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polymer_app_layout_template

polymer_app_layout.behaviors

Element

createFragment

createFragment

polymer_app_layout_template

polymer_app_layout.behaviors

Element

createFragment

method
 createFragment

Source

polymer_app_layout_template

polymer_app_layout.behaviors

Element

Constants

abortEvent

beforeCopyEvent

beforeCutEvent

beforePasteEvent

blurEvent

canPlayEvent

canPlayThroughEvent

changeEvent

clickEvent

contextMenuEvent

copyEvent

cutEvent

doubleClickEvent

dragEndEvent

dragEnterEvent

dragEvent

dragLeaveEvent

dragOverEvent

dragStartEvent

dropEvent

durationChangeEvent

emptiedEvent

endedEvent

errorEvent

focusEvent

fullscreenChangeEvent

fullscreenErrorEvent

inputEvent

invalidEvent

keyDownEvent

keyPressEvent

keyUpEvent

loadedDataEvent

loadedMetadataEvent

loadEvent

mouseDownEvent

mouseEnterEvent

mouseLeaveEvent

mouseMoveEvent

mouseOutEvent

mouseOverEvent

mouseUpEvent

mouseWheelEvent

pasteEvent

pauseEvent

playEvent

playingEvent

rateChangeEvent

resetEvent

resizeEvent

scrollEvent

searchEvent

seekedEvent

seekingEvent

selectEvent

selectStartEvent

stalledEvent

submitEvent

suspendEvent

timeUpdateEvent

touchCancelEvent

touchEndEvent

touchEnterEvent

touchLeaveEvent

touchMoveEvent

touchStartEvent

transitionEndEvent

volumeChangeEvent

waitingEvent

Static methods

isTagSupported

Properties

attributes

baseUri

borderEdge

childNodes

children

classes

className

client

clientHeight

clientLeft

clientTop

clientWidth

contentEdge

contentEditable

contextMenu

dataset

dir

documentOffset

draggable

dropzone

firstChild

hidden

id

innerHtml

isContentEditable

lang

lastChild

localName

marginEdge

namespaceUri

nextElementSibling

nextNode

nodeName

nodes

nodeType

nodeValue

offset

offsetHeight

offsetLeft

offsetParent

offsetTop

offsetWidth

on

onAbort

onBeforeCopy

onBeforeCut

onBeforePaste

onBlur

onCanPlay

onCanPlayThrough

onChange

onClick

onContextMenu

onCopy

onCut

onDoubleClick

onDrag

onDragEnd

onDragEnter

onDragLeave

onDragOver

onDragStart

onDrop

onDurationChange

onEmptied

onEnded

onError

onFocus

onFullscreenChange

onFullscreenError

onInput

onInvalid

onKeyDown

onKeyPress

onKeyUp

onLoad

onLoadedData

onLoadedMetadata

onMouseDown

onMouseEnter

onMouseLeave

onMouseMove

onMouseOut

onMouseOver

onMouseUp

onMouseWheel

onPaste

onPause

onPlay

onPlaying

onRateChange

onReset

onResize

onScroll

onSearch

onSeeked

onSeeking

onSelect

onSelectStart

onStalled

onSubmit

onSuspend

onTimeUpdate

onTouchCancel

onTouchEnd

onTouchEnter

onTouchLeave

onTouchMove

onTouchStart

onTransitionEnd

onVolumeChange

onWaiting

outerHtml

ownerDocument

paddingEdge

parent

parentNode

previousElementSibling

previousNode

scrollHeight

scrollLeft

scrollTop

scrollWidth

shadowRoot

spellcheck

style

tabIndex

tagName

text

title

translate

xtag

Constructors

a

article

aside

audio

br

canvas

created

div

footer

header

hr

html

iframe

img

li

nav

ol

option

p

pre

section

select

span

svg

table

tag

td

textarea

th

tr

ul

video

Methods

addEventListener

animate

append

appendHtml

appendText

attached

attributeChanged

blur

click

clone

contains

createFragment

createShadowRoot

detached

dispatchEvent

enteredView

focus

getAnimationPlayers

getAttribute

getAttributeNS

getBoundingClientRect

getClientRects

getComputedStyle

getDestinationInsertionPoints

getElementsByClassName

getNamespacedAttributes

hasChildNodes

insertAdjacentElement

insertAdjacentHtml

insertAdjacentText

insertAllBefore

insertBefore

leftView

matches

matchesWithAncestors

offsetTo

query

queryAll

querySelector

querySelectorAll

remove

removeEventListener

replaceWith

requestFullscreen

requestPointerLock

scrollIntoView

setAttribute

setAttributeNS

setInnerHtml

toString

            DocumentFragment
            createFragment(

                String html,
{NodeValidator validator,
NodeTreeSanitizer treeSanitizer}

            )

Create a DocumentFragment from the HTML fragment and ensure that it follows
the sanitization rules specified by the validator or treeSanitizer.

If the default validation behavior is too restrictive then a new
NodeValidator should be created, either extending or wrapping a default
validator and overriding the validation APIs.

The treeSanitizer is used to walk the generated node tree and sanitize it.
A custom treeSanitizer can also be provided to perform special validation
rules but since the API is more complex to implement this is discouraged.

The returned tree is guaranteed to only contain nodes and attributes which
are allowed by the provided validator.

See also:

NodeValidator

NodeTreeSanitizer

Source

/**
 * Create a DocumentFragment from the HTML fragment and ensure that it follows
 * the sanitization rules specified by the validator or treeSanitizer.
 *
 * If the default validation behavior is too restrictive then a new
 * NodeValidator should be created, either extending or wrapping a default
 * validator and overriding the validation APIs.
 *
 * The treeSanitizer is used to walk the generated node tree and sanitize it.
 * A custom treeSanitizer can also be provided to perform special validation
 * rules but since the API is more complex to implement this is discouraged.
 *
 * The returned tree is guaranteed to only contain nodes and attributes which
 * are allowed by the provided validator.
 *
 * See also:
 *
 * * [NodeValidator]
 * * [NodeTreeSanitizer]
 */
DocumentFragment createFragment(String html,
    {NodeValidator validator, NodeTreeSanitizer treeSanitizer}) {
  if (treeSanitizer == null) {
    if (validator == null) {
      if (_defaultValidator == null) {
        _defaultValidator = new NodeValidatorBuilder.common();
      }
      validator = _defaultValidator;
    }
    if (_defaultSanitizer == null) {
      _defaultSanitizer = new _ValidatingTreeSanitizer(validator);
    } else {
      _defaultSanitizer.validator = validator;
    }
    treeSanitizer = _defaultSanitizer;
  } else if (validator != null) {
    throw new ArgumentError(
        'validator can only be passed if treeSanitizer is null');
  }

  if (_parseDocument == null) {
    _parseDocument = document.implementation.createHtmlDocument('');
    _parseRange = _parseDocument.createRange();

    // Workaround for Safari bug. Was also previously Chrome bug 229142
    // - URIs are not resolved in new doc.
    var base = _parseDocument.createElement('base');
    base.href = document.baseUri;
    _parseDocument.head.append(base);
  }
  var contextElement;
  if (this is BodyElement) {
    contextElement = _parseDocument.body;
  } else {
    contextElement = _parseDocument.createElement(tagName);
    _parseDocument.body.append(contextElement);
  }
  var fragment;
  if (Range.supportsCreateContextualFragment &&
      _canBeUsedToCreateContextualFragment) {
    _parseRange.selectNodeContents(contextElement);
    fragment = _parseRange.createContextualFragment(html);
  } else {
    contextElement._innerHtml = html;

    fragment = _parseDocument.createDocumentFragment();
    while (contextElement.firstChild != null) {
      fragment.append(contextElement.firstChild);
    }
  }
  if (contextElement != _parseDocument.body) {
    contextElement.remove();
  }

  treeSanitizer.sanitizeTree(fragment);
  // Copy the fragment over to the main document (fix for 14184)
  document.adoptNode(fragment);

  return fragment;
}

                  polymer_app_layout_template 0.1.0 api docs

                •

                •

                  cc license


What happened to the 'Dog Days of August'?
Congress would go on recess, with U.S. senators and House members scattering hither and yon. Out of sight, out of mind.
Oh, and the president would go on vacation, hiding away with his wife and kids; maybe enjoying themselves with extended family members and perhaps a few good friends.
No more, man! Not with this president or this Congress. I want to thank Donald Trump and congressional leadership for providing bloggers such as me and full-time print and broadcast journalists with plenty of grist that will carry us through the era known formerly as the Dog Days.
Senate Majority Leader Mitch McConnell has kept senators on the job through the summer recess. House members have gone about doing whatever it is they when they're not prowling the Capitol Hill halls of power.
As for the president, he hasn't let up one bit while he vacations in New Jersey with his wife and son, Barron.
He's gone after pro football players yet again for protesting police practices against African Americans. He keeps harping on that "witch hunt" that has produced several indictments from the special counsel who's looking for answers to The Russia Thing. He launched creation of the Space Force, the sixth military branch.
There's no let-up. We'll all need to buck ourselves up as we prepare for the home stretch leading toward the highly consequential midterm election.
Let's all get plenty rest. We'll need our strength.


text:
Q: Using Stripe Payments With Angular/Firestore I'm currently having a problem with how to securely implement payments with angular & firstore.
Currently

*

*Items are displayed to the user from a firestore collection.

*User buys item and is redirected to a stripe checkout.

*On the backend webhook listens for checkout.session.completed and saves the order to firestore collection on completion.

My understanding is that with Firestore being accessible to anyone who has access to the webpage, a visitor could easily just write their own order to the collection & even with restricting customers to being logged in as they are directly interacting with the orders collection they do need to be able to update it without maliciously updsting it.
My question: can I mitigate this with security rules alone for example is it possible to pass a secret key with each order & not allowing documents to be written if the key doesn't match. If not are their other options or will I have to restructure the whole app?

redpajama_set_name: RedPajamaStackExchange


text:
Home News Donald Trump Is Already Trying To Skip The Presidential Debates
Donald Trump Is Already Trying To Skip The Presidential Debates
Oh for fuck's sake. Now it's the libruhl's and especially Crooked Hillary's fault that the Presidential debates are scheduled at an inconvenient time for His Holy Cheeto-ness. She is so incredibly gifted and powerful, she was able to rig the debates against him before either of them was even their respective party's nominee. DAYUM That sounds like an amazing endorsement to me! GO HILLARY!
On September 23, 2015, The Nonpartisan Commission on Presidential Debates announced the formats, dates, and locations for the three presidential debates and the one vice-presidential debate.
"The CPD did not consult with any political parties or campaigns in making these decisions," the group said.
According to commission officials, sporting events, religious holidays, and other factors cause scheduling headaches every four years. The group seeks to space out the debates and schedule them for different nights of the week.
The first debate of 2016, slated for September 26, coincides with "Monday Night Football" on ESPN.
The vice presidential debate is scheduled one week later on October 4, a Tuesday.
The second presidential debate, on October 9, coincides with "Sunday Night Football" on NBC.
The final debate is slated for October 19, a Wednesday.
NFL games are played on Sundays, Mondays and Thursdays in the fall. Fridays and Saturdays are effectively ruled out because TV viewing is lower on weekend evenings. That only leaves Tuesdays and Wednesdays.
One complication was Yom Kippur on Tuesday, October 11 and Wednesday, October 12. source
For the love of all that is holy, do NOT let the alt-right know they worked around Yom Kippur! That might just make the folks at /r/the_Donald or Breitbart lose their minds.
Now, if only there were some super secret technological device that would allow us to watch one and record the other. Or watch both at the same time. Bueller? Bueller?
Yes. Absolutely. Agreed on by whom? DC elites. They should try serving the audience and work around sports https://t.co/7LaOIIs8RB
— Hugh Hewitt (@hughhewitt) July 31, 2016
Professional sports are so much more important than Presidential Elections. Goddamn elitists! Trying to teach their children about bullshit civics at the expense of vital and pressing matters like underinflated footballs and head injuries and whether or not domestic violence is more important than touchdowns.
Trump acted as if the dates were news, however. In an interview with ABC News, Trump said of the debate scheduling, "I got a letter from the NFL saying, 'This is ridiculous.'"
The NFL denied that. "While we'd obviously wish the debate commission could find another night, we did not send a letter to Trump," a spokesman said.
On "Reliable Sources," Miller said Trump had been notified of the overlap by a "source close to the league." He dismissed any question over whether Trump had actually gotten a letter from the NFL as "semantics."
Another Trump aide asserted that "millions of voters" will be "disenfranchised by these chosen dates." source
Disenfranchised? Merriam-Webster defines this as "to deprive of a franchise, of a legal right, or of some privilege or immunity; especially : to deprive of the right to vote". The same people who make careers out of actually preventing minorities from voting are complaining that having debates during football games is disenfranchisement?
The real issue here, I believe, is that Election Day is on the same day as NCIS. We have reached out to Mark Harmon for comment, but we are still waiting to hear back. We could avoid most of this bullshittery and make the whole thing a national holiday. Debate days included. Schedule the smackdown for ten o'clock in the morning, and the problem is solved. We all know Secretary Clinton is going to kick his thin-skinned ass in the debates no matter what day or time it is anyway.
Previous articleNewt Gingrich Masterfully Demonstrates Why Republicans Like Feels More Than Facts
Next articleJeff Bezos Revokes Donald Trump's Amazon Shopping Privileges
Virginia Sheriff adds 'In God We Trust' to squad cars
Surprise! Another Case Of Voter Fraud By A Trump Supporter
De Niro Pulls Wakefield's 'Vaxxed' from Tribeca Festival

redpajama_set_name: RedPajamaCommonCrawl


We can relate the Primal Paradox to every stage in the psychological developmental process.
In the Perinatal process, while in utero, we are literally merged with Mother with very little sense of separation. As we come into the physical world, we start with the experience of actually embracing the Paradox of "I" versus "AllElse" during the Perinatal/en Utero process and the birthing process that follows.
The sense of fused union with the cosmos starts to break apart as we emerge through the birth canal. At this point we begin to experience the first tremors of separation from Mother. During the infant years, the "AllElse" is represented primarily by the embracing Mother and Father as well as siblings and others in the family.
This starts to shift around the "terrible twos" when we start to break away from the comfort of Mother and Father and others, on the path to more clearly defining ourselves as an "I." Hopefully this process happens within the context of being able to return to the comfort of the parents. Venturing away from the parents into the "AllElse" becomes an exercise of further defining the sense of self.
In our 20's we solidify our identity or sense of "I" as we graduate, get our first jobs and start to more seriously act to find a partner and start a family.
It's in our 30's and 40's that we have the opportunity to really solidify that personal identity – and the sense of "I" – as the world around and the "AllElse" expands to new dimensions. Our days are filled now with oscillations between our sense of our own importance and needs and our concern with the needs of others we feel a part of.
In our 40's or 50's or later – defined as the mid-life transition — we have another opportunity to broaden and embrace our sense of the Primal Paradox. It is during this mid-life stage that we ponder moving toward what psychologist Abraham Maslow called "self actualization." In self actualization we seek the "right balance" of the terms of the paradox: our self concern is seen as entangled with a concern and a need to put ourselves in service to others.
In his later years, Maslow even talked about the possibility of moving into a stage of what he called "transcending self actualization" where we can start to identify or sense a merging with the Transpersonal or spiritual stages of consciousness.
Finally, at the point of death, we all move back into that ultimate state of union where the paradox of our lives is erased into One.
John Z. Amoroso has maintained a full time transpersonal psychotherapy practice in the Philadelphia area since 1988 with a focus on past life regression. For more than 20 years he has lectured in private venues and taught at the graduate level in the field of transpersonal studies and creativity.
In 2012, he published Awakening Past Lives: A Guidebook to Self Exploration – currently available on Amazon. Having studied painting at the Pennsylvania Academy of the Fine Arts for many years, he is now dedicating most of his time to painting and multi-media art representing the transpersonal or psycho-spiritual dimension of reality.


Rachel Palmén is a researcher at NOTUS. She forms part of the GenPORT consortium which is an FP7 support and coordination action to develop an Internet portal for sharing knowledge and inspiring collaborative action on gender and science. She is the Workpackage 2 leader which involves carrying out a needs assessment with different stakeholders to feed into the development and testing of a classification framework for gender and science resources. She also leads Workpackage 4 which will produce 5 research syntheses and 25 policy briefs. She currently coordinates a civil society dialogue project co-financed by the EU and Turkey concerning Violence against Women in Turkey, Spain and Italy. She completed her doctoral research in 2007 at the Centre for Social and Economic Research at Sheffield Hallam University which looked at community involvement processes in Brazil and the UK. She completed her BA honours degree at Liverpool University in Social and Environmental Studies. She has published various book chapters and articles in a wide range of social science journals. Her main research interests include gender and science, community participation, and public policy. She is interested in cross-national research and works mainly with qualitative data.
Qualitatitive Research, Synthesising information, User needs assessment, facilitating participatory workshops.


12:27 a.m. Traffic stop in the area of N. US Route 31/W. 800 N., Denver. Warning issued.
12:54 a.m. Report of an alarm in the 13000 block of S. US Route 31, Kokomo. No additional information provided.
2:37 a.m. Traffic accident with damage in the area of E. 950 S./S. State Road 19, Amboy. No additional information provided.
3:11 a.m. Report of a suspicious person/circumstance in the 2300 block of S. Dwight Ave., Peru. No additional information provided.
12:37 a.m. An officer provided extra patrol at Maconaquah Park in the 1 block of Park Dr.
1:13 a.m. Traffic stop in the area of N. Broadway St./E. 8th St. Citation and written warnings issued.
1:31 a.m. Officers responded to the report of a suspicious person/circumstance at McClure's South in the 200 block of S. Broadway Ave. Indiana State Police handled this call with the assistance of the Peru Police Department. Unable to locate subject.
12:04 a.m. Traffic stop in the area of W. 100 N./N. US Route 31, Peru. Warning issued.
12:39 a.m. Report of a traffic hazard in the area of S. 150 W./W. 800 S., Bunker Hill. No further information provided.
12:56 a.m. Report of a 911 hang-up at the Heathers Haven MHP in the 4100 block of S. US Route 31, Peru. No further information provided.
1:19 a.m. Traffic stop in the area of W. US Route 24 / N. Eel River Cemetery Rd., Peru. Citation issued.
1:43 a.m. Traffic stop in the area of W. Airport Rd. / S. US Route 31, Peru. Written warning issued.
1:59 a.m. Traffic stop in the area of S. Business 31 / S. US Route 31, Peru. Verbal warning issued.
12:48 a.m. A deputy responded to the 14000 block of N. Macy Millark Rd, Macy for a report of someone in the complainant's barn. The subject was gone upon arrival. The subject told the complainant that he was a recover company representative.
2:51 a.m. Traffic stop in the area of E. 400 S. / S. State Road 19, Peru. Warning issued.
3:23 a.m. Traffic stop in the area of N. 450 E. / E. Wabash Rd., Peru. Warning issued.
1:09 a.m. Traffic stop in the area of W. Main St. / N. Grant St. Written warning issued.
1:23 a.m. Traffic stop in the area of S. Water St. / E. 2nd St. Written warning issued.
1:27 a.m. K9 deployment in the area of S. Water St. / E. 2nd St. Free air sniff.
7:50 a.m. Traffic stop in the area of E. 900 N. / N. Meridian Rd., Denver.
8:01 a.m. Traffic stop in the area of E. 800 N. / N. Meridian Rd., Denver.
10:38 a.m. A report of a juvenile problem at North Miami Schools on E. 900 N., Denver. Assistance was given.


text:
Home/Chief of Staff
Website Massachusetts Department of Energy Resources
Serve as a Counsel II with respect to various energy matters at the Department of Energy Resources' (Department), including administrative proceedings before the Massachusetts Department of Public Utilities as well as other matters involving the energy industry in the Commonwealth.
The Counsel II will work on a variety of complex regulatory matters concerning energy efficiency, renewable and alternative energy, Massachusetts green communities and alternative transportation, as part of a team with other attorneys as well as policy and technical staff (e.g., analysts, engineers and economists).
Proceedings may include adjudications, rulemakings, and other proceedings which may have multiple parties and involve complex legal and policy/technical issues (e.g., rate-setting for public utilities, energy policy and legislation). This position involves a substantial emphasis on legal analysis and writing and requires a minimum of three years of energy and/or public utility law experience.
Responsibilities include, but are not limited to the following:
Serving as a Department Case Manager by tracking procedural schedule and deliverable deadlines, conducting and participating in public hearings and evidentiary hearings and making written and oral legal recommendations:
Working and coordinating with policy and technical staff to ensure the development of thorough pleadings and filings with respect to the Department's positions and findings;
Presenting recommendations and drafting documents on behalf of the Department;
Representing the Department in proceedings before the Massachusetts Department of Public Utilities, including preparing comments, pleadings, briefs, motions, discovery and responses to discovery, witness preparation, cross-examination, and related adjudicatory responsibilities;
Preparing legal research and/or memoranda on various legal issues; and representing the Department in the negotiation of issues or in connection with cases as needed and in proceedings before other state and federal administrative agencies.
Cover letter and resume required.
To apply for this job please visit massanf.taleo.net.
By jessica.ridlen2|2020-08-20T09:44:13-04:00July 2nd, 2020|Comments Off on Chief of Staff


I have 2 more fantastic kits to show you today.
Is called Classic Beauty & was created to match Sebastian Serrano's tube of the same name.
Created in shades of pinks & browns this kit is just gorgeous & says 'elegant' with its papers & elements.
I found this kit enjoyable to work with. The kit contains 17 papers & 95 elements. Another great sized kit to play with.
Don't you just love the dog?
You can also stop by Amy Marie's blog to keep up with the latest going's on. If you do visit please leave some love.
Is called Red Hot & is awesome!!
Created in my favorite colors, red & black, I just loved working with this kit.
FUll of gorgeous elements this kit is a must have.
The kit contains 23 papers & 106 elements.
I have a tag & a timeline set to show you.


.Paradigm is a Whitby 42 ft. center cockpit ketch with a full keel built in 1974 by Whitby Boat Works in Canada, hull # 29. She has a full keel with a 5 ft. draw and her 3blade fixed prop is in a protected aperture. She has a 13 ft. beam and a mainmast height of 55 ft.
Her accommodations include seven berths, two heads, each w/shower, walk through passage to private owner's cabin with its own companionway , and lots of bulk storage plus 4 hanging lockers and many drawers and cabinets throughout.
U-shaped galley, to port, features a three-burner propane stove with oven, double stainless steel sink with three faucets for fresh, filtered, and salt water. Ice box by Technicold, has three large holding plates in two compartments; a 9 cu. ft. refrig. And a 7 cu. ft. freezer.
The walk-through passageway leads to the aft cabin and passes by the workbench with tool storage and the electric control panels across from the large engine room access doors.


text:
Q: Adding image option in custom option magento 1.9 In custom option is a drop-down option with an SKU etc... What I want to know is, is it possible to add image field in magento 1.9

A: you follow this link may be help you :-
https://inchoo.net/magento/add-custom-image-field-custom-options
And second :-
in your comment check this link ---
https://magento.stackexchange.com/a/155310/85907
THANKS.


text:
\section{Introduction}
The absorption of light in thin layers of strongly anisotropic materials has received enormous attention since the dawn of two-dimensional (2D) materials and their heterostructures \cite{Geim2013}. Tremendous progress has been reported using 2D materials for numerous nanophotonic applications such as hybrid graphene-based photodetectors \cite{Gabor2011,Freitag2013}, optoelectronic \cite{He2013,Ross2014} and photovoltaic \cite{Fortin1982,Yu2013} devices employing transition metal dichalcogenide (TMDC) monolayers, enhanced light-matter interaction using photonic integration with optical cavities \cite{Liu2011,Furchi2012}, approaches towards TMDC-based nanolasers \cite{Gan2013,Sobhani2014}, hyperlensing \cite{Dai2015,Ferrari2015} based on hyperbolic polaritons \cite{Ma2018}, bio-sensing \cite{Rodrigo2015}, and thermoelectric applications using black phosphorus (BP) monolayers \cite{Fei2014,Xia2014}. In light of these thriving developments and the great potential entailed in nanophotonic technology, a robust and consistent theoretical framework for the description of light-matter interaction in layered heterostructures of anisotropic materials is of central importance.

In order to understand, analyze and predict the optical response of multilayer structures, the transfer matrix formalism has proven to be of great utility \cite{Scarlat2010,Mounier2011,Ratchford2019}. In isotropic layered media, a $2 \times 2$ transfer matrix fully describes any light-matter interaction, and with knowledge of the local electric and magnetic fields, the optical power flow can be described by the Pointing vector~\SSS~\cite{Chilwell1984,Deparis2011}. However, what already proves intricate in isotropic multilayers, becomes even more cumbersome when the materials are uniaxial or even biaxial requiring a $4 \times 4$ transfer matrix formalism \cite{Berreman1972,Yeh1979,Passler2017a}, as it is the case for many state-of-the-art nanophotonic materials like hexagonal boron nitride (hBN) \cite{Geick1966}, molybdenum trioxide (\ensuremath{\text{MoO}_{\text{3}}}) \cite{Lajaunie2013}, or BP \cite{Liu2017}. In consequence, to the best of our knowledge, previous approaches aiming at the analytical computation of light absorption in anisotropic multilayers are restricted to special cases \cite{Collett1971,GiaRusso1973,Schwelb1986,Ciumac1994}, whereas a fully generalized formalism applicable to any number of layers of media with arbitrary permittivity has not been proposed so far.

In this work, we derive explicit expressions for the layer-resolved transmittance and absorption in stratified heterostructures of arbitrarily anisotropic, birefringent, and absorbing media, using the electric field distribution provided by our previous transfer matrix formalism \cite{Passler2017a,Passler2019b}. Our algorithm is numerically stable, yields continuous solutions, and is implemented in an open access computer program \cite{Passler2020,Jeannin2020}, enabling a robust and consistent framework that is capable of treating light of any polarization impinging at any incident angle onto any number of arbitrarily anisotropic, birefringent, and absorbing layers. To demonstrate the capabilities of our algorithm, we present and discuss simulation results for three nanophotonic device structures, featuring several phenomena such as azimuth-dependent hyperbolic phonon polaritons in a \ensuremath{\text{MoO}_{\text{3}}}~/ aluminum nitride (AlN) / silicon carbide (SiC) heterostructure, layer-selective exciton absorption of molybdenum disulfide (\ensuremath{\text{MoS}_{\text{2}}}) monolayers in a Fabry-P\'{e}rot cavity, and strong light-matter coupling between a cavity mode and an epsilon-near-zero mode in a doped gallium nitride (GaN) multi-quantum well system. Section \ref{sec:theory} summarizes the transfer matrix framework that is used to calculate the electric field distribution and the momenta of the eigenmodes in an anisotropic multilayer system. Based on this theory, section \ref{sec:layer_res_trans_abs} introduces the calculation of the layer-resolved transmittance and absorption. In section \ref{sec:simulations}, the simulation results are presented.

\section{Tranfer Matrix Framework}
\label{sec:theory}
The $4\times 4$ transfer matrix formalism comprising the calculation and sorting of the eigenmodes and the treatment of singularities (Section \ref{sec:matrix_formalism}), the calculation of reflection and transmission coefficients (Section \ref{sec:rt_coefficients}) and of the electric fields (Section \ref{sec:Efield_distribution}) is based on our previous work \cite{Passler2017a}, and therefore is here only briefly summarized in order to provide the necessary framework for the calculation of the layer-resolved absorption (Section \ref{sec:layer_res_trans_abs}).

\subsection{Matrix Formalism}
\label{sec:matrix_formalism}

The incident medium is taken to be non-absorptive with isotropic (relative) permittivity $\ensuremath{\varepsilon}_{\text{inc}}$, while all other media can feature absorption and fully anisotropic (relative) permittivity tensors \ensuremath{\bar{\varepsilon}}. Each permittivity tensor $\ensuremath{\bar{\varepsilon}}_i$ of medium $i$ with principle relative permittivities in the crystal frame $\ensuremath{\varepsilon}_x$, $\ensuremath{\varepsilon}_y$, and $\ensuremath{\varepsilon}_z$ can be rotated into the lab frame using a three-dimensional coordinate rotation matrix \cite{Yeh1979} (Eq.~2 in Ref.~\cite{Passler2017a}). In the following, media with a diagonal permittivity tensor in the lab frame are referred to as non-birefringent, while media with a permittivity tensor that features non-zero off-diagonal elements is called birefringent. Furthermore, all media are assumed to have an isotropic magnetic permeability~$\mu$.

The coordinate system in the lab frame is defined such that the multilayer interfaces are parallel to the $x$-$y$-plane, while the $z$-direction points from the incident medium towards the substrate and has its origin at the first interface between incident medium and layer $i=1$. The layers are indexed from $i=1$ to $i=N$, and the thickness of each layer is $d_i$. Furthermore, $i=0$ refers to the incident medium and $i=N+1$ to the substrate. The plane of incidence is the $x$-$z$-plane, yielding the following wavevector $\ensuremath{\vec{k}}_i$ in layer $i$:
\begin{align}
\ensuremath{\vec{k}}_i=\frac{\omega}{c}(\xi,0,q_i),
\label{eq:kvec}
\end{align}
where $\omega$ is the circular frequency of the incident light, $c$ is the speed of light in vacuum, $\xi=\sqrt{\ensuremath{\varepsilon}_{\text{inc}}} \: \sin(\theta)$ is the in-plane $x$-component of the wavevector which is conserved throughout the entire multilayer system, \ensuremath{\theta}~is the incident angle, and $q_i$ is the dimensionless $z$-component of the wavevector in layer $i$.

In any medium, the propagation of an electromagnetic wave is described by exactly four eigenmodes $j=1,2,3,4$ with different $z$-components \ensuremath{q_{ij}}~of the wavevector. These four \ensuremath{q_{ij}}~can be obtained for each medium $i$ individually by solving the eigenvalue problem of a characteristic matrix \ensuremath{\bold{\Delta}}~(Eq.~11 in Ref.~\cite{Passler2017a}), as has been derived originally by Berreman \cite{Berreman1972}. However, for media with highly dispersive permittivities, the four obtained eigenvalues \ensuremath{q_{ij}}~and their related eigenmodes can switch their order as a function of frequency, and thus have to be identified in an unambiguous manner. Following \etal{Li} \cite{Li1988}, the modes are separated into forward and backward propagating waves according to the sign of \ensuremath{q_{ij}}~(Eq.~12 in Ref.~\cite{Passler2017a}). We assign the forward propagating (transmitted) waves to be described by $q_{i1}$ and $q_{i2}$, and the backward propagating (reflected) waves by $q_{i3}$ and $q_{i4}$. Furthermore, each pair is sorted by the polarization of the corresponding mode, utilizing the electric fields given by the eigenvectors $\Psi_{ij}$~(Eq.~13 in Ref.~\cite{Passler2017a}). In non-birefringent media, the two modes are separated into p-polarized ($q_{i1}$ and $q_{i3}$) and s-polarized ($q_{i2}$ and $q_{i4}$) waves by analyzing the $x$-component of their electric fields. For birefringent media, on the other hand, the sorting is realized by analyzing the $x$-component of the Poynting vector $\ensuremath{\vec{S}}_{ij} = \ensuremath{\vec{E}}_{ij} \times \ensuremath{\vec{H}}_{ij}$, and the modes are separated into ordinary ($q_{i1}$ and $q_{i3}$) and extraordinary ($q_{i2}$ and $q_{i4}$) waves \cite{Passler2017a}.

In the case of non-birefringent media, the four solutions \ensuremath{q_{ij}}~become degenerate, leading to singularities in the formalisms of previous works \cite{Berreman1972,Yeh1979,Lin-Chung1984}. To resolve this problem, we follow the solution presented by \etal{Xu} \cite{Xu2000}. Using the appropriately sorted $q_{ij}$, obtained as described above, the eigenvectors $\vec{\gamma}_{ij}$ of the four eigenmodes in each layer $i$ are:
\begin{align}
\vec{\gamma}_{ij}=\colvec{\gamma_{ij1} \\ \gamma_{ij2} \\ \gamma_{ij3}},
\end{align}
with the values of $\gamma_{ijk}$ given by \etal{Xu} \cite{Xu2000} (Eq.~20 in Ref.~\cite{Passler2017a}), and $k=1,2,3$ being the $x$, $y$, and $z$ components of $\vec{\gamma}_{ij}$. Furthermore, $\vec{\gamma}_{ij}$ has to be normalized:
\begin{align}
\vec{\hat{\gamma}}_{ij}=\frac{\vec{\gamma}_{ij}}{\abs{\vec{\gamma}_{ij}}}.
\label{eq:eigenmodes}
\end{align}
We note that this normalization is essential to ensure a correct calculation of the cross-polarization components of the transfer matrix. The normalized electric field eigenvectors $\vec{\hat{\gamma}}_{ij}$, being free from singularities, replace the eigenvectors $\Psi_{ij}$ for all further calculations in the formalism.

At each interface, the boundary conditions for electric and magnetic fields allow to connect the fields of the two adjacent layers $i-1$ and $i$. Formulated for all four modes simultaneously, the boundary conditions are:
\begin{align}
\AMa{i-1} \ensuremath{\vec{E}}_{i-1}=\AMa{i} \ensuremath{\vec{E}}_i,
\label{eq:A_boundaryShort}
\end{align}
where $\AMa{i}$ is a $4\times4$ matrix calculated from the eigenvectors $\gamma_{ijk}$ \cite{Xu2000} (Eq.~22 in Ref.~\cite{Passler2017a}), and $\ensuremath{\vec{E}}_{i}$ is a dimensionless 4-component electric field vector containing the amplitudes of the resulting electric fields of all four modes. In the following, we refer to $\ensuremath{\vec{E}}_{i}$ as the amplitude vector, and its components are sorted as follows:
\begin{align}
\label{eq:E_xu_sorting}
\ensuremath{\vec{E}}\equiv\colvec{E_{\Rightarrow}^{p/o} \\ E_{\Rightarrow}^{s/e} \\ E_{\Leftarrow}^{p/o} \\E_{\Leftarrow}^{s/e}},
\end{align}
where $\Rightarrow$ ($\Leftarrow$) stands for the forward propagating, transmitted (backward propagating, reflected) modes, and p, s refers to the p- and s-polarized modes in non-birefringent media, while $o$, $e$ indicates the ordinary and extraordinary modes in birefringent media.
By multiplying ${\AMa{i-1}}^{-1}$ on both sides of Eq.~\ref{eq:A_boundaryShort}, we find the implicit definition of the interface matrix \LMa{i}, which projects the amplitude vector in medium $i$ onto the amplitude vector in medium $i-1$:
\begin{align}
\ensuremath{\vec{E}}_{i-1}={\AMa{i-1}}^{-1}\AMa{i}\ensuremath{\vec{E}}_{i}\equiv \LMa{i}\ensuremath{\vec{E}}_{i}.
\end{align}
For the transition between two birefringent or between two non-birefringent media, the projection of a wave of one particular polarization in layer $i$ only yields a finite amplitude in layer $i-1$ of the mode of same polarization, i.e. $s/e \leftrightarrow s/e$, and $p/o \leftrightarrow p/o$. For the transition between a birefringent and a non-birefringent medium, on the other hand, the interface matrix \LMa{i} projects a mode of one particular polarization in layer $i$ onto both polarization states in layer $i-1$. This cross-polarized projection occurs because in birefringent media, the in-plane directions of the ordinary and extraordinary eigenmodes are rotated ($\neq n \pi$, $n \in \mathbb{N}_0 $) with respect to the directions of the p- and s-polarized eigenmodes in the non-birefringent medium.

The propagation of all four eigenmodes through layer $i$ is described by the propagation matrix \PMa{i} \cite{Yeh1979}:
\begin{align}
\PMa{i}\!=\!\fourfourMatrix
{e^{-i\frac{\omega}{c}q_{i1}d_i} & 0 & 0 & 0}
{0 & e^{-i\frac{\omega}{c}q_{i2}d_i} & 0 & 0}
{0 & 0 & e^{-i\frac{\omega}{c}q_{i3}d_i} & 0}
{0 & 0 & 0 & e^{-i\frac{\omega}{c}q_{i4}d_i}}\!,
\end{align}
where the rotation of polarization in birefringent media arises due to a phase difference that is accumulated during propagation through the medium, because of different propagation speeds of the ordinary and extraordinary modes ($q_{i1} \neq q_{i2}$ and $q_{i3} \neq q_{i4}$).

The transfer matrix \TMa{i}~of a single layer $i$ is defined as:
\begin{align}
\TMa{i}=\AMa{i} \PMa{i} {\AMa{i}}^{-1},
\end{align}
and the full transfer matrix \FullTMa{}~of all $N$ layers is
\begin{align}
\FullTMa{}= {\AMa{0}}^{-1} \left( \prod_{i=1}^{N} \TMa{i} \right) \AMa{N+1},
\end{align}
where ${\AMa{0}}^{-1}$ ($\AMa{N+1}$) ensures the correct mode projection between the multilayer system and the incident medium (substrate).

\subsection{Reflection and Transmission Coefficients}
\label{sec:rt_coefficients}

The full transfer matrix \FullTMa{}~projects the amplitude vector in the substrate $\ensuremath{\vec{E}}^+_{N+1}$ onto the amplitude vector in the incident medium $\ensuremath{\vec{E}}^-_0$:
\begin{align}
\ensuremath{\vec{E}}^-_0 = \FullTMa{} \ensuremath{\vec{E}}^+_{N+1},
\end{align}
where $\ensuremath{\vec{E}}^-_{i-1}$ and $\ensuremath{\vec{E}}^+_i$ denote the fields on both sides of the interface between layer $i-1$ and $i$, respectively. Following the equations presented by Yeh \cite{Yeh1979}, the transmission ($t$) and reflection ($r$) coefficients for incident p- or s-polarization can be calculated in terms of the matrix elements of \FullTMa{} as follows:
\begin{align}
r_{pp}&=\frac{\FullTMaEl{31}\FullTMaEl{22}-\FullTMaEl{32}\FullTMaEl{21}}{\FullTMaEl{11}\FullTMaEl{22}-\FullTMaEl{12}\FullTMaEl{21}} &
t_{p(p/o)}&=\frac{\FullTMaEl{22}}{\FullTMaEl{11}\FullTMaEl{22}-\FullTMaEl{12}\FullTMaEl{21}}
\label{eq:reflectionCoefficient_pp}
\\
r_{ss}&=\frac{\FullTMaEl{11}\FullTMaEl{42}-\FullTMaEl{41}\FullTMaEl{12}}{\FullTMaEl{11}\FullTMaEl{22}-\FullTMaEl{12}\FullTMaEl{21}} &
t_{s(s/e)}&=\frac{\FullTMaEl{11}}{\FullTMaEl{11}\FullTMaEl{22}-\FullTMaEl{12}\FullTMaEl{21}} \\
r_{ps}&=\frac{\FullTMaEl{41}\FullTMaEl{22}-\FullTMaEl{42}\FullTMaEl{21}}{\FullTMaEl{11}\FullTMaEl{22}-\FullTMaEl{12}\FullTMaEl{21}} &
t_{p(s/e)}&=\frac{-\FullTMaEl{21}}{\FullTMaEl{11}\FullTMaEl{22}-\FullTMaEl{12}\FullTMaEl{21}} \\
r_{sp}&=\frac{\FullTMaEl{11}\FullTMaEl{32}-\FullTMaEl{31}\FullTMaEl{12}}{\FullTMaEl{11}\FullTMaEl{22}-\FullTMaEl{12}\FullTMaEl{21}} &
t_{s(p/o)}&=\frac{-\FullTMaEl{12}}{\FullTMaEl{11}\FullTMaEl{22}-\FullTMaEl{12}\FullTMaEl{21}},
\label{eq:reflectionCoefficient_sp}
\end{align}
where the subscripts refer to the incoming and outgoing polarization state, respectively. The transmission coefficients describe the transmitted electric field amplitude into p- and s-polarized states in the case of a non-birefringent substrate, and into the ordinary and extraordinary eigenstates in the case of a birefringent substrate. We note that the indices of \FullTMa{} differ from the equations reported by Yeh \cite{Yeh1979} in order to account for a different order of the eigenmodes in the amplitude vector (Eq.~\ref{eq:E_xu_sorting}).

\subsection{Electric Field Distribution}
\label{sec:Efield_distribution}

Employing the interface and propagation matrices \LMa{i}~and \PMa{i}, respectively, an amplitude vector can be projected to any $z$-point in the multilayer system (please note the published erratum that corrects the calculation of the electric field distribution in our original work \cite{Passler2019b}). However, due to the rotation of polarization in birefringent media and thus the mixing of polarization states, in general, the cases of incident p- and s-polarization have to be treated seperately. As a starting point, the transmission coefficients can be utilized to formulate the amplitude vector $\ensuremath{\vec{E}}_{N+1}^+$ for either p- or s-polarized incident light in the substrate at the interface with layer $N$ as follows:
\begin{align}
\begin{split}
\left( \ensuremath{\vec{E}}_{N+1}^+ \right)_{p \text{ in}} &=\colvec{E_{\Rightarrow}^{p/o} \\ E_{\Rightarrow}^{s/e} \\ E_{\Leftarrow}^{p/o} \\E_{\Leftarrow}^{s/e}}=\colvec{t_{p(p/o)} \\ t_{p(s/e)} \\ 0 \\ 0}
\\
\left( \ensuremath{\vec{E}}_{N+1}^+ \right)_{s \text{ in}} &=\colvec{E_{\Rightarrow}^{p/o} \\ E_{\Rightarrow}^{s/e} \\ E_{\Leftarrow}^{p/o} \\E_{\Leftarrow}^{s/e}}=\colvec{t_{s(p/o)} \\ t_{s(s/e)} \\ 0 \\ 0},
\end{split}
\label{eq:Estart}
\end{align}
where the reflected ($\Leftarrow$) components are set to zero, since no light source is assumed to be on the substrate side of the multilayer system. Furthermore, in order to obtain the electric field amplitudes as a function of $z$, the propagation through layer $i$ is calculated by means of the propagation matrix \PMa{i}:
\begin{align}
\begin{split}
\ensuremath{\vec{E}}_{i}(z)&=\PMa{i}(z) \ensuremath{\vec{E}}^-_{i} \\
&=\!\fourfourMatrix
{\!e^{-i\frac{\omega}{c}q_{i1}z}\! & 0 & 0 & 0}
{0 & \!e^{-i\frac{\omega}{c}q_{i2}z}\! & 0 & 0}
{0 & 0 & \!e^{-i\frac{\omega}{c}q_{i3}z}\! & 0}
{0 & 0 & 0 & \!e^{-i\frac{\omega}{c}q_{i4}z}\!}\!\ensuremath{\vec{E}}^-_{i},
\end{split}
\end{align}
with $0<z<d_i$ being the relative $z$-position in layer $i$. Starting from $\ensuremath{\vec{E}}_{N+1}^+$, the interface matrices \LMa{i}~and propagation matrices \PMa{i}~subsequently propagate the amplitude vector towards the incident medium. In the reverse direction, the inverse propagation matrix ${\PMa{N+1}}^{-1}$ allows to calculate the \ensuremath{\vec{E}}-fields in the substrate. As a result, the four mode amplitudes $E_{\Rightarrow}^{p/o}$, $E_{\Rightarrow}^{s/e}$, $E_{\Leftarrow}^{p/o}$, and $E_{\Leftarrow}^{s/e}$ are obtained as a function of $z$ within each layer.

In order to obtain the three-components $\E_x$, $\E_y$, and $\E_z$ of the electric field for each of the four modes $j$, the four mode amplitudes have to be multiplied with their respective eigenmode vector $\vec{\hat{\gamma}}_{ij}$ (Eq. \ref{eq:eigenmodes}). This yields for the electric fields $\vec{\E}$ of the four modes for each layer $i$, as a function of $z$, and for either p- or s-polarized incident light:
\begin{align}
\begin{split}
\left( \vec{\E}_{\Rightarrow}^{p/o} \right)_{p/s \text{ in}} &= \left( E_{\Rightarrow}^{p/o} \right)_{p/s \text{ in}} \colvec{ \hat{\gamma}_{i11} \\ \hat{\gamma}_{i12}  \\  \hat{\gamma}_{i13} }, \quad j=1 \\
\left( \vec{\E}_{\Rightarrow}^{s/e} \right)_{p/s \text{ in}} &= \left( E_{\Rightarrow}^{s/e} \right)_{p/s \text{ in}} \colvec{ \hat{\gamma}_{i21} \\ \hat{\gamma}_{i22}  \\  \hat{\gamma}_{i23} }, \quad j=2 \\
\left( \vec{\E}_{\Leftarrow}^{p/o} \right)_{p/s \text{ in}} &= \left( E_{\Leftarrow}^{p/o} \right)_{p/s \text{ in}} \colvec{ \hat{\gamma}_{i31} \\ \hat{\gamma}_{i32}  \\  \hat{\gamma}_{i33} }, \quad j=3 \\
\left( \vec{\E}_{\Leftarrow}^{s/e} \right)_{p/s \text{ in}} &= \left( E_{\Leftarrow}^{s/e} \right)_{p/s \text{ in}} \colvec{ \hat{\gamma}_{i41} \\ \hat{\gamma}_{i42}  \\  \hat{\gamma}_{i43} }, \quad j=4,
\label{eq:EfieldVectors}
\end{split}
\end{align}
where we have omitted the index $i$ and the $z$ dependence for the sake of readability. The full electric field $\vec{\E}_i(z)$ for either p- or s-polarized incident light in layer $i$ at point $z$ is given by the sum of all four electric field vectors:
\begin{align}
\vec{\E}_i(z) = \vec{\E}_{\Rightarrow,i}^{p/o}(z) + \vec{\E}_{\Rightarrow,i}^{s/e}(z) + \vec{\E}_{\Leftarrow,i}^{p/o}(z) + \vec{\E}_{\Leftarrow,i}^{s/e}(z).
\label{eq:FullE}
\end{align}
The in-plane components of the sum, $\E_x$ and $\E_y$, are continuous throughout the entire multilayer structure, as it is required by Maxwells boundary conditions.

\section{Layer-Resolved Transmittance and Absorption}
\label{sec:layer_res_trans_abs}
The total reflectance \R~of the multilayer system for a given ingoing and outgoing polarization $a$ and $b$, respectively, can be readily calculated from the corresponding reflection coefficient (Eqs. \ref{eq:reflectionCoefficient_pp}-\ref{eq:reflectionCoefficient_sp}):
\begin{align}
\R^{ab} = \abs{r_{ab}}^2, \qquad a,b = p,s.
\label{eq:Rfromr}
\end{align}
The transmittance \T, which is the transmitted power into the substrate, on the other hand, in general is not given by the electric field intensity $\T \neq \abs{t}^2$ (except for the special case if the substrate is vacuum, $\ensuremath{\varepsilon} = 1$). Instead, \T~can be calculated from the time-averaged Poynting vector $\vec{\SSS}$ \cite{Yariv1984,ALBERDI2002,Weber2014}, which describes the direction and magnitude of the energy flux of an electromagnetic wave at any point $z$ in the structure:
\begin{align}
\vec{\SSS}_i(z) = \frac{1}{2} \text{Re} \left[ \vec{\E}_i(z) \times \vec{\HH}_i^*(z) \right].
\label{eq:Svec}
\end{align}
The full electric field $\vec{\E}$ (for incident polarization $a$) as a function of $z$ in each layer $i$ was calculated in the previous section (Eq. \ref{eq:FullE}), and the full magnetic field $\vec{\HH}$~is obtained as follows using Maxwells equations:
\begin{align}
\begin{split}
\vec{\HH}_i(z) = \frac{1}{\omega \mu_i} \Bigl( &\ensuremath{\vec{k}}_{i1} \times \vec{\E}_{\Rightarrow,i}^{p/o}(z) + \ensuremath{\vec{k}}_{i2} \times \vec{\E}_{\Rightarrow,i}^{s/e}(z) \\
                                                         +&\ensuremath{\vec{k}}_{i3} \times \vec{\E}_{\Leftarrow,i}^{p/o}(z) + \ensuremath{\vec{k}}_{i4} \times \vec{\E}_{\Leftarrow,i}^{s/e}(z) \Bigr),
\end{split}
\label{eq:FullH}
\end{align}
where $\ensuremath{\vec{k}}_{ij}$ are the wavevectors in layer $i$ of the four modes $j$, see Eq. \ref{eq:kvec}. Because $\vec{\E}$ and $\vec{\HH}$ are known in each layer $i$ and as a function of $z$ from the transfer matrix formalism, the Poynting vector can be evaluated likewise yielding $\vec{\SSS}_{i}(z)$, which will be used in the following to calculate the transmittance and absorption at any point $z$ in the multilayer system.

It is important to note that while $\vec{\E}$ and $\vec{\HH}$ can be calulcated for each of the four modes $j$ individually, this mode separation in general -- specifically, in the case of birefringent media -- is not possible for the Poynting vector $\vec{\SSS}$. In other words, in birefringent media, the sum of the Poynting vectors of the four modes is not equal to the Poynting vector calculated from the total fields $\vec{\E}$ (Eq. \ref{eq:FullE}) and $\vec{\HH}$ (Eq. \ref{eq:FullH}). The difference arises because in birefringent media, $\vec{\E} \not\perp \vec{\HH}$. Therefore, the cross products $\vec{\E} \times \vec{\HH}$ between different modes $j$ are no longer zero. For the correct calculation of the Poynting vector in birefringent media, it is thus necessary to calculate the cross product of the total fields $\vec{\E}$ and $\vec{\HH}$, as shown in Eq. \ref{eq:Svec}. Interestingly, this means that the energy flux in birefringent media cannot be split up into the ordinary and extraordinary eigenmodes, but has to be considered as a single quantity. In the following, we therefore discuss the transmittance and absorption for s- or p-polarized incident light without differentiating between the eigenmodes anymore.

An exception is the incident medium, which is set to be isotropic. Here, the Poynting vector can be calculated for each mode individually, and for the purpose of normalizing the transmitted power, we calculate the Poynting vector of the incident light $\vec{\SSS}_{\text{inc}}$ (in layer $i=0$ at position $z=0$) for either p- or s-polarization as follows:
\begin{align}
\begin{split}
\vec{\SSS}_{\text{inc}}^{p} &= \frac{1}{2} \text{Re} \left[ \vec{\E}_{\Rightarrow,0}^{p}(0) \times \left( \ensuremath{\vec{k}}_{01} \times \vec{\E}_{\Rightarrow,0}^{p}(0)\right)^* \right] \\
\vec{\SSS}_{\text{inc}}^{s} &= \frac{1}{2} \text{Re} \left[ \vec{\E}_{\Rightarrow,0}^{s}(0) \times \left( \ensuremath{\vec{k}}_{02} \times \vec{\E}_{\Rightarrow,0}^{s}(0)\right)^* \right].
\end{split}
\label{eq:Sinc}
\end{align}
In a stratified multilayer system, the transmitted energy is given by the $z$-component of the Poynting vector. Thus, we note that alternatively to Eq. \ref{eq:Rfromr}, the reflectance \R~can be calculated from the Poynting vector:
\begin{align}
\R^{ab} = \frac{-\SSS_{\text{refl},z}^{b}}{\SSS_{\text{inc},z}^{a}},
\end{align}
where the minus sign accounts for the negative $z$-direction of the reflected light. $\SSS_{\text{refl},z}^{b}$ is the $z$-component of the Poynting vector of the reflected light of polarization $b=p,s$ (in layer $i=0$ at position $z=0$) given by:
\begin{align}
\begin{split}
\vec{\SSS}_{\text{refl}}^{p} &= \frac{1}{2} \text{Re} \left[ \vec{\E}_{\Leftarrow,0}^{p}(0) \times \left( \ensuremath{\vec{k}}_{03} \times \vec{\E}_{\Leftarrow,0}^{p}(0)\right)^* \right] \\
\vec{\SSS}_{\text{refl}}^{s} &= \frac{1}{2} \text{Re} \left[ \vec{\E}_{\Leftarrow,0}^{s}(0) \times \left( \ensuremath{\vec{k}}_{04} \times \vec{\E}_{\Leftarrow,0}^{s}(0)\right)^* \right].
\end{split}
\label{eq:Srefl}
\end{align}
As discussed above, for anisotropic media, a separation of the energy flow into the different eigenmodes of polarization $b$ is not generally possible. Therefore, all following equations calculate the total transmittance or absorption for the respective incident polarization $a$.

\begin{figure*}[!ht]
\includegraphics[width = \textwidth]{fig_testCases.pdf}
\caption{\textbf{Layer-resolved absorption calculations of two simple test structures.} \textbf{a} Sketch of the TMDC heterostructure comprising monolayers of \ensuremath{\text{WSe}_{\text{2}}}, \ensuremath{\text{MoS}_{\text{2}}}, and \ensuremath{\text{WS}_{\text{2}}}~\cite{Jung2019} separated by hBN monolayers on a \nmetr{140} thick \ensuremath{\text{SiO}_{\text{2}}}~film on a Si substrate. \textbf{b} Reflectance (black line) and transmittance (blue line) spectra of the system at visible wavelengths and at an incident angle of \dg{60} for s-polarized incident light. \textbf{c} Layer-resolved absorption spectra of the TMDC monolayers (colored lines), revealing their respective contribution to the overall absorption spectrum (gray line). \textbf{d} Sketch of the polar dielectric heterostructure comprising \nmetr{100} thin GaN, AlN, and SiC films on a Si substrate. \textbf{e} Reflectance and transmittance spectra and \textbf{f} layer-resolved absorption spectra of the second system, enabling identification of the absorbing layer for each of the six absorption lines.}
\label{fig:testCases}
\end{figure*}

The transmittance \T~into the substrate $i=N+1$ at the interface with layer $N$ for incident light of polarization $a$ is given by:
\begin{align}
\T^{a} = \frac{\SSS_{N+1,z}^{a}(D)}{\SSS_{\text{inc},z}^{a}},
\end{align}
where $D=\sum_{i=1}^{N} d_i$ is the thickness of the multilayer system. The full $z$-dependence of the transmittance can be evaluated by using the $z$-component of the Poynting vector $\SSS_{i,z}(z)$ at a certain $z$-position in layer $i$:
\begin{align}
\T_i^{a}(z) = \frac{\SSS_{i,z}^{a}(z)}{\SSS_{\text{inc},z}^{a}}.
\end{align}
With this, the absorption \A~of the entire multilayer system, that is up to the last interface between layer $N$ and the substrate, is given by
\begin{align}
\A^{a} = 1-\R^{a}-\T^{a},
\label{eq:Atotal}
\end{align}
and the $z$-resolved absorption $\A_i(z)$ in each layer $i$ is
\begin{align}
\A_i^{a}(z) = 1-\R^{a}-\T_i^{a}(z),
\label{eq:Aiz}
\end{align}
where $\R^{a} = \R^{ap}+\R^{as}$ is the total reflectance. Note that Eq. \ref{eq:Aiz} describes the total absorption starting from $z=0$ at the first interface up to the specified position $z$ in layer $i$. The layer-$i$-resolved absorption, on the other hand, is given by:
\begin{align}
\begin{split}
\A_i^a &= \T_i^a(d_{1..i-1}) - \T_i^a(d_{1..i-1} + d_i) \\
		&=  \A_i^a(d_{1..i-1} + d_i) - \A_i^a(d_{1..i-1}),
\end{split}
\label{eq:Ai}
\end{align}
where $d_{1..i-1} = \sum_{i=1}^{i-1} d_i$ is the thickness of all layers through which the incident light has propagated before reaching the layer $i$.

Before we study three nanophotonic device structures in the following section, we calculate the layer-resolved absorption for two simple test structures in Fig.~\ref{fig:testCases}. The first is a typical TMDC heterostructure, comprising monolayers of tungsten diselenide (\ensuremath{\text{WSe}_{\text{2}}}), \ensuremath{\text{MoS}_{\text{2}}}, and tungsten disulfide (\ensuremath{\text{WS}_{\text{2}}})~sandwiched between hBN monolayers (sketched in Fig.~\ref{fig:testCases}a), where each TMDC monolayer features individual exciton absorption lines. In Fig.~\ref{fig:testCases}b, the reflection and transmittance of the entire structure is plotted, and Fig.~\ref{fig:testCases}c shows the absorption spectra of each TMDC monolayer. The total absorption (gray line in Fig.~\ref{fig:testCases}c) obtained from the reflectance and transmittance spectra (Eq.~\ref{eq:Atotal}) exhibits three indistinguishable absorption features, whereas the layer-resolved absorption calculations unravel the absorption spectrum, allowing to identify the contribution of each TMDC monolayer.

In Fig.~\ref{fig:testCases}d-f, we show the absorption in a polar dielectric heterostructure of SiC, AlN, and GaN thin films on a Si substrate probed at infrared (IR) frequencies. Polar dielectric crystals feature an IR-active transverse optical (TO) phonon mode, where light is predominantly absorbed, whereas the longitudinal optical (LO) phonon mode is not IR-active and thus featureless in a bulk crystal. Thin films, on the other hand, support the so called Berreman mode in proximity to the LO frequency, leading to a strong absorption feature at oblique incidence \cite{Berreman1963,Vassant2012,Passler2019,Dunkelberger2020}. Thus, the reflectance and transmittance spectra (Fig.~\ref{fig:testCases}e) of the polar dielectric heterostructure are of complicated shape, exhibiting six different features arising from the three different polar crystal thin films. The layer-resolved absorption calculations split these features into three spectra with two absorption peaks each (Fig.~\ref{fig:testCases}f), allowing to identify the respective polar crystal thin film that leads to the absorption at its respective TO and LO frequencies.

\section{Simulations of Nanophotonic Devices}
\label{sec:simulations}
The transfer matrix formalism and the calculation of the layer-resolved absorption and transmittance presented in the previous section can be applied for any wavelength and any number of layers, consisting of birefringent or non-birefringent media described by an arbitrary permittivity tensor $\ensuremath{\bar{\varepsilon}}_i$. As case studies, in this section we describe three selected nanophotonic device structures. The first example discusses the hyperbolic phonon polaritons arising in a \ensuremath{\text{MoO}_{\text{3}}}~/ AlN / SiC system excited evanescently at far-IR wavelengths, highlighting the potential of the formalism for nanophotonic studies in stratified media. The second example describes the freespace response in the visible of a van der Waals heterostructure of monolayers of \ensuremath{\text{MoS}_{\text{2}}}~embedded in a hBN matrix, featuring layer-selective absorption of the \ensuremath{\text{MoS}_{\text{2}}}~excitons. Finally, the third example calculates the mid-IR absorption of strongly coupled modes formed from intersuband plasmons in multi-quantum wells embedded in an optical cavity.

\begin{figure*}[!ht]
\includegraphics[width = \textwidth]{fig_MoO3.pdf}
\caption{\textbf{Azimuthal dependence of the layer-resolved absorption in a \ensuremath{\text{MoO}_{\text{3}}}/AlN/SiC heterostructure.} \textbf{a} Prism coupling setup in the Otto geometry for the evanescent excitation of phonon polaritons in a \nmetr{1200} \ensuremath{\text{MoO}_{\text{3}}}~/ \nmetr{1000} AlN / SiC substrate multilayer sample with p-polarized incident light. The incident angle is fixed to $\theta=\dg{28}$ and the air gap to $d_{\text{gap}}=\mumetr{8}$. \textbf{b} Real value of the principle relative permittivities $\ensuremath{\varepsilon}_x$, $\ensuremath{\varepsilon}_y$, and $\ensuremath{\varepsilon}_z$ of uniaxial SiC and AlN and biaxial \ensuremath{\text{MoO}_{\text{3}}}. Their respective reststrahlen bands located between the corresponding TO and LO phonon frequencies are indicated by the horizontal color bars. \textbf{c} Reflectance $\R^p$ of the entire system as a function of incident frequency $\omega$ and azimuthal angle of the sample $\Phi$. \textbf{d-f} Absorptions $\A_i^p$ (Eq. \ref{eq:Ai}) of the \ensuremath{\text{MoO}_{\text{3}}}, AlN, and SiC layers as a function of $\omega$ and $\Phi$.}
\label{fig:MoO3}
\end{figure*}

\subsection{Hyperbolic Phonon Polaritons in \ensuremath{\text{MoO}_{\text{3}}}/AlN/SiC}
\label{sec:MoO3}

\begin{figure*}[!ht]
\includegraphics[width = \textwidth]{fig_MoS2.pdf}
\caption{\textbf{Cavity-enhanced exciton absorption in \ensuremath{\text{MoS}_{\text{2}}}~monolayers.} \textbf{a} Structure of the Fabry-P\'{e}rot cavity. A \nmetr{140} thin \ensuremath{\text{SiO}_{\text{2}}}~film on the Si substrate serves as a back-reflector, while the cavity with variable thickness $d_c=1.9-\mumetr{2.4}$ is made of three layers of hBN of relative thicknesses $0.465$ $d_c$, $0.035$ $d_c$, and $0.5$ $d_c$, separated by two \nmetr{0.6} thin \ensuremath{\text{MoS}_{\text{2}}}~monolayers. \textbf{b} Real and imaginary values of the isotropic relative permittivity of \ensuremath{\text{MoS}_{\text{2}}}~\cite{Jung2019}. All calculations are based on this permittivity, except the dashed lines in subfigures d and e. \textbf{c} Anisotropic relative permittivity of \ensuremath{\text{MoS}_{\text{2}}}~\cite{Funke2016}. \textbf{d} Absorption spectra of the two \ensuremath{\text{MoS}_{\text{2}}}~monolayers for s-polarized incident light, for a cavity thickness of $d_c=\mumetr{2.15}$ ($d_c=\mumetr{2.128}$) optimized for layer-selective absorption of the two \ensuremath{\text{MoS}_{\text{2}}}~excitons, calculated with the \ensuremath{\text{MoS}_{\text{2}}}~permittivities of \etal{Jung} (\etal{Funke}). \textbf{e} Analogous calculations as in subfigure d but for p-polarized incident light. \textbf{f,g} Reflectance $\R^s$ and $\R^p$ for s- and p-polarized incident light, respectively. \textbf{h,i} and \textbf{j,k} Absorption $\A_i^s$ and $\A_i^p$ of the two \ensuremath{\text{MoS}_{\text{2}}}~monolayers, respectively. \textbf{l,m} Transmitted power $\T^s$ and $\T^p$ into the Si substrate, respectively. The maps of \R, \A, and \T~are all plotted as a function of the incident photon energy and the cavity thickness $d_c$.}
\label{fig:MoS2}
\end{figure*}

Polar crystals such as \ensuremath{\text{MoO}_{\text{3}}}, AlN, and SiC support surface phonon polaritons (SPhP) at frequencies inside their reststrahlen region between the TO and LO phonon frequencies \cite{Caldwell2015}. On smooth surfaces, SPhPs can be excited via prism coupling in the Otto geometry \cite{Otto1968,Neuner2009} as illustrated in Fig.~\ref{fig:MoO3}a, where the air gap width $d_{\text{gap}}$ governs the excitation efficiency and the incident angle $\theta$ defines the in-plane momentum of the launched SPhP \cite{Passler2017}. The system we investigate here is a multilayer heterostructure comprising an \ensuremath{\text{MoO}_{\text{3}}}~and AlN film on a SiC substrate. Employing the presented transfer matrix fomalism, the layer-resolved absorption $\A_i$ of such a structure can be calculated as a function of incident angle $\theta$, incident frequency $\omega$, layer thicknesses $d_i$, and azimuthal angle~$\Phi$ of the sample.

The absorption in the \ensuremath{\text{MoO}_{\text{3}}}, AlN, and SiC layers as a function of $\omega$ and $\Phi$ and for fixed $\theta$ and $d_i$ is shown in Fig.~\ref{fig:MoO3}d-f. The reflectance of the entire system is plotted in Fig.~\ref{fig:MoO3}c. As required by energy conservation, the sum of the absorbed power in the three polar crystals equals the attenuated power visible as absorption dips in the reflectance. However, while the reflectance only yields the total absorption, the layer-resolved calculations allow to identify the exact position of a power drain in a multilayer system.

In particular, the \ensuremath{\text{MoO}_{\text{3}}}/AlN/SiC heterostructure features several sharp absorption lines at $660$, $800$, $920$, and \wavenumber{980} that are mostly independent of $\Phi$, and one prominent absorption line that strongly varies with $\Phi$, indicating that the latter depends on in-plane anisotropy ($\ensuremath{\varepsilon}_x \neq \ensuremath{\varepsilon}_y$) while the former do not. In the multilayer sample, only the \ensuremath{\text{MoO}_{\text{3}}}~layer exhibits in-plane anisotropy, while AlN and SiC are c-cut uniaxial crystals (principle relative permittivities $\ensuremath{\varepsilon}_x$, $\ensuremath{\varepsilon}_y$, and $\ensuremath{\varepsilon}_z$ are shown in Fig.~\ref{fig:MoO3}b). Indeed, the $\Phi$-dependent feature is mostly absorbed in the \ensuremath{\text{MoO}_{\text{3}}}~layer. This feature is the hyperbolic phonon polariton (hPhP) supported by the air/\ensuremath{\text{MoO}_{\text{3}}}~interface arising in the in-plane reststrahlen bands of \ensuremath{\text{MoO}_{\text{3}}}. Its tunability in frequency arises from the large in-plane anisotropy of \ensuremath{\text{MoO}_{\text{3}}}, which leads to a $\Phi$-dependent effective permittivity sensed by the hPhP upon azimuthal rotation.

Notably, below the SiC TO frequencies ($\sim\wavenumber{800}$), a significant part of the hPhP leaks into the SiC substrate, while above \w{TO}{SiC}~the mode is mostly confined to the \ensuremath{\text{MoO}_{\text{3}}}~layer. The high confinement above \w{TO}{SiC}~occurs because of the negative permittivity in the SiC reststrahlen band, while below \w{TO}{SiC}, the mode can penentrate the substrate. Interestingly, this mode penetration happens across the AlN layer, where only a small part of the mode is absorbed. Since AlN features its reststrahlen bands across the entire frequency range of the hPhP supported by the \ensuremath{\text{MoO}_{\text{3}}}, and thus evanescently attenuates all modes, the hPhP appears to tunnel through the AlN layer to be absorbed by the SiC substrate.

The multilayer system presented here only scratches the surface of various possible material configurations and compositions that can potentially be employed for tailoring surface and interface polariton resonances \cite{Jia2015,Dufferwiel2015,Low2017,Wintz2018,Passler2019a,Ratchford2019,Zhang2019}. In particular the emerging field of volume-confined hyperbolic polaritons \cite{Li2015,Ma2018,Dai2018,Ratchford2019}, enabled by the anisotropic permittivity of the supporting media, holds great potential for future nanophotonic applications, such as subdiffraction imaging and hyperlensing \cite{Dai2015,Ferrari2015}. Providing the full layer-resolved absorption, our algorithm paves the way to predict and study hyperbolic polariton modes in any anisotropic stratified heterostructure.

\subsection{Layer-selective absorption of \ensuremath{\text{MoS}_{\text{2}}}~excitons in a Fabry-P\'{e}rot cavity}
\label{sec:MoS2}
TMDC monolayers such as \ensuremath{\text{MoS}_{\text{2}}}~feature strong exciton resonances at visible frequencies \cite{Funke2016,Jung2019}. By inserting these monolayers into van der Waals heterostructures forming a Fabry-P\'{e}rot cavity, the light-matter interaction enabled by the TMDC exciton can be strongly enhanced \cite{Schneider2018}. Here, we embed two \ensuremath{\text{MoS}_{\text{2}}}~monolayers into a $d_c =1.9-\mumetr{2.4}$ thick hBN cavity with a \ensuremath{\text{SiO}_{\text{2}}}~back-reflector on a Si substrate \cite{Benameur2011}, as sketched in Fig.~\ref{fig:MoS2}a. Employing the presented formalism, the absorption $\A_i$ in the two \ensuremath{\text{MoS}_{\text{2}}}~monolayers as a function of photon energy and cavity thickness $d_c$ can be calculated.

In the photon energy range of $1.7-2.2$ eV, \ensuremath{\text{MoS}_{\text{2}}}~features two excitons (A and B) at $\sim 1.9$ and $\sim 2.1$ eV. These excitons are apparent as resonance peaks in the isotropic relative permittivity plotted in Fig.~\ref{fig:MoS2}b and resulting in peaks in an absorption spectrum. However, the Fabry-P\'{e}rot cavity creates a static modulation of the electric field enhancement with peaks and nodes as a function of $z$ position, and thus the resulting absorption in a \ensuremath{\text{MoS}_{\text{2}}}~monolayer not only depends on the photon energy, but also sensitively depends on the $z$ position of the \ensuremath{\text{MoS}_{\text{2}}}~monolayer in the cavity. Taking advantage of this field modulation, we place one \ensuremath{\text{MoS}_{\text{2}}}~monolayer (layer 4) at the center of the cavity where the cavity modes alternate between node and peak with maximal amplitude, and the other \ensuremath{\text{MoS}_{\text{2}}}~monolayer (layer 2) in close proximity where the cavity features a node when there is a peak in the center, and vice versa.

The cavity modes yield the periodic modulation in photon energy and cavity thickness $d_c$ that can be seen in the reflectance (transmittance) maps shown in Fig.~\ref{fig:MoS2}f and g (l and m), for incoming s- and p-polarized light, respectively. The different modulation contrast for $\R^s$ and $\R^p$ ($\T^s$ and $\T^p$) arises from the large incident angle of $\theta = \dg{70}$, which was chosen to optimize the absorption $\A^s$ in the \ensuremath{\text{MoS}_{\text{2}}}~monolayers for s-polarized incident light. For smaller incident angles, the differences for s- and p-polarization decrease, but with a reduction in the absorption $\A^s$.

In Fig.~\ref{fig:MoS2}h,i (j,k) the absorption for s- and p-polarized incident light in the first \ensuremath{\text{MoS}_{\text{2}}}~monolayer, $\A_2^{s,p}$ (second \ensuremath{\text{MoS}_{\text{2}}}~monolayer, $\A_4^{s,p}$), is shown. Due to the choice of the $z$ positions of the two \ensuremath{\text{MoS}_{\text{2}}}~monolayers, each film is sensitive to only every second cavity mode, where layer 2 (first \ensuremath{\text{MoS}_{\text{2}}}~monolayer) absorbs those modes that are not absorbed by layer 4 (second \ensuremath{\text{MoS}_{\text{2}}}~monolayer). Additionally to the absorption modulation imposed by the cavity, the A and B excitons of \ensuremath{\text{MoS}_{\text{2}}}~yield two absorption features at their respective energies, marked by dotted vertical lines in Fig.~\ref{fig:MoS2}h-k. For the optimized case of s-polarized incident light, the \ensuremath{\text{MoS}_{\text{2}}}~monolayers reach cavity-enhanced absorption values of up to 20\% at both exciton energies. At a cavity thickness of $d_c = \mumetr{2.15}$ for s-polarization, layer 2 only absorbs at the energy of exciton A, while in layer 4, absorption only occurs at the energy of exciton B. This layer-selective absorption is further illustrated in the absorption spectra (solid lines) for a fixed cavity thickness of $d_c = \mumetr{2.15}$ shown in Fig.~\ref{fig:MoS2}d and e for s- and p-polarized light, respectively. For both polarizations, a high contrast between the two layers at each exciton absorption line is achieved, yielding efficient layer-selectivity.

Finally, we compare these results obtained for an isotropic permittivity model \cite{Jung2019} with the same calculations performed for an anisotropic model of \ensuremath{\text{MoS}_{\text{2}}}~\cite{Funke2016}. In Fig.~\ref{fig:MoS2}c, the in-plane ($\varepsilon_{x,y}$) and out-of-plane ($\varepsilon_{z}$) permittivity values taken from \etal{Funke} \cite{Funke2016} are plotted. While $\varepsilon_{x,y}$ are qualitatively the same as the values from \etal{Jung} \cite{Jung2019} (Fig.~\ref{fig:MoS2}b), $\varepsilon_{z}$ differs strongly, taking the almost constant value of $\varepsilon_{z}=1+0\: i$. Even though the difference in  $\varepsilon_{z}$ is substantial, the absorption spectra shown in Fig.~\ref{fig:MoS2}d,e (dashed lines) are qualitatively identical to the spectra calculated from an isotropic permittivity model (solid lines). This confirms that spectroscopic measurements of TMDC monolayers are mostly insensitive to their out-of-plane permittivity \cite{Jung2019},  with the exception of cases where $\varepsilon_{z}$ features a zero-crossing, the so-called epsilon-near-zero (ENZ) frequency, giving rise to drastic optical responses such as enhanced higher-harmonic generation \cite{Vincenti2011,Capretti2015a,Passler2019}. However, this is not the case for \ensuremath{\text{MoS}_{\text{2}}}, and therefore the spectra are almost identical.

In this example, the layer-resolved absorption calculations from our algorithm provide the essential information for simulating the layer-selective exciton absorption and optimizing the system parameters. In the thriving field of 2D nanophotonics featuring TMDC van der Waals heterostructures, where structures are optimized for maximal light harvesting \cite{Fortin1982,Yu2013}, optoelectronic devices \cite{He2013,Ross2014} or nanolasers \cite{Gan2013,Sobhani2014}, such layer-resolved absorption calculations promise to be of essential importance. Due to the generality of the presented algorithm, the light-matter interaction in any 2D heterostructure can be readily investigated, highlighting the broad applicability of our approach.

\begin{figure*}[!ht]
\includegraphics[width = \textwidth]{fig_MQW.pdf}
\captionof{figure}{\textbf{Strong coupling between an ENZ mode in a MQW superlattice and a cavity mode.}
	A \mumetr{1} thick GaN cavity is formed between a Au mirror and a \nmetr{500} thick doped GaN slab on a GaN substrate (top sketch).
	The doped GaN slab acts as a low-index mirror.
	\textbf{a} Simulated reflectance $\R^p$ of the cavity mode showing its dispersion relation as a function of
	the angle of incidence $\theta$ inside the GaN substrate.
	\textbf{b,c} Layer-resolved absorption $\A^p$ inside the doped GaN slab 	and in the Au mirror, respectively.
	\textbf{d} Calculated reflectance $\R^p$ of the \mumetr{1} thick cavity partially filled with a  doped GaN/AlGaN
	MQW superlattice (see bottom sketch), featuring an electronic excitation at $\sim \wavenumber{2030}$ (white dotted line).
	The strong light-matter coupling between the cavity mode and QW resonance leads to an avoided crossing and the emergence of two polariton branches.
	\textbf{e-g} Layer-resolved absorption $\mathcal{A}^{p}$ inside the doped GaN mirror, the superlattice and the Au mirror, respectively.
	\textbf{h} Real part of the permittivity of the isotropic layers forming the cavity.
	\textbf{i} Real part of the different components of the permittivity tensor of the doped GaN QW and of the AlGaN barrier,
	showing the ENZ mode frequency of the QW at  $\sim \wavenumber{2030}$.}
\label{fig:MQW}
\end{figure*}

\subsection{Strong Coupling in a Multi Quantum Well - Cavity System}
\label{sec:MQW}
Doped semiconductor quantum wells (QWs) support transitions between consecutive quantum-confined electronic levels,
called intraband or inter-subband (ISB) transitions. Contrary to interband transitions, ISB transitions do not only depend
on the bandgap properties of the semiconductor, but also on the electronic confinement inside the QWs, and thus offer a great
frequency tunability by changing the width, but also the doping level inside the QW.
They play a major role in semiconductor optic devices operating in the IR where
semiconducting materials with a suitable bandgap are lacking, and are the building block of
quantum well IR photodetectors \cite{Levine1987} and quantum cascade lasers \cite{Faist1994}.
They also offer a practical platform to study the optical properties of dense confined electron gases \cite{Vasanelli2016},
which notably lead to the demonstration of the strong \cite{Liu1997,Dini2003} and ultra-strong
light-matter coupling regimes \cite{Ciuti2005, Todorov2010}.
It is remarkable that such fundamental electrodynamical phenomena are directly observable on semiconductor devices \cite{Dupont2003, Jouy2010,Vigneron2019}.
One peculiar aspect of these ISB transitions is that they only couple to the component of the electric field along
the confinement direction of the QW structure. Hence, the optical properties of a doped QW can be described by an effective
permittivity tensor with different in-plane ($\varepsilon_{x,y}$) and out-of-plane ($\varepsilon_{z}$) values, which has been realized by several permittivity models \cite{Ando1982,Zauzny1999,Alpeggiani2014,Pegolotti2014}. The description of light propagation in stratified anisotropic media containing such QWs requires a complex formalism, such as the transfer matrix method the here presented algorithm builds on \cite{Passler2017a}.

We focus here on an existent experimental configuration to further demonstrate the potential of our formalism for the case of strong light-matter coupling between a cavity mode and a collective intersubband excitation in a multi-quantum well (MQW) superlattice. The system is composed of a GaN cavity formed by a
\nmetr{500} thick, Si-doped GaN slab and a Au mirror, as sketched on the top of Fig.~\ref{fig:MQW}.
The doped GaN layer is modeled using the Drude model, and acts as a low-index mirror.
The two mirrors are separated by a \mumetr{1} thick GaN spacer, forming an empty cavity.
The system sustains a guided transverse magnetic (TM) mode, where the electric field is confined mostly between the two mirrors and its out-of-plane component is maximal near the Au mirror. This guided mode can be probed in a reflectance experiment, as discussed in the following. In order to probe large internal angles of incidence experimentally, the sample has to be prepared in a prism shape, for example by cleaving the incident GaN layer facets.
The calculated p-polarized reflectance $\R^{p}$ is shown in Fig.~\ref{fig:MQW}a,
evidencing the dispersion relation of the cavity mode with varying incidence angle inside the GaN substrate.
The layer-resolved absorption spectra as a function of the angle of incidence for this system are reported in Fig.~\ref{fig:MQW}b,c
and reveal that absorption occurs mostly in the doped GaN mirror. Notably, for an internal incidence angle of \dg{48} at the
guided mode frequency, all the light is dissipated in the mirrors, leading to a minimum of zero reflectance in Fig.~\ref{fig:MQW}a.

We now turn to the system's response when the cavity is partially filled with a MQW structure (Fig.~\ref{fig:MQW}d-g).
The superlattice is composed of twenty repetitions of a \nmetr{3} thick GaN QW, Si-doped with a concentration of
$2 \times 10^{13}$~cm$^{-2}$ and twenty-one loss-less \nmetr{10} thick Al$_{0.26}$Ga$_{0.74}$N barriers.
Since the guided mode is a TM mode, it naturally provides a component of the electric field along the $z$-direction
for non-zero angles of incidence, which fulfills the ISB transition selection rule.
In order to maximize the coupling between the ISB transition in the MQW and the cavity mode, the superlattice is placed where the $z$-component of the electric field is the largest, that is just below the Au mirror, as shown at the bottom of Fig.~\ref{fig:MQW}.
The QW dielectric tensor is modeled using a semi-classical approach \cite{Zauzny1999}.
We selected the QW dimension and doping level in a way that it sustains a strong, collective electronic excitation known as an intersubband
plasmon \cite{Vasanelli2016}. The components of the real part of the dielectric permittivity tensor of all the layers
are presented in Fig.~\ref{fig:MQW}h,i. Note that while the ISB transition in the quantum well has a resonance frequency
of $\sim \wavenumber{1600}$, the Coulomb interaction between the QW electrons results in an ENZ mode at
$\sim \wavenumber{2030}$, marked by the dotted line in Fig.~\ref{fig:MQW}i. The ENZ mode dominates the optical
response of the QWs.

We show in Fig.~\ref{fig:MQW}d the calculated reflectance $\R^{p}$ of the cavity containing the MQW. The white dotted line
shows the ENZ frequency. A clear anti-crossing can be seen between the cavity mode and the ENZ resonance, which is characteristic
of the strong light-matter coupling regime, resulting in two polariton branches. The minimal separation between the two branches amounts to a vacuum
Rabi splitting $2\Omega_{R} = \wavenumber{200}$.
The layer-resolved absorption spectra as a function of the angle of incidence are shown in Fig.~\ref{fig:MQW}e-g for the GaN mirror,
the MQW superlattice, and the Au mirror, respectively.
When the cavity mode is far detuned from the ENZ frequency, the absorption occurs mostly in the two mirrors, and especially
in the GaN mirror, as for the empty cavity case. The situation changes dramatically when the cavity mode is tuned near the ENZ frequency.
The absorption then mostly occurs in the MQW. It is however important to note that the absorption is maximal at the frequencies of the two
polaritons and not at the ENZ frequency, as it would occur in the case of a weak coupling between the cavity mode
and the QW resonance. For an internal incidence angle of \dg{48}, the maximum absorption inside the MQW stack is now
0.5 at each of the frequencies of the two polaritons.

The algorithm presented here thus allows to directly
calculate the light absorption in the complex case of a doped MQW superlattice strongly coupled to a cavity mode.
In addition to the known features, such as the avoided crossing in the angle-dependent reflectance spectrum, we can directly
calculate the absorption inside the MQW active region, which can be usefully linked to the detected photocurrent
in the perspective of using such structures in photodetector devices
\cite{Liu1997,Dupont2003,Vigneron2019}.

\section{Discussion}
We have presented three nanophotonic devices based on anisotropic multilayer structures made from metals, polar dielectrics, and TMDC monolayers, covering incident wavelengths from the far-IR up to the visible. Our formalism allows to calculate the transmittance and absorption in any layer, giving unprecedented insight into the physics of light propagation in anisotropic, and even birefringent, stratified systems.

In recent years, the field of nanophotonics has intensely investigated the optical response of 2D heterostructures. A particularly thriving subject has been polaritonic excitations, which can be supported by a broad variety of systems including slabs of metals, doped semiconductors, polar dielectrics, 2D materials such as TMDC monolayers and their stratified heterostructures \cite{Xia2014,Caldwell2015a,Low2017}. Key features of polaritons for nanophotonic technologies are their high spatial confinement and field enhancement, which are driven by the particular design of the multilayer materials, stacking order, and layer thicknesses. Potential applications of such systems range from sensing \cite{Neuner2010,Berte2018} and solar cells \cite{Luk2014}, over optoelectronic devices \cite{Freitag2013} and beam manipulation via metamaterials \cite{Zeng2013}, to waveguiding \cite{Liew2010,Passler2019a}, and ultrafast optical components \cite{Ni2016,Passler2019}. However, due to the lack of a general formalism, the optical response of these polaritonic multilayer systems is often either approximated by effective, isotropic permittivity models \cite{Liu2014,Li2014,Huber2017}, or described by specifically derived formulas \cite{Collett1971,GiaRusso1973,Schwelb1986,Ciumac1994}. Our generalized formalism allows for a precise, layer-resolved study that includes any isotropic, anisotropic or even birefringent response of any number of layers, and thus holds great potential for the prediction and analysis of polariton modes in stratified heterostructures.

This is especially relevant for systems where one or more materials feature anisotropic permittivity with spectral regions of hyperbolicity where the principle real permittivities have opposite signs, such as hBN or \ensuremath{\text{MoO}_{\text{3}}}. Recently, these hyperbolic materials have attracted increasing interest \cite{Li2015,Ma2018,Dai2018,Ratchford2019} due to the existence of hyperbolic polaritons, featuring novel properties for nanophotonic applications such as subdiffraction imaging and hyperlensing \cite{Dai2015,Ferrari2015}. Because of the strong anisotropy of these systems, an effective permittivity approach is not purposeful. Here, our formalism provides the essential theoretical framework that is necessary to model, predict and analyze the optical response of such hyperbolic heterostructures, as we have discussed exemplarily for a \ensuremath{\text{MoO}_{\text{3}}}/AlN/SiC system (Fig.~\ref{fig:MoO3}). Providing the full layer-resolved information about the field distribution and power flow of the excited polaritons, our method allows to readily and concisely model and study hyperbolic polariton modes in any anisotropic stratified heterostructure.

Furthermore, the layer-resolved absorption formalism provides a description for designing optoelectronic devices such as detectors, for which the photoresponse is linked to the light absorption solely in the active region of the device. In the case of the MQW system, optimizing the overall light absorption by minimizing both the reflectance and transmittance of the system would not be sufficient to model the performances of a photodetector in the strong light-matter coupling regime \cite{Dupont2003,Vigneron2019}. Unfortunately, these are the only quantities that can be probed in a reflectance experiment. Calculating the layer-resolved absorption in the structure allows to optimize the cavity and MQW design, aiming at minimizing the light absorption in the cavity mirrors while maximizing the absorption in the MQW. This is well exemplified in Fig \ref{fig:MQW} e,f where we can see that the light is preferably dissipated in either the doped GaN mirror or the MQW superlattice depending on the detuning between the cavity mode and the QW resonance. This behavior cannot be deduced only from the reflectance measurement simulated in Fig.~\ref{fig:MQW}d. Fitting experimental reflectance data using our formalism \cite{Passler2017a} would allow to retrieve the amount of light dissipated inside the active region only from the experimentally observable quantities, and to estimate figures of merit such as the quantum efficiency of the device. The present method thus provides a convenient way to model and optimize complex, optically anisotropic heterostructures for optoelectronic devices.

\section{Conclusion}
In this work, we have derived explicit expressions for the calculation of the layer-resolved transmittance and absorption of light propagating in arbitrarily anisotropic, birefringent, and absorbing multilayer media. The algorithm relies on the electric field distribution computed from a $4 \times 4$ transfer matrix formalism \cite{Passler2017a, Passler2019b}, yielding a robust and consistent framework for light-matter interaction in stratified systems of arbitrary permittivity, which is implemented in an open access computer program \cite{Passler2020,Jeannin2020}. As case studies, we applied the algorithm to simulations of three nanophotonic device structures featuring hyperbolic phonon polaritons in a polar dielectric heterostructure, \ensuremath{\text{MoS}_{\text{2}}}~excitons in a Fabry-P\'{e}rot cavity, and ENZ resonances in a cavity-coupled multi-quantum-well, where we observed azimuth-dependent hyperbolic polariton tunneling, layer-selective exciton absorption, and strong coupling between ENZ and cavity modes. Allowing for a detailed analysis of the layer-resolved electric field distribution, transmittance, and absorption of light in any multilayer system, our algorithm holds great potential for the prediction of nanophotonic light-matter interactions in arbitrarily anisotropic stratified heterostructures.

\section{Acknowledgments}
We thank M. Wolf, S. Wasserroth and R. Ernstorfer (FHI Berlin) for careful reading of the manuscript and M. Wolf and the Max Planck Society for supporting this work.

\bibliographystyle{apsrev4-1}

redpajama_set_name: RedPajamaArXiv


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How To Choose The Best Essential Oil Diffusers


Kahl Audio » Audioquest Sky interconnect cables w/72V DBS. 3m XLR pair. Reference series. $6,300 MSRP.
The Sky is third from the top in Audioquest's range topping series. It sits under the Wild Blue Yonder and the flagship WEL Signature (which incidentally costs $8,900 and $12,900 respectively). So if you are in the market looking for something better than the Niagara but don't want to splash more than $6,000, the Sky is your next best bet. Conductors are pure silver but the cable does not sound overly bright at all. In fact, it has a nice open, transparent sound and almost strain free. Music just flowed freely through the cable with ease and speed. The 3m pair you see here was previously used to mate to two Krell monoblock amps which sat closer to the speakers than the component rack. Cables have some minor scuff marks here and there that are common with use but nothing major.
We don't have the original box but will securely pack them for safe shipping.


Since 1946, MCCJ has awarded the coveted Silver Medallion to outstanding leaders who have exhibited extraordinary humanitarian efforts in the community that mirror MCCJ's mission of creating an inclusive community. Paulo Coelho once said, "In a forest of a hundred thousand trees, no two leaves are alike. And no two journeys along the same path are alike." That is precisely what the theme for the 2019's Humanitarian Awards will encompass and celebrate. The Branches of Inclusion will take place at the InterContinental Miami, and it will embrace the differences and highlight the efforts of six South Florida leaders who have taken extraordinary measures to build a more welcoming, inclusive, just and cooperative society.
More than 400 business and community leaders will attend the 2019 Humanitarian Awards Dinner featuring the presentation of the Silver Medallions to the following outstanding civic leaders: Ruth W. Greenfield, Musical Prodigy and Activist; Saif Yamani Khan lshoof, Florida International University Vice President of Engagement; John C. Sumberg, Managing Partner at Bilzin Sumberg Baena Price & Axelrod LLP; and Alexandra Villoch, Miami Herald President & Publisher. The Reverend Diane Shoaf will be honored with The Clergy Medallion, and The Robert H. Traurig Lifetime Achievement Award will go to Burton Young, Marital and Family Attorney. The black-tie optional event will be hosted by WPLG Local 10's Calvin Hughes, along with Dinner Chairs, Gail and Albert E. Dotson, Jr., and Honorary Chairs Brian Dervishi, Michelle Ramirez-Patricios, and Rev. Dr. Patrick O'Neill.
Sponsorship levels and tables are available. For information and tickets, please call the MCCJ office at 305-755-6096.
MCCJ, Inc. was originally founded in Miami in 1935 as the Miami Branch of the National Conference of Christians & Jews. Later on, the group was renamed the Miami Coalition of Christians and Jews. In 2011, in recognition of its embrace of people of ALL faiths and secular backgrounds, the organization formally changed its name to MCCJ, Inc. Today, it goes beyond its original intent by creating a haven for dialog, training inclusive leaders and promoting understanding for all faiths, races and cultures through education, advocacy and conflict resolution. It hosts the oldest interfaith clergy dialog in the United States and presents prejudice reducing programs in local high schools, among other activities. For more information, please visit www.miamiccj.org.
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text:
Павло I (; *бл. 870 — †921) — жупан (князь) Рашки (Сербії) у 917—921 роках.

Життєпис
Походив з династії Властимировичів. Син Брана, онук Мутімира, князя Рашки. Народився близько 870 року. Був вже православним. У 891 році разом з батьком вимушений був тікати до Хорватії. У 894 році Бран загинув. Згодом перебрався до Болгарії, де панував імператор Симеон I Великий. Останній боровся проти Візантії та його союзника Петро, жупана Рашки.

У 917 році на чолі болгарського загону Павло виступив проти князя Петра, але зазнав поразки. Втім, того ж року болгари перемогли Петра й Павло став новим жупаном Рашки.

На початку свого панування Павло зберігав союзницькі стосунки з Болгарією. У 920 році він захопив свого родича Захарію, якого було спрямовано візантійцями до Рашки для здобуття трону. Захарію було відправлено до Болгарії.

У 921 році невдоволений посилення болгарського царя Павло перейшов на бік Візантійської імперії. У відповідь Симеон I відправив проти Рашки свої військ. В цій боротьбі князь Павло зазнав поразки й загинув. Новим володарем став Захарія.

Джерела
 Fine, John Van Antwerp (1991). The Early Medieval Balkans: A Critical Survey from the Sixth to the Late Twelfth Century. Michigan: The University of Michigan Press. ISBN 0-472-08149-7.
 Živković, Tibor (2006). Portreti Srpskih Vladara (IX—XII Vek) (in Serbian). Belgrade. ISBN 86-17-13754-1.

Князі Сербії
Властимировичі

redpajama_set_name: RedPajamaWikipedia