Polarization Quadrature Interferometer

Transcription

1 Laboraor for dapive Opics Rev 1: 5/11/04 UCO Lic Observaor Rev : 9/1/04 Polarizaion Quadraure nerferomeer Donald Gavel, UCSC Laboraor for dapive Opics This memo describes a Mach-Zehnder inerferomeer seup ha will measure he phase of an opic o he spaial resoluion of he deecor, using a polarizaion based quadraure deecion echnique. The approach was demonsraed b K. Baer a LLNL. We will use he inerferomeer in he MCO bed o characerize he response of Programmable Phase Modulaors (PPM). The Mach-Zehnder laou is shown below. PPM laser Laser λ/4 PBS ref λ/ Q quarer wave plae in he reference leg convers he laser s linearl polarized ligh ino circular polarizaion. When he beams combine, he sine and cosine informaion from he device under (PPM) is encoded on he 45 and 135 componens of he elecric field vecor (see below). fer he beam combiner plae, a half wave plae roaes he coordinae ssem 45 so ha he encoding is on 0 and 90 componens and a subsequen polarizaion beam splier can separae channels. On he channel is 1+cos(φ) and on he Q channel is 1+sin(φ). The daa gahering compuer does a bias subracion and inverse angen o ge he 4-quadran phase map of he PPM. Subsequen processing unwraps his phase. lecric Field Vecor The laser produces he plane wave r laser ˆ 0 x cosω xˆ cosω + ˆ cosω (1)

2 Laboraor for dapive Opics Rev 1: 5/11/04 UCO Lic Observaor Rev : 9/1/04 where is he elecric field vecor, xˆ and ŷ are uni vecors in he plane of he able and rhogonal o he able respecivel, ωπc/λ is he emporal frequenc of he ligh, is o 0 he ampliude of he plane wave and 0. x ˆ and ˆ are uni vecors in a coordinae frame roaed 45 degrees from xˆ and ŷ. This beam afer i his he device under is w here φ( x, ) ( ω + φ) + ˆ cos( ω + φ) r xˆ cos () φ is he phase aberraion of he wavefron. n he reference arm, he plane wave is convered from linear o circular polarizaion afer he λ/4 plae. This produces he wave r xˆ cosω + ˆ sinω ref x ˆ cosω + ˆ sinω Noe ha since he ligh is circularl polarized, i doesn maer ha he coordinae ssem is roaed b 45. (3) he combinaion plae, he waves are added coherenl r r r + ref xˆ cos( ω + φ) + cos( ω) + ˆ cos ω + φ + sin ω ) [ ] [ ( ) ( ] (4) Noe ha he beam exacl overlaps he reference beam; here is no shear or il o inroduce fringes. This is opimal for minimizing non-common pah aberraions hrough subsequen powered opics, since he wo beams follow he same pah. The half wave plae hen roaes he coordinae ssem bac o orhogonal wih he able [ cos( ω + φ) + cos( ω) ] + ˆ[ cos( ω + φ) sin( ω) ] r xˆ +. (5) polarizing beam splier hen separaes he wave ino he x and componens. Deecion The camera deecs he power in he waves according o a square-law:

3 Laboraor for dapive Opics Rev 1: 5/11/04 UCO Lic Observaor Rev : 9/1/04 Q x [ cos ( ω + φ) + cos ( ω) + cos( ω + φ) + cosφ] [ cos ( ω + φ) + sin ( ω) + sin( ω + φ) sinφ] Since we inegrae he deecor over man periods of he ligh wave, he ime-average signal is recorded (6) Q ( 1+ cosφ) ( 1 sinφ) (7) We should noe ha he cameras are opicall imaged o he plane of he device under so as o minimize he effecs of diffracion in his analsis. Processing Deermining he mpliude The compuer processing consiss of subracing he bias, aing he arcangen, hen, finall, unwrapping he phase. To measure he bias, one can bloc he arm of he inerferomeer and measure he reference beam onl, in which case Q. This mehod is sensiive o he assumpion ha he inensi in he wo arms is balanced and is sensiive o an flucuaions in laser power beween inensi-onl and inerferogram measuremens. n alernaive is o r o use he daa iself o find. From (7) 4 sin cosφ sinφ Q φ + 4 cos φ Q 0 Q + + ( Q + ) Q ( ) 4 4 (8) The las equaion is a quadraic form ha can be solved for. The soluion is + Q ± Q (9) Unforunael, here is no wa o brea he ambigui of he sign choice from he daa alone (i.e. here are alwas wo ses of feasible {,φ} soluions). To help deermine he rue value of one can choose he one ha is closes o an iniial esimae of obained using one of he oher mehods menioned earlier. Deermining he phase Once he ampliude is deermined, he phase modulo π is

4 Laboraor for dapive Opics Rev 1: 5/11/04 UCO Lic Observaor Rev : 9/1/04 ( x, ) ( x, ) ( ) ( ) x, x, 1 Q φ ( x, ) an (10) where we ve reinroduced he ransverse spaial coordinaes x and o highligh he fac ha his measuremen can be made a he pixel resoluion of he camera. n sofware, he rcangen implied in (10) is acuall an rctan funcion, wih wo argumens (he numeraor and he denominaor). This preserves he phasor s quadran informaion so ha he oupu spans a full range of - π o +π. Phase unwrapping ssuming ha he phase does no change oo rapidl wih x and, he final sep is o use phase unwrapping o ge he full phase map. We use a echnique (implemened in he DL rouing unwrap.pro) ha is loosel oulined as: Calculae he gradien of phase, φ. This convers π phase jumps ino spies. dd or subrac π o he spies o eep all phase jumps in he (-π,+π] range Reconsruc φ given φ wih a Poisson s equaion solver (described below) The mehod of reconsrucing phase from gradien is based on use of he Fourier Transform, defined as: The gradien φ πi( xx+ ) ( x, ) φ( x, ) e πi( xx+ ) φ( x, ) φ ( x, ) e dxd φ has he following Fourier ransform pair d x d. (11) φ x x φ( x, ) πi φ ( x, ) πiφ φ (1) which can be verified b aing he derivaive of he second equaion in (11). Reconsrucion of φ given φ is hen equivalen o finding φ given φ. The leassquares soluion is φ i ( ) ( πi φ ) (13) π where x +, or

5 Laboraor for dapive Opics Rev 1: 5/11/04 UCO Lic Observaor Rev : 9/1/04 φ i FT x π 1 ( x) FT { φ( x) x where FT represens he Fourier ransform. } (14) Firs order analsis of spli imbalance Real-world beam spliers are no perfec, hus here is a misbalance in he inensiies of he ligh in each of he Mach-Zehnder inerferomeer arms and in each of he camera channels. To ae his ino accoun, Zhenrong Wang has performed a firs-order analsis of he affecs of hese imbalances and has incorporaed he resuls ino he daa processing sofware 1. Le α be he raio of P o S polarizaion oupu inensiies afer he polarizing beam splier, assuming a circular polarized beam inpu. P polarizaion ransmis hrough he splier and goes ino he channel camera and S polarizaion reflecs off he splier and goes ino he Q channel camera. Le β be he raio of he wo inerferomeer arms hroughpus so β ref (which includes reflecing off he SLM in he arm). Nominall, boh raios are equal o one. The resuling fields are ( ω + φ) + ˆ α cos( ω + φ) r xˆ cos (15) r xˆ βcosω + ˆ αβ sinω ref so r r r + xˆ + ref [ cos( ω + φ) + β cos( ω) ] + ˆ [ cos( ω + φ) β sin( ω) ] (16) and he inensiies deeced in he and Q channels are Q x Le ( 1+ β ) and Q Q α ( 1+ β ) 1+ β + β cosφ 1+ β α β sinφ. Then (17) where β ( 1+ β ) m. ( 1+ mcosφ) ( 1 msinφ) Q (18)

6 Laboraor for dapive Opics Rev 1: 5/11/04 UCO Lic Observaor Rev : 9/1/04 Phoomeric measuremens have deermined he following: α β 1. m (19) Since m is ver nearl one, he processing seps can proceed using he modified and Q signals as defined above wih less han 0.4% error, whereas using he unmodified signals would inroduce on he order of 10% error. References 1 Zhenrong Wang, "Characerizaion of Programmable Phase Modulaor wih Polarizaion Quadraure nerferomeer", maser's hesis, UC Sana Cruz, Sep. 004.