# Get the Gist? Using Large Language Models for Few-Shot Decontextualization

Benjamin Kane
University of Rochester
<EMAIL_ADDRESS>
&Lenhart Schubert
University of Rochester
<EMAIL_ADDRESS>

###### Abstract

In many NLP applications that involve interpreting sentences within a rich
context – for instance, information retrieval systems or dialogue systems – it
is desirable to be able to preserve the sentence in a form that can be readily
understood without context, for later reuse – a process known as
“decontextualization”. While previous work demonstrated that generative
Seq2Seq models could effectively perform decontextualization after being fine-
tuned on a specific dataset, this approach requires expensive human
annotations and may not transfer to other domains. We propose a few-shot
method of decontextualization using a large language model, and present
preliminary results showing that this method achieves viable performance on
multiple domains using only a small set of examples.

##  1 Introduction

As large language models (LLMs) improve in capabilities, work in NLP is
increasingly turning towards systems that rely on natural text as a core
representation, and that use LLMs as a foundation for reasoning Bommasani et
al. (2022); Park et al. (2023). In many cases, this requires the ability to
preserve text from an input source in a form that can readily be reused for
downstream tasks.

Figure 1: An example of decontextualization within a conversational context
(indicated in yellow). The types of edits used to produce the decontextualized
sentence – based on the scheme proposed by Choi et al. (2021) – are shown
below the original and decontextualized sentences.

However, sentences in natural corpora are often heavily dependent on the
surrounding context, making it difficult to extract a sentence directly.
Sentences may contain anaphors that refer to other entities in the context, or
they may contain discourse markers that relate to the overall structure of the
embedding document, or they may contain a variety of elided material. Yet, in
many cases it’s possible to map a sentence into a semantically equivalent form
that can be understood in the absence of context. For example, in the
conversation in Figure 1, given the previous two turns of context, an agent’s
utterance can be mapped to a form that can be readily understood without
context – allowing it to be reused by downstream tasks.

Choi et al. (2021) provide a formal definition of this task – known as
decontextualization – and decompose the process into several stages of edits.
However, this approach – based on fine-tuning pretrained coreference models
and language models – relied on the crowdsourcing of large numbers of
decontextualization annotations, which may be infeasibly expensive, and may
not transfer to other domains. A few-shot approach to decontextualization
would allow this technique to be more widely adopted in NLP system design.

In this paper, we present a few-shot approach to decontextualization that uses
an LLM to map a sentence to a decontextualized form through a series of edits.
We present preliminary results involving both automatic and human evaluations
demonstrating that this method can achieve viable performance across multiple
domains, using only a single small set of annotated examples.

##  2 Related Work

The term “decontextualization” was initially introduced by Parikh et al.
(2020), and the concept was further refined and generalized by Choi et al.
(2021). The latter relied on fine-tuning models on large amounts of annotated
data. Shin et al. (2021) demonstrated a few-shot method for deriving
decontextualized canonical forms by using constrained decoding procedure.
However, this approach is only viable within a closed domain where a grammar
for the constrained language can be created.

Decontextualization is closely related to, but not identical to, the task of
text summarization Gambhir and Gupta (2017). In summarization, a sentence need
not be rendered into a form that can stand without context; indeed, a summary
is often retrieved or generated relative to the context provided by a query.
Decontextualization, therefore, is a more constrained problem that involves
resolving complex linguistic phenomena such as anaphora and ellipsis that may
be ignored by common text summarization methods.

##  3 Method

Given a context $C=\\{c_{1},...,c_{n}\\}$ and a sentence $s$, the goal of
decontextualization is to produce a new sentence $s^{\prime}$ such that
$s^{\prime}$ is interpretable in the empty context, and carries the same
truth-conditional meaning as $s$ does given context $C$.

We propose a pipelined approach to few-shot decontextualization using the
gpt-3.5-turbo LLM111https://platform.openai.com/docs/models/overview, based
upon the categorization of possible edits described by Choi et al. (2021). Our
method, diagrammed in Figure 2, consists of several edit nodes that transform
a given sentence $s$ sequentially, each provided the context $C$ and a set of
examples.

We further decompose each edit node into several substeps (shown for the “NP”
node in Figure 2)222We found direct edit prompts to be hallucination-prone..
To ensure that each substep is accurate, we employ validator functions that
compare the input sentence to the output sentence333The same validator
functions are used for each edit node.; if the LLM cannot generate a correct
output after $N$ retries, then the input sentence is returned for that
substep. Each edit step has access to $K$ in-context examples for that edit
type444Each example contains a bracketed and an edited sentence, though the
former may be derived from the latter.. We elaborate on each component below;
full details and prompts can be found in Appendix A).

Figure 2: A diagram of our few-shot pipeline. Given an input sentence,
context, and set of examples, a sequence of edit nodes transform the input
sentence into a decontextualized sentence. Each edit node consists of several
substeps to ensure validity, and some edit nodes may also be optionally
preceded by cutoff checks.

##### Bracket Substep

We first prompt the LLM to bracket each candidate span for that edit type,
using “[” and “]” as special delimiter tokens. For example, during the “NP”
step in Figure 2, the sentence from Figure 1 may be bracketed as “Well,
mainly, I like the complex characters that [he] creates, such as Tyrion.”. The
validator function for this substep ensures that the output string is
identical to the input apart from brackets.

##### Replace Substep

Next, the LLM is prompted to replace each bracketed expression with edits of
the appropriate type. E.g., after bracketing the previous sentence, the
expression “[he]” may be replaced with “[George R. R. Martin]”. We validate
this substep by ensuring that the non-bracketed sections of the output string
match those of the input string. However, we allow for some tolerance by
thresholding based on the Jaccard similarity between the uni-grams of the
input and output sentence, excluding brackets:
$J(S_{I},S_{O})=\frac{|S_{I}\cap S_{O}|}{|S_{I}\cup S_{O}|}\geq 0.5$.

##### Completion Checks

To avoid overmodification, we allow for cutoff checks – “Check” in Figure 1 –
to be optionally introduced prior to certain edit nodes; in our pipeline, we
use cutoff checks for the “Deletion” and “Addition” stages. If the LLM
classifies a sentence as sufficiently decontextualized (given $K$ examples),
the sentence is returned and all subsequent edit nodes are skipped.

##  4 Experiments

###  4.1 Datasets

For our primary experiment, we use the Decontext
dataset555https://github.com/google-
research/language/tree/master/language/decontext created by Choi et al.
(2021). This dataset, oriented towards text summarization, consists of
sentences from the Wikipedia corpus embedded within a context paragraph, each
annotated with decontextualized sentences by up to 5 annotators.

We also evaluate whether the performance of our method transfers to a
conversational dataset, using the same example set from the Decontext dataset.
For this, we use a subset of the validation split from the Switchboard
corpus666https://huggingface.co/datasets/swda Godfrey et al. (1992).
Additional details about our data preprocessing can be found in Appendix B.

###  4.2 Baselines

We consider two baselines for each experiment, following Choi et al. (2021).
First, we consider a method repeat that simply repeats the original sentence
as an output. Second, we consider a human generated sentence, chosen by taking
the median length annotation for each data sample and using the remaining
annotations as references.

###  4.3 Metrics

We use the following automatic metrics as our primary method of comparison.
First, we report the average percentage increase in length of the
decontextualized sentences over the original sentences (Len inc.), as well as
the percentage of items where the decontextualized sentence was not identical
to the original sentences (% edited). Next, we report the percentage of items
where the decontextualized sentence was an exact match to at least one of the
gold annotations, excluding punctuation and stopwords (% match). Since some
items did not require edits, we report this score both for all items, and for
the subset of items where all gold annotations contained an edit.

Finally, we report the (SARI) score Xu et al. (2016), which allows us to
compute precision/recall/F1 scores between the output and gold annotation for
edits relative to the input sentence. We compute this metric separately for
add and delete edits (micro-averaging over all items), looking at unigrams
only and using fractional counts for items with multiple gold annotations.

###  4.4 Decontextualization Experiment

Method | Len | % | % match | SARI add | SARI del
---|---|---|---|---|---
| inc. | edit | all / edit | F1 (P/R) | F1 (P/R)
repeat | 0 | 0 | 36 / 0 | 0 (0/0) | 0 (0/0)
-check | 32 | 92 | 8 / 0 | 24 (21/28) | 8 (5/40)
+check | 24 | 91 | 10 / 5 | 31 (33/29) | 21 (15/37)
human | 24 | 78 | 44 / 30 | 55 (63/50) | 59 (61/58)

Table 1: Automatic evaluation results on the Decontext test set for each set
of decontextualizations.

We first evaluate the performance of our method on a random subset of $1000$
items (approximately 50% of the data) from the Decontext test split777We use a
subset of the data due to cost considerations.. We generate $K=20$ in-context
examples from the annotations in the development split of the Decontext
dataset, filtering for items with 1-2 context sentences and each sentence
having less than 30 words. We use an 80/20 ratio of positive to
negative888I.e., examples where no edit is necessary. examples, and a 50/50
split for cutoff nodes.

####  4.4.1 Automatic Evaluation

We generate decontextualized sentences both with (+check) and without (-check)
the cutoff check nodes in Figure 2. Results for our automated evaluation
metrics are shown in Table 1. We observe that including the cutoff check steps
in the pipeline helps avoid extraneous modifications, leading to an average
length increase that reflects human annotations, and substantially higher SARI
F1 scores.

The scores achieved by our best performing method, while significantly above
the baseline, are still well below human-level annotation; Further performance
gains can likely be achieved by improved validation and filtering of delete
edits (SARI del), which we found have quite low precision relative to recall.
However, we note that our method tends to edit a larger fraction of sentences
relative to the human annotators, likely diminishing its exact match and SARI
add scores despite not being an inherent limitation999We note that whether
more or less information in the decontextualized sentence is desirable may
depend on the particular application.. For this, we turn to a human evaluation
of the decontextualized sentences.

####  4.4.2 Expert Evaluation

| LLM | Either | Human | Sum | % valid
---|---|---|---|---|---
LLM | 6 | 4 | 1 | 11 | 74
Either | 6 | 45 | 11 | 62 | -
Human | 3 | 6 | 18 | 27 | 87
Sum | 15 | 55 | 30 | 100 |
% valid | 75 | - | 88 |  |

Table 2: Preferences between the LLM output and human annotations, as well as
overall % marked as valid, with columns/rows showing judgments of expert A/B.

Due to the wide space of possible “acceptable” decontextualized sentences, we
also ground our automatic evaluation in an expert evaluation of the generated
decontextualizations. We randomly selected 100 examples from our Decontext
test subset; given pairs of randomly shuffled candidate decontextualizations,
two of the authors annotated each pair for (a) whether each candidate is a
valid decontextualization, and (b) which candidate is preferred (allowing for
“either”, i.e., indifference).

Our results are shown in Table 2. On average, the annotators judged 74.5% of
LLM outputs as being sufficiently decontextualized, vs. 87.5% of human
annotations; interannotator agreements measured by Cohen’s kappa were 0.76 and
0.68. The preference annotations indicate that, while human annotations were
slightly preferred to LLM outputs, in the majority of cases the expert
annotators were indifferent between the two decontextualizations.

###  4.5 Conversational Transfer Experiment

One question of interest is whether extending our approach to a new domain or
application – such as extracting “gist clauses” in a conversational system
Razavi et al. (2017) – requires a set of new hand-annotated examples, or
whether the LLM’s performance using the previous set of examples transfers to
the new domain. To test this, we replicate the previous automatic evaluation
on a small annotated subset of the Switchboard corpus, using the same 20
examples from Section 4.4.

####  4.5.1 Annotation Collection

After preprocessing data from the Switchboard validation set, we randomly
select 150 sentences that have been determined by an LLM to be interpretable
within a 2-turn context window for decontextualization by three expert
annotators101010Two of the authors + a PhD student studying NLP. – see
Appendix B for more details about this procedure. Annotator agreement, which
we compute using mean pairwise Jaccard similarity between annotators over sets
of added/deleted unigrams, was significant – about 0.56 and 0.64 respectively.

####  4.5.2 Automatic Evaluation

Method | Len | % | % match | SARI add | SARI del
---|---|---|---|---|---
| inc. | edit | all / edit | F1 (P/R) | F1 (P/R)
repeat | 0 | 0 | 19 / 0 | 0 (0/0) | 0 (0/0)
+check | 34 | 96 | 5 / 5 | 27 (22/34) | 47 (56/41)
human | 8 | 84 | 44 / 37 | 62 (63/61) | 75 (77/73)

Table 3: Automatic evaluation results on the Switchboard annotated subset for
each set of decontextualizations.

We generate results for the annotated Switchboard data using the best
performing method from 4.4 (i.e., using cutoff checks); shown as +check in
Table 3. As before, the LLM tends to edit at a higher rate than human
annotators, leading to a low percentage of exact matches. However, we achieve
a comparable F1 score for SARI add as in Section 4.4, and actually achieve a
significantly higher F1 score for SARI del – potentially due to the prominence
of discourse markers and other removable content in the Switchboard sentences
relative to the Decontext sentences.

###  4.6 Qualitative Error Analysis

Of the items marked as invalid by both annotators in 4.4.2, 15 were due to
missing NP edits (typically unresolved pronouns or definite NPs); 4 were due
to failures to ADD disambiguating postmodifiers; and 2 were due to a failure
to DELETE discourse markers. On inspection of the generated Switchboard
decontextualizations, we found that missing DELETE edits were a relatively
more common form of error – likely due to containing forms of discourse
markers that were not common in the Decontext example set. Some specific
examples for both datasets are shown in Appendix C.

##  5 Conclusion and Future Work

We proposed a few-shot LLM-based pipeline for performing decontextualization
through a series of edits resembling human annotations, and presented
preliminary results showing that this method achieves reasonable performance
across multiple domains using few examples. In the future, we believe that the
performance of our method can be improved by incorporating smaller specialized
NLP models – e.g., a coreference model – as well as by experimenting with
additional edit types for more complex linguistic phenomena, such as Wh-
Question gaps or conversational implicatures.

## Limitations

Our method is based on a “pure” LLM prompting strategy, and achieves lower
performance in automatic metrics than the fine-tuned language models explored
by Choi et al. (2021). While our work aims to demonstrate that viable results
can be achieved in a few-shot setting, in settings where large numbers of
annotations are feasible to collect, it is still likely a better option to use
a model specifically trained for that task. Additionally, due to cost and
resource constraints, we show our results using a GPT-3.5 model; performance
may differ using the more recent GPT-4 model.

The automatic metrics used in our paper may be difficult to intuitively
interpret, due to the open-ended nature of the possible edits that human
annotators can make. While we ground our results for the Decontext dataset in
an expert evaluation, our results should still be considered preliminary –
further human evaluations, ablation studies, as well as user studies for
downstream tasks using the proposed pipeline are likely necessary to fully
assess the utility of this approach.

## Ethics Statement

Since this paper proposes a heavily constrained pipeline for mapping sentences
to a semantically equivalent form (borrowing from a user-provided context), we
do not believe that it presents notable ethical concerns in itself.
Nevertheless, we would suggest that applications of this method in sensitive
domains implement stricter validation functions than those used in this paper,
in order to safeguard against potential hallucinated LLM outputs.

## References

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  * Choi et al. (2021) Eunsol Choi, Jennimaria Palomaki, Matthew Lamm, Tom Kwiatkowski, Dipanjan Das, and Michael Collins. 2021.  Decontextualization: Making sentences stand-alone.  _Transactions of the Association for Computational Linguistics_ , 9:447–461.
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##  Appendix A Method Details

###  A.1 Hyperparameters

We use the gpt-3.5-turbo LLM for all generation. We use the default
hyperparameters, i.e., a temperature of 1, top p 1, frequency penalty 0, and
presence penalty 0. We use $N=2$ retries for substeps that fail validation,
and $K=20$ in-context examples for each step.

###  A.2 LLM Prompts

For each edit node in Figure 2, we show the LLM prompts that are used for
bracketing in Table 4, and the prompts that are used for replacing in Table 5.
For the cutoff checks, we use the following prompt: “Given a context and a
sentence, decide whether the meaning of the sentence can be understood without
the context. Answer “True” if the sentence can be understood without context,
and “False” otherwise.”

NP |  Given a sentence, put brackets around any personal pronouns, definite pronouns, and definite noun phrases that can be replaced with more specific expressions. If there are none, give the original sentence.
---|---
NAME |  Given a sentence, put brackets around any acronyms, nominative pronouns, or proper names that can be replaced with more specific expressions. If there are none, give the original sentence.
DEL |  Given a sentence, put brackets around any discourse markers and connectives that can only be understood in context. If there are none, give the original sentence.
ADD |  Given a sentence, insert empty brackets wherever additional modifiers should be added in order to allow the sentence to be interpretable without context. If there is no need for modifiers, give the original sentence.

Table 4: The LLM prompts that are used for bracketing at each edit node.

NP |  Given a context and a sentence, replace any bracketed expressions in the sentence with a more explicit referring expression from the context or general knowledge. If there are no bracketed expressions, do nothing.
---|---
NAME |  Given a context and a sentence, replace any bracketed expressions in the sentence with a more explicit referring expression from the context or general knowledge. If there are no bracketed expressions, do nothing.
DEL |  Given a context and a sentence, remove any bracketed expressions if they are extraneous or require context to interpret. If there are no bracketed expressions or if there is no need to make any changes, do nothing.
ADD |  Given a context and a sentence, replace any bracketed expressions (which may be empty) with additional modifiers from the context or general knowledge that make the sentence more explicit. If there are no bracketed expressions or if there is no need to make any changes, do nothing. Do not change any content except for replacing brackets.

Table 5: The LLM prompts that are used for replacing at each edit node.

##  Appendix B Experiment Details

###  B.1 Decontextualization Experiment Details

####  B.1.1 Subselection Procedure

We use the same procedure for subselecting data to use for our generation
experiments as Choi et al. (2021) for both the Decontext and Switchboard
datasets, which we reproduce here. First, we remove all examples where three
or more annotators (out of five in the case of Decontext; out of three in the
case of Switchboard) marked decontextualization as “impossible”, and then
discard any remaining annotations that mark the example as “impossible”. To
select the human annotation for comparison, we sort the annotations by length
(in raw bytes) in descending order, take the median output as the human
annotation, and use the remaining annotations as our gold references for the
automatic evaluation.

####  B.1.2 Expert Evaluation Setup

To collect expert annotations of decontextualization validity and preference,
we randomly sample 100 items from the subset for which we’ve generated
decontextualized sentences. For each item, we randomly swap the order of the
generated output and the human annotation.

Each annotator – i.e., two of the authors – annotated each item with the
following:

  1. 1.

Is candidate 1 a valid decontextualization? (I.e., is it in a form that can be
understood without context?) Answer with “y” or “n”.

  2. 2.

Is candidate 2 a valid decontextualization? Likewise, answer with “y” or “n”.

  3. 3.

What is your preference between candidate 1 and candidate 2? Answer with “1”
if candidate 1 is better, “2” if gist 2 is better, or “0” if you are
indifferent between the two.

###  B.2 Switchboard Annotation Details

Since the Switchboard corpus is a fairly noisy dataset containing annotated
transcriptions of telephone conversations, we first clean the data using
gpt-3.5-turbo. We split each full conversation into chunks of utterances such
that each chunk fits within the LLM token limit, and generate cleaned
conversations using the following prompt: “Given an annotated conversation
between two people, clean the conversation by removing all annotations and
backchannels.”. We then re-combine the cleaned blocks and split each turn in
the conversation into multiple sentences. Finally, we create sentence and
context pairs using a sliding window of size 5 over each conversation.

After preprocessing the Switchboard data, we filter all sentences and keep
those that have at least one turn of context, and that have at least 6 words.
For each item, we remove all context turns except for the 2 most recent turns.
Since there are many sentences in the dataset that cannot be decontextualized
with only the 2 most recent turns, we also use gpt-3.5-turbo to rank items by
their quality, according to the following prompt:

> Given a sentence and context sentences, provide a numerical quality rating
> between 1 (worst quality) and 5 (best quality) based on the following
> criteria:
>
> \- Whether every sentence is fluent and natural.
>
> \- Whether the sentences have interesting content.
>
> \- Whether the sentence can be understood given the provided context.
>
> Do not give an explanation. Just give a single integer between 1 and 5.

We filter out all items that have a rating of less than 3, and then randomly
select 150 of the remaining examples to annotate.

The annotators (two of the authors, and one PhD student studying NLP in the
same department) annotated each item with decontextualized sentences (referred
to in the instructions as “gist clauses”, due to the conversational setting) –
the instructions provided to annotators are shown in Figure 3.

###  B.3 Experiment Costs

We estimate that a full generation pass through $1000$ examples from the
Decontext dataset, using $20$ examples for each edit step, cost about $20 and
took about 4 hours to complete. A full generation pass through $150$ annotated
examples from the Switchboard dataset cost about $2.5, and took about 30
minutes to complete.

##  Appendix C Examples

###  C.1 Decontext dataset

Context |  A starting pistol or starter pistol is a blank handgun that is fired to start track and field races , as well as competitive swimming races at some meets .
---|---
Sentence |  Beside sporting events , they are also used in films and in TV or stage shows .
LLM |  Beside sporting events , starting pistols are also used in films and in TV or stage shows .
Human |  Beside sporting events , starting pistols are also used in films and in TV or stage shows .

Table 6: A “good” example from the Decontext dataset (exact match)

Context |  Geronimo Stilton is an Italian children ’s book series published by Edizioni Piemme of Milan , Italy , since 2000 .
---|---
Sentence |  Although the series credits its title character as the author , the idea was originally thought of by Elisabetta Dami .
LLM |  Although the children ’ s book series " Geronimo Stilton " credits its title character as the author , the idea was originally thought of by Elisabetta Dami , who is the actual author of the series .
Human |  Although the Geronimo Stilton Italian children ’s book series credits its title character as the author , the idea was originally thought of by Elisabetta Dami .

Table 7: A “good” example from the Decontext dataset

Context |  “Ms. Jackson” is a song by American alternative hip hop duo OutKast .
---|---
Sentence |  It was released on October 3 , 2000 , as the second single from their fourth album , Stankonia .
LLM |  It was released on October 3 , 2000 , as the second single of OutKast ’ s fourth album “Stankonia” .
Human |  The song “Ms. Jackson” was released on October 3 , 2000 , as the second single from the group OutKast fourth album , Stankonia .

Table 8: A “bad” example from the Decontext dataset (missing highlighted NP
edit)

Context |  On 8 May 1895 , the first North American cat show was hosted at Madison Square Garden in New York City. A female Maine Coon brown tabby , named Cosey , was entered into the show .
---|---
Sentence |  Owned by Mrs. Fred Brown , Cosey won the silver collar and medal and was named Best in Show .
LLM |  Owned by Mrs . Fred Brown , the female Maine Coon brown tabby named Cosey won the silver collar and medal and was named Best in Show .
Human |  Owned by Mrs. Fred Brown , Cosey won the silver collar and medal and was named Best in Show in the first North American cat show in May 1895 .

Table 9: A “bad” example from the Decontext dataset (missing highlighted ADD
edit)

In Tables 6, 7, 8, and 9, we provide some concrete examples of
decontextualizations generated using our method for the Decontext dataset. We
sample two examples of successful decontextualizations, and two examples of
unsuccessful decontextualizations.

###  C.2 Switchboard dataset

In Tables 10, 11, 12, and 13, we provide some concrete examples of
decontextualizations generated using our method for the Decontext dataset. We
sample two examples of successful decontextualizations, and two examples of
unsuccessful decontextualizations.

Context |  What do you think, next to salary do you rank, is the most important job benefit?
---|---
Sentence |  I would say the medical group insurance area as far as covering medical costs for yourself or family members.
LLM |  I would say medical group insurance , as far as covering medical costs for yourself or family members , is the most important job benefit after salary .
Human |  The medical group insurance area is the most important job benefit as far as covering medical costs for yourself or family members.

Table 10: A “good” example from the Switchboard dataset

Context |  I need to know how do you feel about sending an elderly family member to a nursing home?
---|---
Sentence |  Of course, it’s one of the last few things in the world you’d ever want to do, unless it’s really for their own good.
LLM |  Of course , the act of sending an elderly family member to a nursing home is considered one of the last resort options , unless it is genuinely in their best interest .
Human |  You wouldn’t want to send an elderly family member to a nursing home, unless you’re sending them to the nursing home for thier own good.

Table 11: A “good” example from the Switchboard dataset

Context |  A: What kind of hobbies do you have? B: I do a lot of cross-stitching and painting, when I do have spare time.
---|---
Sentence |  A: Really, I like cross-stitch too.
LLM |  A: Really ! I like cross - stitch too .
Human |  A: I like cross-stitching.

Table 12: A “bad” example from the Switchboard dataset (missing highlighted
DEL edits)

Context |  A: Okay, the topic was, should the United States adopt the metric system, and if so, should.
---|---
Sentence |  B: How should they do it?
LLM |  B: How should they do it?
Human |  B: How should the United States adopt the metric system?

Table 13: A “bad” example from the Switchboard dataset (missing highlighted NP
edits) Figure 3: The instructions shown to annotators for annotating
decontextualized sentences, or “gist clauses”, in the Switchboard corpus.


# The Effect of Inhomogeneous Dielectric Loading on Transmission Through a
Slot in an Infinite Thick Metallic Shield

Abdulaziz H. Haddab, and Edward F. Kuester The authors are with the Department
of Electrical, Computer and Energy Engineering, University of Colorado,
Boulder, CO 80309 USA e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Transmission through a slot in a an infinite metallic shield of finite
thickness, inhomogeneously loaded with dielectric material, is modeled using
an analytical approximation based on the slot being small compared to a free
space wavelength. Resonances of relatively broad bandwidth (Fabry-Perot) as
well as of very narrow bandwidth (Fano) are found. We show that by introducing
a gap of different dielectric constant within the slot, we can control the
separation between the broad bandwidth resonances, and the locations and
magnitudes of the very narrow bandwidth resonances. The influence of grooves
on both kinds of resonances is considered using the same analytical model. The
effect of losses in the dielectric slot is also considered, using data for
commercially available dielectrics.

###### Index Terms:

Extraordinary transmission, inhomogeneous dielectric-loaded slot, thick
metallic shield, higher-order slot modes, Fano resonances, metasurfaces.

##  I Introduction

Transmission through a perforated metallic screen where the hole is much
smaller than or comparable to the wavelength of the incident field has been a
subject of intensive study. In particular, it has been shown that strong
transmission can occur through a single slot [1, 2, 3, 4, 5, 6, 7, 8] or an
array of slots [8, 9, 10, 11, 12, 13, 14, 15] under certain conditions. Many
techniques have been used to solve for the transmission through single or
multiple slots in a thick metallic screen: a Green’s function formulation [2],
an integral equation with a mode-series expansion [1, 3, 6, 10], an equivalent
circuit admittance model [4, 5], a surface impedance model [14], and a circuit
theory model [13, 15], to mention only a representative sample of work. In the
articles cited above, there are two main mechanisms responsible for the strong
transmission—a Fabry-Perot resonance [7, 13, 15] that has a relatively broad
bandwidth and is mainly due to resonance within the slot, or what has been
called extraordinary transmission [13, 15], which is due to coupling among the
slots in an array. The latter resonances are usually of very narrow bandwidth
and have the characteristic asymmetric shape of a Fano resonance. Our previous
work on a single dielectric-loaded slot in a thick metallic screen [16, 17]
showed that in addition to previously known resonant (Fabry-Perot)
transmission based on the fundamental waveguide mode of the slot, there exist
extremely narrowband (Fano) resonances associated with the presence of higher-
order modes in the slot, rather than coupling to adjacent slots. Due to the
shape and narrow bandwidth of these resonances, the authors referred to them
as extraordinary transmission—metaphorically, since we are dealing with a
single slot and extraordinary transmission is generally understood to be an
array phenomenon.

In the above-mentioned articles, only a homogeneously loaded slot was
considered. In some articles, the problem of an inhomogeneously shaped slot,
homogeneously loaded [19] or inhomogeneously loaded [18, 20, 21] has been
formulated for purely numerical solution, but only in [21] were results
presented for such a case, the inhomogeneity being an absorbing material so
that any resonances were significantly damped. In the present study, we extend
our earlier analysis to include the effect of introducing a gap region of
different low-loss dielectric permittivity within the slot on resonances due
to both the fundamental parallel-plate TEM mode and to a higher-order mode in
the slot 111A brief preliminary version of the work presented in this paper
has been given in [22]. . Our study reveals how the thickness of the gap
within the slot, as well as the dielectric constant of the substance that
fills the gap, can control the location and magnitude of resonances. We show
that a specified resonance may be targeted and moved to another frequency
without significantly affecting neighboring resonances. This approach opens a
new degree of freedom in design that to the best of our knowledge has not been
recognized before. Our results are validated by comparison with a full-wave
numerical finite element simulation (ANSYS-HFSS) [23].

##  II Derivation of the fields

###  II-A Transmission Factor of $TF$

Figure 1: A metallic shield with an inhomogeneous slot.

Similar to what we have done in previous work [16, 17, 22], let an H-polarized
electromagnetic wave be obliquely incident at an angle $\theta$ to a single
inhomogeneous dielectric-loaded slot of width $2b$ in a perfectly conducting
metallic shield of finite thickness $a_{3}$ as shown in Figure 1. The slot
consists of three layers of different thicknesses, where the 1st and 3rd
layers of thickness $a_{1}$ and $a_{3}-a_{2}$ respectively are filled by a
material of relative permittivity $\varepsilon_{r1}$, and the layer in the
middle of thickness $a_{2}-a_{1}$ is filled by a material with
$\varepsilon_{r2}$. We assume for simplicity that the gap is symmetrically
located ($a_{3}-a_{2}=a_{1}$).

The formulation is based on an integral representation of the field external
to the screen, and a mode-series expansion within the slot, as used in [3, 16,
17, 22]. The magnetic field is expressed as:

$\displaystyle H_{x}(y,z)=$ (1) $\displaystyle e^{-ik(\alpha y-\gamma
z)}+e^{-ik(\alpha y+\gamma
z)}+\int\limits_{-\infty}^{\infty}a(\xi)e^{-ik(\sqrt{1-\xi^{2}}z+\xi
y)}\,d\xi,$ ($z>0$)
$\displaystyle\sum\limits_{m=0}^{\infty}[A_{m}e^{-ih_{m}(z+a_{1})}+B_{m}e^{ih_{m}z}]\cos{\frac{\pi
m}{2b}(y+b)},$ ($-a_{1}<z<0$)
$\displaystyle\sum\limits_{n=0}^{\infty}[C_{n}e^{-ig_{n}(z+a_{2})}+D_{n}e^{ig_{n}(z+a_{1})}]\cos{\frac{\pi
n}{2b}(y+b)},$ ($-a_{2}<z<-a_{1}$)
$\displaystyle\sum\limits_{m=0}^{\infty}[E_{m}e^{-ih_{m}(z+a_{3})}+F_{m}e^{ih_{m}(z+a_{2})}]\cos{\frac{\pi
m}{2b}(y+b)},$ ($-a_{3}<z<-a_{2}$)
$\displaystyle\int\limits_{-\infty}^{\infty}d(\xi)e^{ik(\sqrt{1-\xi^{2}}(z+a_{3})-\xi
y)}\,d\xi,\quad\mbox{ ($z<-a_{3}$)}$

where $h_{m}=\sqrt{k^{2}\varepsilon_{r1}-(\frac{\pi m}{2b})^{2}}$,
$g_{n}=\sqrt{k^{2}\varepsilon_{r2}-(\frac{\pi n}{2b})^{2}}$,
$\alpha=\sin\theta$, $\gamma=\cos\theta$ and $A_{m}$, $B_{m}$, $C_{n}$,
$D_{n}$, $E_{m}$ and $F_{m}$ are mode amplitudes. Applying the boundary
condition of continuity of $H_{x}$ and $\frac{1}{\tilde{k}^{2}}\frac{\partial
H_{x}}{\partial z}$ (where $\tilde{k}=k\sqrt{\varepsilon_{r}}$) at the
interfaces between the dielectrics within the slot ($|y|<b$) at $z=-a_{1}$ and
$z=-a_{2}$, we can obtain the following relations between $A_{m}$, $C_{n}$,
$D_{n}$, $F_{m}$, $B_{m}$ and $E_{m}$:

$A_{m}=\frac{e^{-ih_{m}a_{1}}}{T_{1m}+T_{2m}}\left[E_{m}-T_{3m}B_{m}\right]$
$F_{m}=B_{m}e^{-ih_{m}a_{1}}\left[T_{1m}-T_{2m}+\frac{T_{3m}^{2}}{T_{1m}+T_{2m}}\right]$
$-\frac{T_{3m}e^{-ih_{m}a_{1}}}{T_{1m}+T_{2m}}E_{m}$

where

$T_{1m}=\cos g_{m}(a_{2}-a_{1}),\quad T_{2m}=\frac{i(t_{m}^{2}+1)}{2t_{m}}\sin
g_{m}(a_{2}-a_{1})$ $T_{3m}=\frac{i(t_{m}^{2}-1)}{2t_{m}}\sin
g_{m}(a_{2}-a_{1})\quad\mbox{\rm and}\quad
t_{m}=\frac{g_{m}\varepsilon_{r1}}{h_{m}\varepsilon_{r2}}$

Mode orthogonality shows that modes with $m\neq n$ do not couple with each
other. We next apply the boundary condition $\frac{\partial H_{x}}{\partial
z}=0$ at the top and bottom surfaces of the metallic shield ($z=0$, $z=-a_{3}$
and $|y|>b$) and continuity of $H_{x}$ and $\frac{1}{\tilde{k}}\frac{\partial
H_{x}}{\partial z}$ at the slot ($z=0$, $z=-a_{3}$ and $|y|<b$). Using the
Fourier transform and defining $\tilde{x}^{\pm}(\zeta)=[a(\zeta)\pm
d(\zeta)]\sqrt{1-\zeta^{2}}$ (where a new variable $\zeta$ has been introduced
to avoid confusion with the variable of integration $\xi$), we arrive at the
integral equations:

$\displaystyle\tilde{x}^{\pm}(\zeta)=\frac{4\alpha(kb)^{2}\zeta}{\pi\varepsilon}\sum\limits_{m=0}^{\infty}\frac{h_{m}b\Gamma_{m}^{\pm}}{(1+\delta_{0m})}G_{m}(\zeta)G_{m}(\alpha)$
$\displaystyle+\frac{2(kb)^{2}\zeta}{\pi\varepsilon}\sum\limits_{m=0}^{\infty}\frac{h_{m}b\Gamma_{m}^{\pm}}{(1+\delta_{0m})}\int\limits_{-\infty}^{\infty}\tilde{x}^{\pm}(\xi)\frac{\xi}{\sqrt{1-\xi^{2}}}G_{m}(\zeta)G_{m}(\xi)\,d\xi$
(2)

where

$G_{m}(\zeta)=\frac{\sin(kb\zeta-\frac{\pi m}{2})}{(kb\zeta)^{2}-(\frac{\pi
m}{2})^{2}},\quad\Gamma_{m}^{\pm}=\frac{\Gamma_{2m}^{\pm}}{\Gamma_{1m}^{\pm}}$
$\Gamma_{1m}^{\pm}=\frac{e^{-i2h_{m}a_{1}}}{T_{1m}+T_{2m}}\pm
1\mp\frac{T_{3m}e^{-i2h_{m}a_{1}}}{T_{1m}+T_{2m}}$
$\Gamma_{2m}^{\pm}=\frac{e^{-i2h_{m}a_{1}}}{T_{1m}+T_{2m}}\mp
1\mp\frac{T_{3m}e^{-i2h_{m}a_{1}}}{T_{1m}+T_{2m}}$

and $\delta_{0m}$ is the Kronecker delta.

For simplicity, we now assume normal incidence ($\theta=0$), and further that
$kb\sqrt{\varepsilon_{r1}}<2\pi$; the latter condition ensures that only the
$m=$ 0, 1 and 2 modes of the parallel plate waveguide will be above cutoff and
all other modes will be evanescent (and for normal incidence, the $m=1$ mode
is not excited). Retaining only these modes in (2), we get a degenerate kernel
integral equation whose solution can be found by well-known methods [17]:

$\tilde{x}^{\pm}(0)=\frac{2kb}{\pi\sqrt{\varepsilon_{r1}}}\left[\frac{N_{0}^{\pm}}{1+\frac{2h_{0}h_{2}b^{2}q^{2}N_{0}^{\pm}N_{2}^{\pm}}{\pi^{6}\varepsilon_{r1}^{2}}}\right]$
(3)

where
$N_{0}^{\pm}=\frac{\Gamma_{0}^{\pm}}{1-\frac{kb\Gamma_{0}^{\pm}}{2\sqrt{\varepsilon_{r1}}}I_{00}}$
and
$N_{2}^{\pm}=\frac{\Gamma_{2}^{\pm}}{1-\frac{h_{2}b\Gamma_{2}^{\pm}}{\varepsilon_{r1}}I_{22}}$,
while for $kb\ll 1$ [3]:

$I_{00}\simeq
2+\frac{4i}{\pi}\ln\left(\frac{\phi}{kb}\right)-\frac{(kb)^{2}}{3}\left[1-\frac{i2}{\pi}\left(\ln(kb)-\frac{19}{12}+\gamma_{E}\right)\right]$

and $I_{22}\simeq\frac{2i}{\pi^{2}}Si(2\pi)\simeq i0.2874\ldots$, $Si$ being
the sine integral function [16]. Here, $\ln\phi=\frac{3}{2}-\gamma_{E}$ with
$\gamma_{E}\simeq 0.5772\ldots$ being Euler’s constant.

Now we can determine the magnitude of the transmission through the infinite
dielectric slot. The following integral will be defined as the (amplitude)
transmission factor ($TF$):

$\displaystyle TF$ $\displaystyle=$
$\displaystyle\frac{\int\limits_{-b}^{b}E_{y}(y,-a_{3})\,dy}{\int\limits_{-b}^{b}|E^{inc}(y,0)|\,dy}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\varepsilon_{r1}}}\left[\frac{N_{0}^{+}}{1+\frac{2h_{0}h_{2}b^{2}q^{2}N_{0}^{+}N_{2}^{+}}{\pi^{6}\varepsilon_{r1}^{2}}}-\frac{N_{0}^{-}}{1+\frac{2h_{0}h_{2}b^{2}q^{2}N_{0}^{-}N_{2}^{-}}{\pi^{6}\varepsilon_{r1}^{2}}}\right]$

having used (3) to obtain the final formula.

###  II-B Transmission Factor fundamental mode $TF_{0}$

Analogous to what was done in [16], in this section we will present a
simplified result that takes only the TEM mode into account. We then get a
transmission factor that only contains the effect of the fundamental mode
$m=0$:

$TF_{0}=\frac{1}{\sqrt{\varepsilon_{r1}}}(N_{0}^{+}-N_{0}^{-})$ (5)

The reader may notice the similarity between the formulas for the homogeneous
case [16, 17] and those for the inhomogeneous case presented in this article;
the only difference is in the definition of the $\Gamma_{m}^{\pm}$ parameters
in each case.

##  III Results

Our formula allows us to examine two different problems or geometries. When we
have $\varepsilon_{r1}>\varepsilon_{r2}$ we have what we call the gap
geometry; on the other hand, when $\varepsilon_{r1}<\varepsilon_{r2}$ we have
what we call a groove. The homogeneously filled slot showed two kinds of
resonances: broad resonances due to the TEM mode, and sharp (Fano) resonances
caused by the higher-order mode ($m=2$) [16, 17]. Below, we investigate the
effect of a gap or a groove on each type of resonance. A full description of
the simulation setup for ANSYS HFSS can be found in [16].

###  III-A Gap Geometry

####  III-A1 Fundamental-mode formula ($TF_{0}$)

We assume a shield thickness $a_{3}=4$ mm and a slot width $2b=2$ mm, with
$\varepsilon_{r1}=50$, $\varepsilon_{r2}=1$ and a symmetrically located gap:
$a_{3}-a_{2}=a_{1}$. A comparison for different gap sizes (no gap,
$a_{2}-a_{1}=1$ mm , 2 mm and 3 mm) determined from equation (5) is shown in
Figure 2. A comparison between the result of equation (5), the analytical
result for the case of a homogeneous slot (no gap) [16], and a full-wave
finite-element (FEM) simulation using ANSYS HFSS for gap size
$(a_{2}-a_{1})=2$ mm is shown in Figure 3.

Figure 2: Transmission Factor $TF_{0}$ for slot with $a_{3}=4$ mm, $b=1$ mm
and different gap size= 0, 1, 2 and 3 mm

Figure 3: Transmission Factor $TF_{0}$ of homogeneous and inhomogeneous slots:
$a_{3}=4$ mm, $b=1$ mm and gap = 2 mm

The results show that introducing a gap within the slot will shift the even-
order resonances to higher frequencies while keeping odd-order resonances
almost unchanged. Increasing the size of the gap will shift the even-order
resonances to progressively higher frequencies until they meet the next odd-
order resonances, which begin to be shifted to higher frequencies but at a
rate much slower than for the even-order resonances, eventually forming new
resonances which keep moving to higher frequencies as the size of the gap is
increased. Because the gap is introduced in the middle of the slot in our
example, the electric field of the odd-order resonances is minimum at that
location, but is at a maximum for even-order resonances as can be seen in
Figure 4 for homogeneous and in Figure 5 for inhomogeneous, which explains why
the impact of the gap on the latter is larger.

Figure 4: Electric field distribution within the slot at first (odd, f=5.2
GHz), second (even, f=10.4 GHz) and third (odd, f=15.7 GHz) resonances for the
homogeneous case $a_{3}=4$ mm, $b=1$ mm (no gap).

Figure 5: Electric field distribution within the slot at first (odd, f=5.65
GHz), second (even, f=20.2 GHz), third (odd, f=22.4 GHz) resonances for the
inhomogeneous case $a_{3}=4$ mm, $b=1$ mm and gap= 2 mm, reference figure 3.

####  III-A2 Resonances from higher order modes ($TF$)

In the previous section (also see [22]), we saw that the gap size controls the
frequency intervals between TEM resonances. In [16] and [17], we showed that
including higher order modes in the analysis of the homogeneous slot reveals
new sharp resonances that are always located above the cutoff frequency of the
second parallel-plate waveguide mode ($m=2$). We obtained an analytical
formula to locate these new resonances in the frequency domain.

In this section, we will investigate the effect of introducing a gap within
the slot on higher order mode resonances. For parameter values $a_{3}=2$ mm,
$2b=4$ mm, $a_{2}=1.3$ mm, $\varepsilon_{r1}=50$ and $\varepsilon_{r2}=20$, a
comparison between results of the inhomogeneous gap formula (II-A), the
homogeneous slot (no gap) formula from [16] and [17] is shown in Figure 6.
Also, comparisons between results of the inhomogeneous gap formula (II-A) and
a full-wave ANSYS HFSS simulation near 17.6 GHz and 27.4 GHz are shown in
Figure 7.

Figure 6: Transmission Factor $TF$ of homogeneous and inhomogeneous
symmetrically filled ($a_{3}-a_{2}=a_{1}$) slot: $a_{3}=2$ mm, $b=2$ mm,
$\varepsilon_{r1}=50$, $\varepsilon_{r2}=20$ and gap = 0.6 mm.

Figure 7: Analytical transmission Factor $TF$ of inhomogeneous and numerical
results (HFSS) near 17.6 GHz for symmetrically filled ($a_{3}-a_{2}=a_{1}$)
slots: $a_{3}=2$ mm, $b=2$ mm, $\varepsilon_{r1}=50$, $\varepsilon_{r2}=20$
and gap = 0.6 mm.

The numerical results show excellent agreement with the analytical formula for
the frequency location of the Fano resonances. There is some disagreement
between the amplitudes at the Fano resonant frequency, but this is typical for
full-wave simulations of resonances with such narrow bandwidth [16].

The results show that the higher order resonances’ locations and magnitudes
will be determined by the gap’s size and its dielectric permittivity
$\varepsilon_{r2}$. Increasing the size of the gap will always shift the
higher order mode resonances to higher frequencies. The choice of dielectric
constant will determine the cut-off frequency of the higher-order parallel-
plate waveguide mode within the gap (in our case, the second mode $m=2$) which
represents the threshold between propagation and evanescence of the mode
within the gap. So if we choose a dielectric such that the sharp resonance is
located below that threshold (cut-off frequency), then increasing the gap size
will shift the resonance to a higher frequency. If it doesn’t reach and pass
the threshold, it will be attenuated and vanish, but if it passes this
threshold, it will be sustained and keep shifting to higher frequencies as the
size of the gap is increased.

The electric field distribution of those resonances in both cases homogeneous
and inhomogeneous cases are shown in figures 8 and 9.

Figure 8: Electric field distribution (magnitude) within the slot for higher
order resonances of homogeneous case near 15 and 23.5 GHz (reference 6),
$a_{3}=2$ mm, $b=2$ mm, $\varepsilon_{r}=50$.

Figure 9: Electric field distribution (magnitude) within the slot for higher
order resonances of inhomogeneous case near 17.58 and 27.38 GHz (reference 6),
$a_{3}=2$ mm, $b=2$ mm, $\varepsilon_{r1}=50$, $\varepsilon_{r2}=20$ and
gap=0.6 mm.

It is obvious that the electric field distribution of the first higher order
resonance is disturbed by introducing the gap while the second one is almost
unchanged. The electric field components ($E_{y}$ and $E_{z}$) for the first
and second higher order resonances for homogeneous case are shown in Figures
10 and 11 respectively. The (tangential) $E_{y}$ component will be continuous
while the normal one ($E_{z}$) is not, so introducing the gap in the middle
disturbs $E_{z}$ in the higher order resonance near 15 GHz where the field is
maximum, but has almost no effect near 23.5 GHz where the field is minimum.

Figure 10: Electric field distribution (y and z components) within the slot
for higher order resonances of homogeneous case near 15 GHz (reference 8),
$a_{3}=2$ mm, $b=2$ mm, $\varepsilon_{r}=50$.

Figure 11: Electric field distribution (y and z components) within the slot
for higher order resonances of homogeneous case near 23.5 GHz (reference 8),
$a_{3}=2$ mm, $b=2$ mm, $\varepsilon_{r}=50$.

To see whether realistic materials with loss will alter the foregoing results,
we consider the following commercially available dielectric, for which
$\varepsilon_{r1}=75$ and the loss tangent is $\tan\delta=2.6\times 10^{-4}$
at 3 GHz and $\varepsilon_{r2}=35$ and the loss tangent is
$\tan\delta=2.9\times 10^{-5}$ at 2 GHz [24]. For a fair comparison, we move
the resonant frequency to around 3 GHz by modifying the slot size to $a_{3}=8$
mm and $b=16$ mm.

Figure 12: Transmission Factor $TF$ lossless case, $a_{3}=8$ mm, $b=16$ mm:
(a) Homogeneous: $\varepsilon_{r}=75$. (b) Inhomogeneous:
$\varepsilon_{r1}=75$, $\varepsilon_{r2}=35$ and gap=2 mm

Figure 13: Transmission Factor $TF$ and numercial results (HFSS), lossy case,
$a_{3}=8$ mm, $b=16$ mm: (a) Homogeneous: $\varepsilon_{r}=75(1-i2.6\times
10^{-4})$. (b) Inhomogeneous: $\varepsilon_{r1}=75(1-i2.6\times 10^{-4})$,
$\varepsilon_{r2}=35(1-i1\times 10^{-4})$ and gap=2 mm

Results are shown in Figure 12 for the lossless case and in Figure 13 for the
lossy case and compared with numerical results from HFSS. We see that the
broad TEM mode resonances are barely affected by losses within the dielectric
slot, while the sharp resonances, though still present, are changed
significantly in amplitude.

###  III-B Groove Geometry

####  III-B1 Fundamental-mode formula ($TF_{0}$)

We assume a shield thickness $a_{3}=4$ mm and a slot width $2b=2$ mm with
$\varepsilon_{r1}=1$ (air) and $\varepsilon_{r2}=50$ (symmetrically located:
$a_{3}-a_{2}=a_{1}$). With this choice, we actually form a groove in the slot,
in which we will fix the thickness of the shield (PEC) while varying the
thickness of the dielectric layer. A comparison among the result of equation
(5), the analytical result for the case of homogeneous (no groove) [16], and a
full-wave finite-element simulation using Ansys HFSS for different groove
sizes $a_{1}=1$ and $1.5$ mm (the thicknesses of dielectric layer
($a_{2}-a_{1}$) within the slot being 2 mm and 1 mm respectively) are shown in
Figures 14 and 15, where we compare with the homogeneous case that has
thickness equal to the thickness of dielectric layer in inhomogeneous case.
The electric field magnitude distribution at 10 GHz for a groove size
$a_{1}=1$ mm is shown in Figure 16.

Figure 14: Transmission Factor $TF_{0}$ of homogeneous ($a=2$ mm, $b=1$ mm and
$\varepsilon_{r}=50$) and inhomogeneous ($a_{3}=4$ mm, $b=1$ mm, $a_{1}=1$mm
(groove=1 mm), $\varepsilon_{r1}=1$ and $\varepsilon_{r2}=50$). Note: The
thickness in homogeneous case is 2mm which equal the thickness of dielectric
in inhomogeneous case

Figure 15: Transmission Factor $TF_{0}$ of homogeneous ($a=1$ mm, $b=1$ mm and
$\varepsilon_{r}=50$) and inhomogeneous ($a_{3}=4$ mm, $b=1$ mm, $a_{1}=1.5$mm
(groove=1.5 mm), $\varepsilon_{r1}=1$ and $\varepsilon_{r2}=50$). Note: The
thickness in homogeneous case is 1mm which equal the thickness of dielectric
in inhomogeneous case

The results show that introducing a groove in the slot has less effect on the
first resonance, where it slightly changes its resonance frequency and
magnitude. On the other hand, the groove significantly alters the magnitude of
the Transmission Factor at the higher resonance frequencies and away from
those resonances.

If we introduce the air groove but keep the screen thickness constant, we
actually reduce the thickness of dielectric layer, and that will alter the
ratio ($\frac{a}{b}$), which will change the magnitude of the broad resonances
that can be determined using equation (19) in [16]. We note in Figure 15 that
there appear many closely spaced ”ripples” in the HFSS results between 20 and
40 GHz. By plotting the field distribution in HFSS, we observe standing waves
at the peaks of the ripples in Figure 15. We believe that this artifact is due
to imperfect absorption of the radiation boundary at those particular
frequencies.

Figure 16: Electric field distribution (magnitude) at 10 GHz for inhomogeneous
case: $a_{3}=4$ mm, $b=1$ mm, $a_{1}=1$mm (groove=1 mm), $\varepsilon_{r1}=1$
and $\varepsilon_{r2}=50$.

####  III-B2 Resonances from higher order modes ($TF$)

In this subsection, we investigate the effect of a groove on the higher order
mode resonances. Comparison between results of the homogeneous and
inhomogeneous cases is shown in Figure 17 for $a_{3}=4$ mm, $a_{1}=0.9$ mm
(symmetrically located: $a_{3}-a_{2}=a_{1}$) and a slot width $2b=8$ mm with
$\varepsilon_{r1}=1$ (air), $\varepsilon_{r2}=50$.

Figure 17: Transmission Factor $TF$ of homogeneous and inhomogeneous
symmetrically filled ($a_{3}-a_{2}=a_{1}$) slot: $a_{3}=4$ mm, $b=4$ mm,
$\varepsilon_{r1}=1$, $\varepsilon_{r2}=50$ and groove= 0.9 mm.

As we expected, introducing the air grooves shifts all resonances to higher
frequencies as the high dielectric layer decreases in size, with magnitude
reduction in the broad resonances consistent with the prediction of equation
(19) and Table I in [16]. The comparison with HFSS results for the higher-
order resonance at 10.81 GHz for the inhomogeneous case in Figure 17 is shown
in detail in Figure 18, and the agreement is seen to be good.

Figure 18: Analytical transmission Factor $TF$ of inhomogeneous and numerical
results (HFSS) near 10.8 GHz for sysmmetrically filled ($a_{3}-a_{2}=a_{1}$)
slot: $a_{3}=4$ mm, $b=4$ mm, $\varepsilon_{r1}=1$, $\varepsilon_{r2}=50$ and
groove= 0.9 mm.

##  IV Conclusion

In this study, we have obtained an approximate analytical formula to determine
the transmission through an inhomogeneously loaded dielectric slot in a thick
metallic screen. This analytical model allows us to study two different
loading geometries (a gap and grooves) and analyze the influence on both kinds
of resonances (Fabry-Perot and Fano). By introducing a region of different
dielectric constant within the slot (gap), we can control the separation
between the broad bandwidth resonances (Fabry-Perot) and the locations and
magnitudes of the very narrow bandwidth (Fano) resonances. In addition, the
effect of creating grooves on both kind of resonances is considered, as well
as the losses in the dielectric slot. Our results were validated by comparison
with a full-wave numerical finite element simulation (HFSS).

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