# Improved selectivity of optical transmission through cascaded waveguide-
metal-grating filters

Jérôme Le Perchec<EMAIL_ADDRESS>CEA, LETI, Minatec Campus, Optics
and Photonics Department, 17 avenue des Martyrs, 38054 Grenoble, France

###### Abstract

The resonant transmission of two near-field coupled, cascaded band-pass
filters, based on metallic stripe or patch gratings, is analysed. The
response, in terms both of maximum efficiency and light rejection out of the
resonance, overpasses the simple convolution of the responses of two isolated
filters, while keeping good angular tolerance. Illustrations are given in the
infrared range where sizing and technological integration of such compact
structures are particularly relevant for detection applications.

Guided-mode resonant (GMR) filters are based on the electromagnetic resonance
of a waveguide mode inside a high index dielectric slab coupled to a sub-
wavelength grating. Purely dielectric structures allow getting extremely
narrow transmission or reflection peaks Tibuleac2001 ; Jacob2002 ;
Kawanishi2020 , for highly wavelength-selective applications going from
infrared (IR) to visible ranges. Their sharp spectral responses are generally
very sensitive to the incidence angle and to the finite size of the grating
Bendickson . When the grating consists of thin metallic two-dimensional (2D)
patches (or infinite stripes), broader and high transmission peaks may be
easily obtained, less sensitive to the incidence angle and/or light
polarization Leperchec2011 ; Kaplan2011 . In some extent, such a kind of
structure joins the well-known family of the frequency-selective surfaces
(metal meshes) with, here, a dielectric loading Luebbers1978 . The compactness
of GMR filters and the resonance tunability with the lateral dimensions of the
grating are particularly advantageous compared to thicker multi-layered
structures, whose efficiency and angular-tolerance are also interesting Im2017
; Frey2018 .

In case of high-performance optical detectors with a somewhat large field of
view, like quantum detectors used for IR imaging, the working spectral range
may be rather large (for instance $[3-5.5]\mu$m wavelength for mid-wave IR
applications). Thus, band-pass GMR filters should present a high transmission
peak (at least $>70\%$) around a wavelength of interest, with very low
sideband transmittance ($<0.1\%$ preferred) over the rest of the spectrum. A
single metallic-grating/waveguide filter may not offer sufficient light
rejection out of resonance. Thus, in the present work, we study the selective
transmission properties offered by the near-field stacking of two similar GMR
filters (see Fig.1(a)) like the ones previously studied Leperchec2011 . A
quite similar architecture was already implemented in double-layer metallic
subwavelength slit arrays Chan2006 , but not based on filters with high-index
slabs (not the same mechanisms). We aim at getting significantly improved
light rejection out of the pass-band without altering the maximum peak. Very
sharp resonances (like the ones we may find with coupled dielectric gratings
Song2009 ) are not necessarily sought here as some applications like
hyperspectral imgaging may need to use some part of the spectral window around
a wavelength of interest. We mainly investigate the case of cascaded filters
with realistic materials and geometrical parameters, designed for the mid-
infrared, using Rigorous Coupled Wave Analysis (RCWA) in 1D. The 2D
configuration is also discussed at the end of this paper. Before tackling the
cascaded structure, we describe the essential behavior of the single filter
with the support of practical analytical formulae.

Figure 1: (a) Sketch of two cascaded GMR filters with a spacer layer of low
index $n_{s}$ between them. (b) Reference transmission of the isolated band-
pass filter number 1 (only air above it), by taking $n_{g}=n_{Ge}=4$,
$n_{met}=n_{Alu}$, and $n_{t}$=1 (1D stripe grating, TM-polarization). RCWA
simulations are given for three aperture widths $w$ and compare the real metal
case (red curves, complex metal permittivity $\varepsilon_{Alu}$) to a purely
reflecting but non-absorbing case by simply omitting the imaginary part of
$\varepsilon_{Alu}$ (blue lines). For instance, at $\lambda=4\mu$m,
$\varepsilon_{Alu}=-1543+512i$. Dotted curves result from the simplified
analytical model (see Appendix) with the complex permittivity. The fine peak
at $\lambda=3.6\mu$m is a resonance of metallic antennas that does not exist
with real metals

In a 1D configuration, the resonant mode of interest is found under TM-
polarization (incident magnetic field colinear to the metallic stripes). Other
transmission peaks could be observed in TE-polarization, but generally show
less interesting spectral profiles. Simple design rules of such GMR filters
have already been given Leperchec2011 . We remind that the waveguide is
advantageously of high optical index $n_{g}$ in order to get a better
resonance quality and an acceptable robustness with respect to the incidence
angle $\theta$ ($\pm 10^{o}$ typically, which might be increased by resorting
to a bi-atomic structurationSakat2013 ). Its thickness $h_{g}$ is
preferentially $\sim\lambda_{0}/2$, where $\lambda_{0}$ is the wanted
resonance wavelength. The periodicity $P$ of the thin grating is such that
$\lambda_{0}/n_{g}<P<2\lambda_{0}/n_{g}$ in order to have a propagating first-
order mode inside the slab. Figure1(b) gives a reference example taking a
Germanium/Aluminum structure surrounded by air. It clearly shows that, at
given period, the slit width $w$ should be narrow if we seek spectral
selectivity, and allows tuning the transmission amplitude and bandwidth.
However, it may be to the detriment of the efficiency because of optical
absorption in the metal.

A way to analytically calculate the transmission of a single filter has been
given in a previous work Leperchec2011 , based on a simplified modal method,
for highly reflecting metallic-stripe grating. We give in the Appendix an even
more general expression with arbitrary optical indices
$n_{i},n_{g},n_{d},n_{t}$ of the incidence, intra-slit, waveguide, and
transmission regions, respectively. This formula gives reliable quantitative
results compared to RCWA simulations, as shown in Fig.1(b). Moreover, by
tending to the perfect metal, new key relations were found and presented
hereafter, in particular the explicit resonance condition of such filters.
While $kh_{met}<<1$ ($k=2\pi/\lambda$), at $\theta=0$, the TM-transmission
reads:

$Tr\approx\frac{n_{i}}{n_{t}}\left|\frac{2}{\cos(kn_{g}h_{g})\left[\left(1+\frac{n_{i}}{n_{t}}\right)+\frac{2}{n_{t}}\sum_{m>0}A_{m}\right]-j\sin(kn_{g}h_{g})\left[\left(\frac{n_{g}}{n_{t}}+\frac{n_{i}}{n_{g}}\right)+\frac{2}{n_{g}}\sum_{m>0}A_{m}\right]}\right|^{2}$
(1)

where $j$ is such that $j^{2}=-1$ and

$A_{m}(\lambda)=\sec^{2}(m\pi\frac{w}{P})\left[\frac{n_{i}}{\beta_{m}^{i}}+\frac{n_{g}}{\beta_{m}^{g}}\left(\frac{1-jY_{m}\tan
X_{m}}{Y_{m}-j\tan X_{m}}\right)\right]$ (2)

with $X_{m}=kn_{g}h_{g}\beta_{m}^{g}$ and
$\beta_{m}^{g}=\sqrt{1-(m\lambda/(n_{g}P))^{2}}$, m being a relative integer
(for $\beta_{m}^{t}$, replace $n_{g}$ by $n_{t}$). The terms $k\beta_{m}$ are
wave vectors along the z-direction.
$Y_{m}=(n_{g}\beta_{m}^{t})/(n_{t}\beta_{m}^{g})$ are Floquet modal impedance
ratios. We remark that taking $w=P$, all $A_{m}=0$ and we retrieve the
transmission of a homogeneous slab of index $n_{g}$. When $\lambda<n_{g}P$
($\lambda<6.4\mu$m in the above example), the first-order mode inside the
waveguide is propagating, i.e. $\beta_{1}^{g}>0$ and $A_{1}$ becomes purely
imaginary. We thus find the first transmission resonance when the imaginary
part of the denominator in (1) is zero, say:

$\left(\frac{n_{g}}{n_{t}}+\frac{n_{i}}{n_{g}}\right)\tan(kn_{g}h_{g})\approx\frac{2}{n_{t}}\Im(A_{1})$
(3)

by willingly neglecting high-order diffraction terms ($m\geq 2$). The
resonance condition is actually more precise by taking at least
$\Im(A_{1}+A_{2})$, which gives $\lambda_{0}=4.04\mu$m when $w=600$nm in our
example. Finally, given (3), the transmission peak is:

$Tr_{\lambda=\lambda_{0}}\approx\frac{n_{i}}{n_{t}}\left|\frac{2\cos(k_{0}n_{g}h_{g})}{1+\frac{n_{i}}{n_{t}}\cos^{2}(k_{0}kn_{g}h_{g})+\frac{n_{i}n_{t}}{n_{g}^{2}}\sin^{2}(k_{0}n_{g}h_{g})}\right|^{2}$
(4)

Contrary to what could be thought, the aperture ratio $w/P$ has no impact on
this amplitude (which is maximized when $\lambda_{0}/(2n_{g})\sim h_{g}$).
This is also true if the metal were simply not absorbing (purely negative
permittivity), as shown in Fig.1(b) (blue curve). $w/P$ has actually an impact
on (2) and therofore on the dispersion relation (3): when it diminishes, it
makes $\lambda_{0}$ shift to lower frequencies, whereas the resonance quality
factor increases. But, as the transmission peak becomes sharper, it is more
attenuated in a real metal case because of optical absorption.

Since we cannot overcome the natural trade-off between a high transmission
peak (large ratio $w/P$) and a strong light rejection out of the resonance, we
investigate now a cascaded structure where two identical filters are closely
superposed (see Fig.1(a)), with fairly large opening widths $w$. The spacer
layer separating both filters is characterized by a low optical index in order
prevent or minimize diffraction perturbation ($n_{s}\leq 1.5$, by using e.g.
$SiO_{2}$, $MgF_{2}$, $YF_{3}$) and by a sub-wavelength thickness $h_{s}$
(further explanation given afterwards).

Figure 2(a) shows the transmission profile on the wide interval
$[2.5;5.5]\mu$m, resulting from the cascaded structure based on our reference
filters with $w=600$nm and $h_{met}$=20nm only, separated by an air layer of
thickness $h_{s}=600$nm. The spectral response is now much cleaner and
selective: light rejection is immediately reinforced, without deteriorating
the maximum transmission efficiency. The resonance bandwidth is a bit
tightened and a slight shift of $\lambda_{0}$ is observed, from $4.07$ to
$4.01\mu$m. The responses in logarithmic scale (Fig.2(b)) show that,
surprisingly, rejection is better than the square of the rejection of isolated
filters (see at $\lambda=3$ or $4.5\mu$m, for instance), whereas the resonance
peak is barely lowered ($90.6\%$ versus $93.8\%$ with the single filter).
Thus, the far-field does not derives from a simple convolution of isolated
filters, because of some near-field interaction (we come back to this point
later).

Figure 2: (top) (a) Transmission spectra of two cascaded same Ge/Alu filters
as sketched in Fig.1(a), compared to that the single filter (surrounded by
air), with $P=1.6\mu$m, $w=600$nm, $h_{g}=550$nm, $h_{met}=20$nm, $n_{s}=1$
and $h_{s}=600$nm. (b) Same transmission in logarithmic scale. (bottom)
Spatial maps of the normalized magnetic and electric fields at the resonance
wavelength $\lambda_{0}$: (c) $|H_{y}|$, (d) $|E_{x}|$, and (e)
$|D_{z}|=|\varepsilon(x,z)E_{z}|$. Here, incident light comes from $z<0$, and
the slit aperture is centered at $x=800$nm. (f) Cartography of $H_{y}$ when a
transmission null occurs.

Fig.2(c,d,e) gives maps of the electromagnetic fields ($H_{y}$, $E_{x}$,
$D_{z}$) at the resonance, which are consistent with the excitation of the
first-order hybrid waveguide mode, and shows highly symmetrical patterns, with
light enhancements inside the waveguides and/or within slits (the latter
present an effective dipole momentum, supported by $E_{x}$, with hot spots at
the corners, which is intrinsic to the field scattered by such metallic
apertures in TM-polarization Leperchec2015 ).

It is worth noticing that the resonance peak appears within a spectral window
bounded by two transmission nulls, independent of $w$ and occurring when :

$\tan(kn_{g}\beta_{1}^{g}h_{g})+j\frac{Z+Y_{1}(\beta_{1}^{g}/n_{g})}{\beta_{1}^{g}/n_{g}+ZY_{1}}=0$
(5)

where $Y_{m}$ is already given above, and $Z=1/\sqrt{\varepsilon_{metal}}$
(see Appendix). For instance, (5) gives the wavelengths $\lambda=5382$ and
$2816$nm: the filter exposed to incident light simply behaves as a good
reflector with almost no absorption (no transmission and flat reflectivity
$\sim 95\%$), and then blocks any energy transfer to the following filtering
stage, as shown in Fig.2(f).

The influence of the spacer layer thickness ($h_{s}$) on the far-field
response is illustrated in Fig.3. When filters come closer, an additional part
of energy may be transferred through the surrounding evanescent field, that is
why the transmission peak is greater than the simple square $0.938^{2}$ (see
the isolated filter). Indeed, the resonating first order Rayleigh wave
typically decreases as $exp(-kn_{s}|\beta^{s}_{1}z|)=e^{-|z|/\delta}$ in the
spacer layer, just outside the high-index waveguide, with
$\delta=1/(n_{s}k_{0}\sqrt{(\lambda_{0}/(n_{s}P))^{2}-1})\approx 280$nm in our
example. The best trade-off is when $h_{s}\sim 2\delta$. Out of the resonance,
the coupling plays an unfavorable role for the passage of light, hence a
rejection better than expected. Below $300$nm separation, the near-field
interaction becomes strong and creates a resonance splitting, with a symmetric
and anti-symmetric coupling of filters. Here, antisymmetric coupling means
that $\Re(E_{x}(x,z))$ changes of sign on both sides of the median plane
separating both filters.

Figure 3: Influence of the separation distance $h_{s}$. $\delta$ is the
characteristic damping length of the evanescent 1rst order diffraction mode
outside the waveguides.

The orientation of each grating with respect to the other (gratings above or
under the waveguide, in an aligned stacking or a staggered arrangement) makes
no significant difference on the optical response with an ideal planar
structure. However, realistic technological means may lead to a slightly
perturbed geometry, for instance by using an auto-aligned technological
process. Fig.4(a) gives such an example with a CMOS-compatible technology,
taking amorphous silicon-based materials. The diagram of the transmission as a
function of $(\lambda,\theta)$ is given in log scale in Fig.4(b). The maximum
efficiency ($Tr_{max}=88.8\%$) and rejection out of the resonance are still
very good. If the grating of the upper filter 2 was put on the air side, the
rejection would be a bit deteriorated at shorter wavelengths, and the
transmission curve would lose in symmetry (not shown). Besides, the angular
robustness (transmission level and inertia of the resonance wavelength) is
typically $\theta=0\pm 15^{o}$, which is equivalent to that measured on single
filters Leperchec2011 . At higher angles, an other waveguide mode (existing
even in the perfect metal case) is excited around $\lambda=5\mu$m and is quite
absorbing (contrary to the resonance at $\lambda=4\mu$m). For short
wavelengths, new diffraction orders become propagating inside the high-index
slab.

Geometrical differences between both filters might be willingly introduced to
deform/tailor the transmission profile. The cascaded filter can stand alone in
the air, or be bonded just above a detector. In the latter case, a low-index
layer, whose thickness is adapted in the sense of constructive interference
(quarter-wave plate or an odd multiple), must be inserted between the filter 1
and the detector underneath. If the detector is a matrix of pixels (e.g. with
pixels of $20\mu$m in diameter), it may be useful to conveniently etch
(trench) the cascaded stack along the pixels perimeter to avoid possible
optical cross-talk through the waveguide. Since we are limited by the pixel
area, a dozen of periods is necessary to get the wanted effect as we deal with
a resonance that is not too sharp (out of the scope of this paper).

1D structures make it possible to discriminate the light polarization of the
filtered signals. The conception of a polarization-independent 2D-structure
(square patch grating) is accompanied with additional difficulties. This is
pointed out through Fig.4(c): whereas the high resonance peak of cascaded
2D-filters is still present at $\lambda=4\mu$m, secondary transmission peaks
appear on both sides, which limit the spectral window where strong light
rejection is wanted. These peaks actually correspond to waveguide modes we
would get in $TE$-polarization with a 1D filter. These modes are also quite
absorbing ones, contrary to the TM-resonance we work with (see reflectivity).
A way to limit their excitation strength is to reduce $w$ (with some reduction
of the period to retrieve the same $\lambda_{0}$), as the fundamental TE-slit
mode is strongly evanescent when $\lambda\gg 2w$. Also, 2D metallic patches
with rounded corners should be avoided in order to minimize the appearance of
such perturbations Leperchec2011 . By reducing $w$, the main TM-resonance
obviously loses in amplitude. Nevertheless, rejection levels remain strong: in
the case $w=0.4\mu$m, $P=1.5\mu$m, $Tr\approx 10^{-3}$ when $\lambda\approx
3.5$ or $4.5\mu$m, and $Tr<10^{-5}$ at $\lambda=5.5\mu$m. Another way to get
an equivalent 2D-filter without TE-modes could be to resort to a polarization
converter upstream to cascaded 1D-filters. Optical stages able to efficiently
convert an entire signal into a given linear polarization state have already
been demonstrated Kim2012 .

Figure 4: (a) Possible perturbed filter architecture resulting from a CMOS
technological process with a simple conformal deposition of successive layers
once the first grating is made. (b) Corresponding diagram of the transmission
in function of $(\lambda,\theta)$. All materials permittivities taken from
Palik  ($n_{g}\approx 3.84$ and $n_{s}=n_{t}\approx 1.4$). (c) 2D cascaded
filters (square patch gratings), taking $n_{g}=4$, $n_{met}=n_{Alu}$,
$n_{i}=n_{t}=n_{s}=1$, $h_{g}=550$nm, $h_{met}=40$nm, and $h_{s}=600$nm (to be
compared to spectra of Fig.2). With $n_{s}\sim 1.45$, the resonance peak would
be slightly shifted towards $4.1\mu$m.

In conclusion, thanks to rigorous simulations and with the support of a modal
approach, we have shown how two near-field coupled GMR filters allows
obtaining a bandpass device with a transmission selectivity better than that
expected by a simple convolution of isolated filters. Ideal and realistic
examples have been especially given in the mid-infrared where such filtering
properties (transmission peak $\sim 90\%$, strong rejection out of the
resonance over a large spectral range, good angular tolerance) are demanded
for hyper-spectral imaging and gas detection applications. Those structures,
whose overall thickness remains smaller than the half-wavelength, are compact
and rather easy to do on a technological level.

##  I Appendix: Analytical calculation of the transmission

We give hereafter the main mathematical elements allowing to get the
analytical formulae of the zero-order transmission though a grating-mode-
resonant filter based on a subwavelength metallic slit grating, in TM-
polarization.

Whereas the well-known Rigorous Coupled Wave Analysis is based on Fourier
expansions of the electromagnetic fields and of the spatial distribution of
the electrical permittivities, in every region of the system, we may resort to
a true modal expansion of the field along the one-dimensional grating region,
with some practical simplifications regarding boundary conditions Wirgin1984 ;
Barbara2002 . By this way, we get a quite precise analytical formula giving
the transmission in function of all opto-geometrical parameters, expandable to
the perfect metal case. First, we only consider good (real) metals in the
spectral range of interest, so that relevant surface impedance conditions can
be applied along metallic interfaces. Tangential fields obey the relation
$\textbf{E}^{tan}\propto Z(\textbf{n}\times\textbf{H}^{tan})$ where n is a
vector normal to the metal surface, and $Z=1/\sqrt{\varepsilon}$,
$\varepsilon$ being the metal relative permittivity ($|Z|\ll 1$). Secondly, we
assume the vertical walls of the slit apertures as perfectly reflecting so
that the field eigen modes are perfectly known (surface impedance could also
be applied within the slits Locbilher1993 ; Leperchec2008  but the analytical
calculation becomes a bit less explicit).

The incident light is transverse-magnetic (TM) polarized (magnetic field
parallel to the slits, i.e. along the $y$-direction) and comes from the region
$z<0$. $n_{i},n_{g},n_{t}$ are optical indices of the incidence, waveguide,
and transmission regions, respectively. For sake of generality, we also
consider the dielectric medium filling the grating slits as possibly
different, with an index $n_{d}$. Then, the magnetic field transmitted outside
the waveguide is written as a Rayleigh expansion:

$H_{y}^{t}(x,z\geq
h_{g}+h_{met})=\sum_{m=-\infty}^{+\infty}T_{m}e^{jkn_{t}(\gamma^{t}_{m}x+\beta^{t}_{m}(z-h_{g}-h_{met}))}$
(6)

with $k=2\pi/\lambda$, $\beta_{m}^{t}=\sqrt{1-(\gamma_{m}^{t})^{2}}$, where
each $\gamma_{m}^{t}$ respects the conservation of the tangential wave vectors
$n_{i}\gamma_{m}^{i}=n_{g}\gamma_{m}^{g}=n_{t}\gamma_{m}^{t}$, with
$n_{i}\gamma_{m}^{i}=n_{i}\sin\theta+m\lambda/P$, $\theta$ being the incidence
angle.

The total transmission in TM-polarization will be given by:

$Tr=\Re\left(\sum_{m}\frac{n_{i}\beta_{m}^{t}}{n_{t}\beta_{m}^{i}}|T_{m}|^{2}\right)$
(7)

As we use a sub-wavelength grating period ($n_{t}P<\lambda$) all over our
spectral range of study, transmitted waves of high order are evanescent
($\beta_{m\neq 0}^{t}$ imaginary) and will not contribute to far-field
transmission.

Applying then the matching conditions of the fields at every interface and
using the one-mode approximation ($w<<\lambda/n_{d}$) for the field inside the
slit $h$-thick, we get (after long algebra):

$T_{m}=\frac{(w/P)S_{m}}{(\beta_{m}^{g}/n_{g})q_{m}+Zp_{m}}\Gamma(h)$ (8)

where

$\displaystyle S_{m}=\sec(kn_{i}\gamma^{i}_{m}w/2)$ (9) $\displaystyle
X_{m}=kn_{g}h_{g}\beta_{m}^{g}$ (10) $\displaystyle
Y_{m}=(n_{g}\beta_{m}^{t})/(n_{t}\beta_{m}^{g})$ (11) $\displaystyle
p_{m}=(1-jY_{n}\tan(X_{m}))\cos X_{m}$ (12) $\displaystyle
q_{m}=(Y_{n}-j\tan(X_{m}))\cos X_{m}$ (13)

And

$\Gamma(h)=\frac{[(n_{d}^{-1}-Z)(1-D_{g}^{+})+(n_{d}^{-1}+Z)(1+D_{g}^{-})]V}{(1+D_{i}^{-})(1+D_{g}^{-})\phi^{-1}-(1-D_{i}^{+})(1-D_{g}^{+})\phi}$
(14)

where $\phi=e^{jkn_{d}h}$, and

$\displaystyle D_{g}^{\pm}=(\frac{1}{n_{d}}\pm
Z)\frac{w}{P}\sum_{m=-\infty}^{+\infty}\frac{S_{m}^{2}}{(q_{m}/p_{m})(\beta_{m}^{g}/n_{g})+Z}$
(15) $\displaystyle D_{i}^{\pm}=(\frac{1}{n_{d}}\pm
Z)\frac{w}{P}\sum_{m=-\infty}^{+\infty}\frac{S_{m}^{2}}{(\beta_{m}^{i}/n_{i})+Z}$
(16) $\displaystyle V=\frac{2S_{0}\cos\theta}{\cos\theta+n_{d}Z}$ (17)

$V$ is an excitation term related to the incident plane wave. Subsequently,
while $n_{t}P<\lambda$:

$Tr=\frac{n_{i}\beta_{0}^{t}}{n_{t}\cos\theta}\left|\frac{(w/P)S_{0}}{q_{0}(\beta_{0}^{g}/n_{g})+Zp_{0}}\Gamma(h)\right|^{2}$
(18)

This analytical formula can rightly describe the behavior of the system even
when Fabry-Perot-type resonances may occur within metallic apertures (i.e.
when $n_{c}h\sim\lambda/2$). When $h_{g}=0$ and $n_{i}=n_{t}=1$, we get the
transmission through a sub-wavelength slit grating made in a metallic screen
and surrounded by air. In this work, we only focus us on cases of very thin
gratings.

For real metals ($Z\neq 0$ and $|Z|\ll 1$), (18) is quantitatively reliable
while $h$ remains greater than the metal skin depth $\delta$ (with $h>2\delta$
preferably), as the symmetric or anti-symmetric coupling of surface waves
through metallic parts is not described by this model. Also, the cavity width
$w$ must not be extremely small because surface-plasmons coupling between
vertical walls would become stronger and would modify the effective index of
the waveguide mode inside the slit, with more optical absorption than
predicted (mainly for deep slits and/or at visible frequencies).

###  I.1 Perfect metal case

Now, let $Z=0$. Eq.(14) becomes:

$\Gamma(h)=\frac{V/n_{d}}{\cos(kn_{i}h)(D_{i}+D_{g})-j\sin(kn_{i}h)(1+D_{i}D_{g})}$
(19)

If we assume a normal incidence ($\theta=0$), then $S_{0}=1$, $V=2$,
$\beta_{0}^{g}=\beta_{0}^{t}$=1, and:

$Tr=\frac{n_{i}}{n_{t}}\left|\frac{2wn_{g}/(n_{d}Pq_{0})}{[\cos(kn_{i}h)(D_{i}+D_{g})-j\sin(kn_{i}h)(1+D_{i}D_{g})]}\right|^{2}$
(20)

with

$\displaystyle
D_{g}=\frac{1}{n_{d}}\frac{w}{P}\sum_{m=-\infty}^{+\infty}\frac{n_{g}}{\beta_{m}^{g}}\frac{p_{m}}{q_{m}}\sec^{2}(m\frac{\pi
w}{P})$ (21) $\displaystyle
D_{i}=\frac{1}{n_{d}}\frac{w}{P}\sum_{m=-\infty}^{+\infty}\frac{n_{i}}{\beta_{m}^{i}}\sec^{2}(m\frac{\pi
w}{P})$ (22)

When $kn_{d}h<<1$, $\cos(kn_{d}h)\approx 1$, $\sin(kn_{d}h)<<1$, and we get:

$Tr\approx\frac{n_{i}}{n_{t}}\left|\frac{2}{(q_{0}/n_{g})\sum_{m=-\infty}^{+\infty}A_{m}}\right|^{2}$
(23)

where $A_{m}$ are given by (2). Moreover, $A_{m}=A_{-m}$, $Y_{0}=n_{g}/n_{t}$
and we have the following sequel

$\displaystyle A_{0}=n_{i}+p_{0}(n_{g}/q_{0}),$ (24) $\displaystyle
q_{0}/n_{g}=\cos(kn_{g}h_{g})/n_{t}-j\sin(kn_{g}h_{g})/n_{g},$ (25)
$\displaystyle p_{0}=\cos(kn_{g}h_{g})-j\sin(kn_{g}h_{g})(n_{g}/n_{t}).$ (26)

Finally, for the perfect metal case, at normal incidence, and $kh<<1$, we find
the zero-order transmission given by Eq.(1). By taking $w=P$ (no grating), all
$A_{m}=0$, we retrieve the transmission response of a simple homogeneous slab
of index $n_{g}$, with maximum $Tr=1$ when $\sin(kn_{g}h_{g})=0$ and minimum
transmission when $\cos(kn_{g}h_{g})=0$.

###  I.2 Resonance conditions

The terms $A_{m}$ (2) are complex quantities whose moduli decrease as
$m\rightarrow\infty$. Resonances may occur when the imaginary part of the
denominator in (1) cancels, say:

$\left(\frac{n_{g}}{n_{t}}+\frac{n_{i}}{n_{g}}\right)\tan(kn_{g}h_{g})\approx\frac{2}{n_{t}}\Im\left(\sum_{m>0}A_{m}\right)$
(27)

In particular, once the first waveguide mode becomes propagating inside the
slab, i.e. $\beta^{g}_{1}>0$, $A_{1}$ becomes entirely imaginary, so we can
directly get an approximated resonance condition for the strong TM-
transmission peak we are studying in this work, by neglecting high orders
terms $m\geq 2$. However, neglecting high orders is only relevant for rather
large values of $w$, i.e. when the factors $\sec^{2}(m\pi w/P)_{m>1}$ in (2)
have weak amplitudes. As $w$ decreases, we have to take into account more and
more $A_{m}$ in the resonance condition hereabove. Note that other resonances
can also appear at oblique incidence, related to anti symmetrical surface
waveguide modes, but this will not be discussed in the present paper.

###  I.3 Transmission nulls

Let us come back to the real metal case (18). We can see that transmission may
cancel when (15) diverges ($\Gamma(h)\rightarrow 0$), i.e. when the following
condition is fulfilled:

$q_{m}(\beta_{m}^{g}/n_{g})+Zp_{m}\approx 0$ (28)

which reduces to $j\tan(X_{m})=Y_{m}$ in the perfect metal case. Zeros of
transmission might also occur when $\beta_{m}^{i}/n_{i}+Z=0$ (divergence of
$D_{i}$), corresponding to classical surface-plasmon excitations between the
grating and the incidence region, but such resonances do not happen here as
$P<<\lambda$.

## References

  * (1) S. Tibuleac, R. Magnusson, Opt. Lett. 26, 584 (2001).”Narrow-linewidth bandpass filters with diffractive thin-film layers”.
  * (2) Donald K. Jacob, Steven C. Dunn, and M. G. Moharam, ”Flat-top narrow-band spectral response obtained from cascaded resonant grating reflection filters”, Appl. Opt. 41, 1241 (2002).
  * (3) K. Kawanishi, A. Shimatani,K. Jin Lee, J. Inoue, S. Ura, and R. Magnusson, Opt. Lett. 45, 312 (2020). ”Cross-stacking of guided-mode resonance gratings for polarization-independent flat-top filtering”.
  * (4) J.M. Bendickson, E.N. Glytsis, T.K. Gaylord, and D.L. Brundrett, J. Opt. Soc. Am. A 18 1912 (2001), ”Guided-mode resonant subwavelength gratings: Effects of finite beams and finite gratings”.
  * (5) J. Le Perchec, R. Espiau de Lamaestre, M. Brun, N. Rochat, O. Gravrand, G. Badano, J. Hazart, and S. Nicoletti, ”High rejection bandpass optical filters based on sub-wavelength metal patch arrays,” Opt. Express 19, 15720-15731 (2011).
  * (6) A. F. Kaplan, T. Xu, and L. J. Guo, Appl. Phys. Lett. 99, 143111 (2011), ”High efficiency resonance-based spectrum filters with tunable transmission bandwidth fabricated using nanoimprint lithography”.
  * (7) H. Young Song, S. Kim, and R. Magnusson, Opt. Epress 26, 23555 (2009), ”Tunable guided-mode resonances in coupled gratings”.
  * (8) R.J. Luebbers, and B.A. Munk, IEEE Trans. Antennas Propag. 26, 536 (1978), ”Some effects of dielectric loading on periodic slot arrays”.
  * (9) Chul-Soon Im and Sang-Shin Lee, 25 16038 (2017) Opt. Express, ”Highly efficient and angle-tolerant midinfrared filter based on a cascaded etalon resonator”.
  * (10) L. Frey, L. Masarotto, L. El Melhaoui, S. Verrun, S. Minoret, G. Rodriguez, A. André, F. Ritton, and P. Parrein, Opt. Lett. 43, 1355-1358 (2018), ”High-performance silver-dielectric interference filters for RGBIR imaging”
  * (11) H. B. Chan, Z. Marcet, K. Woo, and D. B. Tanner, D. W. Carr, J. E. Bower, R. A. Cirelli, E. Ferry, F. Klemens, J. Miner, C. S. Pai, and J. A. Taylor, Opt. Lett. 31, 516 (2006), ”Optical transmission through double-layer metallic subwavelength slit arrays”.
  * (12) E. Sakat, S. Héron, P. Bouchon, G. Vincent, F. Pardo, S. Collin, J.-L. Pelouard, and R. Haïdar, Opt. Lett. 38, 425 (2013), ”Metal–dielectric bi-atomic structure for angular-tolerant spectral filtering”.
  * (13) J. Le Perchec, Phys. Rev B 92, 155404 (2015), ”Origin of the Norton-type wave scattered by a subwavelength metallic slit”.
  * (14) E.D. Palik, Handbook of Optical Constants of Solids, Academic Press (1985).
  * (15) J. Kim, R. K. Komanduri, K. F. Lawler, D. J. Kekas, and M. J. Escuti, Appl. Opt. 51, 4852 (2012). ”Efficient and monolithic polarization conversion system based on a polarization grating”.
  * (16) A. Wirgin, T. Lopez-Rios, ”Can surface-enhanced Raman scattering be caused by waveguide resonances?,” Opt. Commun. 48, 416 (1984); 49, 455 (1984).
  * (17) A. Barbara, P. Quémerais, E. Bustarret, and T. Lopez-Rios, ”Optical transmission through subwavelength metallic gratings,” Phys. Rev. B 66, 161403 (2002).
  * (18) H. Lochbihler and R. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32, 3459–3465 (1993).
  * (19) J. Le Perchec, P. Quémerais, and A. Barbara, ”On light addressing within subwavelength metallic gratings,” J. Light. Tech. 26, 638-642 (2008).


#  Complex surfaces and null conformal Killing vector fields

J. Davidov, G. Grantcharov, O. Mushkarov

###### Abstract

We study the relation between the existence of null conformal Killing vector
fields and existence of compatible complex and para-hypercomplex structures on
a pseudo-Riemannian manifold with metric of signature $(2,2)$. We establish
first the topological types of pseudo-Hermitian surfaces admitting a nowhere
vanishing null vector field. Then we show that a pair of orthogonal, pointwise
linearly independent, null, conformal Killing vector fields defines a para-
hyperhermitian structure and use this fact for a classification of the smooth
compact four-manifolds admitting such a pair of vector fields. We also provide
examples of neutral metrics with two orthogonal, pointwise linearly
independent, null Killing vector fields on most of these manifolds.

##  1 Introduction

The notion of self-duality for metrics of split signature $(2,2)$ called also
neutral metrics, has been studied for a long time and related to various
fields like integrable systems [24] and superstring with $N=2$ supersymmetry
[17, 18]. Many results can directly be extended from the positive definite to
the split signature case but there are also some significant differences. Most
of the similarities , as well as the differences, are based on the fact that
the metrics of signature $(2,2)$ are related to the split-quaternions

$\mathbb{H}^{\prime}=\\{q=a+bi+cs+dt\in\mathbb{R}^{4}|s^{2}=t^{2}=1=-i^{2},t=is,is=-si\\}$

in the same way as the positive definite metrics to the quaternions . For
example, the similarities include a twistor space construction [3] and a
spinor approach [9], while the differences come from the fact that the self-
duality equations in split signature are not elliptic but ultra-hyperbolic.
Most of the research has been focused on local study in view of the fact that
for ultra-hyperbolic equations the global properties are harder to understand.

There has also been some interest in compact 4-manifolds admitting neutral
metrics and compatible complex or para-hypercomplex structures since
topological information like the Kodaira classification of compact complex
surfaces allows one to study global properties. Important results in this
direction are the classifications of compact pseudo-Kähler Einstein and para-
hyperkähler surfaces obtained by Petean [21] and Kamada [14], respectively.
These structures appear in [18] as models for superstring theory with N=2
supersymmetry and [11] in relation to deformation spaces of harmonic maps from
Riemann surfaces into Lie groups. Note also that such structures have been
used recently by B. Klingler [15] in his proof of the Chern conjecture for
affine manifolds.

In our previous papers [6, 7] we initiated the study of compact para-
hyperhermitian surfaces, which are the neutral analog of the hyperhermitian
4-manifolds. These surfaces are anti-self-dual as in the positive definite
case, but in contrast to the well-known classification of compact
hyperhermitian surfaces [4], there are many more compact examples of para-
hyperhermitian surfaces. In our study, we noticed also that any two pointwise
independent, null, orthogonal and parallel vector fields on a 4-manifold with
a neutral metric define a para-hyperkäher structure and conversely, any
compact para-hyperkäher surface admits two vector fields with the above
properties [14]. In this note, we weaken these conditions and study the
neutral Hermitian surfaces admitting non-vanishing null conformal Killing
vector fields. We observe first that the existence of a non-vanishing null
vector field on a compact neutral Hermitian surface leads to a topological
restriction which implies a rough classification of these surfaces. Next, we
show that the existence of two point-wise independent, null and orthogonal
conformal Killing vector fields leads to the existence of a para-
hyperhermitian structure. These surfaces were studied in [6, 7] and here we
consider the problem for existence of two Killing vector fields having the
above properties with respect to the corresponding metrics.

Now we describe shortly the content of the paper. In Section 2, some known
facts about almost para-hyperhermitian structures are collected and in Lemma 2
they are characterized in terms of 2-forms. Then in Section 3 we study the
complex surfaces admitting neutral Hermitian metrics with non-vanishing null
vector fields. After proving some useful local properties of these surfaces,
in Theorem 1, we provide a classification in the compact case. In Theorem 2 we
notice a property analogous to the positive definite case - if a neutral
Hermitian surface admits complete null Killing vector field which is not real
holomorphic, then the metric is anti-self-dual. Section 4 is devoted to the
study of the neutral 4-manifolds admitting two orthogonal and null conformal
Killing vector fields which are linearly independent at each point. We show
that any such a manifold admits a para-hypercomplex structure which is
compatible with the given metric (Theorem 3) and provide a classification of
these manifolds in the compact case (Theorem 4). In Theorem 4, we also exhibit
explicit examples of neutral metrics having two Killing vector fields with the
required properties on most of the possible smooth $4$-manifolds.

Acknowledgments.The first and the third named authors are partially supported
by the Bulgarian National Science Fund, Ministryof Education and Science of
Bulgaria under contract KP-06-N52/3.

The second named author is supported by the Simons Foundation Collaboration
grant number 853269. He wants to express his gratitude to the Institute of
Mathematics and Informatics of the Bulgarian Academy of Sciences, Sofia for
the warm hospitality during his visits to work on the project.

##  2 Para-hyperhermitian structures

Recall that an almost para-hypercomplex structure on a smooth manifold $M$ is
a triple $(I,S,T)$ of anti-commuting endomorphisms of its tangent bundle $TM$
with $I^{2}=-Id$ and $S^{2}=T^{2}=Id$, $T=IS$. Such a structure is called
para-hypercomplex if $I,S,T$ satisfy the integrability condition
$N_{I}=N_{S}=N_{T}=0$, where

$N_{A}(X,Y)=A^{2}[X,Y]+[AX,AY]-A[AX,Y]-A[X,AY]$

is the Nijenhuis tensor of $A=I,S,T$. Note that if two of the structures
$I,S,T$ are integrable, the third one is also integrable [14].

If there exists a pseudo-Riemannian metric $g$ for which the endomorphisms
$I,S,T$ are skew-symmetric, $(g,I,S,T)$ is called an almost para-
hyperhermitian structure. Such a metric is necessarily neutral, i.e., has
split signature. We also say that the metric $g$ is compatible with the almost
para-hypercomplex structure $(I,S,T)$ as well as that $(I,S,T)$ is compatible
with $g$. Every almost para-hypercomplex structure on a 4-manifold locally
admits a compatible metric but a globally defined one may not exist. More
precisely, such a structure determines a conformal class of compatible metrics
up to a double cover of $M$ [6]. Examples of almost para-hypercomplex
structures that do not admit compatible metrics are given in [6, 7].

Let $(g,I,S,T)$ be an almost para-hyperhermitian structure on a $4$-manifold.
Then one can define three fundamental $2$-forms $\Omega_{i}$, $i=1,2,3$,
setting

$\Omega_{1}(X,Y)=g(IX,Y),\hskip 7.22743pt\Omega_{2}(X,Y)=g(SX,Y),\hskip
7.22743pt\Omega_{3}(X,Y)=g(TX,Y).$

Note that the form $\Omega=\Omega_{2}+i\Omega_{3}$ is of type $(2,0)$ with
respect to $I$. As in the definite case, the corresponding Lee forms are
defined by

$\theta_{1}=-\delta\Omega_{1}\circ I,\hskip
7.22743pt\theta_{2}=-\delta\Omega_{2}\circ S,\hskip
7.22743pt\theta_{3}=-\delta\Omega_{3}\circ T,$

where $\delta$ is the co-differential with respect to $g$.

It is well-known [4, 10, 13, 14] that the structures $I,S,T$ are intergrable
if and only if $\theta_{1}=\theta_{2}=\theta_{3}$. Thus, for a para-
hyperhermitian structure, we have just one Lee form $\theta$; it satisfies the
identities

$d\Omega_{i}=\theta\wedge\Omega_{i},~{}i=1,2,3.$

When additionally the three 2-forms $\Omega_{i}$ are closed, i.e. $\theta=0$,
the para-hyperhermitian structure is para-hyperkähler ( also hypersympectic or
neutral hyperkähler). When $d\theta=0$ the structure is locally conformally
para-hyperkähler. We note that, in dimension 4, the para-hyperhermitian
metrics are self-dual and the para-hyperkähler metrics are self-dual and
Ricci-flat [14]. It is well-known that every hyperhermitian structure on a
4-dimensional compact manifold is locally conformally hyperkähler [4], but it
has been shown in [7, Theorem 9], that this is not true in the indefinite
case.

An almost para-hyperhermitian structure on a $4$-manifold can be characterized
by means of the forms $\Omega_{i}$ in the following way [14].

###### Proposition 1

Every almost para-hyperhermitian structure on a $4$-manifold is uniquely
determined by three non-degenerate $2$-forms
$(\Omega_{1},\Omega_{2},\Omega_{3})$ such that

$-\Omega_{1}^{2}=\Omega_{2}^{2}=\Omega_{3}^{2},\quad\Omega_{l}\wedge\Omega_{m}=0,\,1\leq
l\neq m\leq 3.$

This structure is para-hyperhermitian if and only if there is a $1$-form
$\theta$ such that $d\Omega_{l}=\theta\wedge\Omega_{l}$, $l=1,2,3$,

Note that given $2$-forms $(\Omega_{1},\Omega_{2},\Omega_{3})$ with the above
properties the endomorphisms $I,S,T$ and the metric $g$ of the almost para-
hyperhermitian structure they determine are defined by

$\Omega_{3}(IX,Y)=\Omega_{2}(X,Y),\quad\Omega_{1}(SX,Y)=\Omega_{3}(X,Y),\quad\Omega_{2}(TX,Y)=-\Omega_{1}(X,Y),$

and

$g(X,Y)=\Omega_{1}(X,IY),\quad X,Y\in TM.$

One can show that

$\Omega_{1}(X,IY)=-\Omega_{2}(X,SY)=-\Omega_{3}(X,TY),$

so $\Omega_{1}$, $\Omega_{2}$, $\Omega_{3}$ are the fundamental $2$-forms of
$I$, $S$, $T$.

We use later on the following characterization of the almost para-
hyperhermitian structures with a given almost complex structure.

###### Lemma 1

An almost complex $4$-manifold $(M,I)$ admits an almost para-hyperhermitian
structure with almost complex structure $I$ if and only if there is a
$(2,0)$-form $\Omega$ and a non-degenerate real $2$-form $\omega$ such that
$\Omega\wedge\overline{\Omega}=-2\omega^{2}$ and $\Omega\wedge\omega=0$. If
$\Omega$ and $\omega$ satisfy these conditions, $\omega$, $Re\,\Omega$,
$Im\,\Omega$ are the fundamental 2-forms of the almost para-hyperhermitian
structure.

Proof. Suppose we are given $2$-forms $\Omega$ and $\omega$ satisfying the
conditions of the lemma. Set $\Omega_{1}=\omega$, $\Omega_{2}=Re\,\Omega$,
$\Omega_{3}=Im\,\Omega$. We have $\Omega\wedge\Omega=0$ since $\Omega$ is a
$(2,0)$-form and $dim\,M=4$. This implies $\Omega_{2}^{2}=\Omega_{3}^{2}$.
Thus,
$2\Omega_{2}^{2}=2\Omega_{3}^{2}=\Omega\wedge\overline{\Omega}=-2\omega^{2}\neq
0$. Hence the $2$-forms $\Omega_{1},\Omega_{2},\Omega_{3}$ are non-degenerate.
The identities $\Omega\wedge\Omega=0$ and
$\Omega\wedge\overline{\Omega}=-2\omega^{2}$ give
$-\Omega_{1}^{2}=\Omega_{2}^{2}=\Omega_{3}^{2}$ and
$\Omega_{2}\wedge\Omega_{3}=0$. Moreover, the identity $\Omega\wedge\omega=0$
is equivalent to $\Omega_{2}\wedge\Omega_{1}=\Omega_{3}\wedge\Omega_{1}=0$. By
Proposition 1, there exists an almost para-hyperhermitian structure
$(g,I^{\prime},S,T)$ such that $\Omega_{1},\Omega_{2},\Omega_{3}$ are the
respective fundamental 2-forms. The form $\Omega=\Omega_{2}+i\Omega_{3}$ is of
type $(2,0)$ with respect to both $I$ and $I^{\prime}$. It follows that
$\Omega_{2}(X,Y)=\Omega_{3}(X,IY)$ and
$\Omega_{2}(X,Y)=\Omega_{3}(X,I^{\prime}Y)$. Hence $I=I^{\prime}$ since the
form $\Omega_{3}(X,Y)=g(X,TY)$ is non-degenerate.

The rest part of the lemma is obvious.

$\Box$

Remark. If $\Omega$ and $\omega$ satisfy the conditions of Lemma 1, then
$\omega$ is of type $(1,1)$ w.r.t. $I$ since $\omega(X,Y)=g(IX,Y)$ for a
neutral metric $g$ compatible with the almost complex structure $I$.

##  3 Neutral Hermitian surfaces with a null vector field

We begin with preliminary algebraic considerations.

Let $M$ be an oriented $4$-manifold with a neutral metric $g$ and let
$(E_{1},E_{2},E_{3},E_{4})$ be an oriented orthonormal frame such that
$g(E_{1},E_{1})=g(E_{2},E_{2})=1$ and $g(E_{3},E_{3})=g(E_{4},E_{4})=-1$. The
Hodge star operator of $g$ acts as an involution on the bundle of $2$-vectors
$\Lambda^{2}TM$ and is given by

$\ast E_{1}\wedge E_{2}=E_{3}\wedge E_{4},\quad\ast E_{1}\wedge
E_{3}=E_{2}\wedge E_{4},\quad\ast E_{1}\wedge E_{4}=-E_{2}\wedge E_{3}.$

Let $\Lambda^{2}_{\pm}TM$ be the subbundles of $\Lambda^{2}TM$ corresponding
the the eigenvalues $\pm 1$ of the Hodge operator. Set

$\begin{array}[]{c}s_{1}^{+}=E_{1}\wedge E_{2}+E_{3}\wedge E_{4},\quad\quad
s_{1}^{-}=E_{1}\wedge E_{2}-E_{3}\wedge E_{4},\\\\[6.0pt]
s_{2}^{+}=E_{1}\wedge E_{3}-E_{4}\wedge E_{2},\quad\quad s_{2}^{-}=E_{1}\wedge
E_{3}+E_{4}\wedge E_{2},\\\\[6.0pt] s_{3}^{+}=E_{1}\wedge E_{4}-E_{2}\wedge
E_{3},\quad\quad s_{3}^{-}=E_{1}\wedge E_{4}+E_{2}\wedge E_{3}.\end{array}$
(1)

Then $\\{s_{1}^{\pm},s_{2}^{\pm},s_{3}^{\pm}\\}$ is an orthonormal frame of
$\Lambda^{2}_{\pm}TM$.

Recall that a $2$-plane in $TM$ is called self-dual or $\alpha$-plane if there
is a basis $a,b$ of the plane such that $\ast a\wedge b=a\wedge b$. This
condition does not depend on the choice of the basis $a,b$. Similarly, a
$2$-plane in $TM$ is called anti-self-dual, or $\beta$-plane if there is a
basis $a,b$ with $\ast a\wedge b=-a\wedge b$.

If $\Pi$ is an isotropic $2$-plane in a tangent space $T_{p}M$, one can find a
complementary isotropic $2$-plane $\Pi^{\prime}$ and bases $a_{1},a_{2}$ of
$\Pi$ and $a_{1}^{\prime},a_{2}^{\prime}$ of $\Pi^{\prime}$ such that
$g(a_{i},a_{j}^{\prime})=\frac{1}{2}\delta_{ij}$. Then
$e_{1}=a_{1}+a_{1}^{\prime}$, $e_{2}=a_{2}+a_{2}^{\prime}$,
$e_{3}=a_{1}-a_{1}^{\prime}$, $e_{4}=a_{2}-a_{2}^{\prime}$ is an orthonormal
basis of $T_{p}M$ such that $||e_{1}||^{2}=||e_{2}||^{2}=1$,
$||e_{3}||^{2}=||e_{4}||^{2}=-1$. We have $4a_{1}\wedge
a_{2}=(e_{1}+e_{3})\wedge(e_{2}+e_{4})=(e_{1}\wedge e_{2}+e_{3}\wedge
e_{4})+(e_{1}\wedge e_{4}-e_{2}\wedge e_{3})$. This shows that $\Pi$ is either
an $\alpha$ or a $\beta$-plane depending on whether the basis
$(e_{1},e_{2},e_{3},e_{4})$ determines the given orientation of $T_{p}M$ or
the opposite one.

###### Lemma 2

[7] Let $M$ be a 4-manifold with a neutral metric $g$ and let $X$ and $Y$ be
orthogonal null vector fields which are linearly independent at every point of
$M$. Then the triple $(g,X,Y)$ determines an orientation and a unique
orientation and $g$-compatible almost complex structure $J$ on $M$ such that
$JX=Y$.

In the proof of this lemma, it has been shown that in a neighbourhood of every
point of $M$, there exist vector fields $Z,T$ such that:

  1. $(i)$

$(X,Y,Z,T)$ is a local frame of the tangent bundle $TM$;

  2. $(ii)$

$g(X,Z)=1$, $g(X,T)=0$, $g(Y,Z)=0$, $g(Y,T)=1$.

Also, the orientation determined by $(X,Y,Z,T)$ does not depend on the choice
of the vector fields $Z,T$ and we shall say that it is determined by the
triple $(g,X,Y)$. Set $a=g(Z,Z)$, $b=g(T,T)$, $c=g(Z,T)$ and

$\begin{array}[]{lll}E_{1}=\displaystyle{\frac{1-a}{2}}X+Z,\quad
E_{2}=\displaystyle{\frac{1-b}{2}}Y+T-cX,\\\\[8.0pt]
E_{3}=-\displaystyle{\frac{1+a}{2}}X+Z,\quad
E_{4}=-\displaystyle{\frac{1+b}{2}}Y+T-cX.\end{array}$ (2)

Then $(E_{1},E_{2},E_{3},E_{4})$ is an orthogonal frame, positively oriented
with respect to the orientation determined by $(g,X,Y)$ and such that
$g(E_{1},E_{1})=g(E_{2},E_{2})=1$, $g(E_{3},E_{3})=g(E_{4},E_{4})=-1$. The
almost complex structure $J$ for which $JE_{1}=E_{2}$, $JE_{3}=E_{4}$ has the
required properties [7].

Now, by (2)

$X\wedge Y=(E_{1}-E_{3})\wedge(E_{2}-E_{4})=s_{1}^{+}-s_{3}^{+}.$

Thus, $span\\{X,Y\\}$ is an $\alpha$-plane. Set

$U=X-bY+2T.$

This vector field is null and orthogonal to $X$. Moreover

$X\wedge U=-bX\wedge
Y+X\wedge(E_{2}+E_{4}+bY+2cX)=(E_{1}-E_{3})\wedge(E_{2}+E_{4})=s_{1}^{-}+s_{3}^{-}.$

Hence $span\\{X,U\\}$ is an $\beta$-plane. Next, $JU$ is a null vector field
and

$\begin{array}[]{c}JX\wedge JU=Y\wedge(bX+2JT)=bY\wedge
X+Y\wedge(JE_{2}+JE_{4}-bX+2cY)\\\\[6.0pt]
=-(E_{2}-E_{4})\wedge(E_{1}+E_{3})=s_{1}^{-}-s_{3}^{-}.\end{array}$

Thus $span\\{JX,JU\\}$ is also a $\beta$-plane. The vectors $X,JX=Y,U,JU$ are
linearly independent at each point since

$X\wedge U\wedge JX\wedge JU=-4E_{1}\wedge E_{2}\wedge E_{3}\wedge E_{4}.$

We sum up these observations in the following

###### Lemma 3

In the notation above:

$(i)$ The vectors $\\{X,JX=Y,U,JU\\}$ are linearly independent at every point.

$(ii)$ $g(X,U)=0,\quad g(JX,U)=2,\quad g(X,X)=g(U,U)=0$.

$(iii)$ The null $2$-planes $span\\{X,U\\}$ and $span\\{JX,JU\\}$ are
$\beta$-planes.

$(iv)$ Every $\beta$-plane containing $X$ coincides with $span\\{X,U\\}$.

$(v)$ Every $\beta$-plane containing $JX$ coincides with $span\\{JX,JU\\}$.

$(vi)$ The plane $span\\{X,JX\\}$ is an $\alpha$-plane and every
$\alpha$-plane containing $X$ coincides with $span\\{X,JX\\}$.

Proof. We have only to prove $(iv)$, $(v)$ and $(vi)$. Suppose that $\Pi$ is a
$2$-plane containing $X$, and let $\widetilde{X}=\gamma X+\lambda Y+\mu U+\nu
JU$ be a vector such that $\\{X,\widetilde{X}\\}$ is a basis of $\Pi$. We have
by (2)

$\begin{array}[]{c}X\wedge JU=X\wedge(Y-bX+2JT)=(1+2c)(X\wedge
Y)+X\wedge(JE_{2}+JE_{4})\\\\[6.0pt]
=(1+2c)(s_{1}^{+}-s_{3}^{+})-(E_{1}-E_{3})\wedge(E_{1}+E_{3})=(1+2c)(s_{1}^{+}-s_{3}^{+})-(s_{2}^{+}+s_{2}^{-}).\end{array}$

Then

$X\wedge\widetilde{X}=[\lambda+\nu(1+2c)](s_{1}^{+}-s_{3}^{+})-\nu
s_{2}^{+}+\mu(s_{1}^{-}+s_{3}^{-})-\nu s_{2}^{-}.$

It follows that $\Pi$ is a $\beta$-plane if and only if $\lambda=\nu=0$, i.e.
$\widetilde{X}=\gamma X+\mu U$ where $\mu\neq 0$. Hence $\Pi$ is a
$\beta$-plane if and only if $\Pi=span\\{X,U\\}$. Also, $\Pi$ is an
$\alpha$-plane exactly when $\mu=\nu=0$, i.e. $\widetilde{X}=\gamma X+\lambda
Y$ where $\lambda\neq 0$ and $Y=JX$. This proves $(vi)$.

The proof of $(v)$ is similar.

$\Box$

The following lemma shows that a fixed null-vector in a complex vector space
with a (2,2) signature Hermitian metric determines a unique para-
hyperhermitian structure.

###### Lemma 4

Let $V$ be a $4$-dimensional real vector space with a neutral scalar product
$g$, and let $I$ be a compatible complex structure. Then for any null vector
$X\neq 0$ in $V$ there is a unique skew-symmetric endomorphism $S$ of $V$ such
that $S^{2}=Id$, $IS=-SI$, $SX=X$.

Proof. Note first that the complex structure determined by the triple
$(g,X,Y=IX)$ as in Lemma 2 coincides with $I$.

Now, we apply Lemma 3, and define $S$ to be the identity on the unique null
$\beta$-plane containing $X$, and minus the identity on the unique null
$\beta$-plane containing $IX$. Then it is easy to check that $S$ has the
required properties.

Conversely, suppose that $S$ is an endomorphism with these properties. Then
$S$ is an involution different from $\pm Id$. Denote the eigenspace of $S$
corresponding to the eigenvalues $+1$ and $-1$ by ${V^{+}}$ and $V^{-}$.
Clearly, the complex structure $I$ is an isomorphism of ${V^{+}}$ onto $V^{-}$
since it anti-commutes with $S$. In particular, $dimV^{+}=dimV^{-}=2$. Note
also that the spaces ${V^{+}}$ and $V^{-}$ are isotropic since $S$ is skew-
symmetric. Hence $V^{+}$ is either an $\alpha$ or a $\beta$-plane, and
similarly for $V^{-}$. The space $V^{+}$ does not contain $IX$, hence it is a
$\beta$-plane by Lemma 3. Similarly, $V^{-}$ is also a $\beta$-plane. This
proves the uniqueness.

$\Box$

Using the above algebraic observations, we provide a classification of compact
neutral Hermitian surfaces admitting a nowhere vanishing null vector field.

###### Theorem 1

Let $(M,g,I)$ be a compact neutral Hermitian surface with nowhere vanishing
null vector field $X$. Then the complex surface $(M,I)$ is one of the
following:

(i) a complex torus;

(ii) a primary Kodaira surface;

(iii) a minimal properly elliptic surface of odd first Betti number;

(iv) an Inoue surface of type $S^{0}$ or $S^{\pm}$ without curves;

(v) a Hopf surface;

Conversely, each of the smooth manifolds underlying the complex surfaces (i) -
(iii), an Inoue surface of type $S^{+}$ or a primary Hopf surface carries a
neutral Hermitian metric with a nowhere vanishing null vector field.

Proof. Setting $T=IS$, we get an almost para-hyperhermitian structure
$(g,I,S,T)$ on $M$. Let $\Omega_{S}$ and $\Omega_{T}$ be the corresponding
fundamental $2$-forms defined by $\Omega_{S}(X,Y)=g(SX,Y)$ and similarly for
$\Omega_{T}$. Then the $2$-form $\Omega_{S}+i\Omega_{T}$ is non-vanishing and
of type $(2,0)$ with respect to $I$ which implies the vanishing of the first
Chern class of $(M,I)$. Now the classification of compact complex surfaces
with vanishing first Chern class provides the types $(i)-(v)$ in the theorem
and also the $K3$\- surfaces, see, e.g., [7, Theorem 8]. But we exclude the
latter from the list since they do not admit non-vanishing global vector
fields due to the fact that their Euler characteristic does not vanish .

That the manifolds underlying the surfaces (i) - (iii), an Inoue surface of
type $S^{+}$ or a primary Hopf surface admit neutral Hermitian metrics with a
null vector field follows from Theorem 4 proved in the next section.

$\Box$

Remark. If $X$ is Killing and $\mathcal{L}_{X}I=0$, then $\mathcal{L}_{X}S=0$,
where $S$ is the endomorphism of $TM$ defined by means of $X$ as in Lemma 4,
hence $X$ preserves the almost para-hypercomplex structure. Indeed, notice
first that the $(+1)$-eigenbundle $S^{+}$ of $S$ is integrable, see [5], [8].
Hence there is a vector field $Y$ in a neighbourhood of each point of $M$ such
that $span\\{X,Y\\}=S^{+}$ and $[X,Y]=0$. Then $span\\{IX,IY\\}$ is the
$(-1)$-eigenspace of the involution $S$. Clearly $[X,SX]=S[X,X]=0$ and
$[X,SY]=S[X,Y]=0$, so $({\mathcal{L}}_{X}S)(Z)=0$ for $Z\in S^{+}$. By
assumption $\mathcal{L}_{X}I=0$, i.e. $[X,IZ]=I[X,Z]$ for every $Z$, so
$[X,IX]=I[X,X]=0$. Then $[X,SIX]=-[X,IX]=0$ and $[S,SIY]=-[X,IY]=-I[X,Y]=0$,
hence $({\mathcal{L}}_{X}S)(Z)=0$ for $Z\in S^{-}$.

It is well known that on a compact Kähler manifold every Killing vector field
is real holomorphic. On general Hermitian manifolds existence of a Killing
vector field which is not real holomorphic, has a significant consequences for
the curvature. We adapt here the relevant considerations for the split-
signature 4-dimensional case.

###### Theorem 2

Let $(M,g,I)$ be a neutral Hermitian surface and $X$ a complete null Killing
vector field, which is not real holomorphic. Then the metric $g$ is anti-self-
dual.

Proof. Let ${\varphi}_{t}$ be the global flow of isometries of $M$ generated
by $X$. Then $I_{t}=(\varphi_{t})_{\ast}\circ I\circ(\varphi_{-t})_{\ast}$,
$t\in{\mathbb{R}}$, are compatible complex structures on $M$. Suppose that the
Lie derivative ${\mathcal{L}}_{X}I\neq 0$ at a point $p\in M$. Then
${\mathcal{L}}_{X}I\neq 0$ in a neighborhood $U$ of $p$. For a point $x\in U$
we can take three different real numbers $t_{1},t_{2},t_{3}$ closed enough to
$0$ such that $I_{t_{i}}(x)\neq\pm I_{t_{j}}(x)$. It follows that the self-
dual component $W_{+}$ of the Weyl tensor vanishes at the point $x$. This fact
has been proved in [22] in the positive-definite case, but the same proof
using spinors works in the split signature case too. Hence $W_{+}$ vanishes on
$U$ and by the proof of [20, Corollary 1.6], $W_{+}$ vanishes identically.
Note that this result is based on [20, Proposition 1.3] which, although stated
in the positive definite case, holds true in the split signature case with the
same proof replacing the usual twistor space by the so-called hyperbolic
twistor space [3]).

$\Box$

Remark. As is well-known [19], on a complete pseudo-Riemannian manifold, every
Killing vector field is complete.

##  4  Neutral $4$-manifolds with two null conformal Killing vector fields

Recall that a vector field $K$ on a pseudo-Riemannian manifold $(M,g)$ is
called conformal Killing if if it satisfies

$g(\nabla_{A}K,B)+g(\nabla_{B}K,A)=\frac{2}{dim\,M}div(K)g(A,B),\quad A,B\in
TM,$

where $\nabla$ is the Levi-Civita connection.

The following lemma is an extension of a result of Dunajsky-West [8, Lemma 1]
and D. Calderbank [5, Lemma 4.1].

###### Lemma 5

Let $K$ and $L$ be two orthogonal null vector fields that are linearly
independent at every point.

$(i)$ If $K$ is conformal Killing, then $\nabla_{A}B\in span\\{K,L\\}$ for
$A,B\in span\\{K,L\\}$.

Suppose that both $K$ and $L$ are conformal Killing vector fields. Then:

$(ii)$ If $K$ and $L$ commute, the distribution $span\\{K,L\\}$ is parallel
and $\nabla_{L}K=\nabla_{K}L=0$.

$(iii)$ If the distribution $span\\{K,L\\}$ is parallel, the vector fields $K$
and $L$ are Killing and commute.

Proof. $(i)$ At every point, the null-plane $\Pi=span\\{K,L\\}$ is a
Lagrangian subspace, so $\Pi=\Pi^{\perp}$. Hence $A\in\Pi$ if and only if
$A\perp\Pi$. We have

$g(\nabla_{K}K,K)=g(\nabla_{K}L,L)=g(\nabla_{L}K,K)=g(\nabla_{L}L,L)=0$

because $g(K,K)=g(L,L)=0$. Moreover,

$0=g(K,\nabla_{L}K)=-g(\nabla_{K}K,L)=g(K,\nabla_{K}L)$

since $K$ is conformal Killing and orthogonal to $L$. Thus, $\nabla_{K}K$ and
$\nabla_{K}L$ are orthogonal to $span\\{K,L\\}$. Moreover, we have

$g(\nabla_{L}K,L)=-g(\nabla_{L}K,L),$

hence

$g(\nabla_{L}L,K)=-g(L,\nabla_{L}K)=0.$

Therefore $\nabla_{L}K$ and $\nabla_{L}L$ are orthogonal to $span\\{K,L\\}$.

$(ii)$ For every tangent vector $Z$

$g(\nabla_{Z}K,L)=-g(K,\nabla_{Z}L)=g(\nabla_{K}L,Z)=g(\nabla_{L}K,Z)=-g(\nabla_{Z}K,L)$

Hence

$g(\nabla_{Z}K,L)=0.$

Therefore $\nabla_{Z}K\in span\\{K,L\\}$. Similarly, $\nabla_{Z}L\in
span\\{K,L\\}$. Moreover,

$g(\nabla_{K}L,Z)=-g(\nabla_{Z}L,K)=g(L,\nabla_{Z}K)=0.$

Hence $\nabla_{K}L=0$ and $\nabla_{L}K=0$.

$(iii)$ If the distribution $span\\{K,L\\}$ is parallel, then for every $Z\in
TM$

$g(\nabla_{L}K,Z)=-g(\nabla_{Z}K,L)=0.$

Thus $\nabla_{L}K=0$. Similarly $\nabla_{K}L=0$. Hence $[K,L]=0$. Also,

$\begin{array}[]{l}0=g(\nabla_{Z}K,L)=-g(\nabla_{Z}L,K)=-\frac{1}{2}div(L)g(Z,K)+g(\nabla_{K}L,Z)\\\\[6.0pt]
=-\frac{1}{2}div(L)g(Z,K)+\frac{1}{2}div(K)g(Z,L)-g(\nabla_{Z}K,L)\\\\[6.0pt]
=-\frac{1}{2}div(L)g(Z,K)+\frac{1}{2}div(K)g(Z,L).\end{array}$

Since $K,L$ are linearly independent at each point, it follows that
$div(L)=div(K)=0$. Thus, $K$ and $L$ are Killing.

$\Box$

It follows from Lemmas 2 \- 4 that a neutral 4-manifold with two orthogonal,
pointwise linearly independent, null vector fields admits an almost para-
hyperhermitian structure. When the vector fields are conformal Killing we have
a stronger result.

###### Theorem 3

Let $(M,g)$ be a neutral 4-manifold with two orthogonal, pointwise linearly
independent, null conformal Killing vector fields $X$ and $Y$. Then $M$ admits
a para-hyperhermitian structure $(g,I,S,T)$, where $I$ is the structure
determined by $\\{g,X,Y\\}$ as in Lemma 2.

Moreover, if the vector fields $X$ and $Y$ commute, then they are real
holomorphic and Killing.

Proof. Consider $M$ with the orientation determined by $(g,X,Y)$ and let $I$
be the compatible almost complex structure from Lemma 2, so $Y=IX$. By Lemma
3, in the vicinity of every point, there is a null vector field $U$ orthogonal
to $X$ such that the vector fields $X,IX,U,IU$ have the properties $(i)$ \-
$(v)$ of this lemma. Then the $(1,0)$-vector fields $X-iIX$ and $U-iIU$
constitute a frame for $(1,0)$-vectors. Hence, for the integrability of $I$,
it is enough to check that $g(\nabla_{Z}V,W)=0$ for any choice of $Z,V,W$
among $2X^{1,0}=X-iIX$ and $2U^{1,0}=U-iIU$. Now, since (1,0)-vectors are
isotropic, for any (1,0)-vector fields $Z,U,W$, we have the following:

1\. $g(\nabla_{Z}V,W)=-g(V,\nabla_{Z}W)$, and, as a consequence,
$g(\nabla_{Z}V,V)=0$.

Since $X,IX$ are conformal Killing, it easy to see that

2\. $g(\nabla_{Z}(X-iIX),V)=-g(\nabla_{V}(X-iIX),Z)$, and, as a consequence,
$g(\nabla_{Z}(X-iIX),Z)=0$.

From here we get the property $g(\nabla_{Z}V,W)=0$ for any choice of $Z,V,W$
among $2X^{1,0}=X-iIX$ and $2U^{1,0}=U-iIU$ by a case by case argument. The
details are as follows.

a) $g(\nabla_{U^{1,0}}X^{1,0},X^{1,0})=0$ (by 1.) and
$g(\nabla_{U^{1,0}}X^{1,0},U^{1,0})=0$ (by 2.)

b) $g(\nabla_{X^{1,0}}U^{1,0},X^{1,0})=0$ (by 2.) and
$g(\nabla_{X^{1,0}}U^{1,0},U^{1,0})=0$ (by 1.)

c) $g(\nabla_{X^{1,0}}X^{1,0},X^{1,0})=0$ (by 1.) and
$g(\nabla_{X^{1,0}}X^{1,0},U^{1,0})=0$ (by 1. and then b) )

d) $g(\nabla_{U^{1,0}}U^{1,0},U^{1,0})=0$ (by 1.) and
$g(\nabla_{U^{1,0}}U^{1,0},X^{1,0})=0$ (by 1. and a) )

Now, we define an endomorphism $S$ of $TM$ as $S=Id$ on $span\\{X,U\\}$ and
$S=-Id$ on $span\\{IX,IU\\}$. Then Lemma 5 implies that the distributions
$span\\{X,U\\}$ and $span\\{IX=Y,IU\\}$ are integrable. It follows that $S$ is
integrable. At the end, we set $T=IS$. Then $T$ is integrable since $I$ and
$S$ are integrable.

Suppose that the vector fields $X$ and $Y=IX$ commute. Then, by Lemma 5, the
vector fields $X$ and $Y$ are Killing. If $\nabla$ is the Levi-Civita
connection of $(M,g)$, we have $\nabla_{X}IX=\nabla_{IX}X$. It follows that

$g(\nabla_{X}Z,V)=g(\nabla_{IX}Z,V)=0$

for $Z,V\in span\\{X,IX\\}$. Moreover

$g(\nabla_{X}IX,U)=-g(\nabla_{U}IX,X)=g(\nabla_{U}X,IX)$

and

$g(\nabla_{X}IX,U)=g(\nabla_{IX}X,U)=-g(\nabla_{U}X,IX)+div(X)$

Hence

$g(\nabla_{X}IX,U)=\frac{1}{2}div(X)$

Then

$\begin{array}[]{c}g([U,X],IX)=g(\nabla_{U}X,IX)-g(\nabla_{X}U,IX)=-g(\nabla_{U}IX,X)-g(\nabla_{X}U,IX)\\\\[6.0pt]
=2g(\nabla_{X}IX,U)=div(X)\end{array}$

since the vector field $IX$ is conformal Killing, $g(U,X)=0$ and
$g(U,IX)=const$. Moreover

$\begin{array}[]{c}g(I[IU,X),IX)=g(\nabla_{IU}X,X)-g(\nabla_{X}IU),X)=g(\nabla_{X},IU)\\\\[6.0pt]
=-g(\nabla_{IU}X,X)+\frac{1}{2}div(X)g(IU,X)=-div(X)\end{array}$

Therefore

$g([U,X]+I[IU,X],IX)=0.$

Next,

$g([U,X],X)=g(\nabla_{U}X,X)+g(U,\nabla_{X}X)=g(\nabla_{U}X,X)-g(\nabla_{U}X,X)=0.$

Also,

$\begin{array}[]{c}g(I[IU,X])=-g(\nabla_{IU}X,IX)+g(\nabla_{X}IU,IX)=g(\nabla_{IX}X,IU)-g(\nabla_{X}IX,IU)\\\\[6.0pt]
=g([X,IX],IU)=0.\end{array}$

Hence

$g([U,X]+I[IU,X],X)=0.$

It is easy to check that

$g([U,X]+I[IU,X],U)=g([U,X]+I[IU,X],IU)=0.$

It follows that ${\mathcal{L}}_{X}I={\mathcal{L}}_{IX}I=0$. $\Box$

Now, we determine the compact 4-manifolds admitting neutral metrics with two
independent null conformal Killing vector fields.

###### Theorem 4

(A). Let $(M,g)$ be a compact neutral 4-manifold with two orthogonal null
conformal Killing vector fields $X$ and $Y$ which are linearly independent at
each point. Then $M$ is the underlying smooth manifold of one of the following
complex surfaces:

(i) a complex torus

(ii) a primary Kodaira surface

(iii) a minimal properly elliptic surface of odd first Betti number.

(iv) an Inoue surface of type $S^{0}$ or $S^{\pm}$ without curves

(v) a Hopf surface

If in addition the vector fields $X$ and $Y$ commute, then in $(iv)$ only
$S^{+}$ is allowed.

(B). Conversely, the smooth manifolds underlying the complex surfaces $(i)$ \-
$(iii)$, an Inoue surface of type $S^{+}$ or a primary Hopf surface admit
neutral metrics with two orthogonal null Killing vector fields which are
linearly independent at each point.

Proof. (A). By Theorem 3, $M$ carries a para-hyperhermitian structure
$(g,I,S,T)$ such that $Y=IX$. It follows from Theorem 1, applied to the
compact Hermitian surface $(M,I,g)$, that $M$ is one of the manifolds $(i)$ \-
$(v)$.

The claim for the commuting Killing fields follows from Theorem 3 and the fact
that in $(iv)$ only the Inoue surfaces of type $S^{+}$ admit holomorphic
vector fields [12].

(B). The proof for the cases $(i)$ \- $(iii)$ follows from the proof of the
second part of [7, Theorem 7]. Here are the details.

$(i)$. Let $M$ be the quotient of $\mathbb{C}^{2}$ by a lattice
$<a_{1},a_{2},a_{3},a_{4}>$. Take a smooth function $\alpha$ on ${\mathbb{C}}$
such that $\alpha(z+a_{3})=\alpha(z)$, $\alpha(z+a_{4})=\alpha(z)$. Then the
metric

$\widetilde{g}=\alpha\,dzd\overline{z}+2Re(dzd\overline{w})$

on $\mathbb{C}^{2}$ descends to a a split signature Ricci flat, Kähler metric
$g$ on the torus $M$ [21]. For this metric, the global vector field $W$ on $M$
given in the standard local coordinates $(z,w)$ on the torus by
$\displaystyle{\frac{\partial}{\partial w}}$ is holomorphic, parallel and
null. Hence if $K=ReW$, the vector fields $K$ and $IK=ImW$ have the properties
stated in lemma.

$(ii)$. A primary Kodaira surface [16] can be obtained in the following way.
Consider the affine transformations $\varphi_{k}(z,w)$ of ${\mathbb{C}}^{2}$
given by

$\varphi_{k}(z,w)=(z+a_{k},w+\overline{a}_{k}z+b_{k}),$

where $a_{k}$, $b_{k}$, $k=1,2,3,4$, are complex numbers such that

$a_{1}=a_{2}=0,\quad Im(a_{3}{\overline{a}}_{4})=b_{1}\neq 0,\quad b_{2}\neq
0.$

They generate a group $G$ of affine transformations acting freely and properly
discontinuously on ${\mathbb{C}}^{2}$ and $M$ is the quotient space
${\mathbb{C}}^{2}/G$ for a suitable choice of $a_{k}$ and $b_{k}$ [16, p.786].
Taking into account the identities

$\varphi_{k\ast}(\frac{\partial}{\partial z})=\frac{\partial}{\partial
z}+\overline{a_{k}}\frac{\partial}{\partial
w},\quad\varphi_{k\ast}(\frac{\partial}{\partial w})=\frac{\partial}{\partial
w}$

we see that every holomorphic vector field on $M$ is proportional to the
vector field $W$ given in the local coordinates $(z,w)$ as
$\displaystyle{\frac{\partial}{\partial w}}$. As in [21], set
$\alpha(z)=f(z)-\gamma\,z-\overline{\gamma}\,\overline{z}$, where $f(z)$ is a
smooth function on ${\mathbb{C}}$ satisfying the identities $f(z+a_{3})=f(z)$,
$f(z+a_{4})=f(z)$. Then the metric
$\widetilde{g}=\alpha\,dzd\overline{z}+2Re(dzd\overline{w})$ on
$\mathbb{C}^{2}$ descends to a split signature Kähler, Ricci flat metric $g$
on $M$ for which the holomorphic vector field $W$ is parallel and null (the
metric $g$ is flat if $f\equiv const$). Then the vector fields $K=ReW$ and
$IK=ImW$ have the required properties.

$(iii)$. Suppose $M$ is a minimal properly elliptic surface of odd first Betti
number. Let $\widetilde{SL(2,R)}$ be the universal covering group of
$SL(2,R)$. By [23, Theorem 7.4], $M$ is the quotient of
$\widetilde{SL(2,R)}\times{\mathbb{R}}$ by a co-compact lattice. Let
$A^{\prime},B^{\prime},C^{\prime}$ be the frame of left-invariant vector
fields on $SL(2,R)$ determined by the matrices
$A^{\prime}=1/2\left(\begin{array}[]{ll}0&1\\\ -1&0\end{array}\right)$,
$B^{\prime}=1/2\left(\begin{array}[]{ll}0&1\\\ 1&0\end{array}\right)$,
$C^{\prime}=1/2\left(\begin{array}[]{ll}1&0\\\ 0&-1\end{array}\right)$. Denote
the lifts of $A^{\prime},B^{\prime},C^{\prime}$ to vector fields on
$\widetilde{SL(2,R)}$ by $A,B,C$. These vector fields satisfy the following
commutation relations

$[A,B]=C,\quad[B,C]=-A,\quad[C,A]=B.$

Hence, if $(\alpha,\beta,\gamma)$ is the dual frame of $(A,B,C)$,

$d\alpha=\beta\wedge\gamma,\quad d\beta=\alpha\wedge\gamma,\quad
d\gamma=-\alpha\wedge\beta.$

Set $V=\displaystyle{\frac{d}{dx}}$, the standard vector field on
${\mathbb{R}}$, and let $I$ be the invariant almost complex structure on
$\widetilde{SL(2,R)}\times{\mathbb{R}}$ given by $IV=C,IA=B$. If $\theta$ is
dual to $V$, $d\theta=0$ and the basis of $(1,0)$-forms for the invariant
almost complex structure $I$ is given by $\theta+i\alpha,\beta+i\gamma$.
Notice also that

$d(\beta+i\gamma)=-i\alpha\wedge(\beta+i\gamma),\quad
d(\theta+i\alpha)=\beta\wedge(\beta+i\gamma),$

so $I$ is integrable. Now consider the form

$\Omega=(\theta+i\alpha)\wedge(\beta+i\gamma)$

This is a $(2,0)$-form and from above it follows
$d\Omega=-\theta\wedge\Omega$. Note that
$\omega=\theta\wedge\alpha-\beta\wedge\gamma$ satisfies
$d\omega=-\theta\wedge\omega$, so the pair $(\Omega,\omega)$ defines a para-
hyperhermitian structure by Lemma 1, which is left invariant (and locally
conformally para-hyperkähler). In fact the metric defined by $\omega$ and $I$
is the bi-invariant metric $g=\theta^{2}+\alpha^{2}-\beta^{2}-\gamma^{2}$. One
can directly check that

$\mathcal{L}_{V}g=\mathcal{L}_{A}g=\mathcal{L}_{B}g=\mathcal{L}_{C}g=0$

which also follows from the fact that $V,A,B,C$ are left invariant. Then the
neutral Hermitian structure $(g,I)$ and the vector fields $K=V+B$ and $IK=A+C$
descend to $M$ giving a neutral Hermitian structure with two orthogonal null
Killing vector fields which are linearly independent at each point.

$(iv)$. In order to discuss the Inoue surfaces $S^{+}$, we recall first their
definition [12]. Take a matrix $N=(n_{ij})\in GL(2,{\mathbb{Z}})$ with
$det\,N=1$ having two real eigenvalues $\alpha>1$ and $\alpha^{-1}$. Note that
$\alpha$ is a irrational number. Choose real eigenvectors $(a_{1},a_{2})$ and
$(b_{1},b_{2})$ corresponding to $\alpha$ and $\alpha^{-1}$, respectively.
Take integers $p,q,r$, $r\neq 0$ and a complex number $t$. Let $(c_{1},c_{2})$
be the solution of the equation

$\varepsilon(c_{1},c_{2})=(c_{1},c_{2})N^{tr}+(e_{1},e_{2})+\frac{b_{1}a_{2}-b_{2}a_{1}}{r}(p,q)$
(3)

where $N^{tr}$ is the transpose matrix of $N$ and

$e_{k}=\frac{1}{2}n_{k1}(n_{k1}-1)a_{1}b_{1}+\frac{1}{2}n_{k2}(n_{k2}-1)a_{2}b_{2}+n_{k1}n_{k2}b_{1}a_{2},\quad
k=1,2.$

Let $G^{+}=G^{+}_{N,p,q,r;t}$ be the group generated by the following
automorphisms of ${\mathbb{C}}\times{\bf H}$, ${\bf H}$ being the upper half-
plane:

$\begin{array}[]{l}g_{0}(z,w)=(\varepsilon
z+\frac{1}{2}(1+\varepsilon)t,\alpha w)\\\\[6.0pt]
g_{k}(z,w)=(z+b_{k}w+c_{k},w+a_{k}),\,k=1,2,\\\\[6.0pt]
g_{3}(z,w)=(z+\displaystyle{\frac{b_{1}a_{2}-b_{2}a_{1}}{r}},w).\end{array}$
(4)

The group $G^{+}$ acts properly discontinuously and without fixed points in
view of (3) and the fact that $(a_{1},b_{1})$ and $(a_{2},b_{2})$ are linearly
independent vectors. The quotient $S^{+}_{N,p,q,r;t}=({\mathbb{C}}\times{\bf
H})/G^{+}$ is a compact complex surface, known as an Inoue surface of type
$S^{+}$.

Set $t_{2}=Im\,t$ and

$\alpha_{1}=dx-\frac{1}{v}(y-t_{2}\frac{\ln
v}{\ln\alpha})du,\quad\alpha_{2}=dy-\frac{1}{v}(y-t_{2}\frac{\ln
v}{\ln\alpha})dv,\quad\alpha_{3}=\frac{du}{v},\quad\alpha_{4}=\frac{dv}{v},$
(5)

where $z=x+iy$ and $w=u+iv$. These forms are linearly independent and
invariant under the action of the group $G^{+}$. Note also that
$\alpha_{1}+i\alpha_{2}$ and $\alpha_{2}+i\alpha_{4}$ constitute a frame of
$(1,0)$-forms. Moreover,

$d\alpha_{1}=\alpha_{3}\wedge\alpha_{2}-\frac{t_{2}}{\ln\alpha}\alpha_{3}\wedge\alpha_{4},\quad
d\alpha_{2}=\alpha_{4}\wedge\alpha_{2},\quad
d\alpha_{3}=\alpha_{3}\wedge\alpha_{4},\quad d\alpha_{4}=0.$

Set

$\Omega_{1}=\alpha_{1}\wedge\alpha_{3}+\alpha_{2}\wedge\alpha_{4},\quad\Omega_{2}=\alpha_{1}\wedge\alpha_{3}-\alpha_{2}\wedge\alpha_{4},\quad\Omega_{3}=\alpha_{1}\wedge\alpha_{4}+\alpha_{2}\wedge\alpha_{3}.$

Then

$-\Omega_{1}^{2}=\Omega_{2}^{2}=\Omega_{3}^{2}=2\alpha_{1}\wedge\alpha_{2}\wedge\alpha_{3}\wedge\alpha_{4},\quad\Omega_{l}\wedge\Omega_{m}=0,\,1\leq
l\neq m\leq 3,\quad d\Omega_{l}=-\alpha_{4}\wedge\Omega_{l}.$

Therefore, by Proposition 1, $\Omega_{1},\Omega_{2},\Omega_{3}$ define an
$G^{+}$-invariant para-hyperhermitian structure on ${\mathbb{C}}\times{\bf H}$
with the neutral metric $g(X,Y)=\Omega_{1}(X,JY)$ which is locally conformally
para-hyperkähler since its Lie form $\theta=-\alpha_{4}$ is closed. This
structure descends to a para-hyperhermitian structure on the Inoue surface
$S^{+}$.

The frame $(X_{1},...,X_{4})$ of $G^{+}$-invariant vector fields dual to the
frame $(\alpha_{1},...,\alpha_{4})$ is given by

$\displaystyle{X_{1}=\frac{\partial}{\partial
x},\>X_{2}=\frac{\partial}{\partial y},\>X_{3}=(y-t_{2}\frac{\ln
v}{\ln\alpha})\frac{\partial}{\partial x}+v\frac{\partial}{\partial
u},\>X_{4}=(y-t_{2}\frac{\ln v}{\ln\alpha})\frac{\partial}{\partial
y}+v\frac{\partial}{\partial v}}$

The non-zero Lie brackets of these vector fields are

$[X_{2},X_{3}]=X_{1},\quad[X_{2},X_{4}]=X_{2},\quad[X_{3},X_{4}]=-X_{3}.$

If $I$ is the standard complex structure on ${\mathbb{C}}\times{\bf H}$,
clearly $IX_{1}=X_{2}$, $IX_{3}=X_{4}$. Set $K=X_{1}$. Then
$\mathcal{L}_{K}I=\mathcal{L}_{K}\Omega_{1}=0$ , so $K$ is Killing and real
holomorphic vector field. One can check that $IK=X_{2}$ is also Killing and
real holomorphic with $[K,IK]=0$.

For any non-negative smooth function $f$ on $M$ with $X_{1}(f)=X_{2}(f)=0$
consider the non-degenerate real $(1,1)$-form
$\omega_{f}=\Omega_{1}+f\alpha_{3}\wedge\alpha_{4}$. It defines a neutral
metric compatible with $I$ and one can check as above that the vector fields
$K,IK$ are null and Killing with respect to this metric. It is easy to check
that the (2,0)-form
$\Omega=(\alpha_{1}+i\alpha_{2})\wedge(\alpha_{3}+i\alpha_{4})$ and the real
$(1,1)$-form $\omega_{f}$ satisfy the identities

$\Omega\wedge\overline{\Omega}=-2\omega_{f}^{2}\ ,\ \Omega\wedge\omega_{f}=0\
,\ d\Omega=-\alpha_{4}\wedge\Omega\ ,\
d\omega_{f}=-\alpha_{4}\wedge\omega_{f}.$

Hence, by Lemma 1, $\Omega$ and $\omega_{f}$ define a para-hyperhermitian
structure on $M$.

$(v).$ Recall that every primary Hopf surface is diffeomorphic to $S^{1}\times
S^{3}$. Consider $S^{3}$ as the Lie group $SU(2)$ and let $X_{2},X_{3},X_{4}$
be left-invariant vector fields defining its Lie algebra with the commutator
relations

$[X_{2},X_{3}]=X_{4},\hskip 14.45377pt[X_{3},X_{4}]=X_{2},\hskip
14.45377pt[X_{4},X_{2}]=X_{3}$

Set $X_{1}=\displaystyle{\frac{\partial}{\partial x}}$ where $x$ is the
standard coordinate $e^{it}\to t$ on $S^{1}$. Denote by $g$ the metric on
$S^{1}\times S^{3}$ for which the frame $X_{1},...,X_{4}$ is orthogonal and
$||X_{1}||^{2}=||X_{2}||^{2}=1$, $||X_{3}||^{2}=||X_{4}||^{2}=-1$. This metric
is Hermitian with respect to the complex structure defined by $IX_{1}=X_{2}$,
$IX_{3}=X_{4}$. It is easy to check that ${\mathcal{L}}_{X_{k}}g=0$,
$k=1,2,3,4$. Then $X=X_{1}+X_{3}$ and $IX=X_{2}+X_{4}$ are two orthogonal null
Killing vector fields.

$\Box$

Remark. We show that the Hopf surface $S^{1}\times S^{3}=S^{1}\times SU(2)$
does not admit any left-invariant metric with two non-collinear, left-
invariant isotropic and orthogonal Killing vector fields that commute. To do
this we will use the notations above. Suppose that $g$ is a left-invariant
metric (of arbitrary signature) on $S^{1}\times SU(2)$ and let
$X=x_{0}X_{0}+x_{1}X_{1}+x_{2}X_{2}+x_{3}X_{3}$ and
$Y=y_{0}X_{0}+y_{1}X_{1}+y_{2}X_{2}+x_{3}X_{3}$ be left-invariant vector
fields satisfying the above properties. Then

$0=[X,Y]=(x_{2}y_{3}-x_{3}y_{2})X_{1}+(x_{3}y_{1}-x_{1}y_{3})X_{2}+(x_{1}y_{2}-x_{2}y_{1})X_{3}$

and therefore the vectors $U=x_{1}X_{1}+x_{2}X_{2}+x_{3}X_{3}$ and
$V=y_{1}X_{1}+y_{2}X_{2}+x_{3}X_{3}$ are collinear. Without loss of generality
we may assume that $V=aU$ and since $X=x_{0}X_{0}+U$ and $Y=y_{0}X_{0}+aU$ are
isotropic and orthogonal we get

$\begin{array}[]{l}x_{0}^{2}\parallel
X_{0}\parallel^{2}+2x_{0}g(X_{0},U)+\parallel U\parallel^{2}=0\\\\[6.0pt]
y_{0}^{2}\parallel X_{0}\parallel^{2}+2ay_{0}g(X_{0},U)+a^{2}\parallel
U\parallel^{2}=0\\\\[6.0pt] x_{0}y_{0}\parallel
X_{0}\parallel^{2}+(ax_{0}+y_{0})g(X_{0},U)+a\parallel
U\parallel^{2}=0.\end{array}$ (6)

Having in mind that $y_{0}\neq ax_{0}$ ($X$ and $Y$ are non-collinear) it
follows respectively from the first and second, and from the first and third
equation in (6) that

$\begin{array}[]{l}(y_{0}+ax_{0})\parallel
X_{0}\parallel^{2}+2ag(X_{0},U)=0\\\\[6.0pt] x_{0}\parallel
X_{0}\parallel^{2}+g(X_{0},U)=0.\end{array}$ (7)

Hence

$(y_{0}-ax_{0})\parallel X_{0}\parallel^{2}=0\Rightarrow\parallel
X_{0}\parallel^{2}=0\Rightarrow g(X_{0},U)=\parallel U\parallel^{2}=0.$

Note that a left-invariant vector field $A$ is Killing with respect to the
left-invariant metric $g$ if and only if

$\begin{array}[]{l}g([A,B],C)+g([A,C],B)=0\end{array}$ (8)

for all left-invariant vector fields $B$ and $C$. In particular, the vector
field $X_{0}$ is Killing since it commutes with all left-invariant vector
fields. Hence $U=x_{1}X_{1}+x_{2}X_{2}+x_{3}X_{3}$ is a Killing vector field
too and substituting $A=U,B=X_{0}$ in (8) we get $g([U,C],X_{0})=0$ for
$C=X_{1},X_{2},X_{3}$. Set $a_{i}=g(X_{0},X_{i}),i=1,2,3.$. Then using the
commutation relations for $X_{1},X_{2},X_{3}$ we obtain

$x_{2}a_{1}-x_{1}a_{2}=x_{3}a_{1}-x_{1}a_{3}=x_{3}a_{2}-x_{2}a_{3}=0.$

These identities together with $0=g(U,X_{0})=x_{1}a_{1}+x_{2}a_{2}+x_{3}a_{3}$
imply that either $x_{1}=x_{2}=x_{3}=0$, i.e. $U=0$ or $a_{1}=a_{2}=a_{3}=0$,
i.e. $X_{0}=0$ since $g(X_{0},X_{0})=0$. In both cases it follows that the
vector fields $X$ and $Y$ are collinear which is a contradiction.

## References

  * [1] V. Apostolov, M. Gualtieri, Generalized Kähler manifolds, commuting complex structures, and split tangent bundles, Comm. Math. Phys. 271 (2007), 561-575.
  * [2] A. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008.
  * [3] D. Blair, J. Davidov, O. Mushkarov, Hyperbolic twistor space, Rocky Mount. J. Math. 35 (2005), 1437-1465.
  * [4] C. Boyer, A note on hyperhermitian foor-manifolds, Proc. Amer. Math. Soc. 102 (1988), 157-164.
  * [5] D. Calderbank, Selfdual 4-manifolds, projective surfaces, and the Dunajski-West construction. SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 035, 18 pp.
  * [6] J. Davidov, G. Grantcharov, O. Mushkarov, M. Yotov, Para-hyperhermitian surfaces, Bull. Math. Soc. Sci. Math. Roumanie 52 (100) (2009), 281-289.
  * [7] J. Davidov, G. Grantcharov, O. Muskarov, M. Yotov, Compact complex surfaces with geometric structures related to split quaternions, Nucl. Phys. B 865 (2012), 330–352.
  * [8] M. Dunajski, S. West, Anti-self-dual conformal structures with null Killing vectors from projective structures, Comm. Math. Phys. 272 (2007), 85-118.
  * [9] M. Dunajski, M. West, Anti-self-dual conformal structures in neutral signature, Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008, pp. 113-148
  * [10] P. Gauduchon, K. P. Tod, Hyper-Hermitian metrics with symmetry, J. Geom. Phys. 25 (1998), 291-304.
  * [11] N. Hitchin, Hypersymplectic quotiens, Atti Accad. Sci. Torino cl. Sci. Fis. Mat. Natur. 124 (1990), 169-180.
  * [12] M. Inoue, On surfaces of class $VII_{0}$, Invent. Math. 24 (1974), 269-310.
  * [13] S. Ivanov, S. Zamkovoy, Para-hermitian and para-quaternionic manifolds, Diff. Geom. Appl. 23 (2005), 205-234.
  * [14] H. Kamada, Self-dual Kähler metrics of neutral signature on complex surfaces, PhD thesis, Tohoku University (2002).
  * [15] B. Klingler. Chern’s conjecture for special affine manifolds. Ann. of Math. (2) 186 (2017) (1) 69 - 95.
  * [16] K. Kodaira, On the structure of compact complex analytic surfaces I. Amer. J. Math. 86 (1964), 751-798.
  * [17] H. Ooguri, C. Vafa, Geometry of N=2 strings. Nucl. Phys. B 361 (1991), 469-518.
  * [18] H. Ooguri, C. Vafa, Self-duality and N=2 string magic. Modern Phys. Lett. A 5 (1990), 1389-1398.
  * [19] B. O’Neill, Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103, Academic Press, Inc. , New York, 1983.
  * [20] M. Pontecorvo, Complex structures on Riemannian four manifolds, Math. Ann. 309 (1997), 159-177.
  * [21] J. Petean, Indefinite Kähler-Einstein metrics on compact complex surfaces, Comm. Math. Phys. 189 (1997), 227-235.
  * [22] S. Salamon, Special structures on four-manifolds, Riv. Mat. Iniv. Parma (4) 17* (1991), 109-123.
  * [23] C. T. C. Wall, Geometric structures on compact complex analytic surfaces, Topology 25 (1986) 119–153.
  * [24] R. S. Ward, Integrable and solvable systems, and relations among them. New developments in the theory and application of solitons. Philos. Trans. Roy. Soc. London Ser. A 315 (1985), no. 1533, 451-457.

Johann Davidov

Institute of Mathematics and Informatics

Bulgarian Academy of Sciences

1113 Sofia, Bulgaria

<EMAIL_ADDRESS>
Gueo Grantcharov

Department of Mathematics

Florida International University

Miami, FL 33199

<EMAIL_ADDRESS>
Oleg Mushkarov

Institute of Mathematics and Informatics

Bulgarian Academy of Sciences

1113 Sofia, Bulgaria

and

Sauth-West University

2700 Blagoevgrad, Bulgaria

<EMAIL_ADDRESS>