect to the wavevector k. Part (b) demonstrates the wavelength dependence of the optic axis angle θ as given by Eq. (5).

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z_{F I P} = l (1 - \frac{1}{n_{2}}) .

The three principal refractive indices $n_{i}$ have wavelength dependencies which can be represented [16] using a form of the Sellmeier equation:

n_{i} (λ) = A_{i} + \frac{B_{i}}{1 - {(\frac{C_{i}}{λ})}^{2}} - D_{i} λ^{2},

where

A_{i},

B_{i}

are dimensionless,

C_{i}

has units of nm, and

D_{i}

has units of

{nm}^{- 2} .

These data have been measured for KGd(WO₄)₂ crystals [16], which are the biaxial materials used in the following experiments. The wavelength-dependent refractive indices

n_{i} (λ)

now give a wavelength-dependent semi-angle of the diffraction cone

A (λ),

which in turn translates to a wavelength dependence in the radius of the ring

R_{0}

given by

R_{0} = A (λ) l .

To achieve conical diffraction light must propagate down the optic axis of the crystal, otherwise double refraction occurs [9]. The optic axis may be shown to make an angle

θ = arc \tan \sqrt{(n_{1}^{- 2} - n_{2}^{- 2}) / (n_{2}^{- 2} - n_{3}^{- 2})}

with the

k_{3}

axis of the dispersion surface [4] as seen in Fig. 1(a). With the introduction of wavelength-dependent refractive indices it is clear that

θ

will vary with wavelength. This variance is shown explicitly in Fig. 1(b). When the crystal is aligned such that a wavelength

λ_{0}

passes down the optic axis, all other wavelengths are misaligned. This produces a gradual transition from conical diffraction to double refraction as the wavelength is shifted further from

λ_{0} .

To describe this effect, it is useful to introduce a misalignment parameter

μ (λ, λ_{0})

which represents the extent of misalignment between the wavelength-dependent optic axis and the beam direction:

μ (λ, λ_{0}) \equiv ρ_{0} (λ) - ρ_{0} (λ_{0}),

where

λ_{0}

is the aligned wavelength. It is also useful to recapitulate some of the theory explored by Berry and Jeffrey when dealing with beams propogating in directions which do not coincide with the optic axis [9]. Such misalignment results in a breaking of symmetry in the plane orthogonal to the beam propagation direction, the

ρ

plane. This leads to the introduction of a new complex coordinate system

ρ = {ξ, η} \to \tilde{ρ} = {ξ + i μ, η}

where the orientation of

η

has been chosen to be orthogonal to both the optic axis and the direction of the incident beam as it enters the crystal.

ξ

is orthogonal to both

η

and the optic axis. Thus, the length and direction of

\tilde{ρ}

may be given by

| \tilde{ρ} | = \tilde{ρ} (ξ, η, λ) = \sqrt{{(ξ + i μ (λ, λ_{0}))}^{2} + η^{2}},

\cos \tilde{ϕ} (ξ, η, λ) = \frac{ξ + i μ (λ, λ_{0})}{\tilde{ρ} (ξ, η, λ)}, s i n \tilde{ϕ} (ξ, η, λ) = \frac{η}{\tilde{ρ} (ξ, η, λ)} .

The electric displacement matrix after the crystal may now be expressed [12–14] in the form

D \propto (\begin{matrix} B_{0} + B_{1} \cos \tilde{ϕ} & B_{1} \sin \tilde{ϕ} \\ B_{1} \sin \tilde{ϕ} & B_{0} - B_{1} \cos \tilde{ϕ} \end{matrix}) d_{0},

where

d_{0}

is the incident polarisation and

B_{0}

and

B_{1}

are the fundamental integrals given by

B_{0} (\tilde{ρ}, \tilde{ζ}; ρ_{0}) = \int_{0}^{\infty} d κ κ \exp [- \frac{1}{2} i \tilde{ζ} κ^{2}] J_{0} (κ \tilde{ρ}) \cos (κ ρ_{0}),

B_{1} (\tilde{ρ}, \tilde{ζ}; ρ_{0}) = \int_{0}^{\infty} d κ κ \exp [- \frac{1}{2} i \tilde{ζ} κ^{2}] J_{1} (κ \tilde{ρ}) \sin (κ ρ_{0}),

where

\tilde{ζ} = ζ - i

and

J_{i} (κ \tilde{ρ})

is the

i^{th}

order Bessel function of the first kind. The intensity profile for circularly polarised or unpolarised input light is thus

I = \bar{D} D = | B_{0} (\tilde{ρ}, \tilde{ζ}, λ) |^{2} + | B_{1} (\tilde{ρ}, \tilde{ζ}, λ) |^{2} (| \cos \tilde{ϕ} (ξ, η, λ) |^{2} + | \sin \tilde{ϕ} (ξ, η, λ) |^{2}) .

In the case where

λ = λ_{0},

Eq. (7) produces the familiar symmetric double ring structure associated with conical diffraction [4].

3. Experimental method

The white light source used was a polychromatic light emitting diode (LED). The source spectrum was measured using a spectrophotometer and is shown in Fig. 2(a). The experimental setup is shown in Fig. 2(b). A 100 µm pinhole was placed in front of the LED and an achromatic biconvex doublet lens of focal length 10cm was used to image the pinhole through a 3cm long slab of KGd(WO₄)₂. An iris was used to control the numerical aperture of the lens such that the image of the pinhole was below the diffraction limited resolution of the system. This effectively created a Gaussian spatial distribution for the input beam, which is required for the theory discussed in this paper. The intensity distribution in the focal image plane was recorded using a Sony ICX204AK colour charge-coupled device (CCD) with pixel size of 4.65 µm mounted on a rail allowing movement in the $z$ direction. In order to produce sharply defined rings a high value of $ρ_{0}$ is required. This was achieved by placing the lens a distance of 50 cm from the pinhole to produce a small beam waist. The $1 / e$ diameter was measured to be $28 \pm 10$ μm giving a beam waist of $w = 14 \pm 5$ μm.

Fig. 2 (a) Spectrum of the white LED used in the experiment recorded using a spectro-photometer. (b) Experimental setup used for white light conical diffraction experiments.

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Using Eq. (2) this value of $w$ gives $ρ_{0} = 42.8$ for light at 500 nm. The paraxial theory is valid for $A ≪ 1,$ which is the case here, and we expect to see well defined ring structures when $ρ_{0} ≫ 1.$ In Fig. 3 the crystal has been aligned using a HeNe laser with peak emission at 632.8 nm, and then the white light source was introduced to produce the image of white light conical diffraction. It can be seen that only the wavelengths close to 632.8 nm demonstrate the full ring structure, with gradual transition to double refraction as the wavelength separation increases. This double refraction is most clear for blue light.

Fig. 3 Experimental image of the conically diffracted beam in the FIP where the crystal is aligned to 632.8 nm and no filters are used. The transition to double refraction for blue light is clear. The image is 1.5 mm $\times$ 1.5 mm.

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4. Variation of ring radius with wavelength

The radius of the ring produced by conical diffraction is dependent on the wavelength of the input light. To observe this dependence the same setup was used as in Fig. 2(b) but bandpass filters were placed in the beam to select a narrow spectrum of the white light. Each bandpass filter had a full-width at half-maximum (FWHM) of 40 nm and each was inserted in front of the CCD in turn. The crystal was then aligned to produce the sharpest and most symmetric ring structure in the FIP for that wavelength. To ensure the highest accuracy possible, the CCD was first moved into the far field as the beam is more sensitive to crystal misalignment there. The CCD was returned to the FIP and mounted on a 25 μm resolution translation stage. The CCD could then be moved in small increments through the FIP and the resulting images were examined in order to determine the one displaying the sharpest rings. This image was used for the measurement of the radius. The process was repeated for each of the bandpass filters, as well as for the HeNe laser. A number of profiles of each image were taken and the average radii of the Poggendorff dark rings were determined.

The results are shown in Fig. 4(a), along with the theoretical predictions calculated from the expression for $A$ given in Eq. (1) with wavelength dependence arising from the Sellmeier equation given by Eq. (4). It is clear that there is a discrepancy of approximately 70 μm between the predicted values of $R_{0}$ (red curve) and the measured values. This discrepancy arises since the value of the semi-angle of the cone of conical diffraction depends very sensitively on the values of $n_{1},$ $n_{2},$ and $n_{3}$ as given by Eq. (1),

A (λ) \propto n_{2}^{2} {[(n_{1}^{- 2} - n_{2}^{- 2}) (n_{2}^{- 2} - n_{3}^{- 2})]}^{\frac{1}{2}},

meaning a very small variation in the value of any of the

n_{i}

values translates to a relatively large variation in

A (λ) .

To demonstrate this, the green curve plotted in Fig. 4(a) uses the

n_{i}

values given by Eq. (4) but with the value of

n_{1}

increased by just

δ n_{1} = 0.007

throughout the visible spectrum. This small change of about 0.34% is sufficient to reconcile the theory with the observations. All of the refractive indices may have slight variations from those calculated using Eq. (4), however the aim of this example is to convince the reader that the discrepancy between the predicted and the measured radius values may be explained within the error of the measurements of the refractive index values.

Fig. 4 (a) Plot of $R_{0} (λ)$ (red) calculated using the formula for $A$ given in Eq. (1) compared with the measured radii (points) for different bandpass filters. The point at 632.8 nm was recorded using a HeNe laser as a light source. The green curve shows the predicted radius with the value of $n_{1}$ increased by 0.007 throughout the visible spectrum. Part (b) of Fig. 4 shows the corresponding experimental images in the FIP for each of the data points in part (a) obtained using bandpass filters (wavelengths labelled). Each image is 1.16 mm in both directions.

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5. Variation of FIP location with wavelength

The dependence of the location of the FIP on wavelength was also investigated. This was achieved by mounting the CCD on a translation stage with a resolution of 25 μm and placing the 650 nm bandpass filter in the beam. The achromatic focusing lens ensured that any wavelength dependece in the FIP location arose from the crystal alone. The crystal was aligned to achieve the most symmetric ring structure and the CCD was gradually translated until the sharpest ring structure was observed. Defining this location to be the FIP for 650 nm, the bandpass filter could then be replaced and the process was repeated for each of the bandpass filters. The location of the FIP was determined by examining the width of the rings in each image. The one containing the narrowest ring structure was determined to be at the FIP. Figure 5 shows these points sit on the predicted curve within error. Note the structure of the conical beam (images inset) when the 500 nm bandpass filter is used. This is misalignment due to the wavelength dependence of the optic axis direction. The crystal was not reoriented for different bandpass filters as moving it may have incurred errors in the location of the FIP.

Fig. 5 Results for the relative location of the focal image plane with respect to wavelength. The data recorded (red dots) agree well with the theoretical plot (blue line) calculated from Eq. (3). The point at 650 nm sits exactly on the curve as all other measurements were taken relative to this point. Image (i) (inset) shows what is observed when the CCD is at the FIP for 650 nm and the 650 nm bandpass filter is replaced with the 500 nm bandpass filter. The CCD in (ii) has been moved 200 μm and is deemed to be at the FIP for 500 nm due to the sharp beam structure.

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6. Far-field white light observations

Since the value of ζ is proportional to $1 / w^{2}$ as shown in Eq. (2), a beam waist of $w = 14$ μm results in a large variation in ζ over a relatively small range of $z .$ In order to observe the far-field evolution of the beam, the first step was to use a HeNe laser at 632.8 nm as the light source. This allowed very precise adjustment of the biaxial crystal to ensure the optic axis was aligned perfectly for conical diffraction at 632.8 nm. The LED was then returned as the source of light and a bandpass filter centred at 632.8 nm was inserted before the CCD. The CCD was moved in intervals of 1 cm starting at the FIP until reaching a value of ζ of approximately 45. The resultant images were stitched together using numerical tri-linear interpolation which allowed the full evolution of beam structure over ζ to be viewed from the side. We observe the spreading out of the rings for $ζ > 0$ with the axial spike [6] becoming apparent between $ζ = 15$ and $ζ = 20 .$ Fig. 6 compares the observed beam evolution over ζ in the $y = 0$ plane with a plot generated from Eq. (12) with $λ = λ_{0},$ and closely matches previous work [17, 18].

Fig. 6 (a) is a composite image stitched together using a series of slices perpendicular to the beam propagation direction. Image (b) is a numerically generated plot using the same parameters as the experiment and agrees very well with what is observed. The FIP occurs at $ζ = 0$ and this is where the double ring structure is at its sharpest. The intensity in the numerically generated plot has been artificially increased to allow the fine structure of the beam to be seen.

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Of greater interest is what happens when the bandpass filter is removed and this process is repeated. We observe a gradually increasing misalignment of the optic axis with wavelength which translates to a transition from perfectly symmetrical conical diffraction for red light to asymmetrical beam evolution for decreasing wavelength described by Eq. (12). Figure 7 displays the observed patterns on the top row with increasing $z$ and hence ζ values, beginning at the FIP. The rows underneath are numerically generated intensity distributions for a given wavelength. The corresponding optic axis angles calculated from Eq. (5) are also shown.

Fig. 7 Comparison of measured white light images (top row) and the theoretical intensity distributions given by Eq. (12). The crystal is aligned such that the optic axis for light at 632.8 nm and the beam direction coincide. The theoretical plots show the expected pattern that would be observed if all but one wavelength of light were blocked. There is good agreement between the predicted distributions and the observations suggesting the theoretical model is accurate. Note that the white light images for $ζ_{632} \geq 4.6.$ have been artificially intensified to make the structure more visible. The theoretical plots at 450 nm and 500 nm were also intensified for $ζ_{632} \geq 9.1.$ Also note the mirroring of the structure evolution for 650 nm. This is a manifestation of the sign reversal of $μ (λ, λ_{0})$ which occurs on opposite sides of the aligned wavelength, which in this case is 632.8 nm. This effect becomes clearer in Fig. 8.

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The crystal was then aligned using a 500 nm bandpass filter and the beam evolution was again recorded as seen in Fig. 8. The beam exhibits a similar evolution as before but this time it is more apparent that light with $λ <$ 500 nm evolves as a mirror image to light with $λ >$ 500 nm. This is a manifestation of the sign reversal of $μ (λ, λ_{0})$ at $λ_{0}$ as given by Eq. (6).

Fig. 8 A bandpass filter at 650nm was placed in front of the CCD for the images on the top row while the optic axis of the crystal remained aligned to 500nm. For the observed images at 450nm, the white light images were deconstructed into RGB values and only the blue values above a certain threshold were chosen. This plot shows very good agreement between theory and experiment.

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While the theory does appear to fit quite well in Fig. 8 it is worth noting that some of the disagreement is due to the theoretical plot being for a single wavelength $λ$ while the images at 650 nm are taken using a bandpass filter centred at $λ$ with a 40 nm FWHM, and the crystal is aligned to $λ_{0} .$ This will inevitably result in more accentuated asymmetry than the theory predicts due to the presence of wavelengths further from $λ_{0}$ than $λ .$ To see the extent of this effect note that if $λ_{0} = 500$ nm with $λ = 650$ nm, then $μ (λ \pm 20 nm, λ_{0}) = - 2.19 \mp 0.2.$

7. Conclusion

Some of the properties of a conically diffracted beam generated using a white light source have been examined and successfully described using theoretical models. Both the radius of the conically diffracted ring and the position of the FIP were shown to depend on the wavelength of the input light. The radius was observed to deviate from theory by a measurable amount, but by a factor sufficiently small to be within the range of errors of the measured refractive index values. The beam evolution beyond the FIP into the far-field was observed to depend strongly on wavelength, with asymmetries arising as a result of the polychromatic light source. A theoretical model was developed and applied to describe this beam evolution and was found to successfully predict the beam structure for the cases examined in the experiments. A method for incorporating explicit wavelength dependency in conical diffraction has been explained and this should provide a useful foundation for future work in this field.

References and links

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3. N. A. Khilo, “Conical diffraction and transformation of Bessel beams in biaxial crystals,” Opt. Commun. 286, 1–5 (2013). [CrossRef]

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7. C. F. Phelan, J. F. Donegan, and J. G. Lunney, “Generation of a radially polarized light beam using internal conical diffraction,” Opt. Express 19(22), 21793–21802 (2011). [CrossRef] [PubMed]

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11. A. Turpin, Y. V. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “Wave-vector and polarization dependence of conical refraction,” Opt. Express 21(4), 4503–4511 (2013). [CrossRef] [PubMed]

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16. M. C. Pujol, M. Rico, C. Zaldo, R. Sole, V. Nikolov, X. Solans, M. Aguilo, and F. Diaz, “Crystalline structure and optical spectroscopy of Er³⁺-doped KGd(WO₄)₂,” Appl. Phys. B 68(2), 187–197 (1999). [CrossRef]

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18. C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17(15), 12891–12899 (2009). [CrossRef] [PubMed]

White light conical diffraction

Abstract

1. Introduction

2. Theoretical method

3. Experimental method

4. Variation of ring radius with wavelength

5. Variation of FIP location with wavelength

6. Far-field white light observations

7. Conclusion

References and links

Cited By

Figures (8)

Equations (13)

Optics Express