Expeiments with the HoloEye LCD spatial light modulato HoloEye model LC00 spatial light modulato The HoloEye (http://www.holoeye.com/spatial_light_modulato_lc_00.html) LC 00 Spatial Light Modulato (SLM) contains a Sony SVGA (800x600 pixels) liquid cystal micodisplay and dive electonics. It can be used with a standad pesonal compute o laptop by plugging the device to a VGA gaphics connecto. The phase distibution of a beam input to the SLM can be modulated by a voltage on the electodes (800x600 pixels) of the LCD. The phase modulation is a side-effect of the liquid cystal mateial and is caused by its biefingence popety. It yields a moe o less linea phase shift with voltage of maximum π adians at 53 nm. This phase shift is used to intoduce a phase modulation of a coheent input beam by setting a voltage on each pixel of the modulato. In this execise thee diffeent phase distibutions ae used leading to a ing-shape intensity distibution, a diffactive optical element (DOE) acting as a positive lens and a phase hologam espectively. These phase distibutions ae calculated with Matlab pogams povided. Also, simulations can be pefomed using LightPipes fo Matlab (Pacticum vesion).
INSTRUCTIONS TO STUDENT Expeiments with the HoloEye spatial light modulato. Aspects: Spatial phase distibution, axicon, lenses, Theoy to be studied befoe the pactical session: Pedotti, Intoduction to Optics, 3 d edition, chapte 17-5: especially Liquid- Cystal Displays(LCD) Equipment: Optical ail HeNe lase Seveal lenses fom the lensboxes in the lab HoloEye LC00 spatial light modulato + powe supply Exta VGA monito VGA splitte + powe supply thick VGA cables to connect the exta monito and the student s laptop to the splitte 1 thin VGA cable to connect the HoloEye LC00 to the splitte White sceen + alignment sceen with hole. Rule + calipe Seveal Matlab pogams LightPipes fo Matlab (pacticum edition) optical toolbox
Diffactive Optics. Diffactive Optical Elements (DOE s) diffe fom the classical optical elements like lenses and mios because they ae based on thei diffactive popeties athe than eflection and efaction. When dealing with the taditional optical elements, diffaction phenomena ae consideed as undesiable featues, which influences the pefomance of an optical system build with these elements and hence should be minimized. DOE s on the othe hand, make use of diffaction to manipulate the wavefom of an incoming beam of light. Because of the natue of diffaction mostly highly monochomatic and coheent light is used with DOE s. The fist application of wavefont manipulation was hologaphy. An intefeence patten of two light beams, a efeence beam and a beam coming fom an object, is ecoded on a photogaphic plate afte which the ecoded infomation of the object in thee dimensions can be extacted fom the hologam by illumination with a lase beam. Hologaphy inspied people to wavefont pocessing in which the suface of a substate was pocessed to change an input wavefont into anothe fom. In pinciple lenses and othe classical optical elements ae all wavefont pocessos, but thei functionality is limited to elative simple actions. DOE s on the othe hand allow fo moe complex wavefont manipulations and esulted in moden optical applications as hologaphic head up displays in fighte jets and ecently in automobiles. Nowadays the complex ecoding techniques using hologaphy ae moe and moe eplaced by diect witing mico-stuctues on substates using compute-calculated diffaction pattens, o as we will apply in this pacticum: using electonically addessed spatial light modulatos (SLM). Initial instuctions: 1. Complex amplitude. A coheent beam of light can be descibed by its complex amplitude ~ distibution: E( x, y) = E0 ( x, y)exp( iϕ( x, y)). E 0 (x,y) and ϕ(x,y) ae the amplitude and phase as a function of the tansvese coodinates. The amplitude can be detemined easily with sceens o CCD cameas by taking the squae oot of the intensity distibution. The phase distibution is much moe difficult to measue. The phase, howeve, plays an impotant ole and can neve be neglected in optics. A lens, fo example, will change the phase distibution but leaves the amplitude distibution unalteed.. Phase distibution of a lens. The phase distibution of a coheent monochomatic light beam afte passing a lens can be calculated by consideing the optical paths of ays though the lens.
0 01 0 R 1 >0 θ Focal point R <0 The phase can be found by calculating the optical path of a ay paallel to and at above the optical axis. It is (glass + ai): ϕ ) = k n ( ) + k ( ) = k + k ( n 1) (, with: ( ) ) ( 0 0 0 0 0 0 = + π ( ) 01 R1 R1 0 ( R ) R, and k 0 =, λ0 the popagation constant in vacuum. This can be appoximated (by expanding 1 1 the squae oots) to: ϕ ( ) = k ( n 1) k 0 = 0 R1 R, with f 1 1 1 f ( n 1), the lens-makes fomula fo the focal length of a R1 R thin lens. Note the sign conventions fo the lens adii. (The constant phase, k n 0 0 is not elevant and can be neglected.) This fomula is used in the Matlab pogams and is executed by the LightPipes fo Matlab F=LPLens(f,0,0,F) command, which insets a lens at the optical axis in the field (complex amplitude) F. The ay-angle with the optical axis is: θ ( ) =. So, as expected, all ays f point to a single point on the optical axis, the focal point of the lens. The appoximation made holds fo ays close to the optical axis (small angles) and is called the paaxial appoximation. If the appoximation is not valid, we ae dealing with (spheical) abeation and the ays to fa fom the axis will not pass though the paaxial focal point. 3. Phase distibution of an axicon. An axicon is a conical optical element as shown in the figue below.
n Φ θ() α To find the phase distibution and the ay-angles, θ(), afte passage though the axicon, a simila calculation as with the lens can be pefomed. Calculate the phase distibution (paaxial appoximation) and show that the ay-angles Φ ae given by: θ ( ) = ( n 1)α, with α = 90, the shap angle of the axicon and n the efactive index of the axicon mateial. 4. Connection of the HoloEye spatial light modulato (SLM) to you laptop. The HoloEye SLM can be opeated by you laptop by simply connecting it to the exta VGA pot on you laptop (nomally used fo beames). The set-up you use is shown in figue 1. HoloEye LC 00 spatial light modulato sceen f=00-600mm f=100-300mm f=10mm HeNe lase laptop exta monito in out 1 out VGA splitte Fig. 1. Set-up of the expeiments with the SLM. You have to set you laptop in dual monito mode. This is pobably done by pessing the F8 key a few times. Futhemoe the second monito must be set to 800 x 600 pixels with 56 bit colo depth. This can be done by unning you
gaphics dive softwae on you PC. Pobably it can be accessed by leftclicking on you desktop. The voltage on the LCD pixels is popotional to the gay levels (0-55) of the image displayed on the HoloEye second monito. Once the HoloEye has been connected to you laptop you should plug-in the powe supply cable of the HoloEye. A splitte and an exta VGA monito is used to display a copy of the pictue send to the HoloEye. Run the Matlab Axicon.m pogam to test the SLM. A ing should appea in the focus of the lens placed behind the SLM. Sometimes it is necessay to un-plug the powe cable and to plug it in again to eset the SLM. Assignment 1. Detemination of the pixel spacing of the LC00. Build the setup shown in figue 1 (lenses mentioned ae an indication. Expeiment to find the optimal configuation of you set-up). 1. Stat with the HeNe lase. It must be aligned paallel to the optical ail.. Place the 10mm lens of the beam expande and adjust its position such that the expanded beam does not deflect. 3. Place the 300mm lens of the beamexpande and veify that the expanded beam does not deflect and is not diveging o conveging. Check the distance between the 300 and 10 mm lenses. What distance do you expect? 4. Place the HoloEye modulato such that the beam fills the whole apetue of the device. 5. Place the 400mm lens behind the HoloEye and place a white o mm sceen in its focal plane. 6. You should obseve an aay of bight spots. What is the cause of the aay of bight spots? Detemine the spacing of the pixels of the SLM. Assignment. Hologaphic Optical Element (HOE). In this assignment we ae going to make a hologaphic optical element which has the same function as that of a positive lens. This hologam can be made in pactice by letting coheent light oiginating fom a point souce and that of a plane wave to intefee as shown in figue. Of couse the initial phases of the two beams must be stationay, which is best oganized by oiginating them somehow fom the same souce.
Plane wave Photogaphic plate spheical wave Fig.. Recoding of a hologam acting as a HOE lens. Once the intefeence patten (intensity distibution) on the photogaphic plate has been tansfomed into a phase patten (this can be done by etching the suface of a glass plate, i.e. tansfoming the gay levels of the intensity patten into hills and valleys in the glass, causing fluctuations in the local optical path lengths and hence in phase fluctuations of an incoming beam of light). As an altenative, the phase distibution of the intefeing beams can also be pogammed in the SLM. The hologam can be compaed to a Fesnel zone phase plate in (see Hecht 4 th ed., chapte 10.3.5) in which the λ / ings ae eplaced by smooth cosine functions. Figue 3 shows a coss section of such a phase plate. 4 3 Phase [adians] 1 0 3 1 0 1 3 adius [mm] Fig. 3. Coss section of a cosine phase plate (ed). The blue cuve is a Fesnel zone plate. (Hee the plot must be consideed as a set of concentic annula sceens blocking the odd Fesnel zones and passing the even zones o annula stuctues poducing a λ/ phase jump fom zone to zone.) f = 1 m, λ = 63.8 nm.
π The phase as a function of the adius is given by: ϕ ( ) = cos π + 1, λ f whee f is the desied focal length of the HOE lens. 1. Use the set-up shown in figue 1 but with the f=400mm lens emoved.. Use the Matlab HOElens.m pogam to veify the theoy. 3. If you like, play with the LightPipes simulation. 4. A disadvantage of the hologaphic lens is that it has low efficiency because thee is not only a conveging beam, but also a diveging and a plane wave emeging fom the hologaphic lens. Modify the Matlab pogam such that the phase distibution of a thin lens is tansfeed to the SLM. Assignment 3. Hologam of a D pictue. Hee you lean how to make a compute geneated D hologam using the so-called Gechbeg-Saxton algoithm. This an iteative pocess in which Fouie and invese Fouie tansfoms ae taken afte substituting the intensity patten of the input wave and the desied patten at each iteation step keeping the phase distibution unchanged. Input wave (hee plane wave: amplitude = 1, phase = 0) Take -D Fouie tansfom Substitute desied intensity distibution, leave phase unchanged Take -D Invese Fouie tansfom Substitute input intensity distibution, leave phase unchanged Fig. 4. Gechbeg-Saxton algoithm to calculate a phase hologam. See the Matlab Hologam.m document fo details. Ty seveal examples included and see what happens when you incease the numbe of iteations. If the HoloEye SLM is connected, you have to pefom the -D Fouie
tansfom (line 58 in the pogam) optically to see the image on a sceen. How can you do that? Assignment 4. Ring focus with an axicon and a positive lens. A ing focus can be made using an axicon followed by a positive lens. In the focal plane of the lens a ing shaped intensity distibution will appea. An axicon is a coneshaped optical element as depicted in figue 4. In this assignment we ae pogamming the phase distibution of an axicon and a positive lens in the SLM. This will be done using (pat) of the LightPipes optical toolbox LightPipes fo Matlab. The Matlab pogam calculates the popagation of a plane monochomatic wave though an axicon and a lens. The phase distibution is extacted fom the complex amplitude and displayed in a Matlab figue. This figue, with gay values fom 0 (black) to 55 (white) is loaded into the SLM and a collimated beam will be tansfomed into a ing at the focal plane of the lens. 1. Use the set-up build in assignment 1.. Remove the 400 mm lens because a lens will be pogammed in the SLM. 3. Run the Axicon.m Matlab pogam. 4. Measue fo vaious values of the top-angle of the axicon and focal lengths of the lens the diamete of the ing. 5. Deive an expession fo the ing diamete fo an axicon with top-angle, φ, and efactive index, n, and a lens with focal length, f. 6. Compae the measuements with the calculations. (Use a gaph to pesent the esults). 7. Place the 400 mm lens back in the set-up and emove the lens in the Matlab pogam (just put a comment (%) in font of the command). 8. Repeat the measuements and compae again with the calculations. Remaks: 1. If the top angle of the axicon is too small, the ing will be lage than the spacing between the odes of the gating due to the pixels of the SLM and ovelap of ings will occu. Can you deive an expession of the minimum allowed axicon top angle fo ou 600 x 600 / 19.mm x 19.mm gid?. You can also simulate the set-up and compae with the expeiments using LightPipes fo Matlab. Un-comment the appopiate lines in the Axicon.m pogam. You must incease the focal length of the lens (f~1000mm) and the top angle (~179 deg) in the pogam because of the limitation of the gid. 3. Besides a phase modulation, ou SLM also changes the local polaization of the beam. You can use a polaize to enhance the contast. Also otate the lase to change and optimize its polaization. Whee must the polaize be placed? Can you give an explanation? 4. Fo those who ae inteested in moden optics: It is possible to simulate with the Axicon.m pogam a so-called Bessel-beam. (See Hecht Optics 4 th edition, chapte 10..7. The axicon method is even moe elegant than the annula slit descibed in Hecht, because no beam powe is wasted by the annula slit). These Bessel beams do not diffact and have a constant beam size duing popagation ove vey long distances (seveal metes) in fee space. Simply emove the lens in the simulation by commenting the appopiate command line (line 30). Now f is the distance whee the Bessel intensity pofile has to be obseved. Use lage top angles (>179.7 deg) to get a easonable esolving powe with ou 600 x 600 gid. Ty to obseve the Bessel beam expeimentally as well by emoving the 400 mm lens fom the set-up.