Roční zpráva o stavu řešení projektu 2015 Dílčí výstup projektu

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Roční zpráva o stavu řešení projektu 2015 Dílčí výstup projektu
Projekt TE01020229 (Centrum digitální optiky) je řešen s finanční podporou TA ČR
Roční zpráva o stavu řešení projektu
2015
Dílčí výstup projektu TE01020229
Pracovní balíček: Management projektu
Datum dosažení výstupu: 31. 12. 2015
Předkládá: doc. Mgr. Jaroslav Řeháček, Ph.D., řešitel projektu
Průběh řešení projektu v roce 2015
Rok 2015 byl čtvrtým rokem řešení projektu. Řešení projektu probíhalo ve všech pěti pracovních
balíčcích, byla prováděna kontrola stavu řešení 2x ročně na jednáních Řídícího výboru Centra, byl organizován Technický seminář Centra, kde byly prezentovány dosažené výsledky. Centrum úspěšně
prošlo hloubkovým hodnocením poskytovatele a mezi poskytovatelem a hlavním příjemcem byla uzavřena nová smlouva o poskytnutí podpory na období 2016-2019 s aktualizovanou strategickou
výzkumnou agendou.
Postup prací.
Řešení projektu v roce 2015 probíhalo v souladu s plánem a poskytovatelem schválenými změnami.
Závazné parametry projektu byly splněny. Podrobný popis postupu prací v roce 2015 je uveden níže v
členění po jednotlivých pracovních balíčcích. Výstupy projektu byly propagovány na významných
konferencích a v přípravě je několik publikací, které budou v roce 2016 zaslány do kvalitních
optických odborných časopisů (Optics Express, New Journal of Physics).
Čerpání prostředků.
V průběhu roku 2015 došlo k některým přesunům nákladů mezi pracovními balíčky a mezi jednotlivými kategoriemi nákladů v mezích povolených poskytovatelem. Tyto změny byly prováděny ve
snaze využít prostředky projektu co nejúčelněji. U účastníka Meopta-optika, s.r.o. bylo žádáno o mírné
snížení výše investic. Změny neměly vliv na celkové náklady projektu a byly poskytovatelem
schváleny. Účastníkům projektu se dařilo získávat další prostředky na řešení projektu nad rámec uznatelných nákladů. V roce 2015 bylo takto ve prospěch projektu využito přibližně 700tis. Kč z neveřejných zdrojů, což vedlo ke snížení efektivní míry veřejné podpory.
Řešitelský tým.
V souvislosti s ukončování jednotlivých etap projektu a zahajováním nových činností docházelo v
průběhu roku 2015 k dílčím změnám v řešitelkém týmu. Jádro týmu tvořené klíčovými osobami je stabilní. Probíhala intenzivní interakce mezi účastníky konsorcia formou pravidelných koordinačních
schůzek řešitelského kolektivu k dílčím úkolům v jednotlivých pracovních balíčcích.
Přínosy projektu.
Kromě obecného přínosu spočívajícího ve vytvoření silného konsorcia na území České Republiky v
dynamicky se rozvíjející oblasti digitální optiky, posílení spolupráce mezi účastníky projektu, navýšení
investic průmyslových partnerů do perspektvních oblastí aplikovaného výzkumu, a vytvoření nových
příležitosti ke kariérnímu růstu pro studenty a začínající výzkumníky, bylo v průběhu roku 2015
dosaženo důležitých výstupů, které budou mít příznivý dopad na konkurenceschopnost průmyslových
partnerů. Mezi nimi stojí za zmínku 10 výsledků kategorie RIV, které umožní firmám zůčastněným v
projektu expandovat v technologicky náročných oblastech infračervené optiky, ultrafialové optiky a
přesného strojírenství.
Plnění plánu – dosažené parametry
Management projektu
Milníky
• Pravidelné zasedání řídícího výboru 2x.
• Pravidelný technický seminář.
Dílčí cíle
• Dosažení deklarovaných cílů a milníků projektu pro rok 2015.
Dílčí výstupy
• Roční zpráva o stavu řešení projektu.
Digitální Ramanova spektroskopie a Ramanova optická aktivita
Milníky
• Návrh a výroba spektrografu s parametry založenými na předchozí analýze.
Dílčí cíle
• Objektivy pro UV spektrograf.
• Objektivy pro přenos UV Ramanova rozptýleného záření ze vzorku na vstup spektrografu.
• Objektivy pro přenos VIS Ramanova rozptýleného záření ze vzorku na vstup spektrografu.
• Objektivy pro spektrometr s minimalizovaným pnutím (dvojlomem) v optických částech.
• Kompletovaný spektrograf pro měření Ramanova rozptylu v UV oblasti.
• Kompletovaný spektrograf pro měření Ramanova rozptylu ve VIS oblasti.
• Realizace motorizovaných posuvů ve spektrometru.
Dílčí výstupy
• Funkční vzorek rotačního posuvu pro kontinuální rotaci polarizačních komponent.
• Protokol ohledně vyvíjení technologie opracování optických materiálů v UV oblasti.
• Protokol z testování polarizačních komponent.
• Funkční vzorek objektivů pro přenos UV Ramanova rozptýleného záření ze vzorku na vstup
spektrografu.
• Funkční vzorek objektivů pro přenos VIS Ramanova rozptýleného záření ze vzorku na vstup
spektrografu.
• Funkční vzorek jednotky určené na měření reziduálního pnutí objektivu (R1) a návrh
minimalizace pnutí v optice objektivů.
• Funkční vzorek kolimátoru svazku pro UV excitační záření.
• Prototyp spektrografu pro měření Ramanova rozptylu v UV oblasti.
• Prototyp spektrografu pro měření Ramanova rozptylu ve VIS oblasti.
• Funkční vzorek lineárních posuvů pro dvoupolohové umisťování polarizačních optických
prvků.
•
•
•
Funkční vzorek rotačního posuvu pro přesnou orientaci polarizačních optických prvků.
Funkční vzorek polohovacího zařízení držáku vzorku.
Funkční vzorek ovladače spektrografu.
Výsledky RIV
• Technologie pro přesné a rychlé motorizované polohování součástek. 1xZ - ověřená
technologie, 1xG - funkční vzorek.
• Sestava objektivů pro spektrograf určený pro měřený Ramanových spekter ve VIS oblasti.
Druh: 1xG - funkční vzorek.
• Technologie opracování materiálu CaF2 pro UV aplikace. Druh: 2xZ (ověřená technologie)
Multi senzorické a hyperspektrální zobrazovací systémy
Milníky
• Konstrukční návrh a realizace funkčního vzorku komplexní multisenzorické hlavice.
Dílčí cíle
• Vyvinout objektiv pro multisenzorickou hlavici pro pásmo SWIR.
• Vyvinout multisenzorickou jednotku do stadia funkčního vzorku.
Dílčí výstupy
• Funkční vzorek objektivu pro SWIR kameru multisenzorického systému.
• Funkční vzorek multisenzorické jednotky.
Výsledky RIV
• Objektiv pro digitální kameru v pásmu SWIR. 1xF (užitný vzor), 1xG (funkční vzorek).
• Funkční vzorek multisenzorické jednotky 1xG (funkční vzorek).
Digitální zobrazování s podporou technologie PMS
Milníky
• Modifikace a optimalizace parametrů digitální zobrazovací soustavy pro použití v optické
mikroskopii.
Dílčí cíle
• Optimalizovaný návrh zobrazovacího systému pro mikroskopii.
• Optimalizace softwaru pro digitální zobrazování.
• Testy funkční sestavy mikroskopu pro digitální zobrazování.
Dílčí výstupy
• Funkční systém mikroskopu využívající prostorové modulace světla.
Výsledky RIV
• Afokální optický systém pro korekci barevné vady difraktivních zobrazovacích prvků 1xF
(užitný vzor), 1xX(přihláška vynálezu).
Zpracování dat S-H senzoru v metrologii a zobrazování
Milníky
• Předložení analýzy možností uplatnění vylepšené S-H detekce v současné technologické praxi.
Definice metodiky pro diagnostiku netradičních optických parametrů.
• Uvedení do provozu stanice pro měření tvaru optických prvků.
Dílčí cíle
• Návrh, realizace a vymezení možností kompaktního systému pro diagnostiku koherentních
laserových svazků.
• Návrh a realizace zařízení pro měření tvaru optických prvků rovinné a sférické optiky.
Dílčí výstupy
• Výzkumná zpráva: Diagnostika koherentních laserových svazků.
• Zařízení pro měření tvaru optických prvků rovinné a sférické optiky.
Výsledky RIV
• Metodika měření tvaru optických prvků rovinné a sférické optiky pomocí S-H senzoru. 1xZ
(ověřená technologie).
Závěr: Bylo dosaženo všech milníků, dílčích cílů a dílčích výstupů a výsledků podle plánu, popř.
podle poskytovatelem schválených změn plánu.
Pracovní balíček: Digitální Ramanova spektroskopie a Ramanova optická aktivita
V předchozím roce byl proveden návrh optické soustavy UV Ramanova spektrografu. Na počátku
letošního roku byl návrh konstrukčně i technologicky optimalizován. V průběhu roku byly zakoupeny
Meopty koupeny nové speciální stroje na výrobu asférické optiky a díky rychlému zvládnutí
problematiky výroby této optiky byly na konci roku vyrobeny dva kusy asférické čočky fokusačního
objektivu spektrografu. Tyto čočky splňují všechny výkresové požadavky.
Byl dokončen návrh mechanické konstrukce UV a VIS spektrografu a proběhla výroba mechanických
částí a úprava povrchových dílů. Jednotlivé prvky sestavy jsou umístěny na společné monolitické
rovinné základně. Analýza a návrh optické sestavy určené pro přenos záření ze vzorku na vstup
spektrografu pro UV a VIS Ramanův spektrometr byla provedena v předchozích letech, v roce 2015
byla provedena výroba optiky a mechanické konstrukce pro tyto optické sestavy.
Pro testování polarizační optiky, zrcadel a dalších komponent, které mají vliv na polarizační stav
záření, byl zkonstruován spektropolarimetr, který měří plnou Muellerovu matici vzorku a to jak pro
vzorky na průchod, tak i na odraz. V průběhu řešení projektu bylo zkonstruováno několik verzí
spektropolarimetru, ve kterých byly použity různé zdroje záření, fázové destičky, polarizátory, rotační
posuvy i spektrální detektory záření.
Dále byly provedeny rozsáhlé práce týkající se vývoje, výroby a testování polohovacích
motorizovaných součástí spektrometru. Veškeré dílčí výstupy umožnily sestavení funkčního vzorku a
technologie pro přesné a rychlé motorizované polohování součástek.
Během roku 2015 došlo k úspěšné výrobě tvarově různých optických prvků z CaF2 s rovinnými i
sférickými plochami, na kterých byl technologický postup odzkoušen a ověřen. V průběhu výroby byla
dozorována především operace finálního leštění doplněná o hodnocení dosažené drsnosti ploch před
vrstvením a operace centrování. Tyto aktivity v závěru roku vedly až k vytvoření dvou ověřených
technologií opracování materiálu CaF2 pro UV oblast a to pro rovinné i sférické plochy, které jsou
výsledkem projektu V011.
Pracovní balíček: Multi senzorické a hyperspektrální zobrazovací systémy
V návaznosti na konstrukci optického systému SWIR 4/250 určeného pro dálková pozorování byly
studovány vhodné mechanické materiály a jejich povrchové úpravy s cílem snížit jejich odrazivost a
konstruktéři navrhli mechanickou část laboratorního vzorku objektivu. Laboratorní vzorek objektivu
byl testován v laboratoři i exteriéru.
Po úspěšném ukončení vývoje technologie opracování klíčových chalkogenních optických skel ZnS,
ZnSe a AMTIR1 se výzkum soustředil na zjišťování vlivu opracování nejen u jednotlivých elementů,
ale ve složených soustavách. K tomu byla využita sada dříve navržených a vyrobených
technologických čoček k sestavení sady optických soustav, které různým způsobem kombinovaly
optické materiály.
Relativně velký počet zařízení, které tvoří funkční vzorek multisenzorické hlavice, vznikal postupně
během posledních dvou let vývoje. Jednotícím prvkem mezi různorodými přístroji je software
MyVector OL Funkční vzorek představuje prostorově rozprostřenou multisenzorickou jednotku, která
umožňuje v terénu získávat a sdílet v reálném čase multimediální data o prostředí, což je účelné pro
neznámá prostředí s aplikací pro vojenský a bezpečností průmysl.
Z hlediska teorie byly souhrnně popsány fyzikální principy šíření optických svazků s cílem ujednocení
matematicko-fyzikálního popisu jednotlivých typů svazků: konkrétně skalární, vektorový a popis v
neparaxiálním a paraxiálním prostoru. Paralelně probíhal aplikovaný výzkum matematicko-fyzikálních
modelů zpracování digitálního obrazu degradovaného průchodem atmosférou.
Pracovní balíček: Digitální zobrazování s podporou technologie PMS
Byly vytvořeny algoritmy, které rozšířily základní paraxiální model používaný pro analýzu a
rekonstrukci korelačních záznamů v metodách nekoherentní holografické mikroskopie. Nově vytvořené
algoritmy pracují s reálnými experimentálními parametry a umožnily zapracování aberačních vlivů a
efektů souvisejících s částečnou časovou koherencí použitého světla.
Proběhla optimalizace zobrazovacích systémů pro využití v mikroskopii. Dosažené výsledky podpořily
přechod od dříve testovaných laboratorních systémů k funkční sestavě mikroskopu s PMS, která
umožňuje precizovat prováděné experimenty. Byla podaná přihláška užitného vzoru na afokální
refraktivní optický systém provádějící achromatizaci PMS při zobrazovacích aplikacích ve viditelné
oblasti spektra a apochromatizaci PMS v blízké infračervené oblasti spektra. V roce 2015 bylo získáno
Osvědčení o zápisu užitného vzoru do rejstříku. Byla podaná přihláška vynálezu na refraktivní optický
systém provádějící apochromatizaci PMS při zobrazovacích aplikacích v blízké infračervené oblasti
světelného spektra.
Optimalizace vybraných zobrazovacích režimů, které byly provedené v uplynulé a v předešlých
etapách řešení projektu, umožnily realizovat přechod od laboratorního uspořádání k funkční sestavě
mikroskopu s PMS. Návrh funkční sestavy mikroskopu je založen na využití komerčně dostupného
mikroskopu Nikon E200. Realizovaný systém umožňuje zachovat standardní režimy diaskopického a
episkopického osvětlení, které jsou dále podporovány pokročilými metodami využívajícími PMS.
Během prvních ověřovacích experimentů byly pomocí biologických vzorků a vhodných objektů
optimalizovány a úspěšně testovány základní a pokročilé zobrazovací režimy.
V rámci činností na tomto balíčku byly prováděny aktivity směřující k rozšíření problematiky balíčku
WP4 do okruhu topografie optických povrchů, zejména povrchů různé drsnosti. Tato problematika se
na globálním trhu ukazuje jako téma s významným potenciálem. Tato problematika bude v r. 2016 dále
rozvíjena v kooperaci s Ústavem přístrojové techniky ČAV v Brně v rámci aktivit evidovaných
v upřesněné SVA projektu.
Pracovní balíček: Zpracování dat S-H senzoru v metrologii a zobrazování
V roce 2015 se činnost zaměřila na výzkum propagace laserových svazků v koherentním limitu. Pro
některé typy laserových svazků obsahujících velké deformace vlnoplochy, kde je obtížné využití
modové dekompozice, může být i tato limita dostatečná pro popis propagace svazku. Za tímto účelem
bylo zkoumáno záření DPSS Q-SWITCHED laseru vlnové délky 266nm, dostupných v laboratořích
Meopty, a s pomocí S-H detekce, kde je získán intenzitní profilu i vlnoplocha z jednoho snímku, byla
testována možnost predikce difrakce svazku.
Dále se pozornost zaměřila na detekci částečně koherentních polí v různých modových reprezentacích.
Zkoumány byly omezení rekonstrukčního prostoru a na to navazující rekonstrukční chyby. Byly
provedeny numerické simulace prokazující možnost navrženého měření. Jako vhodný prvek pro
realizaci tohoto principu byl vybrán DMD prostorový modulátor. V roce 2015 byly započaty stavby
experimentů tento prvek využívající, v roce 2016 budou tyto dokončeny.
Jednotky pro kontrolu optických ploch realizované a testované na PřF UP byly přesunuty do
vývojových laboratoří společnosti Meopta. Obě jednotky byly kalibrovány pomocí interferometrických
kalibrů. Výsledky byly prováděny další úpravy v softwaru MeoSHS, který slouží pro komunikaci
s měřicími stanicemi a ke zpracování výsledků. Software byl přizpůsoben daným stanicím. Byly
provedeny změny v průměrování měřených dat, díky kterým bylo dosaženo lepší opakovatelnosti
měření. Dále byly přidány možnosti úpravy kalibračních bodů, díky kterým je možné měřit na
obecnějších optických prvcích.
Seznam výsledků, prezentací a publikací
Výsledky
•
•
•
•
•
•
•
Technologie pro přesné a rychlé motorizované polohování součástek. 1xZ - ověřená
technologie, 1xG - funkční vzorek.
Sestava objektivů pro spektrograf určený pro měřený Ramanových spekter ve VIS oblasti.
Druh: 1xG - funkční vzorek.
Technologie opracování materiálu CaF2 pro UV aplikace. Druh: 2xZ (ověřená technologie).
Objektiv pro digitální kameru v pásmu SWIR. 1xF (užitný vzor), 1xG (funkční vzorek).
Funkční vzorek multisenzorické jednotky 1xG (funkční vzorek).
Afokální optický systém pro korekci barevné vady difraktivních zobrazovacích prvků 1xF
(užitný vzor), 1xX(přihláška vynálezu).
Metodika měření tvaru optických prvků rovinné a sférické optiky pomocí S-H senzoru. 1xZ
(ověřená technologie).
Prezentace
•
•
B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L.L. Sanchez-Soto, Shack-Hartmann tomography
for multimode optical beam propagation, OSA Digital Holography & 3-D Imaging, Shanghai
2015
J. Rehacek, Z. Hradil, B. Stoklasa, L. Motka, L.L. Sanchez-Soto, Wavefront-sensor tomography
for measuring spatial coherence, SPIE Optics+Photonics, San Diego 2015
Publikace
•
Kapitán J., Barron L. D., Hecht L.: A novel Raman optical activity instrument operating in the
deep ultraviolet spectral region. J. Raman Spectrosc. 2015, 46, 392-399.
•
J. Rehacek, Z. Hradil, B. Stoklasa, L. Motka, L.L. Sanchez-Soto, Wavefront-sensor tomography
for measuring spatial coherence, Proc. SPIE 9617, Unconventional Imaging and Wavefront
Sensing 2015, 961703 (September 4, 2015); doi:10.1117/12.2188040.
Research article
Received: 22 December 2014
Revised: 22 January 2015
Accepted: 23 January 2015
Published online in Wiley Online Library: 4 March 2015
(wileyonlinelibrary.com) DOI 10.1002/jrs.4665
A novel Raman optical activity instrument
operating in the deep ultraviolet spectral region
Josef Kapitán,a,b* Laurence D. Barrona and Lutz Hechtc
Raman optical activity (ROA) has been exclusively observed in the visible (VIS) and near-infrared (NIR) spectral regions to date.
During the last few years, we have designed, constructed and tested the first ROA instrument, operating in the deep-ultraviolet
(DUV) spectral region employing 244-nm excitation. This novel DUV ROA instrument is based on a backscattering geometry
and incident circular polarization modulation (ICP); it makes use of a fast DUV imaging lens-based spectrograph and specially designed DUV grade polarization optics. The performance of this instrument has been evaluated by analysing measured nonresonant DUV ROA spectra of non-absorbing enantiomeric liquid samples and by comparing these with corresponding ROA spectra recorded in the visible spectral region. Copyright © 2015 John Wiley & Sons, Ltd.
Keywords: Raman optical activity; resonant Raman scattering; ultraviolet Raman optical activity
Introduction
392
Raman optical activity (ROA) typically refers to tiny intensity differences measured in Raman scattering from chiral molecules, either
using right and left circularly polarized exciting radiation or
analysing scattered Raman signals for right and left circular polarization states.[1–5] These two different modulation strategies, respectively, measure incident circular polarization (ICP) and scattered
circular polarization (SCP) forms of ROA. They originate in distinct
time-even pseudoscalar molecular polarizability and optical activity
tensor products and are virtually identical for non-absorbing molecules. ICP and SCP modulation consequently measure the same
ROA spectra under non-resonant and otherwise comparable scattering conditions within the Rayleigh limit.[3,4]
ROA exhibits an exquisite sensitivity for the identification of specific conformations and their changes, especially in biopolymers in
aqueous solution, and frequently provides structural information
not obtainable by other analytical techniques.[6,7] However, ROA
measurements, predominantly performed in the visible (VIS) spectral region, may still be occasionally plagued by annoying shortcomings, impeding a more widespread application of the
technique particularly within a biochemical context.
The detection of VIS ROA spectra with an acceptable signal-tonoise ratio (SNR) may suffer from the application of rather high laser
powers and relatively long acquisition and exposure times, may require the use of fairly large sample quantities and rather concentrated and pure samples and may frequently be hampered by
either intensely coloured samples at non-transparent excitation
wavelengths or strongly fluorescing trace impurities, potentially obscuring any Raman scattering due to comparatively much larger
fluorescence cross sections and quantum yields.
To overcome these disadvantages, the measurement of ROA,
employing other than the customary 488, 514.5 and 532-nm
VIS excitations, have consequently been attempted or
suggested.[8,9] The use of longer excitation wavelengths offers
the principal advantage of significantly suppressed fluorescent
emission levels from impure or inherently fluorescing samples,
and laser emission at 780 nm may be used routinely for the
J. Raman Spectrosc. 2015, 46, 392–399
excitation of ROA spectra.[8] Suggestions to utilize 1064-nm excitation for the detection of Fourier transform ROA[9] have never
been realized in practice, presumably due to the lack of suitable
detectors with sufficiently large quantum efficiencies and the
significantly smaller scattering intensities in the near-infrared
(NIR) region due to the dependences of Raman and ROA intensities on the frequency of the scattered radiation.[1,8,10] Furthermore, NIR ROA does not appear to be viable for studies of
biologically relevant molecules due to the associated poor
signal-to-noise ratios (SNRs).
Intensity of Raman scattering is proportional to the fourth
power of the wavenumber eνs (or frequency) of the scattered
radiation.[10] On the other hand, ROA intensity is eν5s dependent
due to the frequency dependence of the electric dipole–
magnetic dipole optical activity tensor G′ and the corresponding
contribution from the electric dipole-electric quadrupole tensor
A.[1] Most modern spectroscopic detectors employed in the
ultraviolet (UV), VIS and NIR spectral regions do not measure radiant
flux (in Watts) directly; instead they operate in the photon counting
regime. Since the relationship between photon flux P (number of
photons per second) and radiant flux Φ is P ¼ Φ=h ceνs, the number
of Raman and ROA scattered photons is proportional to eν3s and eν4s ,
respectively.[11]
The use of significantly shorter excitation wavelengths within the
deep UV spectral region (λ < 300 nm) therefore holds great promise
* Correspondence to: Josef Kapitán, Department of Optics, Palacký University
Olomouc, 17. listopadu 12, Olomouc 77146, Czech Republic.
E-mail: [email protected]
a Department of Chemistry, University of Glasgow, Joseph Black Building, Glasgow
G12 8QQ, UK
b Department of Optics, Palacký University Olomouc, 17. listopadu 12, Olomouc
77146, Czech Republic
c Tracerco Technology Centre, Tracerco Ltd, Belasis Hall Business Park, Pavilion 10,
The Moat, Billingham TS23 4ED, UK
Copyright © 2015 John Wiley & Sons, Ltd.
Raman optical activity instrument operating in DUV spectral region
for the acquisition of ROA spectra with significantly increased SNRs.
On account of their large frequency dependence, considerably
greater ROA intensities are predicted to occur within the DUV in
comparison to other spectral regions. Depending on the detailed absorption characteristics of the sample under study, especially pre-,
post- or rigorously resonant scattering conditions, DUV excitation is
anticipated to generate Raman and ROA signal boosts of potentially
several orders of magnitude.[12,13] Significantly lower laser powers
with smaller exposure and acquisition times may consequently be
employed, and only relatively low sample concentrations may therefore be required for the measurement of DUV ROA spectra. Also, due
to the relatively narrow spread of DUV Raman spectra and relatively
large fluorescent Stokes shifts, excitations using sufficiently short
wavelengths (λ < 260 nm) generally effect a virtually complete spectral separation of Raman and fluorescent emissions in condensedphase samples.[14–18] DUV Raman and ROA spectra are consequently
not prone to SNR deteriorating interferences from fluorescent backgrounds so that sample purity may prove not to be such a critical factor for the detection of DUV ROA spectra. Depending on the extent
of vibronic coupling, DUV ROA spectra of absorbing samples may,
perhaps more importantly, provide a valuable source of novel structural information.
Despite these potential advantages of DUV ROA measurements,
unfavourable scenarios have been identified which may considerably reduce the chances of DUV ROA detection or render it
completely useless as a new source of structural information.[19–23]
Similarly, anti-resonant scattering regimes with excitations approaching electronic transitions and causing largely reduced
Raman and ROA intensities have been discussed.[19–21] The ROA
spectra of absorbing chiral molecules that show contributions predominantly from a single electronic state (SES) are predicted to exhibit a monosignate appearance with the sign of all bands being
opposite to that of the single electronic transition’s rotatory
strength.[22,23] On the other hand, any significant deviation of even
monosignate resonant ROA relative intensities versus the parent
resonant Raman relative intensities is a breakdown of the SES limit
and indicates the presence of intensity contributions of more then
one excited electronic state.[5]
To explore the practical feasibility of DUV ROA measurements
the authors have designed, constructed and tested an appropriate
ICP backscattering instrument, which is the subject of this paper.
Experimental
Samples
All chemicals were purchased from Sigma-Aldrich Corporation and
used without any additional purification. Both enantiomers of menthol and borneol were dissolved in analytical grade CH3OH, and
those of glucose in distilled water. All the samples were studied in
rectangular fused quartz microfluorescence cells with internal cross
sections of 4 × 4 or 4 × 3 mm (Starna Scientific) in both the DUV and
VIS ROA spectrometers.
Non-resonant VIS Raman and ROA spectra were measured
employing a customized commercial ChiralRaman backscattering
SCP instrument (BioTools Inc.) with 532-nm excitation and
~7 cm1 spectral band-width.[24] Non-resonant DUV Raman and
ROA spectra were obtained using our newly constructed backscattering ICP instrument with 244-nm excitation and ~15 cm1 spectral bandwidth.
Current experimental setup
The novel UV ROA spectrometer is based on a backscattering geometry and ICP modulation.[25] This polarization modulation
scheme has been selected mainly because of its relative simplicity,
ease of implementation and the availability of suitable polarizing
components functioning in the DUV spectral region. The layout of
our new DUV ROA spectrometer is depicted schematically in Fig. 1.
A continuous-wave intracavity frequency-doubled Ar+ ion laser
J. Raman Spectrosc. 2015, 46, 392–399
Copyright © 2015 John Wiley & Sons, Ltd.
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393
Figure 1. Schematic layout of the DUV ROA spectrometer. The spatial filter has been omitted for clarity (vide text for a more detailed description of individual
optical components).
J. Kapitán, L. D. Barron and L. Hecht
(Coherent, Model Innova 90C FRED) equipped with a β-barium
borate frequency-doubling crystal for the generation of 244-nm
DUV radiation is used as a radiation source.[26]
Generating circularly polarized incident radiation
394
As in previous ROA instruments, an electro-optic modulator (EOM)
was chosen to perform polarization modulation of the incident radiation between right and left circularly polarized states.[25]
It is of paramount importance that high purity linearly polarized
radiation enters the EOM. This task seems to be trivial, but selecting
a high-quality polarizer has proven to be one of the main obstacles
to the successful realization of our DUV ROA spectrometer. Several
polarizers of different materials and several designs from various
manufacturers have been tested, but a Glan-Taylor type polarizer
made of the highest quality UV grade calcite (Bernhard Halle, model
PGU12) has finally been implemented. Impurity free calcite is transparent above 220 nm. The above mentioned polarizer shows absorption losses <10% for 244-nm excitation radiation and an
extinction ratio <106; it is also sufficiently stable in an incident laser beam of 1-mm waist diameter and radiant flux <10 mW. Linearly birefringent materials exist that are more transparent in the
DUV region than calcite. However, we have found that, for example,
α-barium borate (α-BBO) is not sufficiently stable (or our material
has not been sufficiently pure); although crystal quartz and magnesium fluoride (MgF2) are stable and transparent, they are much less
birefringent than calcite, and a different type of polarizer (Rochon)
has to be used with a potentially higher extinction ratio of 105.
Crystal quartz is also circularly birefringent (optically active), which
complicates polarization issues even further.
The DUV EOM is based on a longitudinal Pockels cell incorporating a potassium dideuterium phosphate (KD*P) crystal (Leysop,
clear aperture 8 mm, in combination with a high voltage linear differential amplifier, model 5000) as in a previous VIS ROA
spectrometer.[25] KD*P is sufficiently transparent above 220 nm provided it does not contain residual impurities. Retardance of the
Pockells cell is inversely proportional to wavelength, and hence
the quarter-wave voltage is smaller in the DUV than in the VIS spectral region.
It has been necessary to develop a procedure for the fast and
precise adjustment of the EOM in order to generate circularly polarized radiation with absolute values of ellipticities higher than 44° for
successful DUV ROA measurements. The EOM crystal has to be precisely positioned (translation in x and y directions; rotations about x,
y and z axes), and voltages for negative and positive quarter-wave
retardation have to be accurately set. The x and y position of the incident laser beam within the EOM aperture is usually set prior to
subsequent fine adjustments. However, five interdependent degrees of freedom remain. Several possible procedures exist for the
setting of the EOM, but it was necessary to use one that avoids rotation of the EOM by a large angle and insertion of additional
waveplates during or after the setup in order to eliminate the influence of other parameters on the final quality of the exiting polarized radiation. The polarization state of the emerging radiation
has been tested by rotating a Glan-Taylor analyser (Thorlabs, model
GL5) and by measuring its transmittance as a function of its rotation
angle with a photodiode based power meter (Thorlabs, model
S120V). This configuration is of course not able to distinguish
between left- and right-elliptical polarization states; it measures
merely the absolute value of the ellipticity, but its robustness and
simplicity have proven to be a great advantage. The analyser is
placed in a motorized rotation stage (Thorlabs model PRM1/MZ7)
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and operated in step-scan mode synchronized with the EOM voltage switching and data readout from the power meter (Thorlabs,
model PM100) connected to a PC via a RS232 interface. Since the
GL5 polarizer operates outside its specified wavelength range, it exhibits solarization effects, namely increasing absorbance upon DUV
irradiation (244 nm, > 2-mW laser power). These effects are fortunately reversible on a relatively short time scale of seconds, and
continuous rotation of the analyser and placement of the laser
beam outside the centre of the analyser aperture enable long-term
and reproducible measurements. The quarter-wave voltage is
switched alternately between positive-negative, negative-positive
polarities etc. in order to exclude systematic offsets in radiant flux
measurements for each voltage settings.
For perfectly circularly polarized radiation (exhibiting ellipticities χ ± 45°) entering the rotating analyser, the output signal Φ0
is independent of the analyser orientation. For general elliptically polarized radiation the output signal is a harmonic function from which the ellipticity can be calculated as χ = arctan
(Φ0 min/Φ0 max), where Φ0 min and Φ0 max are the minimal and
maximal values of the radiant flux detected during complete
analyser rotation by 360°. Critical for a fast and correct EOM
alignment is the observation that the power output signal for
different voltages cross at certain ‘nodes’ and that these ‘nodes’
determine the best possible ellipticity achievable solely by voltage change; i.e. for situations when extremes of the output signal coincide with these nodes. Four nodes per rotation period of
the analyser exist; two above and two below the constant line
Pconts (see Fig. 2). Any two voltages are then sufficient for the
identification of these ‘nodes’ and also for an estimate of the
best possible ellipticity. The position of the nodes should be
as close to the ‘constant line’ as possible, and it can be altered
by fine rotation of the EOM crystal about the x, y and z axes. Rotations about different axes alter the position of the nodes in a
different manner, so that the position of ‘nodes’ also determines
the axis about which the rotation has to be performed in order
to move the nodes close to the constant line. The laborious EOM
alignment procedure was thus shortened from several hours to
a simple reproducible routine lasting approximately 15 min, enabling frequent readjustments. The EOM needs to be readjusted
every 2–3 days during measurements for successful recording of
DUV ROA in order to maintain absolute value of ellipticities
above 44°.
As in previously realized VIS ICP ROA instruments a combination
of rotational and tilt stages for roll, pitch and yaw movements has
been employed for the alignment of the EOM. Roll movement is
achieved by a rotation stage (Newport, model M-UTR80SA), and
since the rotation axis is aligned parallel to the crystal central axis,
it virtually corresponds to a pure rotation about the z axis. Pitch
movement is realized by a tilt stage (Newport, model M-TGN80)
providing rotation about the y axis combined with translation along
the +x and z directions. Yaw movement is accomplished by another rotation stage (Newport, model M-UTR80SA), although the
axis of rotation does not coincide with the crystal centre, so rotation
about the x axis is also accompanied by translation along the y direction. A representative example of the EOM alignment procedure
is illustrated in Fig. 2.
Focusing circularly polarized incident radiation
The circularly polarized radiation emerging from the EOM is focused into the sample cell through holes drilled in a diverting mirror, collimating lens and Lyot depolarizer (Fig. 1).[25] The beam waist
Copyright © 2015 John Wiley & Sons, Ltd.
J. Raman Spectrosc. 2015, 46, 392–399
Raman optical activity instrument operating in DUV spectral region
Figure 2. Detection of ellipticity in the excitation beam emerging from the EOM. Each panel contains four measurements; positive (red) and negative (black)
voltages of two different values (solid and dashed curves). Positions of nodes are marked by solid circles, positions of nodes from previous steps are marked by
empty circles and arrows indicate changes in node position compared to previous steps. a) Initial measurement: positive voltage generates virtually perfectly
circularly polarized radiation (ellipticity 44.1°), but negative voltage generates elliptically polarized radiation with ellipticity 39.3°. b) Changes after EOM rotation
about the z axis by 1.3°: node pairs (above each other) move in the same direction. c) Changes after pitch movements consisting of a rotation about the y axis by
0.6 mrad, a translation along the z axis by 0.06 mm and translation along the x axis by 0.05 mm. Absolute values of ellipticities measured after these fine adjustments
are 43.9° and 43.8° for positive and negative voltages, respectively. The voltages are subsequently finely tuned in order to move the curve maxima to nodes (not
shown), thereby achieving an absolute value of ellipticity of more than 44.1° for both arms. This figure is available in colour online at wileyonlinelibrary.com/journal/jrs
diameter in the focal region should be approximately 25 μm in order
not to overfill the spectrograph entrance slit with a width of
25–50 μm too much (taking into account aberrations and close to
unit magnification of the collection optics). Since a free working distance of at least 15 cm is needed because of the physical dimensions
of the depolarizing and collecting optics and the diverting mirror, the
laser beam diameter has to be expanded twofold prior to focusing in
order to achieve the desired beam waist diameter in the focal region.
The beam expander is realized by two best form lenses of focal
lengths 50 mm and 100 mm (CVI) and a 50-μm diameter pinhole is
placed in their common focus, so that the beam expander works also
as a spatial filter. The spatial filter unit significantly enhances the quality of the incident laser beam and makes it less prone to contamination by scattering particles that are inevitably attracted to optical
element surfaces due to an optical tweezers effect. The expanded
beam is subsequently focused by a another f = 200-mm best form
lens. All lenses are made from fused silica; they are stress free and
do consequently not affect the polarization state of the incident radiation due to any residual linear birefringence.
For measurements with a wider slit (50 μm), a larger beam waist
diameter of 50 μm is used. In this case the spatial filter consists of
two f = 50-mm best form lenses, so that the diameter of the laser
beam remains unchanged, and unit magnification is maintained.
In principle the entire unit may have been omitted. Nevertheless,
the benefits of the spatial filter, significantly improving the laser
beam quality, more than justify the filter’s presence.
Depolarizing, collecting and analysing Raman scattered
radiation
J. Raman Spectrosc. 2015, 46, 392–399
Copyright © 2015 John Wiley & Sons, Ltd.
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395
Mirrors and especially diffraction gratings represent polarizing elements. It was relatively early recognized that in order to accomplish successful ICP ROA measurements in backscattering, a Lyot
depolarizer is a critical optical element which has to be placed
in the diverging beam of the backscattered Raman radiation.[27]
Initially a Lyot depolarizer made from quartz (Leysop, custom
made model, total length of 20 mm) consisting of three plates of
thicknesses in the ratio 2:1:4 has been tested. However, we were
unable to find the optimum rotation (about the z-axis) in order
to achieve a sufficiently low level of polarization artefacts. Quartz
has been selected because of its high DUV transmittance, but its
circular birefringence is probably the main reason for its failure.
A much thinner depolarizer made of UV grade calcite with two
plates of thickness 4 and 2 mm with a 20 × 20 mm octagonal
aperture, cemented with UV transparent cement (manufactured
by Bernhard Halle GmbH.) has subsequently been tested. A small
1-mm hole has been drilled through the Lyot depolarizer. This
Lyot depolarizer is sufficiently transparent, and its much lower
thickness causes also smaller spherical aberration due to its presence in a divergent beam.
The divergent beam of backscattered Raman radiation is collimated by a plano-convex lens of focal length 36 mm and aperture
diameter 25 mm (CVI, model PLCX-25.4-18.0-UV-248-355). The
quasi-collimated beam is subsequently diverted by a pair of 2-inch
diameter circular mirrors with two narrow band dielectric coatings
(CVI, model TLM1) within a ‘polarization neutral’ arrangement (see
Fig. 1).[28] This utilization of two mirrors is in practice not necessary,
but it provides more flexibility in aligning the train of optical elements. Rayleigh scattering in the diverted beam is subsequently efficiently suppressed by an edge filter (Materion, model with OD5 at
244 nm, edge slope <4 nm) and focused onto the spectrograph entrance slit by a f/2 plano-convex lens with a focal length of 25 mm
(CVI, model PLCX-25.4-12.9-UV-248-355). This particular optical system exhibits a relatively large amount of spherical and chromatic
aberrations, because of the use of singlet plano-convex lenses
and the introduction of flat optical elements into a diverging beam
(cell window, Lyot depolarizer). Custom made achromatized
aspherical doublet lenses will be used in future versions of the spectrometer in order to correct for these aberrations.
Spectral analysis and data recording are performed by a
home designed deep DUV lens-based imaging spectrograph with
a large f/2 aperture, equipped with a reflective grating with
3600 grooves mm1 operating at first order, a silicon deep depletion CCD detector (Renishaw plc, Model UV-RenCam) and the data
collection being synchronized with ICP modulation.[26] The DUV
spectrograph and its detection system are described in detail
elsewhere,[26] the only difference being that the individual elements of the compound focusing and collimating lenses inside
J. Kapitán, L. D. Barron and L. Hecht
the spectrograph have been anti-reflection coated for a spectral
range of 225–308 nm (coating provided by CVI) that significantly increased the transmittance of the entire system.
Backscattered DUV Raman intensities
Raman and ROA intensities measured with the DUV ROA instrument employing 244-nm excitation will be compared with data recorded using a commercial VIS ROA instrument with 532-nm
excitation (Biotools, ChiralRaman) described in detail elsewhere.[24]
In order to compare Raman spectra measured with two distinct
spectrographs operating in very different spectral regions (DUV
and VIS), the data must be evaluated using identical units. Detected
signals strengths are usually expressed in ADC counts which may
be converted into electronic charge (e) units by multiplication
with the reciprocal gain of the CCD camera system, which results
in 2.5 and 8 e per ADC count for the DUV[23] and VIS system respectively; it is customary to display the Raman intensity scale in
the e units. Spectral intensities ought to be scaled by the number
of electronic charges per detector element (pixel). However, this
value is not a good measure, since the spectral width of a pixel
may vary not only between different spectrographs, but also within
the same spectrum (see Fig. 3a), and the information about the
pixel spectral width may not always be specified. We will therefore
compare measured intensities in e/cm1 (e × cm) units, as the
number of detected electrons per pixel is divided by the pixel spectral width. This fact has to be taken into account when integral peak
intensities are calculated.
Raman spectra in the DUV and VIS spectral regions are measured with different acquisition times and excitation laser powers.
A much higher radiant flux can be used in the VIS spectral region
without any danger of sample photo-degradation. It is thus suitable to compare intensities per unit excitation energy
(e × cm × J1).[28] As mentioned in the introduction, Raman scattering intensities depend on the third power of wavenumber of
the scattered radiation eνs for photon counting detectors (in case
the irradiance of the exciting radiation is expressed in W/m2).
Measured signals for different excitation wavelengths may therefore be only directly compared if the correction for this dependence is taken into account (see Figs. 3b, 4) so that the
resulting unit for the detected signal reads e × cm4 × J–1. However, the difference in the wavenumber-dependent correction
factor varies by only approximately 20% within the selected
wavenumber range 0–2000 cm–1 and can consequently be safely
neglected in most cases (as in Figs. 5–7).
As is evident from the comparison of Raman spectra displayed in
Fig. 4 the DUV Raman intensities are more than three orders of
magnitude smaller than anticipated for the case of equally efficient
DUV and VIS Raman scattering collection and detection systems.
However, the efficiency and transmittance values of the DUV ROA
spectrometer components are generally significantly smaller than
those of the VIS ROA spectrometer (Table 1), which may easily explain the efficiency difference of approximately two orders of magnitude, with a much lower efficiency of the DUV CCD detector
being the most important factor. The VIS ROA spectrometer also exhibits an efficient cross-section transformer[28] with an effective
large circular aperture of diameter more than 1.5 mm being utilized
at the entrance of the spectrograph. In contrast, a narrow entrance
slit of 25 μm is employed in the DUV ROA spectrograph. The slit
height is 6 mm, but only the central part (~1 mm) is effectively used.
The DUV ROA spectrometer is also impaired by larger aberrations in
the light collection optics, so that an overall remaining factor of ten
from the signal comparison may be easily attributed to these effects. It should also be mentioned that the spectral resolution of
the DUV spectrometer is approximately two times worse than that
for the VIS spectrometer.
DUV ROA backscattering spectra
Tests for the correct functioning of the ROA spectrometer are conducted within a non-resonant scattering regime so that any potential complications associated with pre-, post or rigorous resonant
scattering scenarios may be safely avoided. Since they do not possess any chromophores, which might absorb the exciting or
scattered radiation in the DUV or VIS spectral regions, and since
their VIS ROA spectra have already been recorded and both enantiomers of each are readily available, menthol,[29–31] borneol[32]
and glucose[33] have been selected as ideal candidates for this initial
DUV ROA study.
Results and discussion
The DUV ICP and VIS SCP Raman and ROA spectra of all the samples
(Figs. 5–7) were measured at ambient temperature and are presented as raw circular intensity sums (IR + IL) and (IR + IL), and circular
intensity differences (IR IL) and (IR IL), respectively. IR and IL denote the Raman intensities using right and left circularly polarized
incident radiation; whereas IR and IL are the intensities of right
and left circularly polarized Raman-scattered radiation.
396
Figure 3. a) Pixel spectral widths for the VIS (black) and DUV (red) ROA spectrometers. b) Ratio of third-power wavenumber dependence of Raman scattering
for UV and VIS Raman spectrometers with excitation wavelengths of 244 and 532 nm in the photon counting regime. This figure is available in colour online at
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Copyright © 2015 John Wiley & Sons, Ltd.
J. Raman Spectrosc. 2015, 46, 392–399
Raman optical activity instrument operating in DUV spectral region
Figure 4. Comparison of Raman spectra of a) a 3 mol/l aqueous solution of D-glucose and b) solution of ()-Menthol (2.6 g in 1.0-g methanol) recorded
with VIS (black) and DUV (red) ROA spectrometers. The VIS Raman spectra intensity has been divided by an empirical factor of 1400 in order to display
1
spectra onto the same intensity scale. The cutoff wavenumber for the DUV edge filter is approximately 450 cm . Laser powers measured at the sample
for the DUV spectra are 2.7 mW; those for VIS spectra are 270 mW (glucose) and 40 mW (menthol). Accumulation times for the DUV spectra are 40 s; those
1
for VIS spectra are 1560 s (glucose, scaled to ~1.1 s) and 1410 s (menthol, scaled to ~1.0 s). All spectra have been corrected for spectral pixel width (cm ),
3
3
excitation energy (J) and eνs dependence (cm ). This figure is available in colour online at wileyonlinelibrary.com/journal/jrs
Figure 5. Non-resonant backscattering Raman (top) and ROA (bottom) spectra of (+)-menthol (red) and ()-menthol (black) solution (2.3 g in 1.0 g
methanol) with a) 532-nm excitation, SCP modulation, accumulation time 0.61 h (+), 0.39 h (), laser power measured at the sample 25 mW (+), 40 mW
() and b) 244-nm excitation, ICP modulation, accumulation time 15 h and laser power 3 mW for both (+) and (). The VIS Raman spectrum of (+)-menthol
has been reduced by a factor of four due to strong sample fluorescence. This figure is available in colour online at wileyonlinelibrary.com/journal/jrs
Figure 6. Non-resonant backscattering Raman (top) and ROA (bottom) spectra of (+)-borneol (red) and ()-borneol (black) solution (1.0 g in 1.0-g methanol)
with a) 532-nm excitation, SCP modulation, accumulation time 0.37 h, laser power measured at the sample 80 mW and b) 244-nm excitation, ICP modulation,
accumulation time 28 h and laser power 3 mW. This figure is available in colour online at wileyonlinelibrary.com/journal/jrs
J. Raman Spectrosc. 2015, 46, 392–399
fluorescence does not introduce any major artefacts or distortions into the measured VIS ROA spectrum of contaminated
(+)-menthol. The VIS ROA spectra of both enantiomers of all
the studied molecules still display virtually perfect mirror image
symmetry in the wavenumber region ~ 800–1400 cm1.
Although the DUV Raman spectra have been measured with
noticeably lower spectral resolution than the corresponding VIS
Raman spectra, they do in contrast not show any sign of fluorescence contamination. Presumably due to fluorescence mediated
Copyright © 2015 John Wiley & Sons, Ltd.
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397
All of our commercially available samples have intentionally
not been recrystallized to remove any potential mildly fluorescing impurities. The VIS Raman spectrum of naturally occurring
()-menthol (Fig. 5a) exhibits only residual fluorescence emission
levels, whereas that of the synthetic (+)-menthol and to a lesser
extent also that of D-glucose (Fig. 7a) suffer from a noticeable
fluorescence background. However, due to the superior performance of the commercial VIS SCP ROA instrument, and apart
from a noticeable signal-to-noise degradation, the presence of
J. Kapitán, L. D. Barron and L. Hecht
Figure 7. Non-resonant backscattering Raman (top) and ROA (bottom) spectra of L- (red) and D-glucose (black) solution in water (3 mol/l) with a) 532-nm excitation, SCP
modulation, accumulation time 1.0 h, laser power measured at the sample 240 mW and b) 244-nm excitation, ICP modulation, accumulation time 57 h and laser power
3 mW. All spectra were subjected to mild third-order five point Savitzky–Golay smoothing. This figure is available in colour online at wileyonlinelibrary.com/journal/jrs
Table 1. Comparison of diffraction efficiency, transmittance and reflection of optical elements implemented in the VIS and DUV ROA spectrometers. Polarization optics implemented in the scattered beam
comprise of retardation plates, a liquid crystal retarder and polarization
beam-splitters in the VIS and a Lyot depolarizer in the DUV ROA
spectrometer
Spectrometer
element efficiency
DUV
(%)
VIS
(%)
Factor
VIS/DUV
CCD
Diffraction grating
Lenses
Mirror, vignetting
Edge/notch filter
Polarization optics
Total:
10
40
50
70
60
50
80
80
90
95
90
70
8
2
1.8
1.4
1.5
1.4
90
398
oxidative photon damage, some of the DUV Raman bands of
contaminated (+)-menthol but particularly the two strongly polarized bands observed at ~1045 and ~1458 cm1 exhibit significantly lower intensities than the equivalent DUV Raman band
intensities of less contaminated ()-menthol (Fig. 5b) and a much
higher intensity of the ~1650 cm1 band. However, the DUV ICP
ROA spectra of both enantiomers show almost perfect mirrorimage symmetry in the wavenumber region ~800–1450 cm1.
They also exhibit slightly better SNRs than the corresponding
VIS SCP ROA spectra and, accounting for the different spectral
resolutions and the non-ideal transmission characteristics of the
utilized DUV edge filter,[26] are virtually identical to the VIS SCP
ROA spectra, confirming the validity of the simplifying approximations associated with the non-resonant ROA scattering
regimes.
A direct comparison of Raman and ROA intensities measured
using the DUV ICP and VIS SCP ROA instruments is not easily performed due to the very different characteristics of the two instruments and the significantly larger spectral bandwidth employed
for the detection of DUV spectra. However, the ratios of the ROA
and Raman band intensities referred to as normalized circular intensity differences (CIDs) do not depend on instrumental response.[1]
Consequently, on account of the linear frequency dependence of
the G′ tensor and the corresponding contribution from the A
tensor,[1] the ratio of the DUV and VIS CIDs for individual bands
should be given by the ratio of the corresponding DUV and VIS
wavenumbers (or frequencies) eνs ðDUV Þ=eνs ðVISÞ of scattered
radiation.
wileyonlinelibrary.com/journal/jrs
Table 2. Comparison of selected CID ratios of DUV and VIS CIDs of ()menthol
1
Band position/cm
847
877
926
954
972
1225
1242
1272
1293
3
4
DUV × 10
VIS × 10
DUV/VIS
2.6
1.0
2.2
–0.6
1.7
3.0
3.4
–1.1
4.1
4.2
3.6
8.2
3.0
4.5
6.7
6.9
–2.8
12.2
6.2
2.8
2.7
2.0
3.8
4.4
4.9
3.9
3.4
Those Raman bands of the less contaminated ()-menthol which
do not overlap with any solvent Raman bands have been analysed
(Table 2) in detail. It is evident that the selected CID ratios are larger
than the reference value ≈ 2.2–2.5. However, the error in determination of the CIDs is relatively high and is strongly influenced by the
magnitude of the Raman spectral backgrounds.
For quantitative discussion of SNRs in DUV and VIS ROA spectra,
it is assumed, that the major contribution to noise is statistical (shot)
noise. The Raman scattering signal associated with left and right circularly polarized radiation may be expressed as I = qN ≈ I(R) ≈ I(L),
where N is a number of scattered photons and q is a positive factor
smaller than one corresponding to the efficiency of the spectrometer and detector. ROA signals may be expressed as I(R) I
pffiffiffiffi
(L) ≈ CID × 2I, the corresponding (statistical) noise as 2I and ROA
pffiffiffiffiffiffiffiffiffiffiffi
SNRs as CID qN=2 . Since CIDDUV =CIDVIS ≅ eνs ðDUVÞ=eνs ðVISÞ ,
NDUV =NVIS ≅ eν3s ðDUVÞ=eν3s ðVISÞ and the ratio of spectrometer efficiencies estimated from the intensities of corresponding Raman spectra
is qDUV/qVIS ≅ 1/1400 (Fig. 4), the predicted ratio of signal-to-noise
ratios is SNRDUV/SNRVIS ≅ 0.2.
ROA spectra of non-fluorescing samples of menthol (Fig. 5) and
borneol (Fig. 6) have approximately similar SNRs, but the excitation
energy (the accumulation time multiplied by the laser power) is approximately three times larger for the DUV than for the VIS spectra.
Similarly, the DUV ROA spectrum of glucose (Fig. 7) exhibits SNR approximately three times smaller than corresponding VIS ROA spectrum of glucose, but the excitation energy in both cases is
approximately same. Consequently, it can be estimated from our
data, that SNRDUV/SNRVIS ≈ 0.3. Within experimental errors this result
Copyright © 2015 John Wiley & Sons, Ltd.
J. Raman Spectrosc. 2015, 46, 392–399
Raman optical activity instrument operating in DUV spectral region
is thus in agreement with the predicted value 0.2. The observed difference may easily be attributed to a lower resolution and hence
higher relative signals of the DUV spectra.
Conclusions
A ROA spectrometer for the DUV spectral region has been successfully realized for the first time. Its construction has proved to
be significantly more challenging than that of VIS ROA instruments due to the great difficulties associated with generating
and manipulating polarized DUV radiation. However, a procedure
for the fast and reliable EOM alignment has been developed for
the generation of high purity circularly polarized DUV radiation,
and polarization artefacts were successfully minimized by
employing a Lyot depolarizer in the diverging beam of the
scattered Raman radiation. DUV ROA spectra of several enantiomeric samples in aqueous and non-aqueous solutions have been
measured, which prove to be of high quality. Non-resonant spectra may be recorded more easily and in a shorter time with our VIS
ROA spectrometer predominantly because the advantage of
greater DUV Raman and ROA intensities is largely negated by
the necessity for using a much lower laser power and a worse performance and efficiency of the optical system for the collection
and detection of DUV radiation. However, the situation may be
dramatically different in pre-resonant or rigorously resonant scattering regimes, which may prove to be the main application for
the new DUV ROA spectrometer, as will be reported in our subsequent publications. We have also demonstrated that DUV Raman
spectra are generally not plagued by any residual fluorescence,
which may be a decisive advantage for the study of autofluorescing samples. The design of the collection optics and especially the efficiency of detectors may admittedly be significantly
improved. Nevertheless, we think that our construction of the first
DUV ROA spectrometer represents an important milestone in the
exploration of pre-resonant and rigorously resonant Raman and
ROA scattering phenomena of a wide range of biologically interesting molecules.
Acknowledgements
This work is supported by grants from the UK Engineering and
Physical Sciences Research Council and Technology Agency of
the Czech Republic (project no. TE01020229). We would also like
to thank Götz Zinner from Bernhard Halle and Steve Payne from
Leysop for their help in the development of custom made optics.
References
[1] L. D. Barron, Molecular Light Scattering and Optical Activity, 2nd ed.,
Cambridge University Press, Cambridge, 2004.
[2] L. D. Barron, A. D. Buckingham, Chem. Phys. Lett. 2010, 492, 199.
[3] L. D. Barron, J. R. Escribano, Chem. Phys. 1985, 98, 437.
[4] L. A. Nafie, in Encycl. Spectrosc. Spectrom. (Eds.: J. Lindon, G.E. Tranter,
D.W. Koppenaal), Academic Press, Oxford, 2010, pp. 2397–2405.
[5] L. A. Nafie, Vibrational Optical Activity: Principles and Applications, John
Wiley & Sons, Chichester, 2011.
[6] L. D. Barron, Curr. Opin. Struct. Biol. 2006, 16, 638.
[7] L. D. Barron, L. Hecht, in Compr. Chiroptical Spectrosc. Vol 2 (Eds.:
N. Berova, P.L. Polavarapu, K. Nakanishi, R.W. Woody), John Wiley &
Sons, Inc., Hoboken, New Jersey, 2012, pp. 759–793.
[8] L. A. Nafie, B. E. Brinson, X. Cao, D. A. Rice, O. M. Rahim, R. K. Dukor,
N. J. Halas, Appl. Spectrosc. 2007, 61, 1103.
[9] P. L. Polavarapu, Chem. Phys. Lett. 1988, 148, 21.
[10] D. A. Long, The Raman Effect: A Unified Treatment of the Theory of
Raman Scattering by Molecules, John Wiley & Sons, Chichester,
2002.
[11] R. L. McCreery, Raman Spectroscopy for Chemical Analysis, John Wiley &
Sons, Chichester, 2000.
[12] K. H. Fung, I. N. Tang, Appl. Spectrosc. 1992, 46, 159.
[13] B. Küstner, C. Schmuck, P. Wich, C. Jehn, S. K. Srivastava, S. Schlücker,
Phys. Chem. Chem. Phys. 2007, 9, 4598.
[14] S. A. Asher, C. R. Johnson, Science 1984, 225, 311.
[15] S. A. Asher, Anal. Chem. 1993, 65, 59A.
[16] L. C. T. Shoute, K. J. Schmidt, R. H. Hall, M. A. Webb, S. Rifai, P. Abel,
P. H. Arboleda, A. Savage, J. T. Bulmer, G. R. Loppnow, Appl. Spectrosc.
2002, 56, 1308.
[17] G. R. Loppnow, L. Shoute, K. J. Schmidt, A. Savage, R. H. Hall, J. T. Bulmer,
Philos. Trans. R. Soc. -Math. Phys. Eng. Sci. 2004, 362, 2461.
[18] D. D. Tuschel, A. V. Mikhonin, B. E. Lemoff, S. A. Asher, Appl. Spectrosc.
2010, 64, 425.
[19] S. Hassing, J. Raman Spectrosc. 1997, 28, 739.
[20] R.-H. Zheng, W.-M. Wei, J. Phys. Chem. A 2007, 111, 3652.
[21] S. Luber, J. Neugebauer, M. Reiher, J. Chem. Phys. 2010, 132.
DOI:10.1063/1.3300069.
[22] L. A. Nafie, Chem. Phys. 1996, 205, 309.
[23] L. A. Nafie Theor. Chem. Acc. 2008, 119, 39.
[24] L. D. Barron, F. Zhu, L. Hecht, G. E. Tranter, N. W. Isaacs, J. Mol. Struct.
2007, 834–836, 7.
[25] L. Hecht, L. D. Barron, E. W. Blanch, A. F. Bell, L. A. Day, J. Raman
Spectrosc. 1999, 30, 815.
[26] L. Hecht, J. Clarkson, B. J. E. Smith, R. Springett, J. Raman Spectrosc.
2006, 37, 562.
[27] W. Hug, in Raman Spectrosc. (Eds.: J. Lascombe, P.V. Huong), WileyHeyden, Chichester, 1982, pp. 3–12.
[28] W. Hug, G. Hangartner, J. Raman Spectrosc. 1999, 30, 841.
[29] M. Baranska, K. Chruszcz-Lipska, Nat. Prod. Commun. 2010, 5, 1417.
[30] L. D. Barron, B. P. Clark, J. Chem. Soc. Perkin Trans. 2 1979, 1164.
[31] L. D. Barron, L. Hecht, S. M. Blyth, Spectrochim. Acta Part Mol. Spectrosc.
1989, 45, 375.
[32] L. D. Barron, J. Chem. Soc.-Perkin Trans. 2 1977, 1074.
[33] Z. Q. Wen, L. D. Barron, L. Hecht, J. Am. Chem. Soc. 1993, 115, 285.
399
J. Raman Spectrosc. 2015, 46, 392–399
Copyright © 2015 John Wiley & Sons, Ltd.
wileyonlinelibrary.com/journal/jrs
Wavefront-sensor tomography for measuring spatial coherence
Jaroslav Rehacek, Zdenek Hradila , Bohumil Stoklasaa , Libor Motkaa
and Luis L. Sánchez-Sotob
a
b
Department of Optics, Palacky University, 17. listopadu 12, 77146 Olomouc, Czech Republic;
Departamento de Óptica, Facultad de Fı́sica, Universidad Complutense, 28040 Madrid, Spain
ABSTRACT
Wavefront sensing is an advanced technology that enables the precise determination of the phase of a light field, a
critical information for many applications, such as noncontact metrology, adaptive optics, and vision correction.
Here, we reinterpret the operation of wavefront sensors as a simultaneous unsharp measurement of position
and momentum. Utilizing quantum tomography techniques we report an experimental characterization and 3D
imaging of a multimode laser light.
Keywords: Wavefront sensing, coherence, tomography, parameter estimation, 3D imaging
1. INTRODUCTION
1
Wavefront sensing is an advanced optical technology providing direct access to phase properties of measured
optical fields. In this sense wavefront sensing constitutes an invaluable information source about many aspects
of the observed objects which are inaccessible to standard intensity detection methods but which are essential
in many phase-dependent applications, such as metrology, high-power laser systems, adaptive optics etc.
Despite recent successful utilization of wavefront sensors in coherent light processing their full potential has
yet to be explored. Recognizable analogies between wavefront sensing and tomography methods previously
developed for quantum information processing hint at introducing a conceptually new approach to wave front
sensing based on tomography capable of characterizing general multimode optical fields.2, 3 Direct access to
additional degrees of freedom of light that are reflected in the coherence properties of the measured signal is
interesting from the fundamental point of view but also opens a wide range of potential applications of wavefront
sensing techniques in state-of-art applications dealing with partially coherent light, such as quantum information
processing, digital holography and 3D imaging. This can be illustrated by a simple motivating example of beam
propagation, where in order to predict the intensity at the output of a known optical system, the knowledge of
the input intensity alone is not sufficient4 and information about the second order coherence properties of the
signal needs to be supplied.
2. WAVEFRONT SENSING
The principle of the Shack-Hartmann (SH) wavefront sensor is shown in Fig. 1. Local slopes of the incoming
wavefront1 in the plane of the microlens array (SH plane) determine the positions of the bright spots in the
lens focal plane (CCD plane), from which partial information about the transversal momentum of the signal
can be extracted. This is the standard way of operating wavefront sensors. Unfortunately, should the signal
be only partially coherent, that is, multimode, a single well-defined wavefront would not exist — the signal
would be a statistical mixture of possibly many distinct wavefronts/modes5 — and the standard wavefront
measurement would fail. To circumvent this problem we notice that a detection at a particular pixel implies that
the photon/photons must have passed through the associated subaperture, which amounts to unsharp position
measurement on the incoming beam. Hence we may conclude that SH measurements with finite apertures are
pertinent examples of simultaneous unsharp position and momentum measurements6 that are of fundamental
importance in physics and quantum theory in particular. Such an interpretation of the registered intensity
profile, see Fig. 1 for example, is a key ingredient of our considerations.
Further author information: (Send correspondence to Jaroslav Rehacek)
Jaroslav Rehacek: E-mail: rehacek(at)optics.upol.cz, Telephone: +420 58 5634257
Figure 1. The measurement principle of SH detection and a typical intensity readout.
3. PARTIALLY COHERENT SIGNAL
Let us demonstrate that common wavefront sensors7 can be utilized for measuring mutual coherence and hence
3D imaging of partially coherent fields provided quantum state reconstruction techniques are adopted for data
processing. Taking advantage of the formal correspondence between wave optics and quantum mechanics we
define ρ the coherence matrix describing the second-order coherence of measured signal G(x0 , x00 ) = hx0 |ρ|x00 i =
Tr(ρ|x0 ihx00 |), where the ket |xi is a vector describing a point-like source located at x and Tr denotes matrix
trace. In this sense, mutual coherence function is seen to be a position representation of the coherence matrix and
intensity distribution across a transversal plane becomes I(x) = Tr(ρ|xihx|) and describes a position measurement
realized e.g. by placing a point-like pixel of a CCD camera at the position x in the detection plane. Furthermore
we can describe a coherent beam (mode), with a complex amplitude U (x), by a ket |U i, such that U (x) = hx|U i.
Now consider the Shack-Hartmann measurement. Notation is simplified by setting the focal length of the
microlenses and the effective wave number to unity, and describe 1D geometry of the measurement and detection
only. This is equivalent to considering one row of microlenses and one row of CCD pixels only. Generalization to
2D geometry is obvious. Upon illuminating the microlense array (SH plane) by a coherent signal U (x), assuming
that the axis of ith microlens (lens) is displaced from the SH optical axis by ∆xi , this lens feels the displaced
field U (x − ∆xi ) = hx| exp(−i∆xi p̂)|U i. This field gets filtered and truncated by the lens aperture function
A(x) = hx|Ai and Fourier transformed by the lens prior to being deteted the SH focal plane (CCD plane),
Z
0
U (∆pj ) = hA|xihx|e−i∆xi p̂ |U ie−i∆pj x dx
Z
(1)
= hA|xihx|e−i∆pj x̂ e−i∆xi p̂ |U idx
= hA|e−i∆pj x̂ e−i∆xi p̂ |U i,
where the closure relation is used in the last step, and we assume that the jth pixel is displaced from the SH
optical axis by ∆pj . Adding together the intensity contributions of all the orthogonal modes comprising the
signal coherence matrix, we find that the intensity measured at the jth pixel behind the ith lens is governed by
a Born-like rule
I(∆xi , ∆pj ) = Tr ρ|πij ihπij | , |πij i = ei∆xi p̂ ei∆pj x̂ |Ai,
(2)
and consequently, each pixel of the SH device makes projection of the input signal on the position and momentum
translated aperture state. Some interesting special cases of those aperture states are worth mentioning:
• Small lenses. A(x) → δ(x), |Ai → |x = 0i implies |πij i → |x = ∆xi i — a position eigenstate.
• Large lenses. A(x) → 1, |Ai → |p = 0i implies |πij i → |p = ∆pj i — a momentum eigenstate.
• Gaussian profile. A(x) = exp(−x2 /2), |Ai = |α = 0i implies |πij i → D(αij )|0i = |αij i — a coherent-like
Gaussian state of amplitude α = ∆xi + i∆pj .
Point-like lenses produce broad diffraction patterns in the CCD plane and information about the transversal
momentum in the SH plane is lost. Conversely, very large apertures provide sharp momentum measurement
with the corresponding loss of positional sensitivity. The last case is the most interesting one: SH devices with
Gaussian-apodized microlenses are capable of projecting the signal on a set of coherent states and hence yield
direct sampling of Husimi quasiprobability distribution Q(α) = hα|ρ|αi. This provides a convenient phase space
description of the signal: Different choices of CCD pixels and/or microlenses can be interpreted as particular
translations in that phase space in a very close analogy to translations induced by changing the amplitude of the
local oscilator in unbalanced homodyne tomography. The well known correspondence between the signal phase
space description and the signal state itself demonstrates informational completeness of such an SH measurement.
4. REALISTIC PICTURE
Experimental demonstration of SH tomography principles faces an obstacle that needs to be considered. The
apertures of microlenses comprising a real wavefront sensor do not overlap and unlike the Gaussian profiles
discussed above are spatially bounded structures. Introducing notation Πij = |πij ihπij |, the measurement
operators describing two pixels belonging to distinct apertures are found to be compatible [Πij , Πi0 j ] = 0, i 6= i0 ,
which renders the tomography scheme as informationally incomplete. Signal components passing through distinct
apertures are never recombined and the mutual coherence of those components thus cannot be determined. The
remedy here is to exclude all spatially bounded modes from the search space.8, 9 Indeed, spatially bounded
modes carry infinite energy and for this reason are unphysical and impossible to generate with finite resources.
Such truncation can be done by decomposing the signal in a suitable discreet spatially-unbounded computational
basis realizable by sets of plane waves, Gaussian beams, Laguerre-Gaussian beams, etc. depending on the actual
experimental context.
We denote the basis states |ki, k = 1, . . . , d, assuming orthonormality hk|li = δkl , k 6= k 0 , their complex
amplitudes being hx|ki = ψk (x). With this notation the signal coherence matrix ρ and the measurement
operators Πij live in d-dimensional Hilbert space and are parameterized as d × d non-negative matrices. In
order to reconstruct d2 real parameters of ρ we need to have d2 linearly independent measurements. Convenient
matrix representation of measurement Πij in terms of complex amplitudes can be obtained directly from Eq. (2),
∗
(Πij )kl = ψl,i (∆pj )ψk,i
(∆pj ),
(3)
where ψk,i (x) is the complex amplitude in the CCD plane of the i-th lens generated by the incident k-th basis
mode ψk .
Let us illustrate the idea with a conceptually simple and practically important example of an SH sensor
R 1/2
with square lenses |Ai = −1/2 |xidx and A(x) = rect(x). The signal will be decomposed in a discreet basis set
of plane waves parametrized by transverse momenta pk : |ki = |pk i, ψk (x) = exp(−ipk x). This is Fraunhofer
diffraction on a slit so the measurement matrix is immediately obtained in the form
(Πij )kl = sinc(∆pj + pk )sinc(∆pj + pl )ei(pl −pk )∆xi .
(4)
Wavefront detection with a single square lens can never be informationally complete. Let us analyze the smallest
possible search space consisting of just two plane waves, which is equivalent to a single-qubit tomography. By
considering different pixels j all belonging to the same aperture i, linear combinations of only three out the total
of four Pauli matrices can be generated from Eq. (4). For example, a lens placed on the SH axis having ∆xi = 0
fails to generate σy matrix and at least one more lens with a different displacement ∆xi needs to be added to
the setup to make the tomography complete. This argument can be extended to larger dimensions: The larger
search space, the more microlenses must be used to characterize the signal completely.
5. MULTIMODE LASER BEAM PROPAGATION
In the proposed experiment, a genuinely multi-mode light of a Nd:YAG pulsed laser operating in the deep
ultraviolet region served as an example of partially coherent signal. Since laser light can be advantageously
decomposed into the Hermite-Gaussian (HG) superposition, these were used as a basis for the coherence function
Figure 2. Experimental setup. A multi-mode Nd:YAG laser beam is detected with an SH sensor and characterized in the
SH plane with the help of the SH tomography.6 The calculated coherence matrix is numerically propagated back to the
blue plane, where the inferred transverse beam intensity profile is compared to the actual CCD scan.
Figure 3. Typical experimental SH data. About six thousand pixels are analyzed with our SH tomography technique to
characterize the coherence matrix of the detected Nd:YAG laser beam.
representation. Reconstructed space was restricted to 9 active modes, that is, 81 entries of the coherence matrix
had to be estimated. At least 81 independent measurements are required to accomplish this task. In our
experiment, much larger data sets composed of intensity data from 11 × 11 pixels for 7 × 7 array of microlenses,
i.e. 5929 measurements in total, was used for the SH tomography to suppress reconstruction artifacts.
Shack-Hartmann sensor was realized in Meopta-Optika with 150µm microlens pitch, 4.6µm CCD pixel size
and 7mm microlens to CCD distance. The waist size of the basic HG mode is a prior parameter of the reconstruction and was estimated to 0.3mm.
The reconstructed coherence matrix shown in Fig. 4 clearly shows partially coherent nature of the Nd:YAG
beam. Non-zero diagonal elements indicate many active modes in the beam, while non-zero off-diagonal elements
describe existing correlations between the modes. Due to possibility to express any partially coherent field as a
superposition of coherent modes, singular value decomposition of the coherence matrix can be used to reconstruct
these modes along with their amplitude and phase profiles.
To validate the reconstruction, a free space (back)propagation of the laser beam around the beam waist
was performed, see Fig. 4. The coherence matrix was measured in a plane 250mm behind the waist, where an
intensity profile was reconstructed and compared to a direct intensity scan measured by a CCD camera. Then,
the reconstructed coherence matrix was digitally transported back to the plane located on the opposite side
of the waist 220mm distant from the waist. Again, beam intensity profile was calculated from the coherence
matrix and compared to a direct intensity measurement. Correlation coefficients between the reconstructed and
Figure 4. Nd:YAG laser beam propagation. Left panel: a reconstructed coherence matrix ρ of the beam in the HermiteGaussian (HG) basis, real and imaginary parts are plotted separately. Indexes i, j label HG modes in the following order:
HG0,0 , HG1,0 , HG2,0 , HG0,1 , HG1,1 , HG2,1 , HG0,2 , HG1,2 , HG2,2 . Right panel: comparison of numerical simulation of
intensity propagation and directly measured intensity distributions. The top row contains data of initial plane, when left
is a direct CCD scan and right intensity reconstruction from the coherence matrix. The bottom row contains data from
a plane 550mm away of the initial plane. Left is a direct CCD scan and right intensity reconstruction from propagated
coherence matrix.
measured intensity profiles are 0.93 for the input plane and 0.91 for the output plane. Since free space intensity
propagation requires detailed knowledge of the beam coherence properties, high correlation coefficients witness
successful SH tomography.
6. CONCLUSIONS
We experimentally demonstrated feasibility of the new Shack-Hartmann tomography for intensity propagation
of multimode light by measuring Nd:YAG laser beam. Using the representation of Hermite-Gaussian modes,
we observed nontrivial spatial correlations in the beam which provide us all information needed for successful
intensity propagation. The numerical prediction of the beam propagation was compared with direct intensity
measurements to prove the correctness of the reconstruction. As the method is a single-shot measurement and
the Shack-Hartmann sensor is easy to build cost effective device, there is a great potential for implementing this
technique in general 3-D imaging problems.
ACKNOWLEDGMENTS
This work is co-financed by the European Social Fund and the state budget of the Czech Republic, project No.
CZ.1.07/2.3.00/30.0041 (POST-UP II), and supported by the Grant Agency of the Czech Republic, Grant No.
15-031945.
REFERENCES
[1] Platt, B. C. and Shack, R. S., “History and principles of Shack-Hartmann wavefront sensing,” J. Refract.
Surg. 17, S573-S577 (2001).
[2] Hradil, Z., Rehacek, J. and Sánchez-Soto, L. L., “Quantum Reconstruction of the Mutual Coherence Function,” Phys. Rev. Lett. 105, 010401 (2010).
[3] Waller, L., Situ, G. and Fleischer, J. W., “Phase-space measurement and coherence synthesis of optical
beams,” Nature Photonics 6, 474 (2012).
[4] Goodman, J. W., [Introduction to Fourier Optics], Roberts Publishers, Colorado (2005).
[5] Goodman, J. W., [Statistical Optics], Wiley-Interscience, New York (2000).
[6] Stoklasa, B., Motka, L., Rehacek, J., Hradil, Z. and Sánchez-Soto, L. L., “Wavefront sensing reveals optical
coherence,” Nat. Commun. 5, 10.1038 (2014).
[7] Geary, J. M., [Introduction to Wavefront Sensors], SPIE-International Society for Optical Engineering,
Bellingham, WA (1995).
[8] Hradil Z., Mogilevtsev, D. and Rehacek, J., “Biased tomography schemes: An objective approach,” Phys.
Rev. Lett. 96, 230401 (2006).
[9] Rehacek, J., Hradil, Z., Bouchal, Z., Celechovsky, R., Rigas, I. and Sánchez-Soto, L. L., “Full Tomography
from Compatible Measurements,” Phys. Rev. Lett. 103, 250402 (2009).

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