Jinan Zeidan
email: jinan.zeidan@univ-amu.fr

Are Comparisons Luxuries? Subjective Poverty and Positional Concerns in Indonesia

Keywords: Indonesia, subjective well-being, external e¤ects, positional concerns. JEL Classi…cation : O12, I30, I31

We explore (i) the usual determinants of happiness in Indonesia, with a special focus on the role of various measures of absolute income; (ii) the presence of relativistic concerns or positive external e¤ects in shaping attitudes to subjective well-being; and (iii) whether this potential e¤ect changes sign with income level. Additional evidence o¤ered by our investigation relates to the e¤ect of past income levels as well as to that of aspirations. In line with other literature from poor contexts, we …nd that the subjective well-being of Indonesians is positively a¤ected by the comparison with the income of people around them. This positive in ‡uence is unambiguously more important for the poor than for the rich. This pattern is consistent through di¤erent measures of well-being and holds also when accounting for past income levels, and lagged income expectations.

Introduction

There is a growing acceptance that individuals'welfare heavily depends on their relative position in society. Recent evidence from the happiness literature attests that the income of others a¤ects an individual's own subjective well-being (see, e.g., [START_REF] Clark | Relative Income, Happiness, and Utility: An Explanation for the Easterlin Paradox and Other Puzzles[END_REF]. Another strand of the literature points to the presence of relative concerns with respect to other domains in life such as children's education, family's housing and healthcare, vacations or even the value of a car (e.g., [START_REF] Alpizar | How much do we care about absolute versus relative income and consumption[END_REF]; Guillen-Royo, 2011). While economic interest in positional concerns appears to be increasing, most of the available evidence has been for developed countries, where the income of a reference group negatively a¤ects subjective well-being. This negative in ‡uence underlies the so-called relative deprivation and status e¤ects, re ‡ecting envy and jealousy. However, evidence regarding relative concerns in developing countries has been limited, and results o¤ered are somewhat mitigated. While a number of studies are suggestive of negative comparison group e¤ects (e.g., [START_REF] Carlsson | Keeping up with the Vaishyas Caste and relative standing in India[END_REF][START_REF] Fafchamps | Subjective welfare, isolation, and relative consumption[END_REF][START_REF] Knight | Income, aspirations and the Hedonic Treadmill in a poor society[END_REF], other evidence points in a di¤erent direction, indicating a positive impact of income distribution on subjective well-being (e.g., Ravaillon and Lokshin, 2010; [START_REF] Kingdon | Community, Comparisons and Subjective Well-being in a Divided Society[END_REF].

The positive e¤ect of reference group income might arise through various mechanisms. Firstly, according to the theory of anticipatory feelings, others' income serves as piece of information used to form expectations about one's own perspectives. In this sense, the presence of better-o¤ individuals among the peers can be interpreted as a positive "signal" with respect to likely future income ‡ows, and thus lead to greater satisfaction [START_REF] Caplin | Psychological expected utility and anticipatory feelings[END_REF][START_REF] Hirschman | The changing tolerance for income inequality in the course of economic development[END_REF][START_REF] Senik | When information dominates comparison: learning from Russian subjective panel data[END_REF]. Other theories indicate how poor individuals might share in economic gains to friends and neighbors. The uninsured risks facing poor people and the odds of falling into permanent penury lead to various arrangements for mutual support or risk-sharing. 1 Positive externalities may also be present for the poor living in relatively well-o¤ areas since communities can be important institutions for providing employment opportunities and local-public goods [START_REF] Mansouri | Community-based and -driven development[END_REF]Jalan and Ravaillon, 2002). Positive external e¤ects may occur dependently or independently of one's current income, this being for instance the result of better personal security in the presence of uninsured risks. Finally, positive relative concerns can be explained as the result of altruistic or humanitarian concern for other's well-being. This is especially true in rural and communal societies characterized by kinship relations and high intra-group solidarity (Akay et al., 2012).

Positive and negative relative concerns may coexist. Depending on the context, one may prevail over the other. There is a widely held belief that the positional concerns are more pronounced for the rich than for the poor. In this study, we aim to test the presence and the sign of relative income concerns in Indonesia where, despite the progress made in reducing poverty, many people remain poor and vulnerable. Out of a population of 234 million, more than 32 million Indonesians currently live below the poverty line and roughly half of all households remain clustered around the national poverty line set at the equivalent of $22 per month 2 . Income inequality is also an important issue in Indonesia with a Gini Index around 0.39, with it being greater in urban areas than in rural areas. 3 The richness of our survey and the spacial coverage of the sample allows us to shed light on di¤erent aspects of the relationship between reference group income and subjective well-being. We primarly intented to explore (i) the usual determinants of happiness in Indonesia, with a special focus on the role of various measures of absolute income; (ii) the presence and the sign of relative income e¤ect; and (iii) whether this potential e¤ect changes sign with income level. Additional evidence o¤ered by our investigation relates to the e¤ect of past income levels as well as to that of aspirations which may in ‡uence the extent to which individuals are satis…ed with their actual levels of income. We …nd that the subjective well-being of Indonesians is positively a¤ected by the comparison with the income of people around them. This positive in ‡uence is unambiguously more important for the poor than for the rich. This pattern seems to be consistent through two di¤erent measures of well-being and holds also when accounting for past income levels or more complete geographic e¤ects. Further, our results held when lagged income expectations are accounted for.

An important issue in our study regards the de…nition of the reference group. The reference group is often de…ned as the group of people that an individual compares him/herself with or whose standards he uses to evaluate himself. 4 Only a limited number of studies have access to the reference group as directly de…ned by respondents themselves. 5 Obviously, whenever it is not the case that information on individuals'reference groups has been collected in the data, the common practice implies that the researcher creates the reference group based on usual grouping of "similar" others. We follow this practice.

However, this suggests the presence of a potential bias arising from created reference group. Our results are robust to changing reference groups based on various groupings of both individual and geographical characteristics.

The paper develops as follows. We start by reviewing the literature on the role played by relative concerns on subjective well-being in developing countries in section 2. In Sections 3 and 4 respectively, we describe our data and empirical approach, while in Section 4 we present our …ndings. Section 5 proposes some additional checks. In the …nal section, we summarise and conclude.

Literature review: Income comparisons in developing countries

There is a growing literature questioning the presence of relative income concerns in developing countries. However, there is still paucity of panel data evidence or large samples. Most of the literature appeals to cross-country data, or small samples from rural areas. Results in the literature so far di¤er across countries and following the reference group chosen as well as the methodology used (Hypothetical preferences, direct income comparisons, or direct subjective comparisons of one income vis-à-vis neighbours, parents, etc.). Interestingly, relative concerns seem to have di¤erent relevance in developing and developed countries. Generally, studies regarding developed settings conclude to a negative impact of reference group income whereas those in less developed settings …nd it to be positive, or non signi…cant.

The …rst evidence to …nd positive e¤ect of reference group income on subjective wellbeing was [START_REF] Senik | When information dominates comparison: learning from Russian subjective panel data[END_REF] who test for social e¤ects in self-reported happiness in Russia. The author regresses self-reported life staisfaction on one's own income and the mean income of the area of residence. In marked contrast to the developed-country litearture, she founds evidence of positive external e¤ects of neighbors' income, rather than the negative e¤ect predicted by relative concerns.

The dissimilarity between the two contexts is further established by [START_REF] Corazzini | Reign in hell or serve in heaven? A cross-country journey into the relative vs absolute perceptions of well being[END_REF] who use hypothetical preferences to compare the degree of relative income concerns across four low-income countries (Bolivia, Brazil, Kenya and Laos) and four high-income countries (Italy, Sweden, Switzerland and the UK). They conclude to relative concerns in the perception of well-being, with the strength of relativism being higher for respondents in high-income countries.

Our literature review will be presented in the following three subsections to re ‡ect results from three di¤erent continents: Latin-America, Africa, and Asia.

Evidence from Latin-America

In a continent-wide investigation, [START_REF] Graham | Inequality and happiness: Insights from Latin America[END_REF] found negative e¤ect of regional income on individual well-being across 18 Latin American countries. Similarly, [START_REF] Rojas | Relative income and well-being in Latin America[END_REF] appeals to data from the 2007 Gallup survey in 19 Latin American countries. The author provides evidence that both ladder of worst to best possible life and satisfaction with standard of living rise with the log of own income but fall with the log of the average income in the reference group as de…nied by age, sex and country combinations. Moreover, in the case of satisfaction with standard of living, empirical results show equal and opposite coe¢ cients of the two variables, indicating that an increase in everyone's income would leave no-one in Latin America better o¤. [START_REF] Castilla | Subjective well-being and reference-dependence: Insights from Mexico[END_REF] considers Mexican data including subjective assessments of income adequacy and income satisfaction. Positional concerns are introduced by considering these two subjective welfare measures as a function of both the respondents'absolute level of income and either the income of a reference group, income aspirations by that stage of life, or the level of income the individual had three years earlier. Results indicate that the likelihood to be satis…ed with one's income level increases with own expenditures, but falls with income rating relative to the three reference points 6 . Assessments of income adequacy yield to similar results, except for past levels of income, which do not signi…cantly correlate with the degree to which people consider their current income adequate. Interestingly, these …ndings vary across di¤erent segments of the income distribution with subjective welfare of the poor being more in ‡uenced by the inability to reach income aspirations, whereas income comparisons relative to others is a more prominent concern among the non-poor respondents. Guillen-Royo (2011) analyzes small sample data from seven communities in Peru, and shows that reference group consumption is negatively related to satisfaction with children's education, family's housing and clothes. This is consistent with relative income e¤ects for these domains. However, this is not the case for family healthcare and food domains, for which neither family's own level of consumption nor the average level of consumption in the community signi…cantly in ‡uenced the consumption adequacy assessment. Finally, using small sample data from Venezuela in 2005, Kuegler (2009) …nds a higher life satisfaction of Venezualans who consider themselves as better o¤ than their own siblings. At the same time, this relative wealth e¤ect is stronger for respondents with above-median incomes and those who work in higher-rank professions.

Evidence from Africa

Akay et al. (2012a) use two survey experiments from Northern Ethiopia to test whether there are positional concerns among extremely poor people in the dimensions of income per se and income from an aid package. The authors consider hypothetical preferences where individuals selected from two villages to live in (or two aid packages to choose from) and where their own as well as others'income di¤ered between the two scenarios. The low estimated marginal degree of positional concern (compared to what is found in the literature) suggests that the choices of most Ethiopian subsistence farmers are based on absolute income alone. In a previous work using the same data, Akay and Martinsson (2011) also …nd no signi…cant e¤ect of reference group income on life satisfaction. Bookwalter and Dalenberg (2010) explore a large household survey from South Africa in early 1990s. The authors …nd no signi…cant impact of cluster-level wealth for Whites, but a positive and signi…cant e¤ect of others'income for non-Whites. Di¤erent results, more consistent with income comparisons, are nevertheless obtained when other measures of relative accomplishments are analyzed. Notably, failing to be as wealthy as one's parents has a considerable negative impact on an individual's own assessment of wellbeing. Bookwalter and Dalenberg (2010) stress on the fact that the traditional emphasis on geographic, age, and/or educational peers may not provide the best reference group for comparisons. [START_REF] Ravallion | Who cares about relative deprivation[END_REF] appeal to cross-sectional data from the 2004 Malawi Integrated Househlod Survey administrated under the Living Standards Measurement Study (LSMS) to test the presence of a relative deprivation issue in one of the world's poorest economies. Self-rated life satisfaction is …rstly regressed on objective measures of economic welfare (meausred by own consumption expenditures by person), and an estimate of comparison-group income (given by the mean consumption in the area of residence excluding the respondent), as well as an interaction term between own consumption and that of the reference group. Their results do not suggest that comparisons are an important concern for the majority of the Malawians, and certainly not for the poor. Indeed, unlike evidence from developed countries, the study concludes to a positive external e¤ect of comparison-group income on an individual's life satisfaction, although these in ‡uences are largely con…ned to rural areas. Secondly, the authors regress life satisfaction on a measure of self-assessed economic welfare derived from respondents'answers to the question: "Imagine six steps, where on the bottom, the …rst step, stand the poorest people, and on the highest step, the sixth, stand the rich. On which step are you today? ", as well as their assessment of the economic welfare of their neighbors and their friends. Results are similar to those obtained when modeling individual life satisfaction as a function of the objective measures of own and local neighbourhood consumption. Indeed, positive external e¤ects remain for the rural and the poorest groups whereas subjective assessment of friends'economic welfare are negatively correlated to own self-assessed economic welfare amogst those relatively well-o¤ and living in urban areas. [START_REF] Kingdon | Community, Comparisons and Subjective Well-being in a Divided Society[END_REF] analyze the role of relative income in South Africa and …nd evidence of negative relative income e¤ects within race groups. However, similar to [START_REF] Senik | When information dominates comparison: learning from Russian subjective panel data[END_REF] and Ravaillon and Lokshin (2010), they conclude to positive relative income e¤ects within neighborhoods. Finally, [START_REF] Kenny | Does development make you happy? Subjective well-being and economic growth in developing countries[END_REF] uses data from a survey of 566 households in Tanzania, in which respondents report their actual income grouping as well as the amount of income that they see as necessary to be wealthy. The average income of the area appears to be one key determinant of what people de…ne as whealthy income.

Results from Asia

A considerable amount of the literature in Asia has been dedicated to China where evidence is consistent with the presence of income comparisons in developing countries. Using data from 31 Chinese cities, [START_REF] Smyth | Inequality and Happiness in Urban China[END_REF] found that an individual's happiness is negatively related to monthly income in the city where he/she lives, accounting for own income. Akay et al. (2012b) examine the relative concerns of rural-to-urban migrants in China and …nd that results depend on the reference group chosen. Indeed, the subjective well-being of this group is negatively in ‡uenced by the income of other migrants and workers of home regions while a positive, "signal", e¤ect emerges vis-à-vis urban workers. In addition, [START_REF] Gao | Job satisfaction and relative income in economic transition: status or signal? The case of urban China[END_REF] show that job satisfaction is negatively correlated with reference group income de…ned as either the predicted income of people an individual de…nes as similar, or average income in the …rm in which he/she works. [START_REF] Knight | Income, aspirations and the Hedonic Treadmill in a poor society[END_REF] appeal to Chinese data from the 2002 CHIP national household survey where, besides direct questions on subjective wellbeing, people were asked who individuals considered as their reference group. The responses to this later question show that comparisons in China are local inasmuch as 70% of respondents consider their village as their reference group. The main conclusion of the paper is however that, controlling for both own and village income levels, those who declare that their own income was much above the village were signi…cantly happier. Using the same dataset, Knight and Gunatilaka (2010a and 2010b) highlight the relevance of relative income rather than absolute income as well as the importance of varying reference groups. However, [START_REF] Knight | Subjective well-being and its determinants in rural China[END_REF] …nd, for relatively poor rural Chinese households, that their happiness is positively correlated with the income of other rural households. Asadullah and Chaudhury (2012) use data from rural Bnagladesh to test whether relative income position matters for individual well-being, and whether its e¤ect dominantes that of absolute income. Their results show that conditional on own household income, individuals who report their wealth to be lower than their neighbours in the village also report lower satisfaction with life. The strength of this relative e¤ect depends on the individual's own characteristics and wealth, being stronger for rich respondents. Likewise, those living in villages with higher inequality are likely to be worse o¤. However, when compared to the e¤ect of absolute income, these e¤ects remain modest. [START_REF] Fafchamps | Subjective welfare, isolation, and relative consumption[END_REF] …nd that consumption adequacy in Nepal rises with own wealth and falls with reference group consumption, de…ned as the mean or median consumption of other households living in the same ward as the respondent. Relative income is thus an important predictor of subjective welfare even in mountainous villages of Nepal where households are still isolated from modernity and other urban in ‡uences. [START_REF] Carlsson | Keeping up with the Vaishyas Caste and relative standing in India[END_REF] look at hypothetical preferences over di¤erent absolute and relative income situations in India. People are asked to choose between two hypothetical income outcomes where one choice has greater absolute return while the other is more propitious in relative terms. The authors attribute around half of the e¤ect of income on well-being to some kind of status or relative income concerns, and highlight the resemblance between this …gure and that established in rich countries. They also note that respondents from low social status seem to be more sensitive to relative income. Cojocaru (2010) appeals to cross-sectional data from the 2007 LSMS in Tajikstan and …nds that individual's well-being is strongly and negatively in ‡uenced by their self-positionning in terms of household welfare vis-à-vis neighbours, conditional on the household's own consumption expenditures. However, regional income seems to have no signi…cant e¤ect on one's own life satisfaction and the author suggests this result may be a consequence of choosing an inaccurate reference group.

Data

We use the 2007 round of the Indonesian Family and Life Survey (IFLS). This is a comprehensive, nationally representative survey of households that was administered by the RAND corporation in collaboration with the University of Indonesia. The survey was …rst …elded in 1993 (IFLS1) and later waves were conducted in 1997, 2000 and late 2007. The last wave (2007/2008) is the only wave that provides individual data on the perception of happiness and will therefore be used in a cross-section study. A section of the adult questionnaire asks respondents about their self-evaluations of their own standard of living and their happiness, which are used for the purpose of this study. Our measure of overall well-being is provided by answers to the happiness question contained in the adult questionnaire. This question is: "Taken all things together how would you say things are these days -would you say you were (1) very happy; (2) happy; (3) unhappy; (4) very unhappy?". Our sample consists of 24,154 observations, with the majority of the sample being happy (85% gave (2)). Slightly less than 7% were very happy; 8.43% gave ( 3) as their answer and only 0.32 % gave (4). We interpret this ordinal, discrete variable as a proxy for the ‡ow of utility derived by individuals, assuming a latent continuous utility function. Due to the smallish proportion of people choosing the very unhappy answer, we aggregate the very and unhappy categories. We then reversed the scale to obtain an ascending rating of happiness of one to three, where three being the very happy assessment. We check our results when using alternative measure of subjective well-being as derived from the question: "Concerning your current standard of living, which of the following is true? It is less than adequate for my needs(1); It is just adequate for my needs (2); It is more than adequate for my needs (3)". The resulting variable is called "living-standard adequacy". This variable, along with "consumption adequacy" measurements, are called subjective measures of poverty, and have recently been introduced in some analysis of relative concerns. 7The initial sample size consists of 41, 420 adults aged 15 years and above. In the analysis, we take out observations with missing dependent or independent variables (41%) which gives an baseline working sample of 24, 154 observations.

Methodology

Our …rst test entails checking that the data generate the usual pattern of socioeconomic correlates of happiness established in the literature. Therefore, our basic method is to estimate happiness equation of the form:

H i = + ln Y i + X i + i (1)
where H i is a latent variable of the happiness, ln Y i is the logarithm of level of consumption (income) of the respondent's household and X i a vector of variables that allow to control for observable heterogeneity in household and geographic attributes. X i thus contains variables such as age, age-squared, gender, marital and health status, educational attainment, employment as well as ethnicity and province dummies. A detailed description of the variables and descriptive statistics are provided in Table A.1. Assuming that the error term i is normally distributed with unit variance, and taking account of the inherent ordering in the variable H i , Eq.( 1) is estimated using an ordered probit model. We will use alternative measures of economic welfare as one argument of utility. The IFLS collects data on household consumptions of a wide range of food and non-food items, and detailed information on various sources of income including income in-kind, and individual wages.

The most widely used objective measure of economic welfare in developing countries is consumption expenditure per person (C) given by total household expenditure, including spending on food (purchased and home-produced and food received as gifts), non-food items, estimated ‡ow of services from consumer durables and the actual or self-estimated rental cost of housing. 8 In a …rst speci…cation, we thus include the logarithm of household annual household expenditure. The household expenses were de ‡ated by the number of adult equivalents in order to capture di¤erences by age and economies of scale in consumption. The adopted approach and formula to de…ne the number of adult equivalents are detailed in National Research Council (1995). This is one of the most commonly used equivalence scales as parameters can be set at sensible values following the wealth and development level of each country. We then check, in a second and a third speci…cations respectively, the economic welfare impact when de ‡ating by household size to obtain a per capita consumption measure or when accounting for household income 9 rather than household consumption expenditures.

The estimate of obtained from Eq.( 1) re ‡ects to what extent does absolute income matter for happiness. Additional gains in income are generally expected to rise happiness but only until basic needs are ful…lled. Once basic necessities (such as securing the food supply, shelter, health, and clothing) are taken care of, happiness does not increase with income. It is therefore conceivable that, over the long run, raises in wealth will generate no increase in happiness. This is in line with micro-level evidence from the GSOEP provided by Di Tella, Haisken-De New, and MacCulloch (2010) who …nd that income growth provides only a temporary boost to life satisfaction. 10 This statement needs however to be nuanced as highlighted by recent work of [START_REF] Stevenson | Economic Growth and Subjective Well-Being: Reassessing the Easterlin Paradox[END_REF] who have re-assessed the Easterlin Paradox using international and country-level rich datasets, and have concluded to a clear role for absolute income in determining subjective well-being. The authors showed indeed that there is no evidence of a satiation point beyond which wealthier countries have no further increases in happiness levels, and that economic growth is associated with rising happiness when examining the association between changes in subjective well-being and income over time within countries.

Still, the estimate of may be the outcome of a "focusing illusion". In other words, it can proxy for factors such as comparisons of individual's own actual income and their past levels of income or the incomes of those around them [START_REF] Deaton | Income, health, and well-being around the world: Evidence from the gallup world poll[END_REF]. Hence, we also need to include measures of the economic welfare of relevant groups for social comparison or as generators of external e¤ects. We thus estimate the following equation:

H i = + ln C i + ln C n i + X i + i (2)
where C i is a measure of adult-equivalent per capita annual expenditure, ln C n i the logarithm of the leave-out mean consumption for the reference group 11 and X i the same vector of controls as in Eq. (1).

The reference group is de…ned as the group of people to whom one compares oneself. The empirical literature have de…ned di¤erent reference groups based on characteristics such as age [START_REF] Mcbride | Relative-income e¤ects on subjective well-being in the crosssection[END_REF], or geographical area (Blanch ‡ower and Oswald, 2004; [START_REF] Luttmer | Neighbors as negatives: Relative earnings and well-being[END_REF]. Alternatively, some authors consider reference group as a combination of individual and geographic attributes. For instance, Akay and Marstinson (2011) de…ne the reference group as related to age, geographic area, and the size of land holdings. Following past practice in the literature, we use personal attributes along with geographic structure of the sample design to estimate the mean consumption of "similar" people, that is, age-peers of the same gender, ethinicty, having accomplished the same educational attainment and living in geographic proximity. Age of the individuals was classi…ed into …ve groups, and geographic area was de…ned as the sub-district.

The use of household consumption, rather than household or individual income, as a proxy for wealth is motivated by the better consumption data in household surveys in 10 The authors regress life satisfaction on income and on several lags of income and conclude that life satisfaction adapts completely to income within a four-years time interval. 11 i.e. the mean consumption of all sampled households in the respondent's reference group, excluding the respondent.

developing countries 12 [START_REF] Deaton | Measuring poverty in a growing world (or measuring growth in a poor world)[END_REF]) as well as the common view of welfare in the Indonesian society that is far from being individualistic. We will therefore use the per adult equivalent consumption expenditure variable as a proxy for wealth while introducing household income as well as the total value of assets 13 in additional speci…cations as a robustness checks.

Finally, we also test an extended version of Eq. ( 2) where, similar to Ravaillon and Lokshin (2010), we allow for an interaction e¤ect between own consumption and the mean for the reference group as follows :

H i = + ln C i + ln C n i + ln C i : ln C n i + X i + i (3)
If the external e¤ect changes sign at high levels of income, then not considering this interaction term could lead to little or no relationship, through averaging across the positive and negative e¤ects. Such interaction e¤ects may arise from theoretical models of risk-sharing or in the wake of urbanization in developing countries (Ravaillon, 2008; Ravaillon and Lokshin, 2010). It can be presumed that informal risk sharing arrangements and collective action are more common in rural and village-based societies, and tend to corrode with urbanisation. Neighbourliness and reciprocity are indeed widely thought to be more prevalent in villages, whereas institutionalized social security systems are likely to be more developed and consequently, take over in big cities. 14 5 Results

The usual correlates of happiness

Estimates of Eq. ( 1) are reported in Table 1. In line with the international literature 15 , happiness is U shaped in age and increases with subjective health and educational level. Religious Indonesians enjoy higher well-being. This could be owing to the fact that religious people are more likely to adopt healthy lifestyles, and to derive well-being from religious service attendance and meditative states. Religion also provides mechanisms for 12 Expenditures tend to be less understated than income in less developed countries and most households engage in some consumption smoothing. 13 These include housing, car, land properties, etc. 14 For instance, Ravaillon and Dearden (1988) provide supportative evidence of how a moral economy (i.e. based on voluntary interpersonal transfers of money and goods) can perform the role of a social security system. Considering data from rural Java in Indonesia, the authors show indeed that transfers are taregted towards otherwise disadvatanged groups and donors hold a preference for less inequality. 15 e.g. see [START_REF] Helliwell | Well-being, social capital and public policy: What's new?[END_REF] and Blanch ‡ower (2008) coping with life's hindrances, which in turn may reduce worries (e.g., Clark and Lelkes, 2005; Lim and Putnam, 2010). Also consistent with other developing country studies is the result that happiness is positively correlated with being married rather than divorced or widowed. Further, we …nd that happiness is lower for males, those living in West Java and whose ethinicty is Batak or Minang. Unsurprisingly, satisfaction is higher in employed individuals as opposed to those unemployed, and rises with various measures of household income. Notes. The reference categories are: "very religious" for religiousness, "employed" for employment, "unhealthy" for self-assessment of health, "lower" for education, "married" for marital status, "Javanese" for ethnicity and "West Java" for province. *, **, *** : significant respectively at 10, 5 and 1 percent levels.

0,0696

(3) Dependent variable = Happiness 0,0677 0.0677

(1) (2)

Positive or Negative Relative Income Concerns?

Our estimates of Eq. ( 1) are presented in the …rst two columns of Table 2, using household consumption expenditures and income respectively. Results from Eq. ( 2) are in the next two columns. We provide two sets of results depending on how we consider geographic e¤ects. Column (3) uses province and urban-rural dummy variables, whereas column (4) allows for a more complete accounting of the geographic e¤ects, using a set of dummy variables for each district. (

) 2
Results show that subjective well-being increases with own consumption (income) and with reference group consumption (income) 16 . Moreover, our results are robust to changing the subjective well-being variable. Indeed, similar results are obtained for the regressions of more cognitive and income-related subjective variables such as the standard living adequacy, as shown in Table 3.

The negative sign of the interaction e¤ect obtained in column [START_REF] Akay | Relative concerns of rural-to-urban migrants in China[END_REF] indicates that the external e¤ect decreases as own-consumption rises. In other words, the richer an individual is, the lower this positive e¤ect will be. This is consistent with Ravaillon and Lokshin's (2010) …ndings for Malawi pointing to negative interaction e¤ects between absolute and relative income. However, this interpretation may not be convincing. Actually, there can be various issues as the income within a society rises. The reference group de…nition may change and the income inequality may also be di¤erent 17 . In order to further check for the validity of this statement, we apply an alternative method of splitting the sample into two income groups, based on median household consumption expenditures level. Our hypothesis is therefore that the reference group income has di¤erent e¤ects in the two sub-samples. Columns (1) and ( 2) of Table 4 show that the log of household expenditures is signi…cantly positive for both groups but, whereas the coe¢ cient on reference group expenditures is signi…cant and positive for individuals below median income, it loses its statistical signi…cance for the group above median income. In column (3), we focus on the rich people as de…ned by the top income quartile, where results point towards a negative, though insigni…cant, e¤ect of reference group income. Overall, we conclude that absolute income matters for all groups, whereas relative income positively a¤ects the poor but not the higher income groups. Again, this is consistent with results from other studies using the same methodology of splitting the sample in di¤erent income groups and concluding to di¤erent in ‡uence of relative income among the poor and the better-o¤ households.18 Columns (5) of Tables 2 and3. report estimates when we additionally control for lagged income, that can in ‡uence the extent to which people are satis…ed with their actual absolute level of income. Depsite a fall in the coe¢ cients of household income, the coe¢ cient on current income is still positive and signi…cant. We also still have the positive e¤ect of the reference group income along with the negative sign on the coe¢ cient of the interaction term. The coe¢ cient on the lagged income is positive and signi…cant, which suggests that the initial impact of getting more income appears to grow over time (by 4% = 0.03/0.693 in Table 2). Even though this coe¢ cient is small in magnitude, we cannot reject the hypothesis of no complete adaptation to income during the period seperating the two waves. This result is in line with results on adaptation to status changes provided by Di Tella et al. (2010). The role of information and expectations seems to be of particular importance to our study since expectations may a¤ect happiness by raising the aspiration level. We thus construct a variable re ‡ecting what we see as lagged income expectations, and control for it in some speci…cations. In the previous wave of the survey, people were asked about their self-ranking on a six-rung income ladder, 1 represent the poorest group and 6 the richest. Individuals were also asked where they expect themselves to be in terms of ranking in the future. We take the di¤erence between future and present self-ranking to create a categorical variable that takes the value of 0 if people expect themselves to go down the ladder, 1 if they expect to keep the same level of income, and 2 if they expect to go up the ladder. The inclusion of this variable however considerably reduces the sample size. We thus reproduce our basic speci…cations based on a constant sample size along with the inclusion of the expectations variable. Table 5 clearly shows that the positive e¤ect of reference group income is unchanged after the inclusion of controls for lagged income expectations, both for happiness and subjective measures of poverty. (

) (2) (3) 1
Dependent variable

Changing Reference Groups

As we hinted at above, another important issue when analyzing positional concerns is to check for a potential bias arising from created reference groups when no information on individuals'reference groups has been collected in the data. 3) report the results when individuals are assumed to compare themselves to their age-peers in the same occupation at the sub-district level. This leads to a mean of approximately 5 households in the reference group. Columns ( 2) and (4) use the same combination at the district level (rising the size of the reference group to 9 families). As previously, the respondent is taken out of the construction of the reference group income in both cases. Both speci…cations show a positive and signi…cant e¤ect of relative income on subjective well-being assessments. The unanimous conclusion is that relative income is a source of positive external e¤ect in Indonesia, rather than exerting a negative in ‡uence on individuals'well-being.

Happiness 7 Conclusion

This paper has investigated whether relative deprivation matters for people in developing countries, using data from Indonesia. Ours is the …rst estimate of subjective well-being function for this country. The estimates of the micro-determinants of subjective wellbeing show that relative income positively a¤ects individual well-being. Individuals who report their wealth to be lower than others in their reference groups also report higher happiness and living standard adequacy scores. There are signi…cant interaction e¤ects as well -poorer individuals draw greater satisfaction from external e¤ects compared to the comparatively well o¤. Our results are robust to various potential issues, including the bias arising from the construction of reference group as well as the in ‡uence of past income levels and income aspirations. These results are in line with those obtained in other studies pointing to the fact that people in poor societies attach a higher value to the absolute welfare in their community (e.g., Fafchamps and Shilpi, 2008; Ravaillon and Lokshin, 2010; Akay and Martinsson, 2011; etc.). They are also consistent with the predictions by [START_REF] Clark | Relative Income, Happiness, and Utility: An Explanation for the Easterlin Paradox and Other Puzzles[END_REF] assuming that the impact of relative income on subjective well-being within a country will grow smaller as one moves from richer to more deprived economies. This may notably be attributed to the greater role of kinship and altruistic relations in poor rural communities compared to urban areas in Western countries. The positive external e¤ect is also due to the supporting role played by informal networks and insurance mechanisms (such as the Rotating Savings and Credit Association (ROSCA) called arisan in Indonesia). Future research should continue to examine relative poverty from a social exchange perspective which can provide insights that have yet to be examined. The empirical …ndings in this study provide implications for the development research, especially in de…ning and attenuating poverty and income inequalities, and consequently drawing redistributive policies as recently emphasized by many authors. Also, the understanding of poverty founded on a notion of low income should be opposed to that derived from subjective judgments of welfare [START_REF] Asadullah | Subjective well-being and relative poverty in rural Bangladesh[END_REF]. Assessments of these schemes should look beyond income based indicators and revisit previous evaluations by looking at the social aspects of poverty dynamics. Debates and future research on poverty reduction should thus consider relative notions of deprivation, together with determining the threshold level above which these e¤ects occur and start to in ‡uence well-being.

  standard erros. *, **, *** : significant respectively at 10, 5 and 1 percent levels.

Table 1 :

 1 Basic regressions of happiness (ordered probit estimates)

		Coeff.	S.E	Coeff.	S.E	Coeff.	S.E
	Demographic categories
	Female	0.0830*** (0.0209)	0.0825*** (0.0209)	0.0754***	(0.0211)
	Age/10	-0.446*** (0.0448)	-0.440*** (0.0448)	-0.427***	(0.0450)
	Age square/100	0.0415*** (0.00527)	0.0408*** (0.00526)	0.0390***	(0.00529)
	Urban	0.0404*	(0.0209)	0.0407*	(0.0209)	0.0678***	(0.0206)
	Religiousness
	Religious	-0.326*** (0.0377)	-0.327*** (0.0377)	-0.331***	(0.0379)
	Somewhat religious	-0.470*** (0.0429)	-0.470*** (0.0429)	-0.479***	(0.0431)
	Not religious	-0.598*** (0.0664)	-0.598*** (0.0664)	-0.604***	(0.0666)
	Muslim	0.0826*	(0.0442)	0.0836*	(0.0442)	0.0656	(0.0445)
	Financial categories
	Per adult equivalent household expenditures (log)	0.146***	(0.0128)
	Per Capita household expenditures (log)			0.142***	(0.0125)
	Per adult equivalent household income (log)					0.118***	(0.00936)
	Employment
	Out of labour	-0.0141	(0.0222)	-0.0121	(0.0222)	0.0244	(0.0225)
	Unemployed	-0.216*** (0.0684)	-0.214*** (0.0684)	-0.195***	(0.0687)
	Self-assessed health
	Somewhat unhealthy	0.458***	(0.155)	0.457***	(0.155)	0.447***	(0.155)
	Somewhat healthy	0.752***	(0.153)	0.752***	(0.153)	0.745***	(0.153)
	Very healthy	0.974***	(0.156)	0.974***	(0.156)	0.972***	(0.156)
	Education
	Higher secondary	0.184***	(0.0235)	0.186***	(0.0235)	0.193***	(0.0234)
	Post secondary	0.420***	(0.0354)	0.421***	(0.0354)	0.454***	(0.0346)
	Marital status
	Single	-0.390*** (0.0307)	-0.394*** (0.0307)	-0.384***	(0.0310)
	Seperated	-0.746*** (0.112)	-0.749*** (0.112)	-0.747***	(0.112)
	Divorced	-0.651*** (0.0616)	-0.656*** (0.0616)	-0.627***	(0.0623)
	Widowed	-0.251*** (0.0456)	-0.252*** (0.0456)	-0.233***	(0.0461)
	Ethnicity
	Sundanese	0.0488	(0.0388)	0.0513	(0.0388)	0.0597	(0.0390)
	Bali	0.000960	(0.111)	0.00404	(0.111)	-0.00225	(0.112)
	Batak	-0.311*** (0.0698)	-0.311*** (0.0697)	-0.320***	(0.0707)
	Bugis	0.158**	(0.0727)	0.159**	(0.0727)	0.176**	(0.0730)
	Sasak	-0.0320	(0.0791)	-0.0302	(0.0791)	-0.0274	(0.0794)
	Minang	-0.158**	(0.0794)	-0.156**	(0.0794)	-0.163**	(0.0798)
	Banjar	-0.0110	(0.0924)	-0.0112	(0.0924)	-0.0137	(0.0930)
	Betawi	0.0460	(0.0559)	0.0483	(0.0559)	0.0599	(0.0564)
	Southern Sumatrans	-0.149**	(0.0643)	-0.148**	(0.0643)	-0.141**	(0.0647)
	Others	0.0792**	(0.0349)	0.0810**	(0.0349)	0.0853**	(0.0351)
	Province
	North Sumatra	0.185***	(0.0583)	0.191***	(0.0583)	0.210***	(0.0589)
	West Sumatra	0.0102	(0.0745)	0.0128	(0.0745)	0.0248	(0.0750)
	South Sumatra	0.247***	(0.0606)	0.251***	(0.0606)	0.238***	(0.0609)
	Lampung	0.281***	(0.0530)	0.284***	(0.0530)	0.308***	(0.0532)
	DKI Jakarta	0.0952**	(0.0440)	0.0976**	(0.0440)	0.123***	(0.0442)
	Central Java	0.0588	(0.0425)	0.0618	(0.0425)	0.0733*	(0.0427)
	DI Yogyakarta	-0.0330	(0.0515)	-0.0320	(0.0515)	-0.00514	(0.0521)
	East Java	0.134***	(0.0393)	0.135***	(0.0393)	0.164***	(0.0397)
	Bali	-0.129	(0.104)	-0.129	(0.104)	-0.127	(0.104)
	West Nusa Tenggara	0.223***	(0.0670)	0.224***	(0.0670)	0.240***	(0.0674)
	South Kalimantan	0.347***	(0.0841)	0.349***	(0.0841)	0.311***	(0.0844)
	South Sulawesi	0.242***	(0.0652)	0.246***	(0.0652)	0.265***	(0.0655)
	Pseudo R2
	Observations	24,154		24,154		23,960

Table 2 :

 2 The Impact of Reference income on Happiness (ordered probit estimates)

	Dependent variable = Happiness	(1)		(3)	(4)	(5)
	Log per adult equivalent household expenditures (log C )	0.150***		0.797***	1.032***	0.693**
		(0.0146)		(0.297)	(0.316)	(0.301)
	Log leave-out mean per adult equivalent household expenditures in the locality	0.0360*		0.683**	0.915***	0.582*
		(0.0203)		(0.297)	(0.316)	(0.300)
	Log per adult equivalent household income		0.124***
			(0.0105)
	Log leave-out mean per adult equivalent household income in the locality		0.0236
			(0.0159)
	Interaction of log C with log leave-out mean C			-0.0405** -0.0557*** -0.0345*
				(0.0186)	(0.0197)	(0.0188)
	Log lagged per adult equivalent household expenditures (log C(t-1))					0.0389**
						(0.0156)
	Urban	Yes	Yes	Yes	No	Yes
	Province dummies	Yes	Yes	Yes	No	Yes
	District dummies	No	No	No	Yes	No
	Pseudo R2	0.069	0.071	0.069	0.084	0.070
	Observations	19,888	19,719	19,888	19,888	19,723

Notes: All controls from table 1 are included. Figures in brackets are

standard erros. *, **, *** : significant respectively at 10, 5 and 1 percent levels.

Table 3 :

 3 The Impact of Reference Income on Living Standard Adequacy (ordered probit estimates) Notes: All controls from table 1 are included. Figures in brackets are standard erros. *, **, *** : significant respectively at 10, 5 and 1 percent levels.

	Dependent variable = Adequacy of living standard	(1)	(2)	(3)	(4)	(5)
	Log per adult equivalent household expenditures (log C )	0.292***		1.060***	1.006***	0.760***
		(0.0127)		(0.256)	(0.271)	(0.258)
	Log leave-out mean per adult equivalent household expenditures in the locality	0.0378**		0.805***	0.742***	0.549**
		(0.0173)		(0.255)	(0.271)	(0.258)
	Log per adult equivalent household income		0.240***
			(0.00911)
	Log leave-out mean per adult equivalent household income in the locality		0.0288**
			(0.0136)
	Interaction of log C with log leave-out mean C			-0.0480*** -0.0447*** -0.0322**
				(0.0160)	(0.0169)	(0.0161)
	Log lagged per adult equivalent household expenditures (log C(t-1))					0.166***
						(0.0134)
	Urban	Yes	Yes	Yes	No	Yes
	Province dummies	Yes	Yes	Yes	No	Yes
	District dummies	No	No	No	Yes	No
	Pseudo R2	0.078	0.084	0.079	0.100	0.085
	Observations	19,888	19,719	19,888	19,888	19,723

Table 4 :

 4 The Impact of Reference Income on Subjective Well-Being, by Poverty Status

	(1)	(2)	(3)

Notes: All controls from table 1 are included. Figures in brackets are standard erros. *, **, *** : significant respectively at 10, 5 and 1 percent levels.

Table 5 :

 5 The Unchanged E¤ect of Reference Income on Subjective Well-Being after Controlling for Aspirations

		Happiness			Living standard
						(5)	(6)
	Log per adult equivalent household expenditures (log C )	0.542	0.538	0.160***	0.907***	0.908***	0.303***
		(0.365)	(0.365)	(0.0181)	(0.314)	(0.314)	(0.0157)
	Log leave-out mean hhld expenditures	0.432	0.427	0.0513**	0.633**	0.634**	0.0308
		(0.364)	(0.364)	(0.0247)	(0.313)	(0.313)	(0.0210)
	Interaction of log C with log leave-out mean C	-0.0239	-0.0236		-0.0378*	-0.0379*
		(0.0228)	(0.0228)		(0.0196)	(0.0196)
	Lagged income expecattions
	Expect to keep the same level of economic rank (a)	0.109		0.108	-0.0652		-0.0669
		(0.0886)		(0.0886)	(0.0785)		(0.0785)
	Expect rise in economic rank (b)	0.113		0.112	-0.0423		-0.0442
		(0.0899)		(0.0899)	(0.0795)		(0.0795)
	Observations	13,312	13,312	13,312	13,312	13,312	13,312

Notes: All controls from table 1 are included. (a) Reference category for expectations is "expect a fall in economic rank". Figures in brackets are

standard erros. *, **, *** : significant respectively at 10, 5 and 1 percent levels.

  To this end, the researcher should test the robustness of the …ndings by de…ning di¤erent reference groups(Akay et al., 2012a ; Kingdon et al., 2009 ;[START_REF] Clark | Who compares to whom? the anatomy of income comparisons in Europe[END_REF][START_REF] Senik | Direct evidence on income comparisons and their welfare e¤ects[END_REF]. We therefore test how absolute and relative income in ‡uence subjective well-being in two other cases based on di¤erent groupings of individual and geographical characteristics. We used two di¤erent geographical areas as reference groups: (1) sub-district and (2) district. Results for happiness and living standard regressions are all reported in Table6.

Columns (1) and (

Table 6 :

 6 Estimation Results from Di¤erent Reference Groups Notes: All controls from table 1 are included. Figures in brackets are

	Dependent variable

See, e.g, Ravaillon and Lokshin (2010), Ravallion and Dearden (1988), Murgai et al. (2002), Fafchamps and Lund (2003), etc.

In the case of past income, results are only signi…cant when the individual reports being worse o¤ than three years ago, which is consistent with loss-aversion.

For further information on the subjective measures of poverty, see[START_REF] Pradhan | Measuring poverty using qualitative perceptions of consumption adequacy[END_REF],[START_REF] Ravallion | Identifying welfare e¤ects using subjective questions[END_REF],[START_REF] Ravallion | Self-rated economic welfare in Russia[END_REF].

For details on how the consumption aggregate was formed, see[START_REF] Deaton | Guidelines for Constructing Consumption Aggregates for Welfare Analysis[END_REF]

Household income includes all possible kinds of income, comprising work payment, state and private transfers as well as the value of the home production of food items.

Results also hold when using the assets as a proxy for wealth as shown in Table A.2 .

I thank Alpaslan Akay for this remark.

See, e.g.,[START_REF] Asadullah | Subjective well-being and relative poverty in rural Bangladesh[END_REF] for Bangladesh and[START_REF] Kingdon | Community, Comparisons and Subjective Well-being in a Divided Society[END_REF] for South Africa.

The latest statistics point to a Gini Index that reaches 0.42 in Indonesian rural areas according to the World Bank PovcalNet. Source: http://iresearch.worldbank.org/PovcalNet/index.htm?3

A Appendix


Sébastien Clauzier
email: sebclauzier@gmail.com

Stéphane Avrillon
email: stephane.avrillon@univ-rennes1.fr

Laurent Le Coq
email: laurent.le-coq@univ-rennes1.fr

Mohamed Himdi
email: mohamed.himdi@univ-rennes1.fr

Franck Colombel
email: franck.colombel@univ-rennes1.fr

Erwan Rochefort
email: erochefort@cmn-cherbourg.com.

Slotted waveguide antenna with a near-field focused beam in one plane

Keywords:

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION

The near-field focusing technique is used in several applications such as imagery sensing, characterization of materials or in biomedical devices. To focus a beam at a distance , called the focal distance, from the antenna, the phase has to be adjusted along the radiating aperture to obtain a locally plane wave-front (Gaussian beam property) in the desired spatial location. Today, many antennas allow focusing a beam in the near-field region: planar array [START_REF] Buffi | Design criteria for near-field focused planar arrays[END_REF], horn with a dielectric lens [START_REF] Ongareau | Charts for a quick design of spot-focusing corrugated horn lens antenna[END_REF][START_REF] Piksa | Near-field measurement of Gaussian beam behind dielectric lens[END_REF] or rectangular slotted waveguide [START_REF] Ohtera | Focusing properties of a microwave radiator utilizing a slotted rectangular waveguide[END_REF]. But all these antennas focus a beam in 3 dimensions (spot). In some cases, an antenna which focuses along one plane and provides a broad beam in the orthogonal plane (2 dimensional focused beam), can be interesting, such as in optics for testing a long material with a small width and a long length or in microwave application for feeding a linear reflect-or transmit-array antenna.

In [START_REF] Martinez-Ros | Microwave near-field focusing properties of width tapered microstrip leaky-wave antenna[END_REF][START_REF] Gomez-Tornero | Frequency steerable two dimensional focusing using rectilinear leaky-wave lenses[END_REF][START_REF] Martinez-Ros | Holographic pattern synthesis with modulated substrate integrated waveguide line source leaky-wave antennas[END_REF], the authors design two dimensional focusing antennas using leaky-wave principle considering a tapered microstrip line [START_REF] Martinez-Ros | Microwave near-field focusing properties of width tapered microstrip leaky-wave antenna[END_REF], a rectilinear slotted waveguide [START_REF] Gomez-Tornero | Frequency steerable two dimensional focusing using rectilinear leaky-wave lenses[END_REF] or SIW (Substrate Integrated Waveguide) technology [START_REF] Martinez-Ros | Holographic pattern synthesis with modulated substrate integrated waveguide line source leaky-wave antennas[END_REF]. However, this three geometries use partly or completely a printed circuit technology. In our case, the focused antenna will be used as a feed for a transmit-array antenna in navigation radar technology. In such applications, the antenna has to accept high power with a good radiation efficiency, so we will favour a complete waveguide technology. In [START_REF] Clauzier | Near-field focusing in one plane using a loaded sectoral horn antenna[END_REF], a horn antenna with a biconvex dielectric lens inserted in its aperture has been investigated to provide a two dimensional focused beam. But the addition of the lens in the aperture of the H-plane sectoral horn reduces the beam width in the E-plane and the maximum ratio length/width of the beam at the focal point was equal to 8. This paper presents a slotted waveguide antenna dedicated to the illumination of an array with a length of 1530mm and a width of 100mm (ratio length/width equals to 15.3), located in the near-field region of the feed. This structure focuses the field along one plane in order to illuminate the width of the array and provides a wide beam along the orthogonal plane to illuminate the length of the array. Therefore, the spot size specifications are determined by the dimensions of the illuminated array. As the whole system (array + feed) has been developed for a navigation radar embedded inside a compact integrated mast, the architectural restrictions enforce the focused beam characteristics. For example, applying the transmitarray technology to design the illuminated antenna, the standard beamwidth aperture of a navigation radar leads to a 96 elements linear array in the horizontal plane, which corresponds to a length of 1530mm.The restriction in the dimensions of the mast induces a maximum vertical size of the transmit-array antenna of 100mm, with a maximum distance of 500mm between the feed and the transmit-array antenna. To minimize the spill-over losses, the normalized field level at the edge of the array has to be closed to -10dB. So the focused antenna has to be designed to reach these spot sizes restrictions, which are summed up in table 1. In this paper, we present a focused antenna which reaches these specifications with low losses and high power handling. This paper is organized as follows. In the first part, we carry out the theory of the nearfield focusing technique with a slotted waveguide. In the second part, we present the results of the simulation with CST Microwave Studio. Finally, these theoretical results are compared with measurement.

DESIGN PROCESS OF THE NEAR-FIELD FOCUSING SLOTTED WAVEGUIDE ANTENNA

Presentation of the near-field focusing technique

Let's consider a rectangular antenna aperture (A) of length L as shown in figure 1, with a field distribution over this aperture as expressed in equation ( 1):

With, and respectively the magnitude and phase of the field distribution, and the coordinates of the aperture.

As seen in [START_REF] Clauzier | Near-field focusing in one plane using a loaded sectoral horn antenna[END_REF] and considering the reference plane at z=0, the phase distribution along the aperture needed to focus the beam at a distance must be equal to:

(

(1

)
Where, is the free space constant.

Figure 1: Illustration of near-field focusing theory

Such a phase variation along the aperture can be obtained in different ways: a first solution consists in modifying the field distribution of horn antenna by inserting a dielectric lens inside its aperture as shown in [START_REF] Clauzier | Near-field focusing in one plane using a loaded sectoral horn antenna[END_REF]; a second solution uses the modification of the complex propagation wave number to obtain the require phase variation; another solution is based on the synthesis of such a variation thanks to discrete radiating sources. This second solution is proposed in the paper: a slotted waveguide antenna is designed, optimizing the slots and their positions on the waveguide to obtain the desired field distribution.

It is important to notice that the previous formulation (Equation ( 2)) uses the ray-tracing theory. Of course in the reactive near-field region, the electric-field behaviour is more complex and its computation needs much more efforts. But even if the theory presented in this paper is limited, it has the advantage to be very easy to implement and gives a good approximation of the waveguide design.

Design of the antenna

Slot geometry

The shape of the slots has been chosen using [START_REF] Yee | I-slot characteristics[END_REF]. The I-slot design (Figure 2

Arrangement of the slots on the waveguide

To focus the electrical field radiated by the slotted waveguide, the phase of the field radiated by each slot must be the same at the focal point, called F (Figure 1).

Then, the location of each slot is enforced considering the phase distribution given in equation ( 2), and the propagation characteristics inside the waveguide: each slot location corresponds to a phase of the field inside the waveguide , and must be equal to the desired phase .

The phase of the field, for a mode, inside the waveguide is equal to:

(

) 3
where, is the cut-off frequency of the waveguide for a mode is the width of the waveguide.

The previous equation uses the propagation constant of the mode of an unperturbed rectangular waveguide. However, the assignment of slots on the waveguide can obviously perturbed the field inside the waveguide. This perturbation can lead to an error in the location of the slots and induces to a shift in the position of the focal point. To highlight the impact of the slots assignment on the phase distribution, we compare in the next section the theoretical phase with the simulated one.

It is important to notice that the orientation of the slots allows us to use both the negative and the positive components of the field (Figure 3). This capability will optimize the size of the antenna, considering the location where and also the location where . Moreover, this capability to use both negative and positive components of the field also prevents grating lobes which appear when the distance is higher than . To take advantage of the alternation of the orientation, the slots have been done on the small side of the waveguide (called 'b'). The final structure is a symmetrical centre fed waveguide with 14 slots. A drawing of the antenna is reported in Figure 5.

Figure 4: Determination of the position of each slot on the waveguide

The slots layout is symmetrical according to the feed. The right side slot locations are given in table 2 considering the feed as the reference point.

SIMULATION AND MEASUREMENT RESULTS

Simulation results

The final slotted waveguide has been simulated using CST Microwave Studio (Figure 5). Loads have been added at the ends of the waveguide to suppress reflected energy and quarter wave transformer inserted to obtain good reflection coefficient (Figure 9). As expressed in the previous section, the insertion of slot on the waveguide can modify the field inside the waveguide and leads to a perturbation of the field and a wrong phase distribution. To verify the impact of the slot on the field inside the waveguide, we represent in figure 6, the desired phase variation along z-axis at the plane where the slots are located and the simulated variation. We can notice small differences in the phase distribution which could lead to a small shift in the plane-wave location.

Figure 6: Theoretical and simulated phase along z-axis in the slots cut-plane

Figure 7 shows the -field results in the E-plane in both amplitude and phase at 9.41GHz. We can notice that the -field is focused with a minimal width for the beam at 360mm from the slotted waveguide.

Moreover, at 360mm from the slotted waveguide, we can notice a wavefront locally plane. The difference of focal distance between theory (500mm) and simulation (360mm) can be explained through two different reasons:

• Firstly, as said above and shown in figure 6, the error in the phase distribution due to the wave perturbation inside the waveguide induces by slots insertion, can induce a small difference between the desired and the actual plane-wave position.

• Secondly, the most probable reason of this difference between the theoretical location of the plane-wave and the simulated one is due to the approximation used in the theoretical study. Indeed, the theoretical positions of the slots on the waveguide have been determined through the equation ( 2). This equation is based on the well-known expression of the phase of the field in function of the distance and the wavelength (φ=2πl/λ). But, as said in the part 2, in the near-field zone, the propagation of the field is more complex than the applied ray-tracing theory which does not take into account coupling between I-slot. Moreover, in the theoretical part, we have considered that each slot radiates an isotropic pattern. All these approximations in the theoretical study can explain the difference with the simulation results. However, the approach presented in this paper has the advantage to be easy to implement with rather good results. Indeed, as shown in figure 7, the beam is focused on a wide range of distance and the error in the position of the plane-wave will have no impact on the system performance. Indeed, as expressed in references [START_REF] Hansen | Focal region characteristics of focused array antennas[END_REF][START_REF] Sherman | Properties of focused apertures in the Fresnel region[END_REF], the focus is not a single point, and we can consider that the field is still focus on a large area called the focal depth. Theoretically, the focal depth corresponds to the half power beam size in the axial direction. We can clearly notice in figure 7 that the field is still focused at 500mm from the antenna.

As we can see in figure 8, this focal depth depends on the size of the antenna. When the radiating aperture is very large compared to the wavelength, the focus depth is very small and the focus is closed to a single point. Conversely, when the aperture becomes smaller (around in our case) the focal depth is larger with a maximum of intensity before the desired focal distance [START_REF] Hansen | Focal region characteristics of focused array antennas[END_REF][START_REF] Sherman | Properties of focused apertures in the Fresnel region[END_REF].

As reported in table 1, a 10 dB tapering is required in both vertical and horizontal plane for the focused beam at the focal distance. We specify that for the theoretical calculation of the -10dB beamwidth presented in figure 8, we firstly calculate the field radiated by a discrete array of isotropic sources using the following equation (Equation 4) and then, we extract the beamwidth in function of the distance.

(

) 4
Where, is the number of discrete elements in the aperture, is the wave-number in free space, is the position of the discrete element in the aperture and is the phase of the discrete element in the aperture.

Measurement results

A prototype of this antenna has been manufactured as depicted in Figure 9. Figure 9b shows the load inserted at the two ends of the waveguide. We have used a flexible absorbing material (reference ASI 10 from Siepel) held by a metallic square screwed to the waveguide. This absorbing material provides a reflection coefficient of -25dB at 9.41GHz.

The antenna has been measured at 9.41GHz and Figure 10 Firstly, a slight tilt of the beam can be observed in measurement results in both planes. In the E-plane, this tilt is equal to 2mm at 100mm from the slotted waveguide and 11mm at 500mm. In the H-plane this tilt is equal to 2.4mm at 100mm from the slotted waveguide and 12mm at 500mm. This tilt can be explained by a both prototyping errors and slight misalignment of the antenna in the measurement setup.

Secondly, in the H-plane, we can notice that the -10dB beamwidth obtained in measurement is smaller than simulation results. Indeed, at 500mm from the slotted waveguide, the -10dB beamwidth in the Hplane equals to 1307mm in measurement and 1440mm in simulation. This difference is partly due to the post-process which calculates the near-field mapping.

CONCLUSION

A near-field focused antenna providing a two dimensional focus beam, with a focus beam in the E-plane and a wide beam in the H-plane, has been investigated in this paper. This slotted waveguide can be easily designed with the formulas given in this paper. The simulation and the measurement are in very good agreement. As shown, this near-field focused antenna can be used as a feed for a linear array with a ratio length/width equal to 15, with a focal point located at from the slotted waveguide.

Figure 2 :

 2 Figure 2: Design of the slot (a) and field distribution inside the slot (b)

Figure 3 :

 3 Figure 3: Orientation of the slots to use both positive and negative components of the field

Figure 5 :

 5 Figure 5: Design of the slotted waveguide with CST Microwave Studio

Figure 7 :

 7 Figure 7: Simulated electric field in amplitude and phase in the E-plane at 9.41GHz

Figure 8 :

 8 Figure 8: Theoretical and simulated beamwidth at -10dB for a radiating aperture equals to (a) and (b)

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure 9: The proposed slotted waveguide antenna (a) with a zoom on the load (b)

Table 1 : CHARACTERISTICS OF THE FOCUSED BEAM

 1

	Characteristics	Value
	Beamwidth at -10dB in the horizontal plane	1530mm
	Beamwidth at -10dB in the vertical plane	100mm
	Maximum distance of the focal point	500mm

Table 2 : SLOT POSITIONS

 2

	Slot nº 1 2 3 4 5 6	7
	d(mm) 21 42 61 79 97 114 131

ACKOWLEDGMENTS

The authors warmly thank L. Cronier for the manufacturing of the antenna.