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This paper presents a novel design for broadband zero-order half-wave plates to eliminate the first-order or up to second-order wavelength-dependent birefringent phase retardation (BPR) with 2 or 3 different birefringent materials. The residual BPRs of the plates increase monotonously with the wavelength deviation from a selected wavelength, so the plates are applicable to the broadband light pulses which gather most of the light energy around their central wavelengths. The model chooses the materials by the birefringent dispersion coefficient and evaluates the performances of the plates by the weighted average of the absolute value of residual BPR in order to emphasize the contributions of the incident spectral components whose possess higher energies.
©2011 Optical Society of America
1. Introduction
A wave plate, made from birefringent materials, is an optical device that alters the polarization state of a light wave traveling through it. Usually, there are two types of wave plates: zero order (retardation less than 2π) and multiple orders (retardation greater than 2π) [1]. Because of dispersion, a simple wave plate, even a zero-order wave plate, will impart a wavelength-dependent phase difference to the input light thereby can only be used for a particular range of wavelengths [2]. In many applications, a broad flat retardation vs. wavelength is required, such as polarization spectroscopy, magneto optical experiments, spectroscopic ellipsometry, observational polarimetry and telecommunications, where broadband light sources or multiple laser-line sources are often used [1]. In recent two decades, with the development of ultrashort pulse laser technology, as short as few optical cycle laser pulses duration is available either directly from mode-locked laser oscillators or from laser amplifiers centered at 800 nm[3–5]. Correspondingly, the bandwidth of the pulses shall be up to hundreds of nanometers. In order to manipulate the polarization of this kind of ultrashort pulses, it is desirable for the wave plate to keep the constant retardation over hundreds of nanometers around central wavelength. Charles J. Koesrter realized simultaneously π phase retardation at two or more wavelength by use of two or more identical half-wave plates in series [6]. However, such a method has the drawback that the orientation of its principal axis varies with the wavelength. An alternative method free from this disadvantage is to design the achromatic or apochromatic wave plates by a combination of some pieces of different birefringent materials with appropriate thicknesses. This method improves the wave-independence of the birefringent phase retardation (BPR) by setting the specified BPR of the system at 2, 3, or more selected wavelengths which depend on the number of different birefringent materials [2,7]. The choices of the materials are critical for these kinds of the achromatic or apochromatic wave plates [8,9]. By use of this method, M. Emam-Ismail found the residual BPR dispersion was available as small as ± 0.27% by the gypsum/KDP/quartz combination [1,10]. However, it is theoretically possible that these types of achromatic or apochromatic wave plates have very satisfactory BPR at the selected wavelengths but poor performances at other spectral zones. This paper presents a new model to design waveplates with broad constant BPR by minimizing the BPR dispersions over a wide spectral region centered a selected wavelength. Usually, a broadband light pulse gathers most of its energy around its central wavelength, thus our wave plates are very suitable for controlling the polarizations of the broadband light pulses.
2. Theory
It is well known that the difference between the highest and lowest refractive indices of a birefringent crystal for a given wavelength λ can be defined as the birefringence η(λ) [2,8], that is
where n o and n e are the ordinary and extraordinary refractive indices of the birefringent crystal respectively. Correspondingly, η ( n )(λ) (n = 1, 2, 3…) denotes the nth derivative of η evaluated at the point λ. If a light beam passes through the birefringent crystal with a thickness d, the BPR between the ordinary and extraordinary beam can be expressed as φ(λ) = 2πdη (λ)/λ. For a phase retarder made of two different birefringent crystals A and B with the geometrical thicknesses d A and d B, φ(λ) becomesAccordingly, the Taylor series around the central wavelength λ 0 can be written as
whereφ ( n )(λ 0) (n = 1, 2, 3…) denotes the nth derivative of φ evaluated at the point λ 0. Obviously, in order for a half-waveplate to have a flat wavelength-dependent BPR curve, d A and d B can be chosen so that φ (λ 0) = π and φ (1)(λ 0) = 0, thusIn Eq. (4), the same signs of the d A and d B means that the crystal axes of the materials are aligned in parallel, otherwise, means that one axis is rotated through 90° with regard to the other. The corresponding residual BPR may be determined by
From Eqs. (3) and (4), one can figure out the values of d A and d B thus the birefringent dispersion coefficient σ can be expressed as
Obviously, σ depends only on the central wavelength λ 0 and the birefringent materials. Equation (6) implies that the two birefringent materials shall be chosen so that the |σ| of the combination shall be as small as possible, which is different from the Eq. (9) in reference 9.
Similarly, for an apochromatic zero-order half wave plate consisting of three different birefringent crystals A, B and C with the geometrical thicknesses d A, d B and d C at λ 0, Eq. (7) is required to meet for minimal residual BPR, that is
The residual BPR dispersion can be expressed as
with the coefficientand3. Calculation procedure and discussions
Basing on our theoretical model, the calculation procedure for a broadband zero-order half-waveplate can be outlined as following. Firstly, list out all of the available materials with appropriate optical, mechanical, thermal and chemical properties aiming to the applications of the waveplates. Then calculate the coefficient σ of all the possible combinations of two or three kinds of birefringent crystals to find the combinations with small values of |σ| by use of the Eq. (6) or (9). Next, figure out the thicknesses of the birefringent crystals, e.g. from Eq. (4) for two birefringent crystal based combination or from Eq. (7) for three birefringent crystal based combination. Finally, evaluate the synthetical wave plates by the maximal residual BPR over the considered spectral window, ΔΦ = Max|φ (λ)−φ (λ 0)|.
As an example, here we focus on calcite, quartz, ADP, sapphire, KDP and MgF2 as the candidates of broadband zero-order half-wave plates at the central wavelength λ 0 = 550 nm. Table 1 presents the coefficient σ, the crystal thicknesses, and the corresponding maximal residual BPR of the 2 different material-based combinations over the whole calculated spectral window on the 6 candidate birefringent crystals. One can see, in the spectral range from 540 to 560 nm, the smaller value of |σ|, the smaller corresponding ΔΦ. However, if the calculated spectral range extends from 400 to 700 nm, smaller |σ| doesn’t mean smaller ΔΦ any more. It is easy to explain this difference according to Eq. (3). For the spectral components far enough away from λ 0, the wavelength-dependent residual BPR may not only be determined by the 2nd-order wavelength−dependent term, but also by the 3rd, 4th, even 5th–order wavelength-dependent terms. Table 1 also shows that, by use of the combinations, ADP/MgF2 and KDP/MgF2, the wave plates have very small values of |σ| (0.7177 and 0.4851 separately), and the corresponding ΔΦs are as small as 2.3933° for ADP/MgF2 and 2.6906° for KDP/MgF2 from 400 to 700 nm. Figure 1(a) presents the ΔΦ vs. wavelength of the KDP/MgF2, ADP/KDP or ADP/MgF2−based half-wave plates. All the curves show that the residual BPRs increase with |λ−λ 0|. The inset means that the residual BPRs are quadratic-dependent with regard to λ−λ 0 in the spectral region close enough to λ 0 (540~560 nm), which implies the high-order(>2) wavelength-dependent phase retardations are negligible.
Table 1. Birefringent Dispersion Coefficient σ, Thicknesses, and ΔΦ of 2 Different Material-based Zero-order Half Waveplates from 6 Candidate Birefringent Crystals: Quartz, MgF2, Sapphire, Calcite, ADP, and KDP
Fig. 1 The residual BPR ΔΦ vs. wavelength of some zero-order half-wave plates based on 2 (a) or 3(b) birefringent materials
As we know, 20 different combinations of 3 are available from 6 candidate birefringent crystals: calcite, quartz, ADP, sapphire, KDP and MgF2. Similar with Table 1, in Table 2 one can see the calculating values of |σ|s according to Eq. (9), the thicknesses of the three crystals and the ΔΦs. As predicted, for all the combinations, the calculated maximal residual BPR ΔΦ increases monotonously with |σ| in the spectral range from 500 to 600 nm. Even from 400 to 700 nm, our results still show that the combination with larger |σ| usually leads to larger ΔΦ, which is different from that in Table 1. The 3 combinations with smallest |σ| in Table 2 are Quartz/calcite/ADP, Quartz/calcite/KDP and MgF2/sapphire/ADP, correspondingly ΔΦ = 3.4699°, 3.8066° and 5.1357° respectively from 400 to 700 nm. However, the Quartz/calcite/ADP combination is not our choice because the required ADP plate is so thin thus very difficult to be manufactured, in spite of its smaller |σ| and ΔΦ. In Fig. 1(b), all the curves of the residual BPRs ascend with |λ−λ 0|. The residual BPRs are cubic-dependent with regard to λ−λ 0 from 500 to 600 nm, which implies the 2nd-order wavelength-dependent BPRs are almost eliminated, and the residual wavelength-dependent BPRs are dominated by Eq. (8). Our analyses above show it is appropriate to evaluate the performances of the wave plates by Eq. (6) or Eq. (9) together with maximal residual BPR only in the spectral zone close to λ 0.
Table 2. Birefringent Dispersion Coefficient σ, Thicknesses, and ΔΦ of the 3-Material–Based Zero-order Half-wave Plates from 6 Candidate Birefringent Crystals: Quartz, MgF2, Sapphire, Calcite, ADP, and KDP
In spite of the more complicated configuration of the 3 crystal based waveplates compared with those by the 2 crystal based ones, from our discussions above one maybe gets a paradoxical conclusion that the three crystal based doesn’t present their advantage in minimizing ΔΦ. Figure 2 aims to compare the performances of the zero-order half-wave plates by the combinations: KDP/MgF2 and ADP/MgF2/sapphire. At the zone from 400~510 nm, the 3 crystal based combinations have larger ΔΦ than the 2 material based one. However, in the range from 510 to 700 nm, the 3 material based combinations have much smaller residual BPR. As shown in the inset of Fig. 2, especially in the spectral region around λ 0, the advantage of 3 materials combinations is very apparent. It seems that it is inappropriate to evaluate the performances of the wave plates just by ΔΦ. We think a perfect evaluation is related to both the half-wave plate and the incident optical field.
Fig. 2 The residual BPR vs. wavelength of the zero-order half-wave plates composed of the birefringent crystal combinations: Quartz/calcite/ADP, Quartz/calcite/KDP and MgF2/sapphire/ADP
Maybe the weighted average method is an effective alternate way. If an incident pulse energy is E(λ−λ 0), the evaluating function, called the weighted average of the absolute value of the residual BPR (WAAVRB) from the wavelength λ 1 to λ 2 (λ 2>λ 1), can be defined as
It is apparently that using Δψ instead of ΔΦ means more emphasis on the contributions of the spectral components whose possess higher energies. Suppose
λ 1 = 400 nm, λ 2 = 700 nm, λ 0 = 550 nm, and the bandwidth(FWHM) Δλ = 100 nm, from Eq. (11), it is easy to figure out the WAAVRBs of all the combinations both in Table 1 and Table 2 (see Table 3 ). The WAAVRBs are 0.0322, 0.0160 and 0.0189 for the combinations KDP/MgF2, Quartz/calcite/ADP and Quartz/calcite /KDP, respectively. Table 3 shows it is easier to get smaller WAAVRBs for the half-wave plates by using 3 birefringent materials than those by using 2 birefringent materials.4. Conclusions
In summary, this paper proposes a novel model for broadband zero-order half-wave plates. We apply the Taylor series expansion around λ 0 to the BPR in order to eliminate the 1st-order wavelength -dependent BPR by using 2 birefringent materials, or both the 1st- and the 2nd-orders the wavelength- dependent BPR by using 3 birefringent materials. On our model the residual BPRs of the plates increase monotonously with |λ−λ 0|, which means the plates are applicable to the broadband light pulses which gather most of the light energy around λ 0. The materials can be chosen by the combination with minimal value of the coefficient |σ|. If the wave plates are under the radiation of the ultrashort light pulses, it is better to evaluate the performances by using the WAAVRB instead of the maximal residual BRP. Our discussions show the |σ|, together with WAAVRB, is still valid even for the incident bandwidth up to hundreds of nanometers. Using the birefringent crystals: Quartz, MgF2, sapphire, calcite, ADP and KDP, as candidates, we calculate the performances of all combinations consisting of 2 or 3 birefringent crystals. From 400 to 700 nm, the maximal residual BPR can be available as small as 2.39° for ADP/MgF2 pair or 3.47° for Quartz/calcite/ADP combination, while the corresponding WAAVRBs for 100 nm incident bandwidth is 0.0455 o or 0.0160 o respectively. However, the chosen material combinations shall not only have small residual BPR or the WAAVRB, but also have manufacturable sizes.
Acknowledgments
This work was partly supported by National Natural Science Fund (60878017), the Key Project of Chinese Ministry of Education (107047), the Key Project from Natural Science Fund of Shenzhen Municipality (JC201005250072A), the Application and Development Project of Shenzhen University (201056), and the Open Fund of the State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics.
References and links
1. M. Emam-Ismail, “Retardation calculation for achromatic and apochromatic quarter and half wave plates of gypsum based birefringent crystal,” Opt. Commun. 283(22), 4536–4540 (2010). [CrossRef]
2. P. Hariharan, “Broad-band superachromatic retarders,” Meas. Sci. Technol. 9(10), 1678–1681 (1998). [CrossRef]
3. I. D. Jung, F. X. Kärtner, N. Matuschek, D. H. Sutter, F. Morier-Genoud, G. Zhang, U. Keller, V. Scheuer, M. Tilsch, and T. Tschudi, “Self-starting 6.5-fs pulses from a Ti:sapphire laser,” Opt. Lett. 22(13), 1009–1011 (1997). [CrossRef] [PubMed]
4. S. Ghimire, B. Shan, C. Wang, and Z. Chang, “High-Energy 6.2-fs Pulses for Attosecond Pulse Generation,” Laser Phys. 15, 838–842 (2005).
5. J. Zhu, P. Wang, H. Han, H. Teng, and Z. Wei, “Experimental study on generation of high energy few cycle pulses with hollow fiber filled with neon,” Sci. China, Ser. G G50, 507–511 (2007).
6. C. J. Koesrter, “Achromatic Combinations of Half-Wave Plates,” J. Opt. Soc. Am. 49(4), 405–409 (1959). [CrossRef]
7. J. M. Beckers, “Achromatic linear retarders,” Appl. Opt. 10(4), 973–975 (1971). [CrossRef] [PubMed]
8. P. Hariharan, “Broad-band apochromatic retarders: choice of materials,” Opt. Laser Technol. 34(7), 509–511 (2002). [CrossRef]
9. P. Hariharan, “Achromatic and apochromatic halfwave and quarterwave retarders,” Opt. Eng. 35(11), 3335–3337 (1996). [CrossRef]
10. M. Emam-Ismail, “Spectral variation of the birefringence, group birefringence and retardance of a gypsum plate measured using the interference of polarized light,” Opt. Laser Technol. 41(5), 615–621 (2009). [CrossRef]
