Optical Characterization of Micro Spiral Phase Plates

Sebastian Buettner, Erik Thieme and Steffen Weissmantel

Laserinstitut Hochschule Mittweida, University of Applied Sciences Mittweida,

Technikumplatz 17, 06948 Mittweida, Germany

Keywords: Fluorine Laser, Fused Silica, Micro Structuring, Orbital Angular Momentum, Spiral Phase Plates.

Abstract: The results of our investigations on laser fabricated micro spiral phase plates in fused silica are presented.

Other than in previous investigations we focussed on the optical characterization of the SPPs. As we could

show, the laser-based process enables the generation of spiral phase plates with different topological charges,

handedness, modulation depths and level numbers. Each geometric property influences the property of a

transmitting electro-magnetic field and therefore the orbital angular momentum. For the optical

characterization we observed and analysed the diffraction images of the generated SPPs. Moreover, we

validated our results by calculating the diffraction patterns under variation of certain parameters.

1 INTRODUCTION

Since Maxwell light is understood in terms of

electromagnetic fields. The connection of magnetic

and electrical fields led to a new view to photons and

light in general. The solution of Maxwell’s equations

in absence of matter and charges is an

electromagnetic wave with the properties,

wavelength, polarization, amplitude and phase. All

these are used in modern technologies and can be

influenced for different purposes and applications. In

1909 Poynting showed, that electromagnetic fields

can have spin angular momentum (SAM), which is

associated with the circular polarization of an

electromagnetic field (Poynting, 1909). In 1992 Allen

et. al. proved that Laguerre-gaussian laser modes also

got orbital angular momentum (OAM) (Allen et al.,

1992). The recognition that light can have angular

momentum led to a new understanding and

consequently to many new applications. The total

angular momentum

𝐽

⃗

of an electro-magnetic field can

be calculated from the electrical field strength

distribution 𝐸.

𝐽

⃗

𝜖



2𝑖𝜔

𝐸



∗

(𝑟⃗×∇)

,,

𝐸



𝑑



𝑟













⃗

(1)









𝐸

∗

×𝐸 𝑑



𝑟













⃗

As can be seen from equation (1), the SAM is an

intrinsic property, whereas the OAM results from a

spatial distribution. More precisely, the OAM results

from a phase distribution with an azimuthal

dependency. The modulation of an even wave front

with a helical phase term gives the Poynting vector a

helical trajectory (Allen and Padgett, 2011). The idea

that the total OAM results from the spatial

distribution is correct with respect to the entire field.

Counterintuitively, it can be shown that single

photons can also have orbital angular momentum. In

2005 Anton Zeilinger's group showed that the OAM

states of individual photons can be entangled (Mair et

al., 2001). The entanglement of OAM states of single

photons enables the development of new quantum

technologies with enormous potential (Fickler et al.,

2012,2014; Cardano et al.,2015; Krenn et al., 2017).

Moreover, the OAM become interesting for

several other applications like optical tweezers and

particle manipulation (Grier,2003). However, this

property can be used to encode classical information

for optical data transfer. In the last years, several

methods for creating light and in particular laser

beams with a defined OAM were developed. As

example, there are some micro-optical elements

which can be used to modulate and convert light. One

of them is a q-plate, which converts SAM to OAM

(Marrucci et al., 2006; Karim et al., 2009). But also

phase elements like fork gratings (FGs) and spiral

phase plates (SPPs) can be used to influence the OAM

of electromagnetic waves, as well as spatial light

modulators (SLM) (Zhu et al., 2018; Xie et al., 2018;

Bozinivic et al. 2013). In addition to this, FGs and

Buettner, S., Thieme, E. and Weissmantel, S.

Optical Characterization of Micro Spiral Phase Plates.

DOI: 10.5220/0012376400003651

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 12th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2024), pages 57-64

ISBN: 978-989-758-686-6; ISSN: 2184-4364

SLMs were used to encode data to OAM states and

vice versa. To push forward the idea of OAM based

data communication, we developed a laser-based

technique, which allows the generation of individual

micro SPPs in fused silica. In this investigation we

focused on the measurement of the optical function of

the generated SPPs.

2 SPP PROPERTIES

The OAM of an electromagnetic field is a result of the

modulated phase with an azimuthal dependency. A

description of such a modulated wave is given in

equation (2), where

𝑢

(

𝑥,𝑦,𝑧

)

represents the amplitude

and 𝑒



the phase of the field. The phase depends on

the azimuth angle 𝜑 and the topological charge 𝑙.

𝑢

(

𝑥,𝑦,𝑧

)

=𝑢

(

𝑟,𝜑,𝑧

)

=𝑢



(𝑟,𝑧) ∙𝑒



∙𝑒



(2)

The modulation can be done using SPPs, which got

an azimuthal change in thickness. Due to this, the

wave front of a transmitting even electro-magnetic

wave becomes a helical shape. For SPPs the

topological charge in general represents the number

of 2π phase jumps within the structure. Moreover, one

can show that the modulation depth also influences

the number of intertwined phase fronts in the same

way. Due to this, we defined the number of phase

jumps as a separate quantity 𝑗 , as well as the

modulation depth 𝑚 . The resulting topological

charge 𝑙 is given then by equation (3).

𝑙=𝑚∙

𝑗

(3)

This provides a certain degree of freedom regarding

the generation of the SPPs. According to the Huygens

law, the effect of 𝑗 and 𝑚 differs. Regarding to this, 𝑗

led to the interaction of the elementary waves of one

wave front. Whereas 𝑚1 led to the interaction of

the elementary waves of 𝑚 wave fronts. As result, the

modulation depth can act as time delay, due to the

delay of interaction of elementary waves of the 1

and

the m

wave front. Concluding, the modulation depth

induces a time delay from fractional to full

modulation. It is unlikely, but it could be necessary to

take this in account for some ultra-fast applications.

Due to the delay is in a range of few femtoseconds for

a typically wavelengths in the micron range. An

ideally helical modulated wave front converts a

gaussian to a ring-shaped intensity distribution. In the

centre of the SPPs is a phase singularity, which occur

also in the modulated wave front. Therefore, the

intensity in the range of the singularity approaches

zero. In Fig. 1 and Fig. 2 the calculated electric field

and phase distributions are shown for different

distances to the plane of modulation. The calculations

were done using the Fresnel-Kirchhoff diffraction

integral. The comparison of Fig. 1 and Fig. 2 shows

that the modulation depth 𝑚 and the number of phase

jumps j got the same effect. Due to this, the

topological charge of the field can be controlled by

the modulation depth, the number of phase jumps or

both simultaneously. This is beneficial regarding to

the generation process, which allows the generation

of SPPs with at least two sectors. By stetting 𝑚 to 0.5

an SPP with a 𝑙 of 1 can be generated too.

Figure 1: Calculated electrical field strength (l.) and phase

distribution (r.) of an OAM beam (𝑙=2, 𝑗=1, 𝑚=2).

Figure 2: Calculated electrical field strength (l.) and phase

distribution (r.) of an OAM beam (𝑙=2, 𝑗=2, 𝑚=1).

Our interest in this field of research and expertise in

laser micro structuring of optical materials led us to

develop a process for micro SPP generation. The

process allows the fabrication of those elements in

various versions. For example, Fig. 3 shows one of

PHOTOPTICS 2024 - 12th International Conference on Photonics, Optics and Laser Technology

the generated SPPs in fused silica. A detailed

description of the experimental setup and the process

can be found in a former report (Buettner et al. 2020).

Figure 3: Scanning electron microscope image of a right-

handed micro spiral phase plate (j=3, m=2, l=6).

3 RESULTS AND DISCUSSION

For these investigations, we observe and evaluate the

diffraction images of the SPPs using the measurement

setup that we set up earlier. Due to this, a description

of the basic set-up can also be found in the above-

mentioned paper. Following, it is also briefly

described. The measurement setup consists of a

frequency-doubled Nd:YAG laser (λ=532 nm), a

series of polarisers, two lenses with different focal

lengths (f=175 mm, f=10 mm) and neutral density

(ND) filters. With the improved setup the laser spot

size can be set to a defined value. Reducing the size

of the laser spot requires a reduction in the intensity

of the measuring laser to avoid overdriving the

camera. This is achieved using two polarisers and

additional ND filters. The radius of the frequency-

doubled Nd:YAG laser beam is 0.65 mm. This is

reduced to 37 µm using the two lenses. The substrate

is positioned at the focal point of the second lens. At

this point the wavefront is assumed to be flat.

Moreover, an XY axis is used to position the SPPs

within the measurement laser beam. In this way the

SPP can be adjusted concentric to the laser beam. The

resulting diffraction images were recorded with a

confocal microscope positioned above. The whole

microscope can be moved vertically in micron steps,

which allows to adjust the distance between the SPP

and the plane of observation. This allows the

diffraction images to be recorded at different

distances from the SPPs. The diffraction images are

used to calculate the maximum and root mean square

(rms) radius (𝑟



,𝑟



) of the ring-shaped intensity

distributions too, as well as the divergence angle of

the propagating fields. For a first measurement we

used a laser spot size of 90 µm. Due to the spot size

is in the range of the SPP diameter, the modulated and

unmodulated components of the transmitting field

were interfering strongly, as can be seen in Fig. 4. The

generated interferograms show the number and

symmetry of the twisted phase fronts in the form of

curved intensity maxima. The number of maxima is

equal to the topological charge of the corresponding

SPP. Moreover, the comparison of the interferograms

of SPPs with the same topological charge shows the

same number of maxima, due to the modulation depth

and the number of phase jumps have the same effect.

Figure 4: Interferograms of the SPPs with different

topological charges, 3 mm above the modulating surface.

The alignment of the SPPs was done by observing

the symmetry of the diffraction pattern during the

adjustment. Nevertheless, some of the interferograms

show asymmetries within the intensity distribution.

These deviations result from geometric asymmetries

of the SPPs. Moreover, for SPPs with a topological

charge >12 the intensity maxima cannot be separated

easily. The higher the topological charge, the more

the intensity distribution is distorted. This can be

explained by a change of the local pitch angle within

one sector and the slope edges between the steps of

the structures. The change of the local pitch angle is

a result of a varying ablation depth per laser pulse

during the generation process and it is caused by the

fluctuation of the laser pulse energy. The slope angle

at the edges of the ablation area depends on the

ablation depth per pulse. Due to this, the edges of the

steps got a constant slope angle, which depends on the

ablation depth and therefore the laser pulse fluence.

The local pitch angle of the SPP sector is given by the

topological charge, the modulation depth and the

distance to the optical axis. Consequently, the slope

angle of the step edges mostly does not match the

Optical Characterization of Micro Spiral Phase Plates

required local pitch angle of the SPP sector.

Increasing the modulation depth by the step number

increases the slope area within one sector. These areas

contain a different phase information and do not

contribute to the targeted phase and intensity

distribution. The slope edges got a negative effect on

the resulting wave front and increase the losses and

distortion within the diffraction image. Furthermore,

we observed the diffraction images with a beam

radius of 𝑤



= 37 µ𝑚 (Fig. 5). For the SPPs with a

modulation depth of 𝑚=4 the quality of the

diffraction images was very low and affected by

distortion. In general, the diffraction pattern does not

show the expected ring-shaped intensity distributions

as shown in Fig. 1 and Fig 2. Instead, the images show

intensity maxima, which are surrounded by a ring-

shaped distribution with lower intensity. The reason

for this is an insufficient phase modulation, due to the

total depth of the structures not matching the

equivalent

𝜋, 4𝜋 and 8𝜋 phase shift (Tab. 1). This

also can be shown by the numerical calculations of

the diffraction images considering the deviations

from the ideal phase. The calculated diffraction

images show the same characteristics as the measured

(compare Fig. 5 (a) and (b)). Concluding, the

deviation of the ideal phase in terms of a systematic

deviation from the depth causes a change in the

intensity distribution The intensity maxima rinsing

with the increase of the deviations.

The reason for the deviations of the structure

depths is an insufficient adjustment of the laser pulse

fluence. The adjustment was done measuring the laser

Table 1: Phase shift of the SPPs, calculated from the

measured maximum structure depth.

m / j 2 3 4 5 6

1.65π 1.72 π 1.75 π 1.82 π 1.85 π

3.50 π 3.63 π 3.69 π 3.67 π 3.77 π

6.97 π 6.96 π 6.70 π 6.75 π 6.84 π

power near to the image plane of the laser system at a

constant frequency of 200 Hz. This is necessary

because the laser power must be raised to the

measuring range of the power meter. A constant

operation of the laser at a defined frequency led to the

thermal equilibrium of the laser tube. For the SPP

generation a pulse-on-demand operation is necessary.

In this mode, the effective laser pulse frequency is in

a range of 1 Hz, due to the slow angular velocity of

the rotation stages. By changing the operation mode

and therefore the laser frequency, the laser is not in a

thermal equilibrium anymore and the laser power

drops. As result, we reached only 88.5 ± 3.28 % of

the targeted depth. This systematic deviation can be

avoided improving the measurement for the pulse-on-

demand mode or using a power offset.

However, we calculated the diffraction images for

the ideal SPPs with

𝜋, 4𝜋 and 8𝜋 phase shift (Fig.

7). The calculation was done using a continuous

phase profile. The aperture was set to 150 µm and is

therefore twice as large as the beam diameter. This

reduces the influence of the aperture on the

diffraction image.

Figure 5: Coloured representation of the measured diffraction images (generated by the SPPs) (a) and the calculated (b)

intensity distributions for the appropriate distances (𝑧 = 250 µ𝑚, 500 µ𝑚, 1000 µ𝑚 / 𝑤



= 37µ𝑚 / 𝜆 = 532 𝑛𝑚).

(a)

(b)

PHOTOPTICS 2024 - 12th International Conference on Photonics, Optics and Laser Technology

Figure 6: Coloured representation of the calculated intensity distributions (top) using an ideal and continuous phase

distributions (bottom) in a distance of 250 µ𝑚, 500 µ𝑚 and 1000 µ𝑚 to the modulation plane.

Moreover, the size of the target area is changed with

the distance, due to the divergence of the field and

appropriate increase of the radius of the distribution.

The calculation of the ideal diffraction images shows

the dependency of the rms and maximum beam radius

from the topological charge without the influence of

the aperture and steps. The topological charge was

varied from 1 to 25 and the modulation depth was set

to 𝑚=1 for all calculations. According to the target

values of the generated SPPs the optimum diffraction

images are shown in Fig. 6. The maximum radii 𝑟



were calculated from the 1000 biggest intensity

values, using a circular fit function. The rms radius

𝑟



was calculated by the second moment method.

As can be seen, both, the rms and the maximum radius

increases with the topological charge (Fig. 7.). For

small values of the topological charge the maximum

and the rms radius depend on 𝑙 in a very different

way. With higher values the course of both radii

become more linear.

Figure 7: Root mean square (rms) and maximum radii of

the calculated ideal intensity distributions depending on the

topological charge for different distances z.

Figure 8: Radii 𝑟



of the measured and calculated

intensity distribution for different distances in propagation

direction. The calculation was done using the ideal phase

and determined values for the real SPPs (see Tab. 1).

It can be seen further that the rms radii for 𝑙=1 is

close to the radius of the input beam waist 𝑤



. The

measured maximum radii basically follow the course

of the calculated maximum radii (Fig. 8). However,

there is a small offset of the measured radii to the

calculated, which can be partially explained by the

inaccuracy of the phase of the generated SPPs. Taking

this into account, the calculated radii approach the

measured radii, but a small difference remains. The

reason for this could be a slight misalignment of the

SPPs in the direction of propagation. Therefore, the

wavefront of the measuring beam is not flat, which

increases the radii of the intensity distributions and

the divergence of the fields.

Optical Characterization of Micro Spiral Phase Plates

Figure 9: Calculated intensity distribution of an OAM beam (𝑙=2) along the direction of propagation.

For future investigations, the wavefront of the

measuring system must be characterized using a wave

front sensor. The rms and maximum radii were also

compared to theoretical values. Therefore, Padgett

and colleagues derived equations for the maximum

and rms radius for two different kinds of OAM beams

(Padgett et al., 2015). They distinguish between

modes with a fixed beam waist (e.g. converting

Hermite-Gauss modes into Laguerre-Gauss using a

cylindrical lens mode converter) and modulated

gaussian modes, which can be generated by

illuminating a fork grating. Depending on the mode,

a transmitting field propagates differently. Therefore,

the far field beam divergence scales either with the

square root or linear with the topological charge. Due

to the generation of our OAM beams using SPPs, we

expect that the beam radius and the divergence

depends on the topological charge either in the one or

the other way. But it must be taken in account that our

measurements and calculations were done somewhat

between the near and the far field.

Figure 10: Measured and calculated radii 𝑟



of the

intensity distributions depending on the distances in

propagation direction 𝑧.

The equations given by Padgett et al. are valid for the

far field. Moreover, the relation between the size and

feature size of our generated SPPs and the wavelength

got a significant influence on the propagation of the

OAM beams, due to the propagation is not transversal

at all. This can lead to an abrupt change of the radius

of maximum intensity along the propagation axis (see

𝑟



in Fig. 9). Due to this, the measured results are

valid for these specific structures of this specific size

and the mentioned conditions. But the radii do not

match the values calculated from the equations, due

to the given reason. Fig. 10 shows the measured 𝑟



as a function of the distance to plane of modulation

for OAM beams with different topological charge.

The data fits very well with a linear progression.

However, as can be seen from the calculated data, the

curve is generally not linear and it also differs

depending on the topological charge. Here,

investigations with a more precise analysis of the

divergence are necessary.

Figure 11: Divergence angle depending on the topological

charge calculated from of the measured intensity

distributions (top) and the calculated intensity distributions

(bottom).

PHOTOPTICS 2024 - 12th International Conference on Photonics, Optics and Laser Technology

Nevertheless, we calculated the average divergence

angle for the measured and calculated intensity

distributions (Fig 11.). Both, the measured and

calculated angles show a nearly linear dependency

regarding to the topological charge. Moreover, the

data points are grouped by modulation depth m and

the progression between the points of the groups is

discontinuous. In addition, each group show a

different slope depending on the modulation depth.

This shows the influence of the modulation depth on

the divergence of the propagating fields, and it must

be considered for the design of micro SPPs for a

specific application. Concluding, the propagating

fields diverge differently depending on the

topological charge. In addition to this, the modulation

depth also got an influence on the divergence angle.

In order to find a general rule for the radii and the

divergence of the OAM beams as a function of the

topological charge, the further generation and

analysis of a large number of high-quality SPPs is

necessary. In further investigations, also the influence

of the relationship between feature size and design

wavelength should be analysed in more detail.

4 CONCLUSIONS

We show the results of our investigations on the

optical characterization of laser fabricated micro

SPPs. Therefore, we used two measurement

configurations. The measured results were verified by

numerical calculations of the diffraction images using

the Fresnel-Kirchhoff diffraction integral. The

captured interferograms show the equality of the

modulation depth and number of phase jumps, which

act on the topological charge in the same way.

Moreover, we captured the diffraction images with a

reduced beam radius to supress the interference

between the modulated and unmodulated proportion

of the propagating field. It could be shown that the

calculation in consideration of the geometric

properties led to equal diffraction pattern compared to

the measured one. This enables us to identify

irregularities and asymmetries within the structures

and their optical effect.

Simultaneous, we calculated the diffraction

images for ideal spiral phase distributions to

determine the course of the ideal maximum and root

mean square radius of the ring-shaped intensity

distributions. The comparison of the maximum radii

shows a good fit in course but a small offset of the

measured values, due to a small misalignment of the

SPPs. Moreover, we determined the divergence

angles of the OAM beams, as well as the influence of

the topological charge on their divergence.

Surprisingly, the modulation depth also got an

influence on the divergence of the propagating fields.

However, it must be considered that the

measurements and calculations were carried out on

micro-optics with feature sizes close to the

wavelength range. Moreover, the range of

measurement is somewhat between the near and far

field. Due to this, the far field approximation is not

valid. In further investigations, the region of interest

therefore should be increased. This may connect our

experimental data to the far field approximation.

Finally, the generation process and quality of the

micro spiral phase plates must be improved. In

particular, the precise control of the laser pulse energy

and therefor the ablation depth per pulse can raise the

quality of the SPPs significantly. As we have shown,

the depth of the structures was not exactly equal to the

required equivalent targeted phase shift and the slopes

and edges within the structure increased the distortion

within the diffraction image. Regarding to this, we

currently work on an improved process which will

allow us to generate continuous surfaces with

arbitrary geometry with highest possible quality.

AKNOWLEDGEMENTS

The authors would like to thank the referees for their

careful reading the manuscript and their valuable

comments and suggestions. Furthermore, we

thankfully acknowledge the financial support from

the free state of saxony.

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PHOTOPTICS 2024 - 12th International Conference on Photonics, Optics and Laser Technology