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The development of the dual-band IR imaging polarimetry creates the need for achromatic phase retarder used in dual-band. Dielectric grating with the period smaller than the illuminating wavelength presents a strong form-birefringence. With this feature, the combination of several subwavelength gratings can be used as achromatic phase retarders. We proposed a combination of 4 subwavelength structured gratings (SWGs) used as an achromatic quarter-wave plate (QWP) applied to MWIR & LWIR bandwidths. Design method using effective medium theory and optimization algorithms is described in detail. The simulation results led to the possibility of an dual-band achromatic QWP whose retardance deviates from 90° by <±0.75° with the fast axis unfixed and by <±1.35° with the fast axis fixed over MWIR(3-5μm) & LWIR(8-12μm) bandwiths.
©2010 Optical Society of America
1. Introduction
The development of dual-band focal plane arrays [1] in which each pixel consists of superimposed mid-wave and long-wave photodetectors enables the combination of MWIR and LWIR polarimetric imagers into a single unit. Due to the increased target information obtained from the dual-band sensor, MWIR&LWIR polarimetric imager may have better image contrast, longer detecting range and higher spatial target discrimination than the single-band ones. In view of this, some key dual-band polarization elements such as polarizers and phase retarders are in need. An IR polarizer [2] called metal wire grating whose period is much smaller than the illuminating wavelength can easily provide extinction ratio of 25dB over MWIR&LWIR dual-band. But to our knowledge, dual-band achromatic phase retarder has not yet been reported.
Standard phase retarders usually can be used only at a single wavelength, since the phase retardation strongly depends on the frequency. Achromatic phase retarders can be realized on the basis of several physical principles. Achromatic prism retarders [3], which are based on the phenomenon of phase shift at total reflection, are voluminous and often result in an output beam displaced from the input beam which may disturb the imaging process. Another type of achromatic phase retarder is the combination of several crystal waveplates [4] with different kinds of birefringent materials. The design process is similar to making an achromatic lens, but there are far fewer IR birefringent materials to choose from. Besides, the weak natural birefringence of crystals usually makes a thick combination of multi-order waveplates, which results in a relatively higher absorption of IR radiation. A single piece of dielectric subwavelength grating [5] (SWG) with period comparable to the illuminating wavelength can also be designed as an achromatic retarder, due to its strong birefringence dispersion proportional to the wavelength over a certain spectra bandwidth. However, the grating of this kind designed in the resonance region has nonzero-order diffractions and is sensitive to the variance of incident angle. Further more, the birefringence of a single piece of grating monotonously changes with the wavelength [6]. Its birefringence dispersion curve can’t keep the same proportionality factor over two different wavebands at the same time, indicating that a single piece of SWG is unable to achieve the dual-band achromatism.
In this paper, a combination of four SWGs designed as IR dual-band achromatic quarter-wave plate is proposed for the first time. The period of each grating is chosen small enough to exclude non-zero order diffractions. The depth and the orientation angle of each grating are carefully optimized to achieve the best achromatic performance. In order to reduce volume and increase IR transmittance, the grating structures are supposed to be fabricated on both sides of the substrate.
2. Subwavelength grating
Grating in quasi-static domain has a period much smaller than the illuminating wavelength (d/λ≤1/10). Its birefringence is analyzed by the zero-order approximation effective medium theory (EMT) [7]. A surface-relief grating of a rectangular-groove profile, as shown in Fig. 1 , is equivalent to an anisotropic thin film and the effective refractive indices nTE and nTM can be written as
where superscript 0 indicates the zero-order approximation, f is the filling factor, n1 and n2 are the refractive indices composing the grating. Like crystalline waveplate, the modulation of the polarization state is achieved by use of the form-birefringent properties of the SWG: the TE and the TM components of the incoming light undergo different phase shifts, so the relative phase difference between TE and TM polarizationschanges. Compared with IR crystalline materials, dielectric SWGs exhibit much stronger birefringence. The birefringence dispersion curves of Cadmium Sulfide (CdS), Cadmium Sele-nide (CdSe) and silicon SWG within IR range are plotted in Fig. 2 respectively. The birefringence of SWG is calculated by using Eq. (1) and Eq. (2) with f=0.5, n1=1 and n2=nSi(λ)[8]. The value of form-birefringence provided by SWG () is two orders of magnitude larger than that of natural birefringence (. Consequently, by controlling the relatively shallow grating depth, a single piece of SWG with a small period-to-wavelength ratio could achieve any phase retardance with fast axis parallel to the grating vector (shown in Fig. 1).
Fig. 1 Schematic diagram of a dielectric grating. It has a period of d, groove depth of h and a filling factor of f. The width of the ridge is fd in case n1<n2
Grating in resonance domain has a period comparable to the illuminating wavelength. When the grating’s period approaches the illuminating wavelength, the zero-order EMT becomes inaccurate. The second-order EMT, which includes a second-order correction of the finite ratio d/λ [9], has improved accuracy for 1/10<d/λ<3/2. Although the second-order EMT is not a rigorous analysis for gratings in resonance domain, it leads to an approximate value of the phase of light waves that have passed through the grating. We can use it to estimate the dependence of birefringence on wavelength. The second-order EMT has the form,
where superscript 2 indicates the second-order approximation and d is the grating period. The effective refractive indices and change with the wavelength, sothe birefringence of the grating () has spectra dependence. Figure 3 depicts some birefringence spectra curves with respect to different grating periods. The birefringence is calculated by using Eq. (3) and Eq. (4) with f=0.7, n1=1 and n2=nSi(λ). With the increase of the illuminating wavelength, the decrease of the value of radio d/λ makes the dispersion curve go flat. The flat part of the curve is similar to that plotted in Fig. 2, indicating that the second-order EMT approaches the zero-order EMT when d/λ becomes smaller.The phase retardance introduced by an SWG can be expressed as
where Δn(λ) is the birefringence with respect to the wavelength and h is the groove depth. According to EMT, the wave propagating through a grating region is equivalent to passing through an anisotropic film. So the phase retardance is determined by the effective film thickness-groove depth h and the form-birefringence Δn(λ). To make an ideal achromatic phase retarder, i.e., the phase retardance does not change with the wavelength, the birefringence Δn(λ) must be proportional to the wavelength, that is,Thus, the ideal birefringence dispersion curve should be an oblique line.As shown in Fig. 2, the birefringence dispersion curve for a single piece of grating in quasi-static domain goes like a horizontal line indicating that birefringence has no spectra dependence. So, just like a standard waveplate, the phase retardance introduced by the grating varies with the incident wavelength.
As shown in Fig. 3, for the grating in resonance domain, its birefringence has spectra dependence. Although the actual curve does not agree with the ideal oblique line for all wavelengths, it is possible for these two curves to coincide with each other in a specified wavelength range. We can appropriately adjust the gradient of the ideal line by varying the groove depth, making the oblique line tangent to the actual curve within the target waveband. So the grating of this kind can be used as the achromatic phase retarder for a single-band. But it fails for dual-band, because we can’t make the oblique line tangent to the birefringence spectra curve within the two separate wavebands simultaneously.
3. Achromatic design
Enlightened by the idea from Destriau and Prouteau, i.e., combination of a standard half-wave plate and a standard quarter-wave plate creates a new achromatic quarter-wave plate [10,11], we utilized SWGs with different structural parameters to take place crystalline waveplates and extended the design to a combination of up to four SWGs (Schematically drawn in Fig. 4 ). The principle of achromatism is based on the partial cancellation of the change of retardance from each grating with respect to the wavelength. We performed simulations to combinations of 2 to 4 gratings and their retardance spectra curves are shown in Fig. 5 . As expected, the retardance gets closer to π/2 with the number of gratings.
Fig. 4 Schematic diagram for the proposed combination of 4 SWGs. K is the grating vector and θ is the orientation angle.
A single piece of SWG is equivalent to a standard phase plate which can be described by its corresponding Jones matrix J:
where Φ is the phase retardance determined by the birefringence and the groove depth of the SWG, θ is the orientation angle of the fast axis (grating vector). J is a unitary matrix with a form of:where matrix elements satisfies . It has been shown by Jones [12] that any combination of retardation plates is optically equal to a system containing only two elements: a retarder and a rotator. The product of two or more of the J matrices is still unitary. That iswhere and are the phase retardance and the orientation angle of the ith SWG, is the amount of rotation, and are the phase retardance and the orientation angle of the combination. Since the are functions of wavelength, , and will in general vary with wavelength. For a rotator [13], it only rotates the polarization state of the incident wave at an angle of but does not change it essentially. Thus, we do not take into account in most cases. Usually, there are two types of achromatic design: one only concerns about retardance achromatism, while the other one aims to achieve retardance and orientation achromatisms simultaneously. The former is highly achromatic within the designed wavebands, but it needs to rotate when target wavelength changes. The latter is less achromatic, but the angle of the fast axis remains nearly fixed.The phase retardance and orientation angle can be obtained from Eqs. (9) to (12), that is,
For the combination of SWGs used as achromatic QWP within the MWIR&LWIR wavebands with the fast axis unfixed, the design of achromatism for phase retardance is described by a two-objective optimization model [14] which can be expressed as
where is the phase retardance for the wavelength of λ introduced by the combination, h i is the groove depth and θ i is the orientation angle of the ith piece of SWG. Objective function1 is used to minimize the total phase retardance error, while function2 aims to even the phase retardance error over MWIR&LWIR wavebands. We performed simulation for the combination of 4 SWGs. By use of the optimization algorithm, the parameters of each SWG are calculated (listed in Tab.1.) .The result obtained with the combination of these 4 SWGs is depicted in Fig. 6 .Fig. 6 Calculated results of achromatic QWP with the fast axis unfixed (a) Phase retardance dispersion curve. (b) Orientation angle dispersion curve.
Tab.1. Parameters of QWP with its fast axis unfixed
The designed achromatic QWP exhibits perfect retardance stability. The phase retardance deviates very slightly from ideal π/2(the maximal relative deviation is 0.8%) over both MWIR and LWIR wavebands. Since the fast axis is not involved in the design process, the orientation angle of the fast axis depicted in Fig. 6(b) changes rapidly with different wavebands. The orientation angle is positive for MWIR and negative for LWIR.
For the archromatic QWP with the fast axis fixed, orientation achromatism is as important as retardance achromatism. The value of the standard deviation (STD) for orientation angles is selected as the evaluation criteria for the degree of orientation achromatism. The design of achromatic QWP of this kind is described by a three-objective optimization model, that is,
Still, function 1 and function 2 are used to achieve retardance achromatism. The added function3 aims to control the uniformity of the orientation angles. For a multi-objective optimization problem, it is impossible to get a solution that minimizes the value of each objective function. What we can do is to seek for a compromising solution that satisfies the three objective functions simultaneously. Based on this thought, the optimized parameters of QWP are listed in Tab. 2. The calculated retardance and orientation achromatisms with these parameters are demonstrated in Fig. 7 .Fig. 7 Calculated results of achromatic QWP with the fast axis fixed (a) Phase retardance dispersion curve. (b) Orientation angle dispersion curve.
Tab. 2. Parameters of QWP with its fast axis fixed
As expected, the relative retardance deviation increases from 0.8% to 1.5% with the involvement of function3. But the orientation angle of the fast axis varies from 10.5° to 14.5° over both MWIR and LWIR wavebands. The nominal value of the orientation angle is 11.6° with its standard deviation of 1.27°.
4. Fabrication consideration
The QWP is supposed to be used in imaging polarimetry system. Since higher-order diffractions may sneak into the imaging system and form “ghost images”, the gratings should be zero-order elements. In other words, the period should be small enough to exclude nonzero diffraction orders. For any dielectric grating under normal incidence, as depicted in Fig. 1, the threshold period [15] under which only the transmitted and the reflected zero-order diffractions are nonevanescent is
The threshold period is proportional to the illuminating wavelength and the value of the grating period is specified by the smallest wavelength in the waveband. For MWIR&LWIR dual-band, the minimal wavelength is 3μm and the refractive index of silicon for the wavelength of 3μm is 3.4307. By inserting n 1=1 and n 2=3.4307 into Eq. (17), the threshold period of the grating should be 677nm, which guarantees a zero-order grating. The linewidth of the grating should be 338nm with the filling factor of 0.5 (f=0.5 helps to ease the fabrication). As seen from Tab. 1 and Tab. 2, the largest groove depth is 3072nm, which corresponds to an aspect-ratio of 1:9. This aspect-ratio is a little bit challenging but still in the range of fabrication capabilities [16] of electron-beam writing and ion-beam etching machines.In order to increase the IR transmittance and ease the adjustment, the combination of 4 SWGs are supposed to be fabricated on 2 silicon wafers, that is, 4 gratings are orderly on the front and back sides of the two substrates. A 200μm-thick silicon wafer has MWIR transmittance of 55% and LWIR transmittance of 45%[17]. Then the total thickness of the QWP is 400μm and the total transmittance would be around 25%. Of course, other IR materials with higher transmittance, such as ZnSe and BaF2, may also be selected. But compared with Si, other IR materials have smaller refractive indices which brings a larger aspect-ratio to the grating, making the fabrication even more difficult.
Each grating is designed with a groove depth precision of better than 10nm, and the relative angular adjustment between the two silicon wafers is better than 1°. With these fabrication tolerances, the relative retardance deviation will be limited in the range below 3%.
5. Conclusion
In this paper, we discussed the limitations of the currently existing achromatic waveplates applied to dual-band. A combination of 4 SWGs used as MWIR&LWIR dual-band achromatic QWP is proposed . The grating period is specified to exclude nonzero order diffractions, while the groove depth and the orientation of the grating vector are optimized to achieve retardance achromatism. The designed dual-band achromatic QWP has potential application in MWIR&LWIR imaging polarimetry where crystalline retarders are not available.
Acknowledgement
This work was supported by the National Basic Research Program of China under Grant No 2007CB935303 and by Beijing Municipal Commission of Education Project under Grant No KM200910772005.
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