S

<EMAIL_ADDRESS>if different from Sanchez del Rio

[a]European Synchrotron Radiation Facility, 71 Avenue des Martyrs F-38000
Grenoble France

# X-ray focusing by bent crystals: focal positions as predicted by the crystal
lens equation and the dynamical diffraction theory

Manuel

###### Abstract

The location of the beam focus when monochromatic x-ray radiation is
diffracted by a thin bent crystal is predicted by “crystal lens equation”. We
derive this equation in a general form valid for Bragg and Laue geometries.
This equation has little utility for diffraction in Laue geometry. The
focusing effect in the Laue symmetrical case is discussed using concepts of
dynamical theory and an extension of the lens equation is proposed. The
existence of polychromatic focusing is considered and the feasibility of
matching the polychromatic and monochromatic focal positions is discussed.

Dedicated to the memory of Claudio Ferrero

A crystal lens equation is deduced to address the location of the focus when
monochromatic x-ray radiation encounters a bent crystal. It is extended using
dynamical theory of diffraction for Laue symmetrical diffraction. Combination
of polychromatic and monochromatic focusing is also discussed.

##  1 Introduction

The use of curved crystals to diffract and focus x-rays comes as a natural
extension of the mirror and grating technology for radiation of longer
wavelength. Some fundamental concepts, like the Rowland circle, date back to
the 19${}^{\text{th}}$ century [rowland1882].

The fundamental setups using bent crystals to focus X-rays were proposed in
the early 1930’s. Some systems use meridional focusing (in the diffraction
plane), like i) Johann spectrometer [Johann1931], using a cylindrically bent
crystal, ii) Johansson spectrometer [Johansson1933] using a ground and
cylindrically bent crystal and iii) the Cauchois spectrometer [cauchois1933]
in transmission (Laue) geometry. The von Hamos spectrometer [V.Hamos1933]
applies sagittal focusing in the plane perpendicular to the diffraction plane.

With the advent of synchrotron radiation, the concepts of “geometrical
focusing” were applied to design instruments such as polychromators for
energy-dispersive extended x-ray absorption fine structure (EXAFS)
[Tolentino:ms0206], monochromators with sagittal focusing for bending magnet
beamlines [Sparks1980], or several types of crystal analyzers used at
inelastic x-ray scattering beamlines. Bent crystals in transmission or Laue
geometry are often employed in beamlines operating at high photon energies.
The crystal curvature is used for focusing or collimating the beam in the
meridional [Suortti1988, SuorttiShulze] or sagittal [Zhong2001] plane, or just
to enlarge the energy bandwidth and improve the luminosity. The crystal
bandwidth was optimized and aberrations reduced thanks to the high collimation
and small source size of synchrotron beams. Curved crystal monochromators work
in off-Rowland condition, whereas crystal analysers for inelastic scattering
studies work in the Rowland setting.

A “Crystal Lens Equation” (CLE) was indeed formulated by [CK] for the focusing
properties of a cylindrically bent crystal plate diffracting monochromatic
x-rays or neutrons, in Laue (transmission) or Bragg (reflection) geometries.
The crystal is bent around an axis perpendicular to the diffraction plane
(meridional focusing). This CLE is based on a purely geometric approach in
which multiple Bragg scattering (dynamical effects) is neglected. The CLE is
revisited in Section 2, in order to correct errors found in [CK] for the Laue
geometry. A new formula valid in Bragg and Laue geometry is obtained, using
the same geometrical approach as in [CK].

The CLE has wide applicability in Bragg geometry. However, its use for Laue
geometry is limited to very thin crystals, because it ignores a basic
dynamical focusing effect also found in flat crystals, as described in section
3. The applicability of the lens equation in symmetrical Bragg geometry is
discussed in appendix D. The CLE concerns the focusing of monochromatic
radiation, and is in general different from the condition of polychromatic
focusing. The particular cases where these two different focusing conditions
coincide are discussed in section 4. A final summary is given in section 5.

##  2 The crystal lens equation revisited

The lens equation will be derived in Bragg or Laue geometry, with source $S$
and focus $F$ in real or virtual positions (see Fig. 1). Consider a
monochromatic x-ray or neutron beam from a real or virtual point-source $S$.
The origin of coordinates $O$ is chosen at the point of the crystal surface
such that the ray $\overline{SO}$, of wavevector $\vec{k}_{0}$, is in
geometrical Bragg incidence. It gives rise outside the crystal to a diffracted
ray of wavevector $\vec{k}_{h}=\vec{k}_{0}+\vec{h}$, where $\vec{h}$ is the
reciprocal lattice vector in $O$, and $|\vec{k}_{h}|=|\vec{k}_{0}|$ (see Fig.
2). This is valid in both transmission geometry (Laue) or reflection geometry
(Bragg) for both plane and curved crystals111Because of refraction effects,
this choice implies that $\overline{SO}$ is, in general, not exactly in the
direction of the diffraction profile peak, except for the symmetric Laue
case..

Figure 1: Schematic representation of the different diffraction setups with
real or virtual source in Bragg or Laue cases: a) real source, real focus
(red) in Laue case or virtual focus (blue) in Bragg case, b) real source,
virtual focus (red) in Laue case or real focus (blue) in Bragg case, c)
virtual source, real focus (red) in Laue case or virtual focus (blue) in Bragg
case, d) virtual source, virtual focus (red) in Laue case or real focus (blue)
in Bragg case. $L_{0}=\overline{SO}$ is the distance source to crystal and
$L_{h}=\overline{OF}$ is the distance crystal to focus.

The inward normal to the crystal surface in $O$ is $\vec{n}$, and
$\varphi_{0}=(\vec{n},\vec{k}_{0})$ is the oriented angle from the vector
$\vec{n}$ to the vector $\vec{k}_{0}$. Similarly,
$\varphi_{h}=(\vec{n},\vec{k}_{h})$. Without loss of generality $\varphi_{0}$
is positive; $\theta_{B}$ is the Bragg angle (always positive). In the case of
symmetric geometry(asymmetry angle $\alpha=0$) we find
$\varphi_{0,h}=\pm\theta_{B}$ in Laue or $\varphi_{0,h}=(\pi/2)\mp\theta_{B}$
in Bragg. Otherwise, the asymmetry angle $\alpha$ is defined as the angle of
rotation of the vector $\vec{h}$ from its direction in the symmetrical case.
In Laue case $\varphi_{0,h}=\alpha\pm\theta_{B}$; in Bragg case
$\varphi_{0,h}=\alpha\mp\theta_{B}+\pi/2$, therefore
$2\theta_{B}=|\varphi_{0}-\varphi_{h}|$ in both cases,
$2\alpha=\varphi_{0}+\varphi_{h}$ in Laue case and
$2\alpha=\varphi_{0}+\varphi_{h}-\pi$ in Bragg case.

When moving the point of incidence $O$ to $P$ over an arbitrary small distance
$s$ along the curved crystal surface (see Fig. 2), $\vec{h}$ and $\vec{n}$ are
changed into $\vec{h}^{\prime}$ and $\vec{n}^{\prime}$, respectively. The
incident wavevector $\vec{k}^{\prime}_{0}$ has the direction of
$\overline{SP}$. It is diffracted into $\vec{k}^{\prime}_{h}$. The projections
of the vectors $\vec{k}^{\prime}_{h}$ and
$\vec{k}^{\prime}_{0}+\vec{h}^{\prime}$ on the crystal surface are equal
(conservation of the parallel components of wave-vectors). $\varphi_{0,h}$ are
changed into $\varphi^{\prime}_{0,h}=\varphi_{0,h}+\Delta\varphi_{0,h}$.
Furthermore, in the present case of cylindrical bending of very thin crystal,
the surface projection of $\vec{h}^{\prime}$ is constant (the angle between
$\vec{h}$ and $\vec{n}$ is constant). This implies that
$(\sin\varphi_{h}-\sin\varphi_{0})$ is invariant, therefore

$\Delta\varphi_{h}\cos\varphi_{h}=\Delta\varphi_{0}\cos\varphi_{0}.$ (1)

Figure 2: Schematic view of the relevant parameters in focusing by a bent
crystal in Bragg geometry.

The source distance $L_{0}=\overline{SO}$ is set as positive if the source is
on the incidence side of the crystal (real source) or negative if the source
is on the other side (virtual source) (see Fig. 1). The radius of curvature
$R_{c}$ is set as positive if the beam is incident on the concave side of the
bent crystal. The focus distance $L_{h}$ is set as positive if the (real or
virtual) focus $F$ is situated on the incidence side on the crystal. With
these conventions, $(\vec{n},\vec{n}^{\prime})=s/R_{c}$,
$\epsilon_{0}L_{0}=s\cos\varphi_{0}$, $\epsilon_{h}L_{h}=s|\cos\varphi_{h}|$,
where $\epsilon_{0,h}$ are the angles between $\vec{k}_{0,h}$ and
$\vec{k}^{\prime}_{0,h}$. Using the relationship

$\varphi^{\prime}_{0,h}=(n^{\prime},\vec{k}^{\prime}_{0,h})=(\vec{n}^{\prime},\vec{n})+(\vec{n},\vec{k}_{0,h})+(\vec{k}_{0,h},\vec{k}^{\prime}_{0,h})=-\frac{s}{R_{c}}+\varphi_{0,h}+\epsilon_{0,h},$
(2)

we obtain

$\Delta\varphi_{0}=-\frac{s}{R_{c}}+s\frac{\cos\varphi_{0}}{L_{0}}$ (3)

and

$\Delta\varphi_{h}=-\frac{s}{R_{c}}+s\frac{|\cos\varphi_{h}|}{L_{h}}.$ (4)

The crystal lens equation valid in both Bragg and Laue cases, is finally
obtained by inserting these expressions in equation (1)

$\frac{|\cos\varphi_{h}|\cos\varphi_{h}}{L_{h}}-\frac{\cos^{2}\varphi_{0}}{L_{0}}=\frac{\cos\varphi_{h}-\cos\varphi_{0}}{R_{c}}.$
(5)

In the Laue symmetrical case ($\cos\varphi_{h}=\cos\varphi_{0}$) it predicts
$L_{h}=L_{0}$ (for a real source, the focus is virtual at the same distance as
the source) and, in the particular case of $L_{0}=+\infty$, a plane incident
wave is diffracted into a plane wave.

The crystal lens equation (5) obtained here is different from the equation
given in [CK]222The CLE given in [CK] is
$\cos^{2}\varphi_{0}/L_{0}+\cos^{2}\varphi_{h}/L_{h}=(\cos\varphi_{0}+|\cos\varphi_{h}|)/R_{c}$.
We think this is due to mistakes in their calculations, specially sign errors
in their equation (9) as compared to our equation (3). . Both equations are
equivalent for the Bragg case ($\cos\varphi_{h}<0$), which is also considered
by snigirevkohn1995. They are not equivalent in the Laue case.

Note that we used in this section the same notation as [CK], where $R_{c}$ is
positive for a concave surface, used to focus in Bragg case. For the rest of
the paper, we also use the notation: $p\leftarrow L_{0}$, $q\leftarrow-L_{h}$
$R\leftarrow-R_{c}$, $\theta_{1}\leftarrow\varphi_{0}$ and
$\theta_{2}\leftarrow\varphi_{h}$, which is more convenient for Laue crystals,
because real focusing is obtained when the beam coming from a real source is
incident on the convex side of the bent crystal (with positive $R$).

Equation (5) is obtained here using a geometrical ray optics approach. It can
also be deduced from a wave-optics approach as shown in Appendix A.

##  3 Dynamical focusing in Laue geometry

The applicability of the CLE for the Laue case is limited to very thin
crystals. The dynamical theory (see book [authierbook]) predicts “new” focal
conditions, even for flat Laue crystals. This is analyzed here in the
framework of the Takagi-Taupin equations, hereafter TTE [Takagi1962, Takagi,
Taupin, Taupin1967].

Section 3.1 deals with the derivation of the “influence functions” (Green
functions) which represent the wavefield generated in the crystal by a point-
source on the crystal entrance surface.

In section 3.2, the approach to dynamical focusing in the symmetric Laue case
[kushnir, GuigayFerrero2013] is extended to asymmetric geometry. The effects
of anomalous absorption (Borrmann effect) are obtained in parallel. The new
concept of “numerically determined focal length” of a flat crystal, denoted as
$q_{dyn}$, is introduced.

In section 3.3, a lens equation for a bent Laue symmetrical crystal of finite
thckness, expressed in terms of $q_{dyn}$ is established. Its predictions are
shown to be in agreement with numerical calculations.

In section 3.4, we make the verification that the formulation for the Laue
asymmetric case by [GuigayFerrero2016] is in agreement with the CLE (equation
(5)) in the limit of vanishing crystal thickness.

###  3.1 Influence function derived from Takagi-Taupin equations

The x-ray wavefield inside the crystal is expressed as the sum of two
modulated plane waves

$\Psi(\vec{x})=D_{0}(\vec{x})e^{i\vec{k}_{0}.\vec{x}}+D_{h}(\vec{x})e^{i\vec{k}_{h}.\vec{x}},$
(6)

with slowly varying amplitudes $D_{0,h}(\vec{x})$. The spatial position
$\vec{x}$ is expressed in oblique coordinates $(s_{0},s_{h})$ along the
directions of the $\vec{k}_{0}$ and $\vec{k}_{h}=\vec{k}_{0}+\vec{h}$ vectors,
which are the in-vacuum wave-vectors of modulus $k=2\pi/\lambda$, where
$\lambda$ is x-ray wavelength. $\vec{h}$ is the Bragg diffraction vector of
the undeformed crystal. In such conditions, the differential TTE are

$\displaystyle\frac{\partial D_{0}}{\partial s_{0}}=$
$\displaystyle\frac{ik}{2}\left[\chi_{0}D_{0}(\vec{x})+c\chi_{\bar{h}}e^{i\vec{h}.\vec{u}(\vec{x})}D_{h}(\vec{x})\right];$
(7a) $\displaystyle\frac{\partial D_{h}}{\partial s_{h}}=$
$\displaystyle\frac{ik}{2}\left[\chi_{0}D_{h}(\vec{x})+c\chi_{h}e^{-i\vec{h}.\vec{u}(\vec{x})}D_{0}(\vec{x})\right],$
(7b)

where $\chi_{0}$, $\chi_{h}$, and $\chi_{\bar{h}}$ are the Fourier
coefficients of order 0, $\vec{h}$ and $-\vec{h}$ of the undeformed crystal
polarisability. The polarization factor $c$ ($c=1$ for $\sigma$-polarization
and $c=\cos 2\theta_{B}$ for $\pi$-polarization) is omitted from now on.
$\vec{u}(\vec{x})$ is the displacement field of the deformed crystal. In the
case of cylindrical bending we have

$\vec{h}.\vec{u}=-As_{0}s_{h}+\phi_{1}(s_{0})-\phi_{2}(s_{h})$ (8)

where $A$ and the $\phi_{1,2}$ functions are defined in Appendix C. This a
“constant strain gradient” case [authierbook] meaning that
$\partial^{2}(\vec{h}.\vec{u})/(\partial s_{0}\partial s_{h})$ is constant. In
terms of the functions $G_{0,h}(s_{0},s_{h})$ defined by

$\displaystyle D_{0}(s_{0},s_{h})$
$\displaystyle=G_{0}(s_{0},s_{h})\exp[i\frac{k}{2}\chi_{0}(s_{0}+s_{h})-i\phi_{2}(s_{h})]$
(9a) $\displaystyle D_{h}(s_{0},s_{h})$
$\displaystyle=G_{h}(s_{0},s_{h})\exp[i\frac{k}{2}\chi_{0}(s_{0}+s_{h})-i\phi_{1}(s_{0})+iAs_{0}s_{h}],$
(9b)

the TTE have a simpler form

$\displaystyle\frac{\partial G_{0}}{\partial s_{0}}$
$\displaystyle=i\frac{k}{2}\chi_{\bar{h}}G_{h}$ (10a)
$\displaystyle\frac{\partial G_{h}}{\partial s_{h}}$
$\displaystyle=i\frac{k}{2}\chi_{h}G_{0}-iAs_{0}G_{h}.$ (10b)

An incident monochromatic wave of any form can be expressed as a modulated
plane wave $D_{inc}(\vec{x})\exp(i\vec{k}_{0}.\vec{x})$ defining a continuous
distribution of coherent elementary point-sources on the crystal surface,
according to the general Huyghens principle in optics. The “influence
functions” or Green functions, hereafter IF, are the TTE solutions for these
point-sources. The IF for point-sources of oblique coordinates
$(\sigma_{0},\sigma_{h})$ are derived in [GuigayFerrero2016] by formulating
the TTE as integral equations in the case of an incident amplitude of the form
$D_{inc}=\delta(s_{h}-\sigma_{h})$. The calculations (see appendix B) result
in the diffracted amplitude333the result for the transmitted amplitude
$D_{0}(s_{0},s_{h})$ is not necessary for our results and is not presented
here, but it is easily obtained using equation (11) in (10).

$D_{h}(s_{0},s_{h})=\frac{ik}{2}\chi_{h}e^{(ik/2)\chi_{0}(s^{\prime}_{0}+s^{\prime}_{h})}e^{-i\vec{h}.\vec{u}(s_{0},\sigma_{h})}M(\frac{i\Omega}{A},1,iAs^{\prime}_{0}s^{\prime}_{h})$
(11)

where the first exponential term stands for the effects of refraction and
normal absorption, $s^{\prime}_{0,h}=s_{0,h}-\sigma_{0,h}$;
$\Omega=k^{2}\chi_{h}\chi_{\bar{h}}/4$ and the $M$-function is the Kummer
function (a confluent hypergeometric function) defined by the convergent
infinite series

$M(a,b,z)=1+\frac{a}{b}z+...+\frac{a(a+1)...(a+n-1)}{n!b(b+1)...(b+n-1)}z^{n}+...$
(12)

This type of TTE solution was already obtained by different methods
[Petrashen1974, Katagawa1974, Litzmann1974, Chukhovski1977].

It is noticeable that the term $\exp[-i\vec{h}.\vec{u}(s_{0},\sigma_{h})]$ in
equation (11) is the phase shift acquired by scattering at the point of
coordinates $(s_{0},\sigma_{h})$ along the incident ray. We can say that the
kinematical (single-scattering) approximation of equation (11) is

$D_{h,kin}(s_{0},s_{h})=\frac{ik}{2}\chi_{h}e^{(ik/2)\chi_{0}(s^{\prime}_{0}+s^{\prime}_{h})}e^{-i\vec{h}.\vec{u}(s_{0},\sigma_{h})}$
(13)

and the full multiple scattering is $D_{h}=D_{h,kin}M$.

###  3.2 Dynamical focusing and Borrmann effect in a flat, asymmetric, Laue
crystal

Dynamical focusing by flat Laue crystals (without bending) was predicted by
AfanasevKohn1977 and verified experimentally by [Aristov1978,
Aristov1980PhysStatSol, Aristov1980] in the case of symmetrical geometry. The
theory was extended to the asymmetric case by Kohn2000. The application of
dynamical focusing to high-resolution spectrometry was proposed by
KohnGorobtsov2013.

Figure 3: Schematic representation of the relevant parameters in Laue
asymmetrical diffraction.

The basic case of dynamical focusing is that of a point-source in $O$
($\sigma_{0}=\sigma_{h}=0$) on the crystal entrance surface of the crystal of
thickness $t$. $O^{\prime}$ is the middle of the basis of the influence region
(Borrmann fan) on the exit surface (see Fig. 3). The amplitude of the
diffracted wave along the axis $O^{\prime}\xi\perp\vec{k}_{h}$ is the value of
the IF at the point of coordinates

$s_{0}=\frac{a+\xi}{\sin 2\theta_{B}};\>\>s_{h}=\gamma\frac{a-\xi}{\sin
2\theta_{B}},$ (14)

with $a=t\sin 2\theta_{B}/(2\cos\theta_{1})$ and
$\gamma=\cos\theta_{1}/\cos\theta_{2}$.

The amplitude $D_{h}(\xi)$ is zero outside the interval $-a<\xi<a$, and is
proportional to the Bessel function
$J_{0}(k\sqrt{\chi_{h}\chi_{\bar{h}}s_{0}s_{h}})=J_{0}(Z\sqrt{a^{2}-\xi^{2}})$
in this interval [kato1961], with $Z=k\sqrt{\gamma\chi_{h}\chi_{\bar{h}}}/\sin
2\theta_{B}$. In the case $|Za|\gg 1$ the asymptotic approximation

$J_{0}(Z\sqrt{a^{2}-\xi^{2}})\approx\left(\frac{2}{\pi
Z\sqrt{a^{2}-\xi^{2}}}\right)^{1/2}\cos(Z\sqrt{a^{2}-\xi^{2}}-\pi/4)$ (15)

can be used in the central region $|\xi|\ll a$ where
$\sqrt{a^{2}-\xi^{2}}\approx a-\frac{\xi^{2}}{2a}$. We thus obtain in this
central region the approximation

$J_{0}(Z\sqrt{a^{2}-\xi^{2}})\approx\left(\frac{2}{i\pi
Za}\right)^{1/2}\left(e^{iZa-
iZ\frac{\xi^{2}}{2a}}+ie^{-iZa+iZ\frac{\xi^{2}}{2a}}\right),$ (16)

where the two exponential terms are related to the two sheets of the
dispersion surface. The function $\exp(-iZ\xi^{2}/(2a))$ represents a
converging wave if $\operatorname{Re}(Z)>0$ (divergent if
($\operatorname{Re}Z<0$). A double, real and virtual, focusing effect is thus
expected at opposite distances $\pm q_{0}$ from the crystal, with

$q_{0}=\frac{ka}{|\operatorname{Re}(Z)|}=\frac{a\sin
2\theta_{B}}{|\operatorname{Re}(\sqrt{\gamma\chi_{h}\chi_{\bar{h}}})|}$ (17)

This equation is present in [Kohn2000, KohnGorobtsov2013] in a different form
and from a different point of view. These authors consider a point-source at a
finite distance and their equation determines the value of the crystal
thickness needed to focus the diffracted wave on the back crystal surface. A
noticeable difference is that our equation is expressed in terms of
$\chi_{h}\chi_{\bar{h}}$ without approximations concerning the real and
imaginary parts of the crystal polarizability. In the works cited above
$\operatorname{Re}(\sqrt{\chi_{h}\chi_{\bar{h}}})$ is approximated by
$|\chi_{hr}|$ or $|\chi_{h}|$.

The moduli of the two terms in equation (16) are proportional to $\exp(\mp
a\text{Im}~{}(Z))$, respectively. This is the expression of anomalous
absorption (Borrmann effect). Two focal positions will be observed for small
absorption, but only one for strong absorption, as shown in Fig. 4.

The reflected amplitude at any distance $q$ from the crystal can be calculated
numerically, without the approximations used above, by the Fresnel diffraction
integral

$D_{h}(\xi;q)=(\lambda
q)^{-1/2}\int_{-a}^{a}d\xi^{\prime}\,e^{ik\frac{(\xi-\xi^{\prime})^{2}}{2q}}J_{0}(Z\sqrt{a^{2}-\xi^{\prime
2}}).$ (18)

The “axial intensity profile” $|D_{h}(0,q)|^{2}$ shows in general two strong
maxima at distances $q_{1,2}=\pm q_{dyn}<q_{0}$ (Fig. 4). This difference is a
cylindrical aberration effect related to the approximations used to obtain
equation (17). The parameter $q_{dyn}$, which depends on the crystal
thickness, is the “dynamical focal length” obtained numerically, thus non-
approximated (contrary to $q_{0}$). As an example, some numerical values are
given in Table 1.

Table 1: Parameters for symmetrical Laue silicon crystal in 111 reflection and thickness $t$ = $250\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Photon energy (keV) |  $\theta_{B}$ (deg) | $\chi_{0}$ | $\chi_{h}\chi_{\bar{h}}$ |  $a$ ($\text{\,}\mathrm{\SIUnitSymbolMicro m}$) |  $q_{0}$ (mm) |  $q_{dyn}$ (mm)
---|---|---|---|---|---|---
8.3 | 13.78 |  (-14.24 + 0.317 i) 10-6 |  (58.06 - 3.416 i) 10-12 | 59 | 3615 | 2860
17 | 6.68 |  (-3.36 + 0.018 i) 10-6 |  (3.20 - 0.046 i) 10-12 | 29 | 3753 | 2535
Figure 4: Numerical evaluation of on-axis intensity for a
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick flat Si111 crystal
($R=\infty$) with source at the crystal entrance surface ($p=0$) calculated
using equation (18). a) Simulation for a photon energy of 8.3 keV. b)
Simulation for a photon energy of 17 keV. Numerical values of these
simulations are in Table 1.

The focusing condition for a source at a finite distance $p$ from the crystal
can be obtained by considering that propagation in free-space and propagation
in the flat crystal are space-invariant, therefore expressed as convolutions
in direct space or simple multiplications in reciprocal space. Therefore, they
can be commuted. This allows to merge the free-space propagation before and
after the crystal. The focusing condition is therefore

$p+q=q_{dyn}.$ (19)

On the contrary, propagation through a bent crystal is not space-invariant
because the IF is not only dependent on the variables
$(s^{\prime}_{0},s^{\prime}_{h})$, but also on the variables
$(\sigma_{0},\sigma_{h})$ because of the factor
$\exp[-i\vec{h}.\vec{u}(s_{0},\sigma_{h})]$ in equation (11).

###  3.3 A new lens equation for a bent crystal of finite thickness in
symmetrical Laue geometry

In symmetrical Laue geometry, the factor
$\exp(i\chi_{0}(s^{\prime}_{0}+s^{\prime}_{h}))$ in equation (11) is constant
on the crystal exit surface and will be omitted. Equation (11) is (see
Appendix B)

$D_{h}(s_{0},s_{h})=\frac{ik}{2}\chi_{h}e^{-i\vec{h}.\vec{u}(s_{0},\sigma_{h})}J_{0}(2\sqrt{\Omega
s^{\prime}_{0}s^{\prime}_{h}}).$ (20)

Let us consider the incident amplitude $D_{inc}(\tau)=\exp(ik\tau^{2}/(2p))$,
where $\tau$ is a coordinate along the axis $O\tau$ normal to $\vec{k}_{0}$
(see Fig. 3). On the exit surface, using $s_{0}=(\xi+a)/\sin 2\theta_{B}$ and
$\sigma_{h}=-\tau/\sin 2\theta_{B}$, and the notation
$R^{\prime}=R\cos\theta_{B}$ we obtain from equations in appendix C, in the
case $\alpha=0$

$\vec{h}.\vec{u}(s_{0},\sigma_{h})=k\frac{\tau(\tau+a)-\xi(\xi+a)}{2R^{\prime}}.$
(21)

Using the integration variable $\eta=\xi-\tau$, the amplitude along the
$\xi$-axis is, with omission of $i(k/2)\chi_{h}$,

$D_{h}(\xi,0)=\int_{-a}^{+a}\frac{d\eta}{\sqrt{\lambda
p}}e^{\frac{ik}{2}\left[\frac{(\xi-\eta)^{2}}{p}+\frac{\eta^{2}-2\eta\xi-a\eta}{R^{\prime}}\right]}J_{0}(Z\sqrt{a^{2}-\eta^{2}}).$
(22)

The wave amplitude at a distance $q$ downstream from the crystal is obtained
using a Fresnel diffraction integral similar to equation (18). We thus have a
double integral over $\eta$ and $\xi^{\prime}$. The $\xi^{\prime}$ integration
is performed analytically [GuigayFerrero2013] and it turns out that

$D_{h}(\xi,q)=\frac{e^{ik\frac{\xi^{2}}{2L}}}{\sqrt{\lambda
L}}\int_{-a}^{+a}d\eta e^{\frac{ik}{2}[\frac{\eta^{2}}{L_{e}}-\eta(\frac{2\xi
q_{e}}{qL_{e}}+\frac{a}{R^{\prime}})]}J_{0}(Z\sqrt{a^{2}-\eta^{2}}),$ (23)

where $L=p+q$, $p_{e}^{-1}=p^{-1}+R^{\prime-1}$,
$q_{e}^{-1}=q^{-1}-R^{\prime-1}$ and $L_{e}=p_{e}+q_{e}$. The focal positions
are given by $L_{e}=\pm q_{dyn}$. This can be written as

$\frac{R^{\prime}}{R^{\prime}-q}-\frac{R^{\prime}}{R^{\prime}+p}=\pm\frac{q_{dyn}}{R^{\prime}}.$
(24)

Translating equation (24) in the notation of section 2 ($p\to L_{0}$, $q\to-
L_{h}$, $R\to-R_{c}$), we obtain

$\frac{1}{L_{h}-R_{c}\cos\theta_{B}}-\frac{1}{L_{0}-R_{c}\cos\theta_{B}}=\pm\frac{q_{dyn}}{(R_{c}\cos\theta_{B})^{2}}.$
(25)

If $q_{dyn}$ is set to zero, we obtain $L_{h}=L_{0}$, the same result as the
lens equation (5). Equation (25) can be considered as a “modified lens
equation” which takes dynamical diffraction effects into account in symmetric
Laue geometry. We do not know an equation like equation (25) for the general
case of asymmetrical Laue diffraction. However, numerical simulations can be
done to obtain the focal positions [Nesterets, GuigayFerrero2016].

Examples of numerical calculations using equation (23) are shown in Fig. 5,
for the case of the 111 reflection of a
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick cylindrically bent symmetric
Laue silicon crystal, with a curvature radius of $R$ = $1\text{\,}\mathrm{m}$,
at a source distance $p$ = $30\text{\,}\mathrm{m}$ and for x-ray photon
energies of 8.3 keV and 17 keV.

Alternatively, provided that the parameter $q_{dyn}$ has been previously
determined numerically by a plot similar to Fig. 4, the focal positions can be
given directly by equation (25). The results are in very good agreement with
the focal positions obtained obtained numerically in Fig. 5. An important
advantage in using the new CLE is that the same value of $q_{dyn}$ can be used
for any value of the radius of curvature and for any value of source distance.

We are often interested in real focusing ($q>0$) of an incident beam from a
very distant real source, for instance in dispersive EXAFS beamlines. Suppose
$0<R^{\prime}\leq q_{dyn}$. When $p$ increases from zero to infinity, $q_{1}$
decreases from $q_{1}=R^{\prime}q_{dyn}/(q_{dyn}+R^{\prime})$ to
$q_{1}=R^{\prime}q_{dyn}(q_{dyn}-R^{\prime})$. Simultaneously, $q_{2}$
decreases from $q_{2}=R^{\prime}q_{dyn}/(q_{dyn}-R^{\prime})$ to
$q_{2}=R^{\prime}q_{dyn}(q_{dyn}+R^{\prime})$. For very large $p$-values, we
have the simple relation $q_{1}+q_{2}\approx 2R^{\prime}$ in good agreement
with the numerical results in Fig. 5.

Figure 5: Numerical evaluation of diffracted intensity by a
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick Si 111 symmetric Laue
crystal calculated using equation (23) for a bent (R = $1\text{\,}\mathrm{m}$)
crystal and $p$ = $50\text{\,}\mathrm{m}$. a) on-axis intensity for a photon
energy of 8.3 keV. Inset: transverse profile at the focal distances (maximum
values): $q_{1}$ = $651\text{\,}\mathrm{mm}$ (blue), and $q_{2}$ =
$1330\text{\,}\mathrm{mm}$ (red). b) on-axis intensity for a photon energy of
17 keV. Inset: transverse profile at the focal distances (maximum values):
$q_{1}$ = $625\text{\,}\mathrm{mm}$ (blue), and $q_{2}$ =
$1372\text{\,}\mathrm{mm}$ (red).

It can be seen from equation (23) that the intensity function
$|D_{h}(s_{0},s_{h})|^{2}$ as a function of $\xi$ is symmetric around
$\xi_{c}=-aqL_{e}/(2q_{e}R^{\prime})$. This denotes a lateral shift of the
intensity profile from its position for the unbent crystal (the axial
intensity profiles of Fig.5a and 5b are actually plotted as a function of
$(\xi-\xi_{c})$.

###  3.4 Semianalytical approach in asymmetric Laue geometry and its CLE limit

The generalization of equation (22) to asymmetric Laue geometry is
[GuigayFerrero2016]

$D_{h}(\xi,0)=\int_{-a}^{a}\gamma\frac{d\eta}{\sqrt{\lambda
p}}e^{ik\gamma^{2}\frac{(\xi-\eta)^{2}}{2p}+i\phi(\xi,\eta)}M(\frac{i\Omega}{A},1,igk\frac{a^{2}-\eta^{2}}{R}).$
(26)

Here, $\phi(\xi,\eta)$ is calculated from the term
$\exp(-i\vec{h}.\vec{u}(s_{0},\sigma_{h}))$ in equation (11) with
$s_{0}=(a+\xi)/\sin 2\theta_{B}$ and $\sigma_{h}=\gamma(\xi-\eta)/\sin
2\theta_{B}$, giving

$\phi(\xi,\eta)=\frac{k}{2R}[-\mu_{2}\gamma^{2}(\xi-\eta)^{2}+a_{2}\gamma(\eta-\xi)\\\
-\mu_{1}(a+\xi)^{2}+a_{1}(a+\xi)-2g(a+\xi)(\xi-\eta)],$ (27)

with parameters $\mu_{1,2}$, $a_{1,2}$ and $g$ given in Appendix C. The
reflected amplitude $D_{h}(\xi,q)$ at distance $q$ downstream from the crystal
is again obtained as in equation (18), therefore by double integration over
$\eta$ and $\xi^{\prime}$. The $\xi^{\prime}$-integration can be again
performed analytically. The remaining $\eta$-integration involving the Kummer
function is carried out numerically [GuigayFerrero2016]. We consider this
approach as semi-analytical, in contrast to the approach based on a numerical
solution of the TTE [Nesterets].

It is interesting to study analytically the limit of this semi-analytical
formulation in the case of vanishing crystal thickness ($a\rightarrow 0{}$)
because the comparison with lens equation (5) represents a validity test of
the semi-analytical formulation. In the limit ($a\rightarrow 0{}$), the Kummer
function is equal to unity in equation (26), and the integral can be replaced
by $2a$ times the integrand evaluated at $\eta=a=0$, therefore

$D_{h}(\xi,0)=\frac{2a\gamma}{\sqrt{\lambda
p}}e^{\frac{ik\xi^{2}}{2}(\frac{\gamma^{2}}{p}-\frac{\mu_{2}\gamma^{2}+\mu_{1}+2g}{R})}.$
(28)

This is the expression of the amplitude of a cylindrical wave focused at the
distance $q$ such that

$\frac{1}{q}+\frac{\gamma^{2}}{p}-\frac{\mu_{2}\gamma^{2}+\mu_{1}+2g}{R}=0.$
(29)

Using the identity

$\mu_{1}+\gamma^{2}\mu_{2}+2g=\frac{\cos\theta_{2}-\cos\theta_{1}}{\cos^{2}\theta_{2}},$
(30)

which is derived in Appendix C, the focusing condition is

$\frac{1}{q}+\frac{\gamma^{2}}{p}+\frac{\cos\theta_{1}-\cos\theta_{2}}{R\cos^{2}\theta_{2}}=0,$
(31)

or,

$\frac{\cos^{2}\theta_{2}}{q}+\frac{\cos^{2}\theta_{1}}{p}+\frac{\cos\theta_{1}-\cos\theta_{2}}{R}=0,$
(32)

which is the CLE (equation (5)) for the Laue case, with the correspondence
$p\rightarrow L_{0}$, $q\rightarrow-L_{h}$, $R\rightarrow-R_{c}$,
$\theta_{1}\rightarrow\varphi_{0}$ and $\theta_{2}\rightarrow\varphi_{h}$

##  4 Polychromatic geometric focusing

As pointed out by [CK], the monochromatic focusing condition must not be
confused with the polychromatic focusing condition [handbook, Caciuffo1987,
Schulze1998, Martinson, martinson2017], obtained by varying the wavelength of
the reflected rays in order to satisfy the exact Bragg condition on the whole
crystal surface. The equation $\varphi_{0}+\varphi_{h}=2\alpha$ in Laue, or
$\varphi_{0}+\varphi_{h}=2\alpha+\pi$ in Bragg case, implies
$\Delta\varphi_{0}+\Delta\varphi_{h}=0$. Using equations (3) and (4) we obtain

$\frac{{\cos{\varphi_{o}}}}{{{L_{0}}}}+\frac{{\left|{\cos{\varphi_{h}}}\right|}}{{{L_{h}}}}=\frac{2}{R_{c}}.$
(33)

Equation (33) is usually referred to as the ”geometric focusing” condition for
bent crystals. It is also applied in the case of flat crystals
[sanchezdelrio1994]. Like in equation (5), the crystal thickness does not
appear in equation (33). The combination of equations (5) and (33) gives

$\frac{\cos\varphi_{0}}{L_{0}}(\cos\varphi_{h}+\cos\varphi_{0})=\frac{|\cos\varphi_{h}|}{L_{h}}(\cos\varphi_{h}+\cos\varphi_{0}),$
(34)

which is verified either in the symmetric Bragg case
($\cos\varphi_{h}+\cos\varphi_{0}=0$), or if
$\cos\varphi_{0}/L_{0}=|\cos\varphi_{h}|/L_{h}=1/R$, which is the Rowland
condition. The Rowland condition is therefore necessary for the coincidence of
equations (5) and (33) in Laue geometry. 444This is different from the
statement of [CK] that the coincidence is always realised under symmetrical
reflection or the Rowland condition.

A narrow energy band is reflected in Rowland condition, because the angle of
incidence on the local reflecting plane does not change along the bent crystal
surface.

On synchrotron dispersive EXAFS beamlines, the use of a Bragg symmetric
reflection by a bent polychromator at a large distance from the source
guarantees the focusing of a broad bandwidth (up to $~{}1$ keV) on a small
spot [Tolentino:ms0206] at a distance close to
$L_{h}=(R_{c}\sin\theta_{B})/2$.

Laue polychromators are also used in synchrotron beamlines. In symmetric Laue
geometry, condition (5) should be replaced by equation (25), which is
$L_{h}\approx R_{c}\cos\theta_{B}+(R_{c}\cos\theta_{B})^{2}/q_{dyn}$ if the
source distance is very large. Coincidence with (33) is then obtained if
$R_{c}=-q_{dyn}/(2\cos\theta_{B})$, which means real focusing at the distance
$|L_{h}|=q_{dyn}/4$ with beam incidence in the crystal convex side
($R_{c}<0$). If $|L_{h}|$ is fixed, the required conditions are
$|R_{c}|=2|L_{h}|/\cos\theta_{B}$ and $q_{dyn}=4|L_{h}|$. The last condition
should be fulfilled by choosing the crystal thickness, as in [Mocella2004,
Mocella2008].

Another polychromatic condition for Laue geometry has been introduced more
recently [Martinson, PengQi, PengQi2021]. The energy components of a
polychromatic ray traversing a bent Laue crystal with finite thickness meet
the Bragg condition at different positions along the ray path. They are
diffracted with different Bragg angles, therefore they exit in different
directions, giving raise to a polychromatic focus from a single ray. The
“magic condition”, under which single ray focusing and geometric focusing
(equation (33)) would coincide, is achieved by the adequate choice of the
asymmetry. The magic condition is independent of the crystal thickness
[PengQi2021]. We observe that the magic condition (equation (19) in
[PengQi2021]) and the modified lens equation (25) are both satisfied in the
particular case of symmetric Laue geometry in Rowland configuration.

##  5 Conclusions and future perspectives

The crystal lens equation (CLE, equation (5)) based on the conservation of the
parallel component of the wavevector in the diffraction process has been
revisited. It includes all cases of symmetric and asymmetric Laue and Bragg
geometries. It differs from the previous formulation [CK] in the Laue case.
However, in Laue geometry, the lens equation can be only applied if the
crystal is so thin that important effects resulting from the dynamical theory
of diffraction, like the focusing of the Borrmann triangle, can be neglected.
We derived the modified lens equation (25) which overcomes this restriction in
the Laue symmetric case. Consistently, it converges to the CLE if the crystal
thickness tends to zero. The generic case of arbitrary asymmetry is left for a
future investigation. The fact that dynamic focusing cannot be achieved in
Bragg case (see Appendix D) justifies in some way the larger applicability of
the CLE in Bragg case.

The application of the CLE (equation 5) is restricted to monochromatic
focusing. Polychromatic focusing, as used in the polychromators of dispersive
EXAFS beamlines, happens when the wavelength of the reflected rays changes to
exactly match the Bragg angle. This condition is given by a different lens
equation (33). This implies a specular reflection of the rays on the Bragg
planes that is, in general, incompatible with the CLE or the results of
dynamical theory, except for the Bragg symmetric case. It has been
demonstrated that focii predicted by monochromatic and polychromatic focusing
conditions coincide if the source is situated on the Rowland circle. Moreover,
such coincidence is also true for any source position (off-Rowland) in
symmetric Bragg geometry, but not in symmetric Laue geometry. Here, for the
Laue symmetric case, both polychromatic and monochromatic focii can match if
the modified lens equation (25) is used instead, but requires a particular
choice of the crystal thickness. The additional effect of focusing a
polychromatic ray [PengQi2021] gives the “magic condition” for Laue focusing,
which implies geometric and single scattering. Further studies would be
required to match the magic condition (which does not depend on the crystal
thickness) with monochromatic focusing. This could be done by optimizing
numerically the crystal thickness using the formulation in section 3.4.

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##  Appendix A Derivation of the Lens Equation from the phase-factor of the
Takagi-Taupin equations

Under a deformation field $\vec{u}(\vec{r})$, the crystal polarizability is
taken as $\chi(\vec{r}-\vec{u}(\vec{r}))$, where $\chi(\vec{r})$ is the
polarizability of the non-deformed crystal. The Fourier components of the
electric susceptibility $\chi_{h,\bar{h}}$ are multiplied by the phase factors
$\exp(-\vec{h}.\vec{u}(\vec{r}))$ and $\exp(\mp i\vec{h}.\vec{u}(\vec{r}))$,
respectively, in the TTE. In the case of a very thin crystal, the ray
reflected at position $x$ on the bent crystal surface is simply affected by
the phase factor

$e^{-i\vec{h}.\vec{u}(x)}=e^{ik(\cos\varphi_{h}-\cos\varphi_{0})\frac{x^{2}}{2R_{c}}},$
(35)

which is obtained using $\vec{u}(x)=-(x^{2}/(2R_{c}))\vec{n}$ and
$\vec{n}.\vec{h}=\vec{n}.(\vec{k}_{h}-\vec{k}_{0})=k(\cos\varphi_{h}-\cos\varphi_{0})$.

In the case of the undeformed crystal, the incident amplitude
$\exp[ik\tau^{2}/(2L_{0})]$, along the axis $O\tau$ is translated into

$D_{inc}(\xi)=e^{ik\frac{\xi^{2}}{2L_{0}}\left(\frac{\cos\varphi_{0}}{\cos\varphi_{h}}\right)^{2}},$
(36)

along the axis $O^{\prime}\xi$ (see Fig. 3). This is combined with equation
(35) to obtain the amplitude of the Bragg-reflected wave along the axis $O\xi$

$D_{h}(\xi)=e^{ik\frac{\xi^{2}}{\cos^{2}\varphi_{h}}\left(\frac{\cos\varphi_{h}-\cos\varphi_{0}}{2R_{c}}+\frac{\cos^{2}\varphi_{0}}{2L_{0}}\right)},$
(37)

corresponding to a real or virtual focus if the phase of this function is
negative or positive respectively.

Using the convention defined in section 2 (see Fig. 1), a real focus requires
$L_{h}<0$ in Laue case and $L_{h}>0$ in Bragg. A cylindrical converging wave
has then the form $\exp(ik\xi^{2}/(2L_{h}))$ in Laue and
$\exp(-ik\xi^{2}/(2L_{h}))$ in Bragg (we arrive to the same result considering
a virtual focus). For both Bragg and Laue cases we can write

$D_{h}(\xi)=e^{ik\frac{\xi^{2}}{2L_{h}}\frac{|\cos\varphi_{h}|}{\cos\varphi_{h}}}.$
(38)

Comparing equations (37) and (38), we finally obtain

$\frac{\cos\varphi_{h}-\cos\varphi_{0}}{R_{c}}+\frac{\cos^{2}\varphi_{0}}{L_{0}}=\frac{|\cos\varphi_{h}|\cos\varphi_{h}}{L_{h}},$
(39)

which is equivalent to the lens equation (5).

##  Appendix B Derivation of the influence functions equation (11) from the
integral form of the Takagi-Taupin equations

For a point source in position $(\sigma_{0},\sigma_{h})$ on the crystal
surface, the incident amplitude has the form
$D_{inc}=\delta(s_{h}-\sigma_{h})$. The refracted amplitude is
$D_{ref}(s_{0},s_{h})=\exp(ik\chi_{0}s^{\prime}_{0}/2)\delta(s^{\prime}_{h})$
using $s^{\prime}_{0,h}=s_{0,h}-\sigma_{0,h}$. According to equation (9a),
this is transformed in $G_{ref}=E\delta(s^{\prime}_{h})$, with

$E=\exp[-i\frac{k}{2}\chi_{0}(\sigma_{0}+\sigma_{h})+i\phi_{2}(\sigma_{h})].$
(40)

Considering $G_{0,h}(s_{0},s_{h})$ as functions of $s^{\prime}_{0}$ and
$s^{\prime}_{h}$, the TTE (10) are

$\displaystyle\frac{\partial G_{0}(s^{\prime}_{0},s^{\prime}_{h})}{\partial
s^{\prime}_{0}}=$ $\displaystyle
i\frac{k}{2}\chi_{\bar{h}}G_{h}(s^{\prime}_{0},s^{\prime}_{h})$ (41a)
$\displaystyle\frac{\partial G_{h}(s^{\prime}_{0},s^{\prime}_{h})}{\partial
s^{\prime}_{h}}=$ $\displaystyle
i\frac{k}{2}\chi_{h}G_{0}(s^{\prime}_{0},s^{\prime}_{h})-iA(s^{\prime}_{0}+\sigma_{0})G_{h}(s^{\prime}_{0},s^{\prime}_{h}).$
(41b)

We define the functions $F_{0,h}(s^{\prime}_{0},s^{\prime}_{h})$ such that

$G_{0,h}(s^{\prime}_{0},s^{\prime}_{h})=e^{-iA\sigma_{0}s^{\prime}_{h}}F_{0,h}(s^{\prime}_{0},s^{\prime}_{h}).$
(42)

The equations (41) are rewritten as

$\displaystyle\frac{\partial F_{0}}{\partial s^{\prime}_{0}}=$ $\displaystyle
i\frac{k}{2}\chi_{\bar{h}}F_{h}$ (43a) $\displaystyle\frac{\partial
F_{h}}{\partial s^{\prime}_{h}}=$ $\displaystyle
i\frac{k}{2}\chi_{h}F_{0}-iAs^{\prime}_{0}F_{h},$ (43b)

The refracted amplitude is $F_{ref}=E\delta(s^{\prime}_{h})$. Equations (43)
can be written in the form of integral equations:

$\displaystyle F_{0}(s^{\prime}_{0},s^{\prime}_{h})=$ $\displaystyle
E\delta(s^{\prime}_{h})+i\frac{k}{2}\chi_{\bar{h}}\int_{0}^{s^{\prime}_{0}}d\xi_{0}F_{h}(\xi_{0},s^{\prime}_{h}),$
(44a) $\displaystyle F_{h}(s^{\prime}_{0},s^{\prime}_{h})=$ $\displaystyle
i\frac{k}{2}\chi_{h}\int_{0}^{s^{\prime}_{h}}d\xi_{h}F_{0}(s^{\prime}_{0},\xi_{h})-iAs^{\prime}_{0}\int_{0}^{s^{\prime}_{h}}d\xi_{h}F_{h}(s^{\prime}_{0},\xi_{h}).$
(44b)

We can combine equations (44) into a single integral equation for $F_{h}$ only

$F_{h}(s^{\prime}_{0},s^{\prime}_{h})=i\frac{k}{2}\chi_{h}E-\Omega\int_{0}^{s^{\prime}_{h}}d\xi_{h}\int_{0}^{s^{\prime}_{0}}d\xi_{0}F_{h}(\xi_{0},\xi_{h})-iAs^{\prime}_{0}\int_{0}^{s^{\prime}_{h}}d\xi_{h}F_{h}(s^{\prime}_{0},\xi_{h}),$
(45)

where $\Omega=k^{2}\chi_{h}\chi_{\bar{h}}/4$ is used. By iteration starting
from $F_{h}=i\frac{k}{2}\chi_{h}E$, we obtain

$\displaystyle
F_{h}(s^{\prime}_{0},s^{\prime}_{h})=i\frac{k}{2}\chi_{h}E[1-(\Omega+iA)s^{\prime}_{0}s^{\prime}_{h}+...+$
$\displaystyle(\Omega+iA)(\Omega+2iA)...(\Omega+niA)\frac{(-s^{\prime}_{0}s^{\prime}_{h})^{n}}{n!n!}+...]$
(46)

According to the definition of the Kummer function (equation (12)), the series
in brackets is equal to
$M(\frac{\Omega}{iA}+1,1,-iAs^{\prime}_{0}s^{\prime}_{h})$. Using the known
relation $M(a,b,z)=e^{z}M(b-a,b,-z)$, together with equations (40) and (42) we
obtain

$G_{h}=i\frac{k}{2}\chi_{h}\exp\left[-i\frac{k}{2}\chi_{0}(\sigma_{0}+\sigma_{h})+i\phi_{2}(\sigma_{h})-iA\sigma_{0}s^{\prime}_{h}-iAs^{\prime}_{0}s^{\prime}_{h}\right]M(i\frac{\Omega}{A},1,iAs^{\prime}_{0}s^{\prime}_{h})$
(47)

Using equation (9b) and
$s_{0}s_{h}-\sigma_{0}s^{\prime}_{h}-s^{\prime}_{0}s^{\prime}_{h}=s_{0}s_{h}-s_{0}s^{\prime}_{h}=s_{0}\sigma_{h}$,
we get

$D_{h}=i\frac{k}{2}\chi_{h}\exp\left[i\frac{k}{2}\chi_{0}(s^{\prime}_{0}+s^{\prime}_{h})-i\phi_{1}(s_{0})+i\phi_{2}(\sigma_{h})-iAs_{0}\sigma_{h}\right]M(i\frac{\Omega}{A},1,iAs^{\prime}_{0}s^{\prime}_{h}),$
(48)

which, including equation (8), is equation (11).

In the symmetric Laue case, for which $A=0$, we obtain from equation (B)

$F_{h}=i\frac{k}{2}\chi_{h}EJ_{0}(2\sqrt{\Omega
s^{\prime}_{0}s^{\prime}_{h}}),$ (49)

consequently, equation (11) becomes

$D_{h}=i\frac{k}{2}\chi_{h}\exp\left[i\frac{k}{2}\chi_{0}(s^{\prime}_{0}+s^{\prime}_{h})-i\vec{h}.\vec{u}(s_{0},\sigma_{h})\right]J_{0}(2\sqrt{\Omega
s^{\prime}_{0}s^{\prime}_{h}}).$ (50)

In the opposite case when $A>>\Omega$, an approximated solution could be
obtained by considering the simplified integral equation

$F_{h}(s^{\prime}_{0},s^{\prime}_{h})=i\frac{k}{2}\chi_{h}E-iAs^{\prime}_{0}\int_{0}^{s^{\prime}_{h}}d\xi_{h}F_{h}(s^{\prime}_{0},\xi_{h}),$
(51)

with solution
$F_{h}(s^{\prime}_{0},s^{\prime}_{h})=i\frac{k}{2}\chi_{h}E\exp[-iAs^{\prime}_{0}s^{\prime}_{h}]$
from which equation (13) of section 3 is obtained.

##  Appendix C Expression of the phase factor in Laue geomety and derivation
of equation (30)

.

The components of the displacement field, in the case of meridional bending of
radius $R$ are [Nesterets]:

$u_{x}=-\frac{x(z-t/2)}{R};\,u_{z}=\frac{x^{2}+\rho(z-t/2)^{2}}{2R},$ (52)

with $\rho=\nu/(1-\nu)$, and $\nu$ the Poisson ratio. Note that
$h_{x}=k(\sin\theta_{2}-\sin\theta_{1})$,
$h_{z}=k(\cos\theta_{2}-\cos\theta_{1})$. In terms of the oblique coordinates
$(s_{0},s_{h})$ along $\vec{k}_{0,h}$, such that
$z=s_{0}\cos\theta_{1}+s_{h}\cos\theta_{2}$ and
$x=s_{0}\sin\theta_{1}+s_{h}\sin\theta_{2}$ , it is found, by lengthy but
simple calculations and with omission of a constant term, that
$\vec{h}.\vec{u}=-As_{0}s_{h}+\phi_{1}(s_{0})-\phi_{2}(s_{h})$, with the
following definitions

$A=-(2k\sin\theta_{B}/R)\sin\alpha[1+(1+\rho)\cos\theta_{1}\cos\theta_{2}]$
(53) $\displaystyle\phi_{1}(s_{0})$
$\displaystyle=\frac{k}{2R}[\mu_{1}(s_{o}\sin 2\theta_{B})^{2}-a_{1}s_{0}\sin
2\theta_{B}]$ $\displaystyle\phi_{2}(s_{h})$
$\displaystyle=-\frac{k}{2R}[\mu_{2}(s_{h}\sin 2\theta_{B})^{2}-a_{2}s_{h}\sin
2\theta_{B}],$ (54)

where $\gamma=\cos\theta_{1}/\cos\theta_{2}$,
$\theta_{1,2}=\alpha\pm\theta_{B}$,

$\displaystyle g$ $\displaystyle=\frac{A\gamma
R}{k\sin^{2}2\theta_{B}}=-\sin\alpha\frac{\gamma+(1+\rho)\cos^{2}\theta_{1}}{\sin
2\theta_{B}\cos\theta_{B}},$ $\displaystyle\mu_{1,2}$
$\displaystyle=\frac{\sin\alpha(\sin^{2}\theta_{1,2}+\rho\cos^{2}\theta_{1,2})+\cos\alpha\sin
2\theta_{1,2}}{\sin 2\theta_{B}\cos\theta_{B}},$ $\displaystyle a_{1,2}$
$\displaystyle=t\frac{\cos\alpha\sin\theta_{1,2}+\rho\sin\alpha\cos\theta_{1,2}}{\cos\theta_{B}}.$

The parameter $\rho$ is eliminated in the following expressions

$\displaystyle\mu_{1}+g=$ $\displaystyle\frac{\cos\alpha\sin
2\theta_{1}-\sin\alpha(\gamma+\cos 2\theta_{1})}{\sin
2\theta_{B}\cos\theta_{B}},$ (55) $\displaystyle\gamma^{2}\mu_{2}+g=$
$\displaystyle\frac{\gamma^{2}\cos\alpha\sin
2\theta_{2}-\sin\alpha(\gamma+\gamma^{2}\cos 2\theta_{2})}{\sin
2\theta_{B}\cos\theta_{B}}.$ (56)

Using

$\mu_{1}+\gamma^{2}\mu_{2}+2g=\frac{\sin(\alpha+2\theta_{B})\cos^{2}\theta_{2}+\sin(\alpha-2\theta_{B})\cos^{2}\theta_{1}-2\sin\alpha\cos\theta_{1}\cos\theta_{2}}{\sin
2\theta_{B}\cos\theta_{B}\cos^{2}\theta_{2}},$ (57)

by some cumbersome algebraic manipulations, the numerator of this last
expression is reduced to $(\sin\alpha\sin^{2}2\theta_{B})$ from where it is
easy to get equation (30).

##  Appendix D Relevance of the lens equation in the symmetric Bragg case

Let us consider the case of a flat non-absorbing crystal plate (without
bending), in symmetrical Bragg geometry. The fact that experimental results
and also numerical calculations [Honkanen2018] of Bragg diffraction with plane
crystals do not show any focusing effect (contrary to Laue case), can be
loosely explained by the following intuitive approach. Consider that any
geometrical ray emitted from a real distant point-source produces a reflected
ray at the point of incidence on the crystal surface, with a reflectivity
coefficient $r$ equal to the complex reflectivity of the incident plane wave
having the same glancing angle of incidence $\theta=\theta_{B}+\Delta\theta$
as the geometrical ray under consideration

$r(\Delta\theta)=\sqrt{1-\left(\frac{\Delta\theta\sin
2\theta_{B}}{|\chi_{h}|}\right)^{2}}+i\frac{\Delta\theta\sin
2\theta_{B}}{|\chi_{h}|}=e^{i\arcsin{\frac{\Delta\theta\sin
2\theta_{B}}{|\chi_{h}|}}}.$ (58)

Note that $|r(\Delta\theta)|^{2}$ is the usual diffraction profile. Taking the
origin of coordinates at the point corresponding to $\theta=\theta_{B}$, the
reflected wave-amplitude along an axis $O\xi$ situated in the diffraction
plane and perpendicular to the reflected direction, at negligible distance
from the crystal, may be approximated by setting $\Delta\theta=\xi/p$ in
equation (58):

$D(\xi)=e^{i\arcsin{\frac{\xi\sin 2\theta}{p|\chi_{h}|}}}.$ (59)

No focusing effect is expected from this amplitude distribution, because the
phase function $\arcsin(\xi\sin 2\theta/(p|\chi_{h}|))$ is an odd function of
$\xi$, thus it does not have a second-order term characteristic of a focusing
effect. The first-order term produces a lateral shift of the image. There is
no equivalent to the dynamical focusing length $q_{dyn}$ introduced in the
Laue case. The reflected beam is indeed divergent, as in the case of a usual
mirror.

The focusing properties of cylindrically bent crystals in symmetric Bragg
geometry were simulated by [sutter2010], using a finite-difference method, and
by [honkanen2017, Honkanen2018], using a finite-element method for numerical
solution of the TTE. The obtained phase distribution of the reflected
wavefront shows a parabolic shape, with concavity inversion as compared to the
parabolic phase distribution of the incident wavefront. This is a clear
indication of a single real focusing effect, which is indeed confirmed by
simulating the reflected wave propagation. The obtained focusing distances are
indeed in good agreement with the CLE which is
$L_{0}^{-1}+L_{h}^{-1}=2/(R_{c}\sin\theta_{B})$ in this case.


# High-power and narrow-linewidth laser on thin-film lithium niobate enabled
by photonic wire bonding

Cornelis A.A. Franken  John A. Paulson School of Engineering and Applied
Sciences, Harvard University, Cambridge, MA 02138, USA  Laser Physics &
Nonlinear Optics, Department of Science and Technology, MESA+ Institute of
Nanotechnology, University of Twente, Enschede, NL  These authors contributed
equally to this work.  Rebecca Cheng  John A. Paulson School of Engineering
and Applied Sciences, Harvard University, Cambridge, MA 02138, USA  These
authors contributed equally to this work.  Keith Powell  John A. Paulson
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA
02138, USA  Georgios Kyriazidis  John A. Paulson School of Engineering and
Applied Sciences, Harvard University, Cambridge, MA 02138, USA  Victoria
Rosborough  Freedom Photonics, 41 Aero Camino, Goleta CA 93117, USA  Juergen
Musolf  Freedom Photonics, 41 Aero Camino, Goleta CA 93117, USA  Maximilian
Shah  John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, MA 02138, USA  Williams College, Williamstown, MA
01267, USA  David R. Barton III  John A. Paulson School of Engineering and
Applied Sciences, Harvard University, Cambridge, MA 02138, USA  Department of
Materials Science and Engineering, Northwestern University, Evanston, IL
60208, USA  Gage Hills  John A. Paulson School of Engineering and Applied
Sciences, Harvard University, Cambridge, MA 02138, USA  Leif Johansson
Freedom Photonics, 41 Aero Camino, Goleta CA 93117, USA  Klaus-J. Boller
Laser Physics & Nonlinear Optics, Department of Science and Technology, MESA+
Institute of Nanotechnology, University of Twente, Enschede, NL  Marko Lončar
John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, MA 02138, USA  Corresponding author:
<EMAIL_ADDRESS>
(July 5, 2024)

Thin-film lithium niobate (TFLN) has emerged as a promising platform for the
realization of high-performance chip-scale optical systems, spanning a range
of applications from optical communications to microwave photonics. Such
applications rely on the integration of multiple components onto a single
platform. However, while many of these components have already been
demonstrated on the TFLN platform, to date, a major bottleneck of the platform
is the existence of a tunable, high-power, and narrow-linewidth on-chip laser.
Here, we address this problem using photonic wire bonding to integrate optical
amplifiers with a thin-film lithium niobate feedback circuit, and demonstrate
an extended cavity diode laser yielding high on-chip power of 78 mW, side mode
suppression larger than 60 dB and wide wavelength tunability over 43 nm. The
laser frequency stability over short timescales shows an ultra-narrow
intrinsic linewidth of 550 Hz. Long-term recordings indicate a high passive
stability of the photonic wire bonded laser with 58 hours of mode-hop-free
operation, with a trend in the frequency drift of only 4.4 MHz/h. This work
verifies photonic wire bonding as a viable integration solution for high
performance on-chip lasers, opening the path to system level upscaling and
Watt-level output powers.

## Introduction

Integrated photonics shows promise for the realization of low-cost, energy-
efficient, and scalable solutions for numerous applications across
communications, computation, and sensing. In the past decade, the thin-film
(TF) lithium niobate (LN) platform has positioned itself as a promising
alternative to silicon on insulator, boasting many material properties that
other platforms lack: its large electro-optic coefficient has made it an
excellent host for efficient high-speed, wide-bandwidth modulators; its large
second order nonlinearity and ability to be periodically poled allows for
efficient nonlinear three-wave mixing processes; and its wide transparency
window allows for low-loss propagation over a wide range of wavelengths [1].
Recently, a slew of TFLN components have been demonstrated, including high-
performance electro-optic modulators [2, 3], broadband supercontinuum [4, 5]
and Kerr comb [6, 7] generation, nonlinear frequency conversion [8, 9], and
more, confirming it as an excellent platform for large-scale and diverse
photonic systems. Before fully-integrated systems on TFLN can be realized, a
key element to drive on-chip systems remains an outstanding challenge: a
tunable, narrow-linewidth and high-power laser that removes the need for an
off-chip, bulk-optic source.

Figure 1: Photonic wire bonded TFLN-based extended cavity laser. a, Photograph
of the fully integrated laser realized using an InP chip with two amplifier
waveguides and a TFLN feedback circuit, with photonic and electrical wire
bonds. The temperature is monitored and stabilized using a thermistor and
Peltier cooler. The laser output is collected using a fiber array. b,
Microscope image of the InP-LN interface, where 1 and 2 denote the two
amplifiers in the laser cavity, with photonic wire bonds and waveguides angled
to mitigate Fresnel reflections. c, Scanning electron microscope image of a
photonic wire bond. with anchoring structure, to a TFLN waveguide.

As integrated photonic systems continue to scale up, co-integration of lasers
and various TFLN devices would eliminate the large cumulative insertion losses
from using bulk components, increasing signal-to-noise ratio and resulting in
more compact, energy efficient, and higher performance on-chip systems [10].
Due to the lack of a direct bandgap in lithium niobate, different laser
integration approaches have been pursued. Aside from rare-earth doped lithium
niobate lasers [11, 12], the majority of TFLN laser development has been
focused on integration methods with semiconductor gain materials, including
heterogeneous [13, 14] and hybrid integration [15, 16, 17, 18]. However,
significant improvement is still needed to achieve the combination of high
performance with scalability. Devices using heterogeneous integration
approaches require complex, wafer scale fabrication strategies and have yet to
show high-power operation [19]. On the other hand, traditional hybrid
strategies relying on precise, sub-wavelength alignment accuracy of components
have shown promising performance, but have yet to be effectively adopted at
the wafer scale due to these stringent alignment requirements [20]. While
required performance metrics are application-specific, as a benchmark for
laser performance we use current commercial bulk-optic systems, which offer in
the order of 100 mW fiber-coupled output power, with gain-wide tunability and
linewidths in the sub 100-kHz range. Besides the performance metric, here
“scalability” refers not only to a fabrication and integration process that
allows for high-volume production, but also to the capability of
straightforward integration of many separate components, from optical
amplifiers and modulators to fiber arrays.

An emerging integration technology that could address scaling issues without
compromising performance is photonic wire bonding (PWB), which enables three-
dimensional and low-loss optical connections between different photonic chips
via polymer waveguides [21]. Importantly, the PWB formation process not only
can take into account the relative position of different facets, but also can
be adapted to match different optical mode profiles. Therefore, it can
overcome large misalignment and mode mismatch between a whole range of
components within a system, including but not limited to amplifiers,
waveguides, and fibers. Previous work has already heralded the strength of PWB
combined with silicon photonics, showing its applicability for high
confinement platforms [22, 23, 24]. However, the TFLN platform, with its many
functionalities, enables PWB to tailor to a much wider range of applications.

Here, we present a photonic wire bonded, extended cavity laser that integrates
indium phosphide (InP) amplifiers with a lithium niobate feedback circuit
(Fig. 1a). As an initial demonstration of the flexibility of the photonic wire
bonding integration process, we realize upscaling in performance and
integration through PWB of a second amplifier waveguide to one cavity (Fig.
1b), allowing for power scaling in such lasers with possibility of adding
additional amplifiers as needed.

## Design, fabrication and integration

Figure 2: Lithium niobate extended laser cavity. a, Schematic showing the
design of the laser, which comprises of two InP amplifier waveguides, an LN
feedback circuit and a fiber array, with all components integrated by photonic
wire bonding. The main feature of the feedback circuit is the Vernier filter,
consisting of two coupled micro-ring resonators, which provides frequency
selective feedback and cavity extension to the laser. The effective length of
laser cavity, center frequency of the filter, and outcoupling is finely
adjusted using thermo-optic tuning of the phase shifter, Vernier filter and
tunable outcoupler, respectively. b,  Schematic of the tunable outcoupler,
realized through the use of two tunable Mach-Zehnder interferometers (MZI). c,
d, and e, Microscope images of one of the MZIs, the phase shifter, and Vernier
filter, respectively.

A major advantage of hybrid integration is the separate fabrication of optical
amplifiers and photonic feedback circuits. This benefit remains for our laser,
allowing the amplifiers to be fabricated in a dedicated semiconductor foundry
for uncompromised performance. Likewise, our lithium niobate fabrication
process needs no adaptation for the laser integration process. After
fabrication of the individual components, the integration process starts by
coarsely placing and aligning the individual amplifier, TFLN, and fiber array
on a common submount. The tool applying the photonic wire bonds (Vanguard
Sonata 1000) offers a large write field, allowing for a wide tolerance of the
optical interface alignment of approximately 320 x 325 x 250 $\mu$m for the
$x$-, $y$-, and $z$\- direction, respectively (PWB is written along the
$x$-direction, and $z$ is height). The position and angle of the optical
components are detected using a confocal microscope and machine vision
algorithm. The topside of the InP is partially obscured due to a metal layer
used for applying current to the amplifier, reducing the detection accuracy
when compared to the TFLN waveguide and fiber core. Therefore, photonic wire
bond alignment on the amplifier side typically requires a refined calibration
of the photonic wire bond position using experimentally obtained transmission
data. Such a calibration is required only once for a set amplifier material
stack and amplifier waveguide geometry. Using the location of the facets and
user input on the mode profiles at each interface, the PWB structure is
consequently written in a negative photoresist using two-photon polymerization
with a pulsed near-infrared laser. The PWB structure is routed to maximize the
transmission, by ensuring a large bend radius ($\mathrm{>}$ 60 $\mu$m) and
using sufficiently long tapers ($\mathrm{>}$ 100 $\mu$m per taper). After
development of the resist, for endured stability and increased transmission,
all PWB interfaces are cladded with a UV-cured epoxy. A scanning electron
microscope (SEM) image of a photonic wire bond to a TFLN spot size converter
[25] before cladding is shown in Fig. 1c. The single-pass PWB loss between
amplifier and LN is measured at 4 dB, while the LN to fiber PWB loss is 3 dB.
To our knowledge, these are the lowest photonic wire bonding losses reported
so far for the thin-film lithium niobate platform interfacing with a fiber or
semiconductor amplifier.

The TFLN waveguide circuit provides the cavity extension and frequency
selection in the laser using a Vernier filter consisting of two tunable,
micro-ring resonators [26]. A full schematic of the dual-gain laser design is
shown in Fig. 2a, which depicts the Vernier filter with two sequentially
coupled micro-ring resonators with a difference in free spectral range (FSR)
of approximately 2 GHz. The Vernier filter allows for coarse tuning of the
wavelength while, the effective length of the laser cavity, hence cavity
resonance, can be fine-tuned using the thermo-optic phase shifter. Optimizing
the output coupling of the laser allows for maximum extraction of optical
power at a set current and Vernier filter setting, this is achieved using a
tunable outcoupler. In our design, this outcoupler is realized through two
cascaded tunable Mach-Zehnder interferometers (MZI), giving full control to
the splitting of feedback to the amplifier and power to the output waveguide
(Fig. 2b). The rings, couplers, and cavity length are actively tuned using Ti-
Pt thermo-optic heaters (Fig. 2c,d,e). Separate characterization on a Mach-
Zehnder test structure shows a phase shifting efficiency of about 125 mW/$\pi$
and full control over the power splitting between 0 and 100%. Heaters at the
ring resonators allow for shifting the resonance frequency over the full FSR
(measured at around 113 GHz), with an efficiency of about 150 mW/$\pi$. Using
measured transmission spectra of ring resonators fabricated on the same chip,
we infer a low propagation loss in the TFLN circuit of 5.8 dB/m. The low loss
is mainly attributed to the high confinement of the optical mode and low
sidewall roughness. After photonic wire bonding all optical components, the
packaging is completed using gold wire bonds for the electrical connections
and a Peltier element with on-chip thermistor for temperature control.

Figure 3: Laser power, side mode suppression, and wavelength tuning. a, On-
chip optical power as a function of current applied to each amplifier. With a
laser threshold of 100 mA, the laser shows a linear power increase as a
function of current. Residual variations to the linear trend are caused by
longitudinal mode hops. The maximum power is obtained by setting the
amplifiers at different currents, 200 mA and 800 mA, respectively. b, Optical
spectrum showing single-frequency operation with 61 dB side mode suppression.
c,  Superimposed optical spectra obtained by varying the heater setting for
one of the ring resonators in the Vernier filter, showing a wide wavelength
tunability of 43.7 nm.

## Experimental results

With the laser in operation, we find a laser threshold at 100 mA and an
approximately linear increase of output power with current (Fig. 3a). At each
current, the Vernier filter central wavelength is optimized together with the
cavity phase section to obtain single-mode oscillation with maximum output
power and side mode suppression. To achieve the highest power measurement,
different currents were applied to the amplifiers, 200 mA and 800 mA
respectively, at which a high, on-chip output power of 78 mW was achieved.
Optical spectra (Yokogawa AQ6370) confirm that, at all currents reported here,
the laser oscillates single longitudinal mode with a high side mode
suppression ratio up to 61 dB (Fig. 3b). The laser wavelength can be tuned
coarsely by varying the effective length of one the ring resonators in the
Vernier filter using a thermo-optic heater. Here, the laser is widely tuned in
wavelength around 1530 nm over 43.7 nm, which is beyond the 3-dB gain
bandwidth of both amplifiers. Superimposed spectra collected at different
settings for the Vernier filter central wavelength is given in Fig. 3. A high
optical power, 15 mW on average, and single mode operation with
$\mathrm{>}$52.7 dB side mode suppression across the entire tuning range is
measured.

Figure 4: Laser frequency stability over short and long timescales. a,
Frequency noise power spectral density acquired using delayed self-heterodyne
measurement, showing an ultra-narrow 550 Hz intrinsic linewidth. Inset: beat
note recorded in the frequency domain with a Voigt fit yielded a similar
intrinsic linewidth. b, As part of a multi-day long term recording, the
optical spectrum was recorded. Here, a cumulative spectral plot shows the
laser spectrum during the 58 hours of mode-hop-free operation with a
sustained, high side mode suppression ratio of 55.5 $\mathrm{\pm}$ 0.4 dB. c,
On-chip optical power recording during the long measurement. We show the
region of mode-hop-free operation and denote the observed mode hop in grey. d,
Long term frequency drift of the laser collected using a high-resolution
wavelength meter. The laser exhibits mode-hop-free operation for 58 hours.
After initialization and before the mode hop, we observe a trend in the
frequency drift of 4.4 MHz/h. The observed mode hop is highlighted in grey.

Equally important to output power and wavelength tunability is the passive
long and short-term frequency stability of the laser to enable low-noise
devices, necessary for example in microwave photonics applications [10]. Here,
the short term frequency stability can be characterized by measuring the
intrinsic linewidth of the laser [27, 26]. Due to the long, extended cavity of
our laser design we expect a low intrinsic linewidth, calculated to be in the
order of 100 Hz. However, the resolution of the optical spectrum analyzer (20
pm or 2.5 GHz) used for the measurements shown in Fig. 3 is not sufficient to
resolve the linewidth of the laser. To resolve such an ultra-low intrinsic
linewidth a delayed self-heterodyne set-up is employed [28] with a 19.6 m
length delay. The measured frequency noise power spectral density, computed
from five 100-ms long oscilloscope (Keysight UXR1102B) traces of the
heterodyne beat note [29], is shown in Fig. 4a. For this measurement, the
amplifiers of the laser are set to respectively 225 and 150 mA and the laser
produces an on-chip power of 21.3 mW at a wavelength of 1527.8 nm. The
frequency noise flattens at the quantum-limited noise around 2 MHz, at a level
that corresponds to an ultra-narrow intrinsic linewidth of 550 Hz. A similar
linewidth, within the experimental error, is measured by recording the beat
note in the frequency domain using a long delay of 20 km and performing a
Voigt fit to the line shape. We note that for both cases, we are not limited
by the noise floor of our measurement.

In contrast to mechanical, acoustic and thermal alignment drifts of bulk-optic
extended cavity lasers, integrated photonic circuits typically show excellent
passive and long-term stability, an essential benefit for the scaling to
large, fully-integrated systems. Previous demonstrations of chip-based
external cavity diode lasers on silicon nitride have realized stable mode-hop-
free operation of 5.7 hours for stage-coupled lasers [30] and up to 120 hours
for actively stabilized, integrated lasers [28]. To investigate the passive,
long-term frequency stability of the laser without any active stabilization,
we operate and monitor its performance continuously over multiple days. The
current on both amplifiers is set just above threshold at 125 mA current to
minimize thermal fluctuations, corresponding to an initial on-chip output
power of 2.6 mW. After the thermo-optic heaters are set, the laser output is
monitored continuously using an optical spectrum analyzer (Yokogawa AQ6370),
power meter (Thorlabs S155C), and a high resolution wavelength meter (High
Finesse WS6-200 IR, resolution of 5 MHz) to assess the spectral
characteristics, optical power, and frequency drift, respectively. Following
initial start up and when thermal equilibrium is reached, the laser maintains
stable and mode-hop-free over a 58-hour period (Fig. 4b, c ,d). The laser
emission spectrum over time is shown in a cumulative spectral plot in Fig. 4b,
with a sustained, high side mode suppression ratio of 55.5 $\mathrm{\pm}$ 0.4
dB during the mode-hop-free operation. The on-chip optical power over time is
given in Fig. 4c, with less than $\pm 0.2$ mW variation. Finally, Fig. 4d
shows the frequency drift of the laser extracted from the wavemeter
measurement. We note excellent passive frequency stability, with a trend in
the frequency drift of 4.4 MHz/h.

## Discussion and conclusion

In conclusion, we demonstrate a high-power, passively stable, and ultra-narrow
linewidth extended cavity laser integrated with photonic wire bonding. The
maximum on-chip output power exceeds existing TFLN extended cavity diode
lasers by 1-2 orders of magnitude [15, 13], due to the power scaling of the
dual amplifier design enabled by photonic wire bonding. To our knowledge, our
on-chip power exceeds all single-gain ECLs across all photonic platforms, with
performance comparable to or approaching that of other dual-gain ECLs
demonstrated on the more mature silicon and lower-loss silicon nitride
platforms [26, 31, 32]. Currently, our laser shows the lowest intrinsic
linewidth of any on-chip laser source demonstrated using TFLN. We attribute
the narrow linewidth mainly to the low-loss and long extended cavity, with an
effective length of about 24 cm. The highly frequency selective Vernier design
allows the laser wavelength to be tuned over twice the range of what has been
reported before [15]. As far as we know, this work is a first demonstration of
long-term frequency stability for TFLN-based extended cavity diode lasers. Our
lithium niobate laser also shows comparable or improved passive stability to
similar Vernier-based lasers demonstrated using silicon nitride [28, 33].

Although all of the measurements presented here show single frequency
operation with high side mode suppression, we found that at higher currents
($\mathrm{>}$ 525 mA) the laser is more likely to oscillate multi-mode,
similar to what has been observed in other recent work [34]. The reason for
the decreased stability regime at higher powers is likely due to the coupling
loss between the InP and TFLN waveguides, which could be improved upon in
future iterations. Furthermore, it became apparent that the laser performance
had degraded with time, seen as a lower maximum power for measurements taken
after the initial 78 mW measurement. We attribute this to gradual moisture
diffusion into the top oxide cladding leading to increased absorption and
optical loss. Currently, high-quality LPCVD oxides are difficult to integrate
with the lithium niobate fabrication process due to the high temperatures
required for deposition. As a consequence, we resort to a lower temperature
PECVD $\mathrm{SiO_{2}}$. To suppress the moisture diffusion into the top
oxide, an aluminum oxide sealing layer, deposited by ALD, was used to slow the
degradation process down, though it does not eliminate it fully. To assess the
quality of the oxide cladding and the efficacy of the sealing layer, optical
$Q$ measurements on the TFLN chip are used to evaluate loss. Immediately after
processing, the inclusion of the PECVD oxide cladding leads to a 20% reduction
in the optical $Q$ compared to the uncladded case, pointing to reduced optical
quality of our current oxide. Moreover, with the ALD layer, the optical $Q$
degrades by 15% over two weeks of aging in environment compared to
$\mathrm{>}$50% with no sealing layer. A long hot plate bake
($\mathrm{>}$85∘C) before integration can recover any degradation of the $Q$,
but is currently unfeasible for the fully integrated device. We believe that
higher optical powers could be achieved reproducibly with the implementation
of higher density oxides that are compatible with the lithium niobate
fabrication process.

Looking forward to higher power applications, the slope efficiency and maximum
laser power can be improved significantly by increasing the PWB transmission
and reducing the LN propagation loss to $\mathrm{<}$3 dB/m [35, 36, 37], which
would lower the intra-cavity loss. We expect that in future demonstrations,
PWB loss can be improved through further optimization of the bond alignment
and improvement of the facet quality. Moreover, losses could be reduced
further with the improvement to the TFLN coupler design. In further
experiments, the PWB showed no degradation with up to 1 W of optical power in
the bond, verifying that high intra-cavity powers could be supported.
Furthermore, photonic wire bonding to tapered amplifiers could open the path
towards optical powers in the Watt-level regime.

In addition to completing the TFLN platform with a tunable, narrow-linewidth
and high-power laser, our work verifies photonic wire bonding as a scalable
solution for performance and integration. The power scaling through photonic
wire bonding can be used to enable different nonlinear processes, while a
combination with fast modulation can be useful for microwave photonic
applications [10]. Future work involves integration with optical isolators
[38] for a fully protected laser cavity, as well as other functionalities of
the TFLN platform, such as high performance electro-optic modulators [2, 39],
electro-optic frequency comb sources [40, 41], and high-speed on-chip
detectors [42].

## Acknowledgments

The authors thank Jeremiah Jacobson for the photograph in Fig. 1a, Donald Witt
for the image in Fig. 1c, Xinrui Zhu and Yaowen Hu for providing a
characterization chip for the LN-fiber PWB loss, Allen Chu and Henry Garrett
for initial PWB characterization, and Yunxiang Song, Lisa V. Winkler, Letícia
Magalhães, and Amirhassan Shams-Ansari for support during the project. Lithium
niobate fabrication was performed in the Center for Nanoscale Systems at
Harvard University.

## Funding

Defense Advanced Research Projects Agency (DARPA) HR0011-20-C-0137, Office of
Naval Research (ONR) N00014-22-C-1041, National Science Foundation (NSF)
EEC-1941583 and OMA-2137723.

## Disclosures

V.R., J.M., and L.J. are involved in developing III-V technologies at Freedom
Photonics. R.C. and M.L. are involved in developing lithium niobate
technologies at HyperLight Corporation.

## Data availability

Data available upon reasonable request.

## Disclaimer

The views, opinions and/or findings expressed are those of the authors and
should not be interpreted as representing the official views or policies of
the Department of Defense or the U.S. Government.

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