1. INTRODUCTION

The upper bound ${\eta }_{\mathrm{u}}$ of diffraction efficiency [1,2] is one of the key concepts in the paraxial theory of diffractive optics [3,4]. It can be easily calculated for any fixed complex-amplitude signal, without designing the element that generates the signal, and can never be exceeded by the efficiency $\eta$ of an actually designed element. Apart from the one-point signal, ${\eta }_{\mathrm{u}}<100\%$,and $\eta ={\eta }_{\mathrm{u}}$ only for one-point and two-point signals. The upper bound is a useful concept also if only the intensity of the signal is fixed and its phase is free: the design is good only if $\eta$ is close to ${\eta }_{\mathrm{u}}$ computed for the resulting complex-amplitude distribution of the signal. Methods for finding upper bounds have been presented for discrete [5] and continuous [6] signals in the case of this phase freedom being available. Other attempts have also been developed to define the upper bound [7,8]. Finally, it should be noted that the statements made above apply only to scalar signals. In the (paraxial) electromagnetic case, where the element modulates the polarization state of incident light and the state of polarization of the signal is free, $\eta ={\eta }_{\mathrm{u}}=100\%$ is possible for many but not all signals [9].

In this paper, we present a generalization of the upper bound theorem [1] in a form that is applicable to thin paraxial-domain diffractive elements with arbitrary constraints and output fields, within scalar theory. The discussion is based on [10]. In particular, we apply the theory to elements that exhibit phase-dependent absorption and produce single- and two-point signals. Examples of such elements include transmission-type diffraction gratings and diffractive lenses with the modulated region made of a lossy material. In the special case of single- and two-point signals, i.e., interested in single or two diffraction order output, our approach actually allows not only the determination of the upper bound of efficiency but also the design of a surface profile that produces the highest possible efficiency under the given constraints.

We begin with the problem description in Section 2, where we also define the general concept of signal-relevant efficiency (SRE). In Section 3 we proceed to derive the new upper bound theorem for SRE. The theory is then applied to elements with phase-dependent loss, discussed in Section 4 on a general level. In Section 5 we apply the new upper bound theory to elements that generate a single-point signal. Finally, some conclusions are drawn in Section 6.

2. PROBLEM DESCRIPTION AND KEY DEFINITIONS

For the task ahead we require that two assumptions hold and consider these for transmission-type diffraction elements. The first assumption is that the thin-element approximation can be applied and that the scalar theory is adequate to describe the element, i.e., no space-variant polarization modulation takes place. If we denote by $U_{\text{in}}(x,y)$ the field that illuminates the element with transmission function $t(x,y)$, the field after the element is given by

 

Fig. 1. General setup under consideration. The field incident on the thin element with transmission function $t(x,y)$ is labeled as $U_{\text{in}}(x,y)$. The field $U_{\text{out}}(\tilde{x},\tilde{y})$ at the output plane of the system, where the desired field is defined only within a finite region $W$ (the signal window), can be obtained through a linear operator $L$.

Download Full Size | PDF

Let us consider optical fields within the Hilbert space ${\mathcal{L}}_2^{\mathbb{C}}({\mathbb{R}}^2)$ of functions $U\text{:}{\mathbb{R}}^2\to \mathbb{C}$ with finite energy. We thereby define the usual inner product as

Our basic task is to find a diffractive element that would produce the specific output field distribution $U_{\text{desired}}$ that lies in a confined region in the $(\tilde{x},\tilde{y})$ space, called the signal window $W$. Depending on the constraints imposed on the element, some sacrifices must nearly always be made. These constraints may arise from a variety of reasons, the most notable being fabrication restrictions imposed on the height profile. However, regardless of the imposed constraints, one can compose the output field as a combination of the desired field $U_{\text{desired}}$ with some error field $U_{\text{error}}$, along with a field outside the signal window, termed $U_{\text{freedom}}$. Let $⟨U_{\text{desired}}⟩=\mathbb{C}{U}_{\text{desired}}$ denote the one-dimensional subspace spanned by the desired field $U_{\text{desired}}$. This subspace is by definition orthogonal to its complement ${⟨U_{\text{desired}}⟩}^{\perp }$ of all functions $U\in {\mathcal{L}}_2^{\mathbb{C}}({\mathbb{R}}^2)$ with $⟨U|U_{\text{desired}}⟩=0$. The subspace ${⟨U_{\text{desired}}⟩}^{\perp }$ may again be divided into the subspaces ${\mathcal{H}}_{\text{error}}$ of functions identical to zero outside the signal window and ${\mathcal{H}}_{\text{freedom}}$ of functions identical to zero within the signal window. Thus, the Hilbert space ${\mathcal{L}}_2^{\mathbb{C}}({\mathbb{R}}^2)$ may be divided into these orthogonal subspaces:

We define the signal-relevant efficiency ${\eta }_{\mathrm{SRE}}$ to be the proportion of the incident field power that ends up in the desired output signal:

3. UPPER BOUND THEORY

It is assumed that the transmission function $t$ is limited to an (arbitrary) set of values $A_c$ so that $t(x,y)\in A_c\forall (x,y)$. The task is to find the SRE upper bound for a given desired field $U_{\text{desired}}$ under this constraint for $t$.

From Eqs. (9) and (11) it follows that the upper bound is given by the output field $U_{\text{out}}$ that will have largest projection onto the subspace spanned by the desired field. This particular output field will be denoted by $U_{\mathrm{opt}}$ and its projection is abstractly shown in Fig. 2. Therefore, by definition the upper bound for the SRE is given by

 

Fig. 2. Cross section in Hilbert space that contains the field after the grating and the desired field, represented as vectors. The projection of the output field onto the desired field is given by the normalized inner product between the two, and the size of this projection represents the SRE. The gray area shows all possible output fields, and of those the field $U_{\mathrm{opt}}$ yields the largest projection upon $U_{\text{desired}}$.

Download Full Size | PDF

The projection of the output field $U_{\text{out}}$ onto $U_{\text{desired}}$ is independent on the norm of the latter, including the limit

 

Fig. 3. Left: the point that gives the largest projection can be obtained by letting the length of $U_{\text{desired}}$ approach infinity and finding the point in the allowed distribution closest to it. Right: a property of the Hilbert space is that the projection is not influenced by applying a linear operation [Eq. (9)]. Finding the SRE upper bound is therefore equivalent to finding the $U_{\text{in}}t$, $t\in A_{\mathrm{c}}$ that lies closest to the inverse of the desired distribution located at $\underset{r\to \infty }{\lim }{\mathbf{L}}^{-1}\{r{U}_{\text{desired}}\}$.

Download Full Size | PDF

This equation cannot be directly evaluated without running through all possible permutations of $t\in A_c$. Using that the smallest norm is also the smallest after applying the linear operator, Eq. (17) can be rewritten as

From where the ideal transmission function $t_{\text{ideal}}$ will be defined as

It should be noted that the desired output field typically has a constant phase factor ${\phi }_c\in [\mathrm{0,2}\pi )$ that can be chosen arbitrarily, i.e., $e^{i\phi }{U}_{\text{desired}}$. For the upper bound the choice of ${\phi }_c$ does only matter if neither $A_c$ nor the phase values of $t_{\text{ideal}}$ form a rotationally symmetric pattern. In that case the upper bound for the SRE should be evaluated for all ${\phi }_c$ so that the largest resulting SRE will represent the upper bound.

A flowchart to find the upper bound is shown in Fig. 4. To summarize, first the desired signal distribution is defined in the output plane, along with the linear invertible operator that propagates the field after the transmission function to the output plane. The second step is to define the constraints $A_{\mathrm{c}}$ imposed on the transmission function and compute $t_{\text{ideal}}$ with Eq. (19). The last step is to compute the “optimal” transmission function with Eq. (20) so that the upper bound of the signal relevant efficiency can be determined by inserting $t_{\mathrm{opt}}$ into Eq. (14).

 

Fig. 4. Flowchart describes how to obtain the upper bound for the SRE.

Download Full Size | PDF

In the specific case that the input is a plane wave ($U_{\text{in}}(x,y)=1$), the material is phase only ($A_c\text{:}|t(x,y)|=1$) and the linear operator is the Fourier transform, Eqs. (19) and (20) can be rewritten as

4. ELEMENTS WITH PHASE-DEPENDENT LOSS

As an example of constraints $A_{\mathrm{c}}$ we consider a diffractive element with a periodic surface profile $h(x,y)$ and refractive-index distribution

The amplitude modulation and the phase delay are now coupled and we have the constraint

 

Fig. 5. Representation of the transmission constraint $A_{\mathrm{c}}$ for elements with complex refractive index $n=1+\mathrm{\Delta }n+i\kappa$ with (a) $\kappa /\mathrm{\Delta }n=0.1$ and (b) $\kappa /\mathrm{\Delta }n=0.8$. The allowed transmittance values are located on the spirals.

Download Full Size | PDF

5. EXAMPLE: LOSSY SINGLE-ORDER ELEMENTS

Let us now consider thin two-dimensionally periodic diffractive elements (gratings) illuminated by a unit-amplitude plane wave $U_{\text{in}}(x,y)=1$. Assuming that the output plane is at infinity and considering the paraxial case, we have the following direct and inverse relations between the field immediately after the element and in the Fourier domain:

Here we have normalized (without losing generality) the periods in $x$ and $y$ directions equal to unity.

We assume in particular that the desired output field consists of only a single diffraction order:

In the examples below we look for grating profiles that maximize the efficiency of diffraction order $(\overline{m},\overline{n})=(\mathrm{1,0})$; hence $t_{\mathrm{opt}}(x,y)=t_{\mathrm{opt}}(x)$.

Figure 6 shows a visualization of how $t_{\mathrm{opt}}$ is obtained for the constraint Eq. (29) with $\varphi \in [\mathrm{0,2}\pi )$ and $\kappa /\mathrm{\Delta }n=0.2$. In this figure the black lines connect the circle at infinity to the closest point that lies closest to it in $A_{\mathrm{c}}$. Note that the two black lines right next to the “gap” are running (almost) parallel, which means that they arrive at almost the same angle from the distribution at infinity. There are no lines in between because there is no point in the transmission constraint $A_{\mathrm{c}}$ that lies close enough to connect to the distribution at infinity. As a result, the optimum phase profile has no values in the range $1.5\pi \lesssim \varphi <2\pi$, where the absorption is highest; these are replaced by zero phase and thereby a fully transparent “slit” in the lossy grating results. Inserting the found $t_{\mathrm{opt}}$ into Eq. (34) yields a ${\eta }_{\mathrm{SRE}}^{\max }={\eta }_{(\mathrm{1,0})}=0.44$ in this particular case.

 

Fig. 6. Determination of $t_{\mathrm{opt}}(x)$ for $(\overline{m},\overline{n})=(\mathrm{1,0})$. (a) The constraint $A_{\mathrm{c}}$ under consideration with $\kappa /\mathrm{\Delta }n=0.2$. (b) Finding the value of $t_{\mathrm{opt}}$ by search of the point closest to $A_{\mathrm{c}}$ for a single point $x=0$ in the circle $r\exp (i2\pi x)$, $r\to \infty$: the black line connects these two points. (c) Values of $t_{\mathrm{opt}}(x)$ constructed for all $x$. (d) The resulting phase profile ${\varphi }_{\mathrm{opt}}(x)$ used to define $t_{\mathrm{opt}}(x)$ by inserting into Eq. (29).

Download Full Size | PDF

We also considered other constraints, such as a phase-only element with $A_c$ constrained by $|t(x,y)|=1$, a quantized phase element with $A_{\mathrm{c}}=[1,i,-1,-i]$, and a binary-amplitude element with $A_{\mathrm{c}}=[\mathrm{0,1}]$. The construction of $t_{\mathrm{opt}}$ in these cases is illustrated in Fig. 7. Let us first look at the phase-only constraint, which leads to the well-known result of 100% efficiency with a triangular profile. From the SRE computation point of view, this profile is obtained because for a point at infinity under a certain angle, the closest point in the constraint set lies at the same angle.

 

Fig. 7. Designs with different types of constraints. Left: phase only. Middle: quantized phase. Right: binary-amplitude. Top row: retrieval of $t_{\mathrm{opt}}$. Bottom row: resulting phase/amplitude profiles.

Download Full Size | PDF

When the constraint is that of a four-level quantized phase, $A_{\mathrm{c}}=[1,i,-1,-i]$, the results in the middle of Fig. 7 are obtained. The gaps that are visible in the top middle plot are a mathematical consequence of the projection from infinity. The result is the staircase profile shown in the middle below plot. The binary-amplitude constraint is shown on the right; the well-known binary profile with 50% fill factor is obtained from the SRE considerations.

When the phase-only constraint of the transmission function is quantized to allow $Z$ equally spaced levels, the highest possible efficiency of (off-axis) diffractive elements can be approximated as [12]

 

Fig. 8. Phase-only grating with various quantized phase levels $Z$. The dots denote the efficiencies computed by the SRE theory for the indicated number of phase levels. The line is given by Eq. (36).

Download Full Size | PDF

Finally, we return to the case of phase-dependent loss and consider systematically various levels of absorption $\kappa /\mathrm{\Delta }n$. In Fig. 9, the computed results show the efficiency of the gratings obtained by maximizing the SRE. The blazed profile represented the effect of phase-dependent absorption in the efficiency of a blazed grating with a linear phase in the range $[\mathrm{0,2}\pi ]$. The efficiency of such a grating approaches zero at high levels of absorption, and becomes inferior to the efficiency (10.13%) of a binary-amplitude grating with a 50% fill factor when $\kappa /\mathrm{\Delta }n>0.48$, where the blazed and binary profile have equal efficiency. The efficiencies obtained by the SRE theory are above those of either the triangular grating or the binary amplitude grating for all finite levels of absorption.

 

Fig. 9. Effect of absorption in the efficiency of gratings with one-point signal. Compares the computed profile obtained by SRE theory with the gratings with standard triangular surface profile and binary amplitude gratings.

Download Full Size | PDF

Figure 10(a) shows some of the phase profiles optimized by the SRE theory. At small levels of absorption the results are of the type already seen in Fig. 6, and the width of the transparent gap increases with $\kappa /\mathrm{\Delta }n$. With higher levels of absorption the optimum profile starts to lose its sawtooth character and finally approaches the binary case (leading to a binary-amplitude transmission function) in the limit of high absorption.

 

Fig. 10. Phase and amplitude transmission profiles of DOE from SRE theory for selected levels of absorption.

Download Full Size | PDF

6. CONCLUSIONS AND OUTLOOK

We have introduced the concept of signal-relevant efficiency and presented a procedure to determine it for an arbitrary signal and any constraint set for the complex transmission properties of the associated diffractive element. In particular, we applied the theory to gratings with a single diffraction order, establishing that the SRE upper bound theory provides the well-known results for several standard constraints. In addition, we demonstrated that the SRE concept can be used to derive optimal grating profiles in the presence of phase-dependent loss.

The upper bound theory provides significant insight into diffractive optical element (DOE) design and can be used to reduce the complexity of these designs. We will investigate and discuss the DOE design for DOEs with phase and amplitude constraints in forthcoming papers.