2.1. MLDOE Design in Dual-Infrared Waveband
The MLDOE consists of different optical materials with different dispersion characteristics, and the separated double-layer DOE (DLDOE) is the mostly common structure of MLDOE. It consists of two single layer DOEs, which are closely stacked together with a thin air gap between them, in which the complex amplitude transmittance functions are
t1(
x,
y) and
t2(
x,
y), respectively. As shown in
Figure 1, the incoming and outgoing fields for each layer are
,
and
,
for each layer, separately.
Since the two layers are closely attached, it can be considered that the outgoing field
of the first layer for DLDOE does not propagate through the space, and directly becomes the incoming field
of the second layer, as
. Based on the scalar diffraction theory, the total transmittance of the DLDOE can be obtained as [
9]
From Equation (1), we can see that the DLDOE that are closely attached can be regarded as one DOE, and its equivalent transmittance is equal to the product of the transmittance of each single-layer DOE. Similarly, results can be derived for DOE with more than two layers. Therefore, the DLDOE can replace the traditional single-layer DOE in D-RHIOS.
In some special cases, the ambient temperature of the two substrate materials of the DLDOE is different, but in most cases, processes covering design, manufacture and assemble are always at the same ambient temperature. Considering the influence of ambient temperature on diffractive micro-structure heights is much greater than that of the longitudinal dimensions, the effects of the diffractive micro-structure heights on its diffraction efficiency caused by ambient temperature is more obvious than that of the longitudinal dimension changes [
10]. The micro-structure height for each layer of DLDOE is ten times or more that of the traditional single-layer DOE, so the micro-structure heights of the DLDOE and the refractive index of the base material are changing with ambient environment, which will significantly affect its diffraction efficiency, resulting in the inability to achieve high-quality imaging at different ambient temperatures. When the ambient operating temperature changes, the surface profile of the DLDOE changes, as shown in
Figure 2.
In
Figure 2, the red area in the left indicates that when the ambient temperature is higher than the designed temperature, the micro-structure height for the first layer of DLDOE expands from the theoretical value
H01 to
Ha1, and the second layer of DLDOE expands from the theoretical value
H02 to
Ha2. Similarly, the blue area in the right indicates that when the ambient temperature is lower than the designed temperature, the micro-structure height of the first layer of DLDOE shrinks from the theoretical value
H01 to
H′
a1, and the second layer of DLDOE shrinks from the theoretical value
H02 to
H′
a2.
Here, in order to discuss the effects on diffraction efficiency caused by ambient temperature changes of the DLDOE, the effects of the manufacturing errors and the incident angle are not considered. In addition, it is assumed that the ambient temperature of DLDOE changes slowly, regardless of the temperature gradient effects. Then, based on the scalar diffraction efficiency, phase delay for DLDOE is expressed as [
11]
where
Φi is the designed phase delay,
H0k is the designed micro-structure height of the
kth layer of DLDOE,
nk is the refractive index of the substrate material of the
kth layer of DLDOE,
n0 is the refractive index of the medium between the two layers, usually it is with air gap, as
n0 = 1.
After some derivations, we can get the real phase delay caused by ambient temperature, expressed as
where,
represents the temperature coefficient of refractive index of the substrate materials corresponding to the
kth layer of DLDOE, and
represents the temperature coefficient of refractive index of air. When the ambient temperature changes slightly, the heights of the micro-structure of the
kth layer can also changes. The ratio
of this change to the design value of the micro-structure heights is called the linear expansion coefficient, which means the change of the
kth layer of DLDOE in the ambient temperature.
When we use DLDOE in dual-infrared waveband, the optimal design of the bandwidth-integrated average diffraction efficiency (BIADE) of dual-band DLDOE in different wavebands is
ωi = 0.5, achieved by using the bandwidth integral average diffraction efficiency (BIADE) weight factor of different wavebands. The BIADE of the dual-band DLDOE is [
12]
In Equation (4), , and ω1 and ω2 are the weighting factors of the BIADE of the two wavebands, respectively, , and , represent the minimum and maximum wavelengths of the two wavebands, respectively. Using the BIADE weighting factors of different wavebands, the intrinsic relationship between the design wavelength and the BIADE of the dual-band DLDOE is established, and the designed micro-structure heights can be calculated to ensure the maximum diffraction efficiency.
What’s more, for D-RHIOS, the real optical transform function (OTF) is affected by the BIADE, expressed as [
13]
where,
stands for the OTF when the diffraction efficiency reaches 100%, and it can be obtained after optimization form the design software such as ZEMAX or CODE V. For Equation (5), we can see that the OTF of D-RHIOS is the function of both diffraction efficiency and OTF in theory. What’s more, compared with the accurate result, the approximate design results and accurate design results differ greatly in the low-frequency, and the high-frequency is very close.
When the micro-structure height calculated, caused by ambient temperature, the real micro-structure height
is
According to Equation (6), the diffraction efficiency of the first diffraction order can be expressed as
When the ambient temperature changes, the actual diffraction efficiency expression of the DLDOE is obtained as
The BIADE for DLDOE directly affects the OTF of D-RHIOS. When the ambient temperature changes, as the diffraction efficiency of the DLDOE changes, the BIADE of the corresponding waveband changes at the same time, so that the imaging quality of the system deviates from the design value. In the design waveband, the actual BIADE after the ambient temperature changes is expressed as
From Equations (7) and (8), the actual diffraction efficiency and BIADE for DLDOE can be calculated under different working wavebands and ambient temperatures. This is of great significance for the image quality evaluation of the D-RHIOS using DLDOE at different ambient temperatures.