Narcissus of Diffractive OpticalSurfaces. Jonathan B. Cohen. ElOp E1ectrooptics Industries Ltd. P.O.B , Rehovot , Israel LINTRODUCTION

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1 Narcissus of Diffractive OpticalSurfaces Jonathan B. Cohen ElOp E1ectrooptics Industries Ltd. P.O.B , Rehovot , Israel 1. ABSTRACT Narcissus is usually approximated by means of a paraxial ray trace through the optical system in the ordinary direction of light travel An accurate calculation involves tracing real rays backwards from the detector to the reflecting surface, and back to the detector after reflection. It will be shown that the diffractive order of maximum efficiency for a diffractive optical surface is different for transmitted and reflected radiation. This precludes the use of the paraxial approximation for calculating the effect ofnarcissus. The real ray method of calculation must be used with specific orders of diffraction based on their efficiencies. keywords: diffracive optics, narcissus, thermal imaging, gratings, optical design. 2.1 Narcissus calculation LINTRODUCTION Narcissus is a problem encountered in ER imaging systems. As such systns are designed to detect thermal energy, there is an inherent problem of radiation originating within the system, which may be specularly reflected into the detector by optical surfaces. This is problematic in scanned systems with a cooled detector, which can be expected to see its own cryogenic surroundings near the center of the scan, and the system ambient temperature during the remainder of the scan period. It tends to produce a troublesome cold "spike" in the image towards the center of the field of view. Figure 1 shows a schematic representation of narcissus. The dashed lines show retro-reflected detector radiation at the center of the scan. The effect of narcissus on image quality depends on both the concentration of reflected radiation at the image plane, and the extent of variation of the reflected intensity over the scan period. The approximate method of calculating narcissus1 takes only the first factor into account The calculation is based on the fact that the beam emanating from the detector plane up to the reflecting surface is essentially identical to the beam of outside radiation imaged on the detector, except that the direction of light travel is reversed. An ordinary marginal ray trace contains information concerning both the primary characteristics of the narcissus beam, and the first-order characteristics of the optical surfaces which it traverses. It therefore contains sufficient information for calculating the size of.the spot produced on the detector by the reflected narcissus beam, to a first approximation. There is an inherent assumption that the reflective optical power of the surface responsible for narcissus is related to its transmissive power according to the difference between refractive indices on two sides of the surface. Amore accurate calculation of narcissus involves tracing an entire beam of radiation from the detector through the scanner to the surface under investigation, and then tracing the reflected beam back to the detector. This calculation is performed for a large number of rays within the envelope of the beam. A real ray trace is used, thus achieving greater accuracy. In addition, the effect of variation of the reflected radiation during the scan period may be taken into account. 2.2 Diffractive surfaces The diffractive surface here referred to (also called a kinoform surface) is a phase grating produced as a surface relief profile on an optical material2 In common with dispersion gratings, the optical characteristics are determined by two 380 / SPFE Vol O /95/$6.OO

2 factors: the grating frequency governs the diffraction angle, and thus the diffractive optical power of the surface, while the blaze determines the diffraction efficiency in the nominal (and other) orders. Telescope Scanner Detector Lens Narcissus Spot Fig. 1. Narcissus in a typical IR system A diffractive optical element (DOE) used as a lens is commonly designed to operate in the first order. In this case, for a surface ofnominal diffractive optical power F, the power at the mth order will be J m P. In the general case, for a DOE designed to operate at order m, the diffractive optical power at order m' is given by: plp (1) Figure 2 shows the effect of diffractive order on optical power in a typical first-order DOE lens (with positive nominal power). The optimal blaze for an mth-order diffractive surface is one where the phase difference at the boundary between two facets is equal to m wavelengths. This implies a depth of d = ma,0/(n i) (2) where n is the refractive index of the optical material, and is the design wavelength.3 An important variation on the above formula applies to reflective surfaces. In this case, the basic considerations are identical, but the formula for optimal depth is: I m/2, front - surface mirror d=c (3) m0/2n, back -surface mirror SP1EVo!. 2426/381

3 -1st order (virtual focus) 3rd order focus focus focus Fig. 2. Effect of diffractive order on focusing 3. MONOCHROMATIC CONSIDERATIONS In accordancewith formulas(2) and (3), it canbe seen that the order ofdiffraction of maximal efficiency for a reflected narcissus beam will differ from the design order of the nominal transmitted beam. The ratio between reflective and transmissive power of the diffractive surface depends not on the refractive index of the optical material, but on the order of diffraction, in accordance with formula (1). The approximate method of calculating the effect of narcissus is thus not applicable, and the real ray trace method must be used. The spot produced on the detector surface by narcissus is a combination of the reflections of numezous orders, weighted according to their efficiencies. The point spread function can be expressed as a sum of the point spread functions of the various orders: PSF(x,y) = 'I1m#I'mi(X,Y) (4) In practical terms the effect could be simulated by performing spot calculations at several orders in the vicinity of the order of maximal efficiency m, weighted according to their individual efficiencies nm" The order of maximal efficiency for reflected narcissus radiation can be determined from equations (2) and (3) above, noting that the first surface of a lens becomes a back-surface mirror for narcissus radiation, while the second surface becomes a front-surface mirror. An additional consideration concerns the sign of the order. A positive-power transmissive diffractive surface is profiled so that the incident wavefront undergoes a smaller retardation at the axial edge of the facet, thus producing a converging wavefront. The facets are therefore sloped outwards at their axial edges. For reflected 382 ISPIE Vol. 2426

4 radiation, facets which are sloped towards the incident wavefront at their axial edge will induce divergence in the reflected wavefront, while those sloped away from the incident wavefront will induce convergence. It is thus evident that the sign of the order of diffraction is reversed for narcissus radiation when the diffractive surface is the second surface of a lens (see Figure 3). In accordance with the above considerations, the order of maximum efficiency for narcissus radiation m is related to the design order m as follows: = ' I... n-4 m, 1stsurface 2nd surface (5) 1st surface DOE 2nd surface DOE Trasnitted radiation Reflected radiation Fig. 3. Narcissus diffractive order for 1st & 2nd surfaces The order m is not usually an integer and the reflected energy will be distributed in orders in the vicinity of m in accordance with their efficiencies. The efficiencies Tml of equation (4)can be calculated from the orders m', as though the surface proffle had been designed for reflection at the nominal order m. The shift in nominal order between the transmissive and reflective modes is m' m = where the positive sign is correct for a first-surface DOE, and the negative sign for a second-surface. Since the optical power of a diffractive surface is directly proportional to the order, this means that the optical power shift has the same value for first and second surfaces, but with opposite sign. Table 1 gives approximate values of - m for several widely used IR optical materials. Material ;-- Im -tnj ZnS 2.7 ZnSe Si Ge Tab. 1. Maximum efficiency order for narcissus radiation 4. Polychromatic systems In a polychromatic imaging system, the diffractive surface is blazed for maximum efficiency at the design order m and at a specific wavelength, and the efficiency falls off at other wavelengths. The optimal depth is given by equation (2), and the order of maximum efficiency for reflected radiation as calculated in equation (5)is correct for the nominal wavelength SPIE Vol / 383

5 2. The diffraction efficiency for the reflected radiation is a function of both order m'and wavelength. The point spread function of the reflected radiation can be written as: PSF(x,y) (6) 1m' is the efficiency of the diffractive surface for reflected radiation at order m and for wavelength.s is a spectral weighting factor which would account for the spectral distribution of thermal radiation, the spectral reflectance of the optical surface, detector spectral sensitivity, etc. In practical terms, the effect would be simulated by performing a series of spot calculations for various orders and wavelengths, taking the efficiencies lm'?,, into account. Using the scalar diffraction approxiniation,3 the efficiencies will be: 2 I itm, m) 1' (7) As a practical example of the effect of the efficiencies, let us take a first-order diffractive surface designed for maximum efficiency at lop.m, on the first surface of a germanium lens, in an 8 12p.m system. We have m = 1, A0 = 10pm, n = 4.003, and in accordance with formula (5), the order of maximum efficiency for reflected radiation at 10pm is: = ( /3.003) = The relative efficiencies of various orders at the central and extreme wavelengths are given in Table 2. 8pm 10pm 12pm Order 'f DiffraCtion, m' Tab 2. Efficiencies 11mi. forreflected radiation for a 1st-order, 1st-surface DOE on Ge In this case one would probably perform narcissus calculations for the 3rd and 4th orders at a wavelength of 8pm, and for the 2nd and 3rd orders at lopmi and 12pm. 5. CONCLUSION It has been shown that the paraxial approximation to narcissus spot size cannot be used for reflections from diffractive surfaces. The accurate method of tracing real rays from the detector to the surface and back must be used, and the calculation should be performed for one or several orders of diffraction, based on their efficiencies. The order of maximal efficiency m for narcissus radiation at the design wavelength is given by formula (5). The scalar diffraction approximation to the efficiencies for arbitrary order m' and wavelength? can be calculated according to formula (7). An example was presented showing that for a first-surface first-order DOE on germanium in an 8 12pm system, narcissus effects should be calculated for the 2nd, 3rd, and 4th orders. 384 / SPIE Vol. 2426

6 6. REFERENCES 1. James W. Howard and Irving R. Abel, "Narcissus: reflections on retroreflections in thermal imaging systems," AppL Opt. 21, (1982). 2. Kenro Miyamoto, "The Phase Fresnel Lens," J. Opt. Soc. Am. 51, (1961). 3. G. J. Swanson, "Binary Optics Technology: the Theory and Design of Multi-level Diffractive Optical Elements," Tech. Rep. 854, MIT Lincoln Laboratory (Massachusetts Institute of Technology, Cambridge, Massachusetts, 1989). SPIE Vol / 385