Meadowlark Optics LCPM-3000 Liquid Crystal Polarimeter Application Note: Determination of Retardance by Polarimetry Tommy Drouillard

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1 Meadowlark Optics LCPM- Liquid Crystal Polarieter Application Note: Deterination of Retardance by Polarietry Toy Drouillard 5 Meadowlark Optics, Inc.. Introduction: The iediate purpose of a polarieter such as the LCPM- liquid crystal polarieter fro Meadowlark Optics is to easure the vector coponents that cobine to describe the polarization state of light. A polarieter can also be used as a precision diagnostic tool; not only is it useful for characterizing light signals and sources, it is also effective at precisely characterizing optical coponents through their effect on the polarization state of light. This application deonstrates a technique by which a polarieter is used to easure the retardance of a waveplate. A retardance easureent is desirable under circustances in which retardance ust be known ore precisely than the anufacturer s specification, or in which the retarder is to be used at a wavelength other than that specified by the anufacturer. For exaple, a waveplate specified as having a quarter-wave of retardance at 5 n ight actually have a retardance that varies by several percent at the specified wavelength, and oreover it will exhibit a significantly different (and generally unknown) retardance for 55-n or 488-n light. A retarder, or waveplate, is an optical coponent consisting of a birefringent aterial that varies the polarization state of light passing through it. A retarder is characterized by a fast axis orthogonal to the direction of light propagation, and a slow or optic axis in the sae plane and orthogonal to the fast axis. Coercial retarders are usually arked to indicate the orientation of the fast axis. The orientation of the fast axis is denoted by ρ. The following procedure deonstrates the application of a Meadowlark Optics liquid crystal polarieter to position a linear retarder such that its fast axis ρ is at 9, and then to precisely easure the retardance δ of the retarder. The LCPM- is a tokes polarieter, therefore the Mueller atrix convention for optic eleents is used in the following discussion.. Retardance Modeling The polarization state of light is odeled by a tokes vector consisting of four values: Page of 5

2 = total light intensity intensity difference between horizontal & vertical intensity difference between + 45 & 45 intensity difference between right & left circular () An optic that changes the polarization state of light is odeled by a Mueller atrix (4x4) such that the inner product of the tokes vector that odels the polarization state of incident light Ŝ with the Mueller atrix that odels the optic M results in a tokes vector that represents the polarization state of light exiting the optic. = M Ŝ () =,,, 4,,,, 4,,,, 4,,4,4,4 4,4 ˆ ˆ ˆ ˆ () The Mueller atrix for a linear retarder with an undeterined retardance δ and arbitrary rotational orientation (indicated by the fast axis angle ρ is given by Kliger et al. as M = cos(4ρ)sin ( δ / ) + cos sin(4ρ)sin ( δ / ) sin(ρ)sin ( δ ) ( δ / ) cos(4ρ)sin sin(4ρ)sin ( δ / ) ( δ / ) + cos cos(ρ)sin( δ ) ( δ / ) sin(ρ)sin( δ ) cos(ρ)sin( δ ) (4) Careful selection of an incident polarization state and observation of the detected polarization vector coponents as the retarder is rotated akes it possible to deterine the fast axis angle ρ. For this application, incident light that is linearly polarized at +45 [Ŝ = (,,, ) T ] is applied. The retarder is rotated fro ρ = 9 to ρ = +9. The coponent of the detected tokes Page of 5

3 vector varies fro a axiu value at ρ = ±9 to a iniu value at ρ= as shown in Fig.. The retarder can therefore be rotated to ρ = 9, indicated by axiizing. Detected fro linearly-polarized +45 incident light π π π 4 4 π Figure. Detected for 9 < ρ < 9 with linearly polarized incident light at +45 Fig.. suggests that the retardance δ can be deterined as the Arcsine of the axiized ; while this is one technique by which to calculate the retardance, the following procedure has less uncertainty than siply calculating Arcsine( ). The Mueller atrix for a linear retarder with an undeterined retardance δ, oriented at ρ = 9, is given by Kliger et al. as M = (5) Calculating the right side of Eq. (4) using the expression for M in Eq. (5) gives = Ŝ = Ŝ (6i) (6ii) Page of 5

4 ˆ cos( ) ˆ = δ ˆ cos( ) ˆ = δ + (6iii) (6iv) Eqs. (6iii) and (6iv) cobine to give ˆ = ˆ ˆ ˆ cos( δ ) (7) which can be solved for tan(δ) by applying Craer s rule tan( ˆ ˆ δ ) = (8) ˆ + ˆ The expression in Eq. (8) can be applied to tokes vectors easured with a Meadowlark Optics LCPM- to deterine the retardance δ of a retarder oriented with its fast axis at ρ = 9.. Laboratory Apparatus and Procedure The apparatus used to easure retardance consists of a light source, the retarder to be easured, and a Meadowlark Optics LCPM- tokes Polarieter, as shown in Fig.. Incident light should be polarized; the specific state of polarization is arbitrary, although Ŝ and Ŝ should not both be zero, nor should Ŝ and Ŝ be configured such that and (tokes coponents of light eitted fro the retarder) are zero. This constraint is to prevent the denoinator in Eq. (8) fro approaching zero, thereby always giving a retardance of 9. A suggested light source configuration is a laser and a polarizer positioned with its fast axis at +45 to give Ŝ =, which is the sae polarization state that is used to align the fast axis of the retarder. Page 4 of 5

5 Light ource ˆ = Polarized light incident on retarder 9 ρ = Retarder δ = retardance ρ = angle of fast axis (9 in this exaple) F H G Light eitted fro retarder I K J LCPM- tokes Polarieter Figure. Optics Configuration for Retardance Measureent The procedure for retardance easureents is as follows: Configure the polarization state of the light source to be linear, with a polarization angle of +45 [Ŝ = (,,, ) T ]. This is easily accoplished by rotating a polarizer at the source and detecting the polarization state with the Meadowlark Optics LCPM- liquid crystal polarieter (Fig. with the retarder reoved). Record the precise polarization state Ŝ of the incident bea detected by the Meadowlark Optics polarieter. Place the retarder in the bea path. Rotate the retarder and watch detected by the Meadowlark Optics polarieter. Maxiizing orients the fast axis of the retarder at 9. Record the precise polarization state of the retarded bea detected by the Meadowlark Optics polarieter. With Ŝ and precisely easured, use Eq. (8) to calculate the retardance δ. 4. Reference D.. Kliger, J. W. Lewis, and C. E. Randall. Polarized Light in Optics and pectroscopy. Acadeic Press, Inc.: an Diego, CA. 99. Page 5 of 5