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Molecular Crystals and Liquid Crystals
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Achromatic Liquid Crystal Phase Plate
for Short Laser Pulses
Ru-Pin Pan a , Cheng-Wei Lai a , Chia-Jen Lin a , Cho-Fan Hsieh a & Ci-Ling Pan b
a
Department of Electrophysics , National Chiao Tung University , Hsinchu, Taiwan, Republic of China
b
Department of Physics and Inst. of Photonic Technologies , National Tsing Hua University , Hsinchu, Taiwan, Republic of China
Published online: 19 Aug 2010.
To cite this article: Ru-Pin Pan , Cheng-Wei Lai , Chia-Jen Lin , Cho-Fan Hsieh & Ci-Ling Pan (2010)
Achromatic Liquid Crystal Phase Plate for Short Laser Pulses, Molecular Crystals and Liquid Crystals, 527:1, 65/[221]-71/[227], DOI: 10.1080/15421406.2010.486632
To link to this article: http://dx.doi.org/10.1080/15421406.2010.486632
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Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China
2
Department of Physics and Inst. of Photonic Technologies, National Tsing Hua University, Hsinchu, Taiwan, Republic of China
This work demonstrates the feasibility of reducing pulse broadening by using liquid crystal achromatic half and quarter wave plates for ultrafast pulsed lasers. Each wave plate is made of two and three liquid crystal cells stacked together. Total phase retardation and its broadening reducing of the achromatic half wave plate have been measured. The experimental results are in good agreement with theoretical calculation.
Keywords Achromatic; liquid crystal; pulse width; ultrafast pulsed lasers; wave plate
1. Motivation
Liquid crystal has been widely used for phase retarders among which the quarter wave plates and half wave plates are the most common application [1,2]. Quarter and half wave plates can change a linearly polarized light into a circularly light and the polarization direction of a linearly polarized light, respectively. The phase retardation of a wave plate is strongly dependent on the wavelength of incident light and then the phase dispersion can cause pulse width broadening on the laser pulses. Considering a very short light pulse, its spectrum bandwidth (Dk) is inversely pro-portional to its pulse width (Dt). In other words, short pulses have large wavelength bandwidth. Figure 1. shows the specturm of our Ti:Sppphire ultrafast laser with cen-tral wavelength around 800 nm. The bandwidth is 40 nm and the pulse width is less than 100 fs.
In order to maintain the short pulse width, the waveplates must be achromatic. Achromatic wave plates had been suggested by Kohns et al by using liquid crystal mixture which has an birefringence (Dn) varies proportional to wavelength [3]. On the other hand, an equivalent achromatic wave plate can be obtained by combining
Address correspondence to Ru-Pin Pan, Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan 30010, Republic of China. Tel.:þ886-3-5731989; Fax: þ886-3-5725230; E-mail: [email protected]
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several wave plates. Similar idea had been used in reflection liquid crystal displays [4,5], and terahertz wave plates [6]. In this work, we use several electric tunable liquid crystal cells to create achromatic wave plates. The effect on reducing the pulse width broadening is studied. The phase retardation of each cell can be easily modulated by controlling the applied voltage.
2. Theory and Design
We use Stokes representation and Poincare` sphere [7] to predict the variation of polarization state of laser pulses after passing through our devices. A conventional quarter wave plate which is made of dispersionless material and designed for 550 nm has a phase retardation of 0.5p only for this wavelength. For wavelength 450 nm and 650 nm the phase retardations are 0.61p and 0.42p, respectively. When the slow axis of the quarter wave plate is 45 to x-axis, the horizontal polarization
of incident light will be changed into circular polarization only for the wavelength of 550 nm. The polarization state of output laser pulses will not be circular at 450 nm and 650 nm [see Fig. 2].
In our design for achromatic quarter wave plate, a half wave plate and a quarter wave plate are stacked with an angle of 60between their slow axes. The achromatic
half wave plate is formed with three half wave plates. The outer wave plates are aligned with their slow axes parallel while the middle wave plate has its slow axis at an angle of 60with respect to the outer ones. As shown in Figure 3, the polariza-tion states of output laser pulses are very close to circular polarized waves and ver-tical polarized waves after passing through an achromatic quarter and half wave plate even for 450 nm and 650 nm, respectively.
We also use Jones matrix method [6] to calculate the wavelength dependency of the devices. Each plate in our designed can be described by a corresponding Jones matrix Ji,
Jiðdi;hiÞ ¼ cos di sin 2hi=2 i cos 2hisin di=2 i sin 2hisin di=2 isin di=2 cos di=2þ i cos 2hisin di=2
; ð1Þ
Figure 1. The spectrum of Ti:Sapphire ultrafast laser.
66=[222] R.-P. Pan et al.
where diis the phase retardation di¼ 2p(ne-no)di=k and hi is the angle between the
slow axis and x-axis. Since Jones matrix is a unitary matrix, the combined Jones matrix has a form of
J ¼Y i Ji¼ A B B A ð2Þ
Figure 3. (a) The horizontal polarization (H) changes to circular polarization (C) after passing through an achromatic quarter wave plate and the slow axes of the quarter and half wave plates are 15and 75to x-axis, (b) the horizontal polarization (H) changes to vertical polar-ization (V) after passing through an achromatic half wave plate and the slow axes of each half plates are 15, 75and 15to x-axis.
Figure 2. Poincare´ sphere representation of polarzation state evolution through a quarter wave plate.
and the total phase retardation d is obtained by tan2d 2¼ Im A j j2þ Im Bj j2 Re A j j2þ Re Bj j2: ð3Þ By using Jones matrix calculation, we can find out the relationship between phase retardation and wavelength of our design. Figure 4 shows that there is a wave-length region with the phase retardation is insensitive to wavewave-length (solid line). The biggest error in200 nm are 7.96% and 8.00% at 600 nm for an achromatic quarter wave plate (AQWP) and an achromatic half wave plate (AHWP), respectively. The dashed lines are non-achromatic QWP and HWP.
Our wave plates are rubbed antiparallel cells filled with nematic liquid crystal (E7). The direction of slow axis is the same as rubbing direction and the phase retardation are controlled by applied voltage.
3. Phase Retardation Measurement
The phase retardation of our devices was measured with a setup as shown in Figure 5 [8]. The light source is a Ti:Sapphire laser working at continuous wave mode.
The linearly polarized light passing through a modulator cell M (a nematic liquid crystal cell) is incident on a beam splitter BS and is divided into two parts: the reference beam and the test beam. The reference beam passes through an analy-zer Ar, then is detected by photodetector Dr. The test beam passes through a tested
achromatic wave plate (AWP) and an analyzer Atand is detected by another
photo-dector Dt. The polarization axis of polarizer is along the x-direction, while both
analyzers are perpendicular to the polarizer. The z axis is in the propagation direc-tion. The slow axis of M and the equivalent slow axis of AWP are both at 45to the
x axis. The normalized intensity of the reference beam and the test beam obtained by applying Jones matrix method are as following:
Ir¼ Sin2
CM
2 ; ð4Þ
Figure 4. Relationship between phase retardation and wavelength: (a) solid line: AQWP; dashed line: QWP, (b) solid line: AHWP; dashed line: HWP.
68=[224] R.-P. Pan et al.
and It¼ Sin2
CMþ d
2 ; ð5Þ
where CM is the phase retardation introduced by the modulator cell and d is the
phase retardation to be measured. We varied the applied voltage on M from 0.1 V to 10 V a range in which CM had a total change slightly over 2p as shown by the
red circles in Figure 6. Then we also observed a intensity detected by At varies
accordingly as shown by the black square dots. The phase retardation d was then deduced following Eqs. (4) and (5).
The measured phase retardation of the AHWP are shown in Figure 7 with the theoretical calculated results. The solid line and dashed line repersent the theoretical results for an AHWP and a 0th order non-achromatic HWP, respectively and the square dots represent the experimental data. It is clearly shown that the phase retar-dations for the light with wavelength between 740–850 nm are remained at p after passing through the AHWP, which agrees with the theoretical prediction.
Figure 6. Normalized intensity versus the applied voltage on M at 800 nm: square dots, the test beam; dots, the reference beam.
A, analyzer; D, photodetector.
4. Application to Ultrafast Laser Pulses
We used interferometric autocorrelation [9] to measure the full width half maximum (FWHM) of the pulses and their broadening caused by phase retarders. The FWHM of incident pulse is 61.48 fs. The FWHMs are 63.34 fs and 74.52 fs after passing through the achromatic quarter and half wave plate while the FWHMs are 67.03 fs and 83.83 fs after passing through the 0th order non-achromatic wave plates, respectively. The device we designed can reduce the effect of pulse broadening due to the wavelength dispersion in regular wave plates. Next, we consider the pulse broadening caused by material dispersion of glass substrates and liquid crystal. The relation of pulses width between the incident beam and output beam is given by [10], Dt0¼ Dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 4 ln 2 K 00 Dt2 2 s ; ð6Þ
Here, Dt and Dt0are the FWHMs of the incident and the output pulses, respectively and K00is the group dispersion delay (GDD) of the glass substrates and liquid crys-tal. We have measured the pulse width of the incident pulse and the pulse after pass-ing through a glass substrate and 24 mm thick liquid crystal cell. By uspass-ing Eq. (6), we determine that the GDD of one glass substrate is 97.48 fs2and the GDD of a 3 mm layer of liquid crystal is 1.28 fs2. For a liquid crystal sample including two glass sub-strates and a 3 mm layer of liquid crystal, the GDD of liquid crystal is thus 0.66% that of the substrates. The pulse broadening is entirely contributed caused by the material dispersion of glass substrates.
5. Conclusion and Future Work
We have succeessfully demonstrated a new type of achromatic half wave plates and quarter wave plates. Comparing to comercial AWP, our achromatic region can be
Figure 7. Phase retardation versus wavelength of the half wave plate. Solid line: theoretical data of the AHWP; Dashed line: theoretical data of the 0th order HWP; Square dots: experi-mental data of the AHWP.
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