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Research Article

Vol. 61, No. 22 / 1 August 2022 / Applied Optics

Apophyllite waveplates
Nathan Hagen,*

Kentaro Yoshida, Yuejin Shan, AND Yukitoshi Otani

Department of Optical Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-0954, Japan*Corresponding author: nh@hagenlab.org
Received 16 May 2022; revised 1 July 2022; accepted 5 July 2022; posted 7 July 2022; published 21 July 2022

We investigate the polarization characteristics of apophyllite crystals in an attempt to evaluate their potential use
for achromatic waveplates. Among the 50 plates that we extracted and polished from natural apophyllite crystals, a
few show the sought-for characteristic of birefringence that increases linearly with the wavelength. However, we alsofind that the crystals among our samples exhibit a sectored structure in their polarization properties, as well as anundesirable degree of spatial nonuniformity. © 2022 Optica Publishing Grouphttps://doi.org/10.1364/AO.463809

1. INTRODUCTION
For optical systems working over a wide spectral range, thestrong wavelength dependence of common waveplates is a
nuisance that either limits their range of operation to narrowspectral regions or limits their performance over wide spectral
ranges. To compensate for this retardance dispersion, a common method is to combine multiple retarder materials in an
achromat sandwich, widening the spectral range over which
the compound retarder element is approximately achromatic[1–3]. These achromatic components are now widely commercially available, but require precision in their manufacture andtherefore are somewhat expensive.
Because the retardance equation,
δ = 2π t1n(λ)/λ,(1)

includes a factor of 1/λ for a waveplate of thickness t, anotherapproach to achieve achromaticity is to use a material whosebirefringence 1n increases linearly with the wavelength. West
and Makas showed that this can be approximately done withstretched sheets of cellulose nitrate [4].
Looking into the literature on waveplate materials and theirwavelength dependence, however, one encounters a few curious
references to waveplates made of apophyllite that demonstrate
a surprising degree of achromaticity, as shown in Fig. 1. The
first paper of which we are aware that shows measurements of anapophyllite waveplate are from Françon et al. in 1965 [6]. After
this paper, one finds a few scattered and exasperatingly shortcomments from several authors [5,7–10], but one gets the distinct impression that all of these researchers seem to be referringto the same individual waveplate, if not the same measurement
data, and that only one like this has ever been made. The lack of
research in this direction is particularly odd because apophyllite
has additional promising characteristics:

1559-128X/22/226518-09 Journal © 2022 Optica Publishing Group

Fig. 1. Quarter-wave plate retardance spectra of different materials(crystal data from [5]; polymer data from [4]).

• Unlike quartz, calcite, and almost every other crystal used
for manufacturing retarders, apophyllite has a very low birefringence, which means that one can manufacture zero-order
quarter-wave and half-wave plates at thicknesses that are quite
practical for cutting and polishing.
• Apophyllite has excellent basal cleavage, much like mica,so that it tends to fracture along planes parallel to the crystal axis,
just as one would want for a standard linear waveplate.
• Apophyllite is neither too hard (a problem for sapphireand MgF2 waveplates) nor too soft (a problem for calcite andmica waveplates) to cause major difficulties with polishing.
• In its natural mineral form, highly transparent crystals of
apophyllite are readily available.
• Apophyllite is nontoxic.
With these promising characteristics in mind, we looked into
the mystery of why apophyllite is not more widely used, how tomake apophyllite waveplates from scratch, and then measure theresulting apophyllite waveplate characteristics.

Research Article
2. APOPHYLLITE CRYSTAL STRUCTURE
Apophyllite was first mentioned in 1784 by Carl Rinman as a“zeolite from Hellesta” in Sweden, and was first shown to be an
independent mineral (not a zeolite) in 1800 by José Bonifáciode Andrada e Silva [11]. Andrada e Silva named the mineral“ichthyophthalm” as a reference to the silvery white luster ofreflections from the crystal surface, which to him resembled theappearance of a cooked fish eye. In 1806, the French mineralogist René-Just Haüy renamed the mineral “apophyllite” fromthe Greek word apophylliso (exfoliate) because of its tendency toflake apart when heated in a furnace. [12–14]
The modern name apophyllite actually does not stand for asingle mineral but rather for a group of five related minerals: [15]• Fluorapophyllite-(K)(formerly
fluorapophyllite)
KCa4 Si8 O20 F · 8H2 O;• Hydroxyapophyllite-(K) (formerly hydroxyapophyllite)
KCa4 Si8 O20 (OH) · 8H2 O;• Fluorapophyllite-(Na)(formerly
natroapophyllite)
NaCa4 Si8 O20 F · 8H2 O;• Fluorapophyllite-(NH4 )(ammonianapophyllite)
NH4 Ca4 Si8 O20 F · 8H2 O; and• Fluorapophyllite-(Cs), (cesiapophyllite) CsCa4 Si8 O20 F ·
8H2 O.
These five, together with the micas and many other minerals that have a sheet structure, belong to the phyllosilicateclass of minerals. In addition, the apophyllites are solid
solutions; in other words, while they have a single crystal
structure, they possess a range of compositions. For example,
fluorapophyllite and hydroxyapophyllite are two ends of acontinuum at which any number of fluorine atoms can be
replaced with an (OH) at the corresponding lattice position
within the crystal [i.e., Fx (OH)1−x ], so that the chemical formula for the first two members of the group is often written as
KCa4 Si8 O20 (F, OH) · 8H2 O [16]. Similarly, sodium atoms
can be substituted for potassium (giving natroapophyllite). The
more recently discovered forms have ammonium or cesium inplace of potassium.
Of the five forms of apophyllite, Fluorapophyllite-(K) is by
far the most common, and comprises all of the samples that weanalyze here. Thus, when we use the short form apophyllite, wewill actually be referring to Fluorapophyllite-(K) specifically,
implying that it has only a minimal fraction of F atoms replacedwith OH. In our measurements, we obtained apophyllite
samples from three locales: the Pune and Jalgaon districts of
Maharashtra, India; and the Kamioka district in Gifu, Japan.
We used a Rigaku ZSX Primus II spectrometer to make X-ray
fluorescence measurements to look for differences in elemental
composition, and the results are shown in Table 1. The far rightcolumn of Table 1, labeled “F-Apo Thy” gives the elemental
mass composition of pure fluorapophyllite derived from thechemical formula and atomic masses. The measurement results
show that each of the three samples are Fluorapophyllite-(K),with trace quantities of Fluorapophyllite-(Na). Note that while
the two Pune samples are almost indistinguishable, the Gifu
sample shows a higher fluorine content, lower calcium, and trace
impurities of Al. A similar XRF measurement for a sample from
Jalgaon is not available. The aluminum concentrations shown

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Table 1. X-Ray Fluorescence Measurementsof Apophyllite Samples’ Elemental Composition
by % Mass

O
Si
Ca
K
C
F
Al
Na
H

Gifu

Pune #1

Pune #2

F-Apo Thy

51.1
23.3
17.8
4.18
1.82
1.70
0.057
0.041
0
99.998

51.1
23.4
18.2
4.13
1.78
1.18
0.122
0.074
0
99.986

51.0
23.4
18.1
4.16
1.75
1.31
0.114
0.097
0
99.931

56.5
22.0
15.7
3.84
0
1.86
0
0
1.59
100

here are also a common feature of apophyllites, in which somefraction of silicon atoms are replaced with aluminum within the
crystal lattice [17].
The crystal form of apophyllite is shown in Fig. 2, for various
degrees of truncation of the pyramid tips. The c (basal) face indicated in the figure contains the optic axis, and so these faces arewhat one generally wants to use for the retarder optical surfaces.
The middle drawing in the figure displays the characteristic ditetragonal dipyramidal form of the fully developed crystal, andwith four-sided pyramid tips placed atop prismatic columns,
with the pyramid faces rotated 45◦ from the prism faces.
In its natural environment, apophyllite appears most often
in the form of amygdules or druse—a coating of small crystals
attached to a surface; in some cases, however, the crystals havegrown large enough to produce optically useful sizes. While
crystal twinning can be a problem for quartz, it occurs only
rarely in apophyllite.
Finally, the characteristic that gives apophyllite its name indicates that the crystal must be kept away from very hot environments to avoid damage. At high temperatures, the water encapsulated within the crystal lattice begins to escape when temperatures reach as low as 200◦ C, and its structure will be completely
destroyed if exposed to temperatures above 400◦ C [18,19].
3. OPTICAL PROPERTIES OF APOPHYLLITE
The first researcher to analyze the birefringence of apophyllite
in detail appears to be David Brewster in 1817, who likened
the unusual optical structures that he found there to an artistic
mosaic [20]. Jean-Baptiste Biot, Brewster’s contemporary,later said [21], “No substance thus far known has been more
exasperating for crystallographers, nor richer in optical effects
for physicists.” These comments show that apophyllite has
been a material of interest to crystallographers, making it
all the more mysterious that it has seen so little quantitative
analysis in modern times. One can, however, find that dataon the refractive index of fluorapophyllite, possibly due to its
range of chemical compositions, is generally listed as varying from 1.530 ≤ n o ≤ 1.536 for the ordinary index, and1.532 ≤ n e ≤ 1.538 for the extraordinary index [22].
The procedure to prepare an apophyllite waveplate involves
the following steps. We first extract a single crystal from anapophyllite cluster (such as the one shown in Fig. 3), then use asharp blade pressed against the side of the crystal (i.e., face a or p

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Research Article

Fig. 2. Various crystal forms of apophyllite, shown with the prism faces increasingly prominent from left to right. The stripes on face a illustrate
the striations that characterize this surface.

Fig. 3.

Example crystal cluster of raw apophyllite.

in Fig. 2) to force it to cleave at approximately the desired thickness. (See Visualization 1 for a short video that offers an exampleof this cleaving process.) The cleaved slab is then checked for
optical clarity; if it has sufficient clarity, then we polish bothsides, using progressively smaller grit until we achieve a surface
with no apparent scatter. We did not polish the plates down to
any specific target sample thickness but instead left each sample
at the thickness obtained after surface polishing. At this point,
locating sample waveplates with 10 mm × 10 mm transparent
regions without visible crystal defects was not difficult.

The hardness of apophyllite is generally given as
4.5 ∼ 5 mohs [22], indicating a material that is harder thanmica and calcite (2 and 3 mohs, respectively), but softer thanquartz (6 ∼ 7 mohs). Its softness allows the polishing process
to proceed quickly, but care must be taken to not generate deep
scratches when doing the initial polishing work because thesescratches can be difficult to remove in the final polishing.
The characteristics that most fascinated David Brewster
about apophyllite derive from its unusual sectored optical structure. Although this sectoring is not visible with the naked eye,Fig. 4 shows that it becomes apparent when viewed between apair of polarizers. Inside each of the five sectors indicated in thepolarization contrast image of Fig. 4 are drawn circles showing the locations where we measured Mueller matrix spectraof the sample. In the case of the sample #1, the central sector(sector #1) is too small for measuring spectral data, but for othersamples this sector can be large.
Although the crystal structure of fluorapophyllite follows thesymmetry of space group P4/mnc, the crystal’s optical properties do not follow this symmetry; hence, its categorization as an“optically anomalous” crystal [21]. Note that the classification
as an optically anomalous crystal is not directly related to theproperty of anomalous birefringence, since the latter simply
means that the birefringence increases with the wavelength.
To image the polarization characteristics of the apophyllite
sample waveplates, we illuminated the samples with circularly

Fig. 4. Measurements of a 0.47 mm thick apophyllite sample (sample #1) from Pune. (Top row) the light intensity, polarization contrast, retardance δ, and retardance orientation θ . (Bottom row) Histograms of δ and θ obtained from each of the indicated regions in sectors 2 through 5.

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Fig. 5.
Optical layout for imaging apophyllite samples.
Visualization 2 shows a video sequence for the process of imaging
the sample in white light, when illuminated by a circular retarder andviewed through a rotating analyzer between angles 0◦ and 135◦ . In
the visualization video, one can see how the sectors become visible ordisappear from view as the analyzer angle changes.

polarized light (i.e., a white light LED followed by a linear polarizer and an achromatic quarter-wave plate oriented at 45◦ tothe polarizer). We then imaged the samples through an analyzer
oriented successively at azimuth angles of 0◦ , 45◦ , 90◦ , and135◦ , as illustrated in Fig. 5. With this setup, estimates for theretardance δ and retardance orientation θ at each pixel proceedfrom [23]I (x , y ) = 21 (g 0 + g 45 + g 90 + g 135 ),1/2 
1 (g 0 − g 45 )2 + (g 90 − g 135 )2
,
I


1
g 0 − g 90
θ (x , y ) = arctan,2
g 135 − g 45

δ(x , y ) =

(2)

(3)

(4)

where I (x , y ) is the estimated intensity, and g i (x , y ) is the raw
camera image obtained when the analyzer is oriented at angle
i. For clarity, the (x , y ) dependence of the individual equation
parameters is not shown explicitly. The advantage of this measurement approach is its simplicity and its ability to handle any
birefringence orientation. Its disadvantage is that it requires ahighly accurate circular input state when attempting to measure
low retardance values, as we attempt to do below.
From the imaging polarization characteristics shown in
Fig. 4, we can see that the various sectors have different retardances and different orientations, but that opposing sector
pairs (i.e., sectors 2 & 3, or sectors 4 & 5) tend to share similar
retardances and orientations. For all measurements, we use the“orientation” to refer to the azimuth of the crystal’s fast axis.
To evaluate the apophyllite samples’ spectral dependence, weused an Axometrics Axoscan Mueller matrix spectropolarimeter.
Because of the sectoring of each sample, we used the imaging
results to create a template of the sectors of each plate, and centered the Axoscan beam on the appropriate region to measureeach sector individually. The circles drawn in the polarization
contrast image shown in Fig. 4 indicate the approximate location of the Axoscan beam. For smaller samples, the ∼4 mm
Axoscan beam diameter is too large, and in this case we place anaperture over the measurement region to reduce the effective
beam size. Due to the difficulty of matching the reference axis
between the imaging setup and the Axoscan setup, the imaging
measurements can only be used to establish the spatial variationof polarization characteristics and can not easily be used to6521

match up with the azimuth angles from the Axoscan spectral
measurements.
Figure 4 also shows histograms of the pixels inside the indicated circular regions, from which we can see the variation in theretardance and orientation angles within each sector. To establish the baseline variation due to our measurement approach,independent of any variation in the sample, we also analyzed
the variation within the region outside the sample, coming
directly from the circularly polarized light source. Ideally, thisregion should show zero retardance and random orientation,
but the images clearly show that there is a residual nonzero
retardance of about 5◦ , and that the orientation is not random
from pixel-to-pixel in these background areas, but rather showa preferred orientation that depends on the small errors in thefour polarizer angles used in Eqs. (2)–(4). A careful calibrationof the polarization components used in the imaging measurements can reduce these errors, but the current measurements
are useful to establish the approximate spatial variation. The
cause of the wider histogram for δ in sector #5 is evident in the δimage, where we can see a sharp transition that occurs in about
the middle of the encircled region. This kind of crystal defectis common in natural minerals, and is not visible to the naked
eye without the use of polarizers. Because sector #1 is so small,its histogram is noisy, but shows (δ, θ ) values near 3◦ and 15◦ ,respectively.
Another of the samples for which we show detailed imaging
data is sample #5, in Fig. 6. While not easily apparent in theprinted figure, zooming into the polarization contrast image
shows the familiar cross-pattern in the lower left corner of thecrystal. As a result, sectors 1 through 4 are small, while sector 5comprises almost half the crystal face. Unlike sample #1, thissample shows achromatic behavior, as we discuss further below.
From an initial set of about 50 plates cleaved from different
apophyllite crystals, we selected 24 samples that looked clear
enough to polish: 13 plates cut from crystals obtained from Gifu
prefecture in Japan, 10 from crystals obtained from the Pune
region in India, and 1 obtained from India’s Jalgaon region. The
sample thicknesses range from 0.4 mm to 5.7 mm, where theIndia-sourced samples typically allow greater thickness due totheir larger crystal sizes (up to 10 mm diameter regions withoutvisually apparent defects). A selection of measurement spectra from this set of 24 samples is shown in Fig. 7. Overall, thespectra show that none of the Pune or Jalgaon samples exhibit
achromaticity. Sample #1 was obtained from a Pune crystal,and therefore exhibits normal birefringence. Among the Gifu
samples, a subset of plates show achromatic behavior (samples
4, 5, and 8). The Gifu samples tended to be smaller (<7 mm indiameter), with sectors <3 mm in diameter. The smaller size of
the Gifu samples also led to generally thinner plates, so that theirretardance also tends to be lower.
One of the results we had not expected when starting thisresearch is shown in the birefringence spectra of Fig. 7. Whereas
the few references in the optics literature report apophyllite’sbirefringence as 0.0018 ∼ 0.0028, [5,6] we found a birefringence value two orders of magnitude lower than what had beenreported. In the mineral science literature, many resources
report apophyllite birefringence as 0.000–0.002, and statethat it “can be uniaxial or biaxial, and can be positive or negative in the sign of its birefringence [25].” This birefringence6522

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Fig. 6. Measurements of a 0.78 mm thick apophyllite sample (sample #5) from Gifu. (Top row) the light intensity, polarization contrast, retardance δ, and retardance orientation θ . (Bottom row) Histograms of δ and θ obtained from the indicated region.

Fig. 7. Spectrally resolved optical properties of sectors within the apophyllite samples, estimated from Mueller matrix measurements obtained byan Axoscan spectropolarimeter. Estimating optical properties from the Mueller matrix requires a physical model of the sample, for which we use a
Lu–Chipman decomposition, and assume a homogeneous material. [24] In the legend (inset within the bottom right plot), the first number is the
sample number, while the number after the dash is the sector number of the crystal. The color of each curve encodes the sample number, while thelinestyle encodes the sector number. The birefringence spectra B(λ) were obtained from the linear retardances and the plate thicknesses t via the equation B(λ) = λδ(λ)/(2π t).

range agrees with our own results, although we have not tested
whether the samples are uniaxial or biaxial. The sample in our
set that exhibited the strongest birefringence has B in the range(4 ∼ 5) × 10−5 . Thus, while we had thought that the low birefringence would be an asset for ease of manufacturing zero-orderplates, a birefringence this low means that the waveplates haveto be inconveniently thick. Using the mean birefringence of
our samples, 1.0 × 10−5 , a quarter-wave plate at 600 nm would
require a thickness of t = 15 mm. From the birefringence spectra of Fig. 7, we can also see that the majority of samples havenormal birefringence (B decreasing slightly with λ). However,the spectra for samples 4, 5, and 8 show anomalous birefringence (B increasing with λ). Figure 7 also shows that, for mostof the samples, the circular retardance is not quite zero, but smallenough to be close to the measurement noise floor. For sample#2, however, the circular retardance reaches as high as 4◦ at the
shortest wavelengths. Since sample #2 is also the thickest of
the waveplates measured here (at 5.73 mm thick), it is not clear
whether this sample is unusual or whether this result indicatesthat all of the samples exhibit some circular birefringence. If thelatter, then circular retardance will become significant if a crystal
is sufficiently thick.

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6523

Fig. 8. Selection of spectra from Fig. 7, showing only the results for samples 4, 5, and 8. In the linear retardance and orientation angle figures, both
the raw data and third-order polynomial fitted curves are shown for each sample.

Figure 8 shows a selection of the results from the many spectra of Fig. 7, showing only the three samples (all from Gifu)
that exhibit reduced chromatic dependence. The linear retardance and orientation plots in the figure each show third-orderpolynomial fits superimposed on the raw spectral data. These
fitted curves are used to clarify the differences among samples
and the general wavelength dependence when scaling the data
for the scaled linear retardance plot, or when shifting the data
for the relative linear retardance orientation plot. The scaled
linear retardance is used to show the spectral dependence thatwould occur if we either increased the thicknesses of the current samples to achieve quarter-wave retardance at 600 nm,
or if we were able to find equivalent apophyllite samples withhigher birefringence. For reference, the scaled linear retardance
curves also include two curves for zero-order quartz and mica
retarders, using data obtained from [7]. The reduced chromatic
dependence of the Gifu samples is clear, particularly for sample#8’s sector #3, and for sample #5. Note that only sample #5’ssector 5 is shown in the figure because the other four sectors are
too small to measure with the spectropolarimeter’s beam. The
imaging measurements of sample #5 are shown in Fig. 6, while

those for samples #3, #4, and #8 are shown in Figures S1–S3 in
Supplement 1.
Figure 8 also includes a plot of the sample diattenuation
spectra. With the exception of sample #8’s sector #4 (“8-4”), theresults show negligible diattenuation. The low-contrast fringes
present in the diattenuation curves might indicate the presenceof an internal cleavage inside the crystal that is not apparent to
the naked eye.
Since one of the difficulties encountered with commercial
achromatic waveplates is the dispersion of their fast axis angle,
Fig. 8 also shows the relative orientation angle spectra of theapophyllite samples. Since the absolute values of the orientation
angle only correspond with the angles at which the samples
happened to be oriented for the spectral measurements, it isprimarily useful in analyzing the overall distribution of angles. If
we select the orientation angle at a wavelength of 600 nm as ourreference point, then we can take the third-order polynomialfitted curves of the orientation angles and shift them so thatthe 600 nm orientation is zero. The results are the “relative”
orientation angle curves shown, allowing a clear view of the fastaxis dispersion. While sample #8’s sector #4 shows very little
orientation dispersion, we can see that the remaining curves

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Fig. 9. (a) The retardance spectra for several commercially available achromat and superachromat waveplates, obtained from the various manufacturers’ websites, plus four achromat designs obtained from Fig. 10 of [3]. (b) The retardance azimuth spectra for the four achromat designs of [3],using the azimuth angle at 600 nm as the reference point. Note that while the Kraemer2019-10a and Kraemer2019-10b designs use a stack of threeplates to achieve achromaticity, the Kraemer2019-10c design uses a stack of five plates.

exhibit about 10◦ of dispersion across the 420–800 nm range;or, in the case of sample #8’s sector #2, 20◦ of dispersion.
The concept of measuring the angle of a crystal’s fast axis
implies that there is only one such angle; a dispersion of thisangle therefore implies that the crystal cannot be consideredhomogeneous. As with commercial achromatic waveplates thatuse a sandwich of different elements at different orientations,
the wavelength variation in the apophyllite samples’ fast-axis
orientation angle is an artifact of modeling the crystal as a single
homogeneous element rather than as an inhomogeneous one.
To compare the apophyllite sample results with commercial
achromatic waveplates, Fig. 9 shows spectra obtained frommanufacturers. These show that the achromats designed for the
VIS or VIS-NIR spectral range maintain a retardance of roughly
90◦ ± 5◦ across the 400–800 nm region.
Another concern with achromatic waveplates is the dispersion of their fast axis. Since manufacturer websites do notgive the azimuth angle dispersion of their designs, the azimuth
spectra of Fig. 9 can only show the designs obtained from [3].
From these spectra, we can see that the standard three-platecommercial achromats can maintain the azimuth angle within±2.5◦ within this spectral range. The apophyllite samples show
a similar range of azimuth angle dispersion.
To quantify the comparison between commercial waveplates
and apophyllite samples, we use three different metrics to evaluate performance. Metric M1 simply uses the full 400–800 nm
spectral range and reports the total range of retardance over that
region. Metrics M2 and M3 look for regions in the spectrum
over which the variation in retardance is minimal, reporting thewidest spectral range for which the variation is within ±5% of
the nominal retardance for M2 , or within ±1% of the nominal
retardance for M3 . Thus, we should want M1 to be as small
a number as possible, but M2 and M3 should be as large as
possible. Table 2 shows the results of applying these metrics
to each of the various apophyllite samples and commercial
achromatic waveplates. Note that the results shown here forthe apophyllite samples use the polynomial-fitted curves rather
than the raw data to prevent excessive noise from affecting theresults. From the table, we can see that samples 5-5 and 8-4 havea performance that is within the same range as that achieved by
the commercial achromats. These, however, are the highestperforming apophyllite samples, and so any randomly selected
sample is unlikely to have performance near this level.

Table 2. Metrics Comparing Achromaticities of theaApophyllite Samples and Various Commercial Designs
SourceApo 4-1
Apo 5-5
Apo 8-3
Apo 8-4
Kraemer 5aKraemer 10a
Kraemer 10b
Kraemer 10c
Meadowlark SA
Newport
10RP54-1B
Newport
ACHRO14W-S
Thorlabs
AQWP05M-600
Thorlabs
SAQWP05M-700

M1
δrangeM2
λmin − λmax : 3

M3
λmin − λmax : 3

17.0◦
12.7◦
7.6◦
28.4◦
9.4◦
10.2◦
8.9◦
0.8◦
12.1◦
14.2◦

412–633: 221
412–696: 284
412–800: 388
652–800: 148
408–800: 392
410–780: 370
400–800: 400
400–800: 400
420–702: 282
400–726: 326

412–511: 99
418–562: 144
478–634: 156
746–799: 53
542–788: 246
488–639: 151
495–654: 159
400–800: 400
683–800: 117
436–579: 143

11.2◦

400–755: 355

469–608: 139

27.0◦

466–773: 307

523–647: 124

4.5◦

400–800: 400

620–800: 180aAfter scaling their retardances to 90◦ .

4. CONCLUSION
Looking through the optics literature, there are comments frommultiple authors [5,9] that optical quality apophyllite is rare.
From our own samples of apophyllite, we find that this problem is not at all apparent to the naked eye, since many samples
exhibit a clarity that is comparable to high-quality samples ofquartz. However, when one looks at the polarization propertiesof the apophyllite samples, one finds a large degree of spatial
variation; not only due to the sectored structure of the crystal
but also within each sector. This is likely to pose serious problems for using these minerals to construct waveplates. Although
it was easy to purchase raw specimens of fluorapophyllite that
had relatively large (over 1 cm × 1 cm) regions without visible
internal defects, the wide variation in birefringence dispersion
from sample to sample is disappointing. While it is true thatnatural crystals of quartz and calcite often exhibit substantial
spatial variation within samples, as well as sample-to-sample
variation such as that exhibited by the apophyllite samples
here, the nonuniformity exhibited by our apophyllite set seems
particularly egregious.

Research Article
Many crystals exhibit anomalous dispersion of birefringencein the narrow spectral regimes near their absorption peaks,so that anomalous birefringence itself is not specially rare.
Examples include cadmium sulfide (515–540 nm), magnesium
fluoride (115–150 nm), and sapphire (143–153 nm). [26]However, what is unusual is to find a material that exhibits
anomalous birefringence over such a broad spectral range asapophyllite does. Another anomalously birefringent material
that currently finds commercial use is stretched polymer, whose
retardance curves are drawn in Fig. 1. It may be that similarinternal stresses frozen into some apophyllite crystals during
their natural growth are responsible for the anomalous dispersion that we see, as suggested by Taylor and Náray-Szabó[18].
Overall, our investigation into the potential uses of apophyllite for achromatic waveplates revealed the following difficulties:
1. It has a birefringence and a dispersion of birefringence thatvaries strongly from sample to sample;
2. Among the samples we obtained, the birefringence wasfrustratingly low;
3. It appears to have serious nonuniformity problems, as aresult of its sectored structure as an anomalous crystal,and also of irregularities in its growth (a problem sharedby all natural crystals, but perhaps worse than usual forapophyllite);
4. Apophyllite crystals tend to be somewhat small; in fact,
we have not yet encountered crystals whose cross-sections
exceed about 20 mm × 20 mm at their base; and5. Apophyllite’s vulnerability to high temperatures [12,27]
means that care must be taken when polishing to keep thepolishing surface from heating. After polishing, apophyllite’s sensitivity to heat >100◦ C is likely not a serious
problem, since many AR coatings applied to waveplatesalso have difficulty withstanding those temperatures.
With these difficulties in mind, we were able to find several
apophyllite samples whose retardances demonstrated a promising level of achromaticity. What remains to be seen, and is a task
for our future work, is whether it is possible to locate and polishsufficiently large samples of these crystals at reasonable expense
to be useful for the community.
One community that has tried using apophyllite crystals fortheir optical characteristics, and reported on the results, is thelapidary community. According to their efforts, apophyllite
gemstones are unusually colorless, giving their sparkle a silvery
appearance. Although it is valued as a gemstone, apophyllite isalso known to be difficult to cut and facet on planes other thanthe basal plane due to its perfect basal cleavage [28]. Its sensitivity to heat also means that one cannot use a jeweler’s torchnear the mineral without causing cracking and flaking. This also
implies that it may be difficult to cut out individual sectors froma single plate sample without causing defects. While the small
sizes of natural apophyllite crystals could perhaps be overcome
by growing crystals in a lab, synthetic apophyllite is currently
unknown [28].
Looking into the literature discussing apophyllite crystals,
it is surprising to see just how little quantitative optical data isavailable. The imaging polarization characteristics and spectropolarimetric characteristics reported here are, to the best ofVol. 61, No. 22 / 1 August 2022 / Applied Optics

6525

our knowledge, the first to appear in the literature, and provide
a step toward understanding the properties of this material.
Before this paper, the only polarization images available in theliterature are contained in the qualitative illustrations given by
Brewster more than 200 years ago [20] and by Sahama in 1965
[29]. Qualitative discussions, however, leave a lot of room forinterpretation that quantitative data does not. For example,
Brewster reports that sector 1 in apophyllite samples is uniaxial,
while sectors 2–5 are biaxial. Other sources, however, disagreeand declare unequivocally that apophyllite is uniaxial. In fact, ina widely used reference, Dana and Ford state (without supporting data) that some apophyllite crystals can be positive uniaxial
at one end of the visible spectrum, while negative uniaxial at
the other, such that at one intermediate wavelength the crystal
becomes isotropic [30]. However, not only do none of our samples demonstrate this behavior, but Dana and Ford do not say
anywhere that the crystal can be biaxial, which implicitly contradicts Brewster. The problem with taking all of these researchers’comments at face value, however, is that apophyllite’s sectored
structure creates large spatial variations in polarization properties. Anyone who discusses apophyllite’s optical properties
without specifying the differences between sectors makes one
suspect that they either measured the entire crystal at once (ifusing a large optical beam, combining all sectors together) or
measured just one of the sectors randomly (using a narrowmeasurement spot). In either case, without specifying how theindividual sectors were treated in the measurement, it is very
difficult to use this data.
Funding. Japan Society for the Promotion of Science (20K04516)
Disclosures. The authors declare no conflict of interest.
Data availability. Data underlying the results presented in this paper are
not publicly available at this time but may be obtained from the authors upon
reasonable request.
Supplemental document. See Supplement 1 for supporting content.

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