1 INTRODUCTION

In the past 30 years, arrayed waveguide gratings (AWGs) have been actively employed as multiplexers/demultiplexers in optical communication and telecommunication systems. With improvement of AWGs, devices on their basis have found application in many fields of science and technology, e.g., development of optical sensors [1, 2], Raman spectrometers [3], and multifrequency lasers [4]. In addition, AWGs are used in astronomy [5] and laser physics [6, 7].

The use of AWGs in real devices faces a number of serious problems. One of them is temperature drift of the central wavelength of the spectral characteristics, which can be no less than 0.1 nm/°C, unacceptably large for real applications. Therefore, development of AWGs with temperature drift compensation, so-called athermal AWGs, has become an in-demand research problem in the past years. Several approaches to performing thermal compensation have been proposed, such as active cooling [8, 9]; usage of materials with specially selected refractive indices, thermal expansion coefficient, and temperature dependence of the refractive index [10]; and passive thermal compensation methods [11–19].

One of the simplest and most effective ways of compensating the temperature drift is active cooling of an AWG [8, 9]. Devices of this kind require power supply, and, in addition, the cooling element considerably increases the size of the demultiplexer. Therefore, AWGs with passive thermal compensation capable of functioning independently arouse particular interest. Quite a lot of methods for passive thermal compensation have been developed, in which the temperature drift is decreased by using polymers with certain refractive indices as materials for waveguide arrays [10], by making polymer-filled grooves in the input planar waveguide [11, 12], by making a cut or a cascade of cuts in the waveguide array [13–15], and by making controlled cuts in the input/output planar waveguide [16–19]. These methods allow the central wavelength temperature drift to be decreased to 0.01 nm/°C, but these technical solutions are quite difficult to implement because of strict requirements on the precision of the AWG chip fabrication and on the circuit calculation.

Calculations of circuit designs for most athermal AWGs can be found in the literature. The method of thermal compensation using a controlled cut in the input planar waveguide is considered, to the best of our knowledge, in the patent [19], where several technical designs are proposed, but no method for calculation of their optical and thermal characteristics is given. In this work, a similar demultiplexer is analyzed, relations needed for its development are obtained, and recommendations and restrictions on the choice of parameters ensuring its functioning are given.

2 UNPERTURBED DEMULTIPLEXER

Let us first consider the initial demultiplexer, i.e., the device without a cut in its input planar waveguide (Fig. 1).

Fig. 1.
figure 1

(Color online) Structure of the AWG demultiplexer.

Full size image

The optical path length of a wave propagating in the channel with number j (j = –M, …, 0, …, M; the total number of channels is N = 2M + 1) and entering the receiving waveguide with number q is defined by the relations

$$H_{{jq}}^{{\left( 0 \right)}} = {{n}_{{{\text{pl}}}}}{{R}_{{{\text{in}}}}} + {{n}_{{{\text{ch}}}}}{{L}_{j}} + {{n}_{{{\text{pl}}}}}{{R}_{{{\text{ex}}}}}{{f}_{{jq}}},$$
(1)
$${{L}_{j}} = {{L}_{0}} + j\Delta L,$$
(2)

where Lj is the length of the jth channel, ΔL is the difference of the lengths of the neighboring channels, Rin and Rex are the circle radii at the demultiplexer input and output (see Figs. 2 and 3), npl and nch are the refractive indices of the planar and channel waveguides, and the dimensionless factor fjq is calculated by the formula

$${{f}_{{jq}}} = 1 - \frac{1}{8}\beta _{q}^{2} - \frac{1}{2}{{\beta }_{q}}{{\alpha }_{j}},$$
(3)

where the meaning of the angles αj and βq is clear from Fig. 3. Since αj = jΔα, where Δα is the step of variation of the angle αj, formula (3) can be rewritten as

$${{f}_{{jq}}} = {{f}_{{0q}}} - \frac{1}{2}{{\beta }_{q}}{\kern 1pt} j\Delta \alpha ,$$
(4)

where \({{f}_{{0q}}} = 1 - \frac{1}{8}\beta _{q}^{2}\). Now formula (1) is reduced to the form

$$H_{{jq}}^{{(0)}} = H_{{0q}}^{{(0)}} + {{(\Delta H)}_{q}}j,$$
(5)

where

$$H_{{0q}}^{{(0)}} = {{n}_{{{\text{pl}}}}}\left( {{{R}_{{{\text{in}}}}} + {{R}_{{{\text{ex}}}}}{{f}_{{0q}}}} \right) + {{n}_{{{\text{ch}}}}}{{L}_{0}},$$
(6)
$${{(\Delta H)}_{q}} \equiv {{n}_{{{\text{ch}}}}}\Delta L - \frac{1}{2}{{n}_{{{\text{pl}}}}}{{b}_{{{\text{ex}}}}}{{\beta }_{q}},$$
(7)

and the following relation is used (see Fig. 3):

$${{R}_{{{\text{ex}}}}}\Delta \alpha = {{b}_{{{\text{ex}}}}}.$$
(8)
Fig. 2.
figure 2

(Color online) Input planar waveguide (Free Propagation Region, FPR) with a cut.

Full size image
Fig. 3.
figure 3

Output planar waveguide in the Rowland circle geometry [20].

Full size image

3 DEMULTIPLEXER WITH A CUT

Now we consider a demultiplexer (see Fig. 2) in which the input planar waveguide has a cut with width h and a part of the waveguide is rotated by an angle θ (|θ| \( \ll \) 1). The cut is filled with a plastic dielectric with refractive index nx. As a result, in (5) there appears perturbation δHj

$${{H}_{{jq}}} = H_{{0q}}^{{(0)}} + {{(\Delta H)}_{q}}j + \delta {{H}_{j}}.$$
(9)

In the Appendix, it is shown that calculation of δHj leads to

$${{\delta }}{{H}_{j}} = {{n}_{j}}{{h}_{j}} - {{n}_{{{\text{pl}}}}}h~{\text{cos}}{{{{\Phi }}}_{j}} + \frac{{n_{x}^{2}}}{{{{n}_{j}}}}\left[ {\left| {{{h}_{j}}} \right| - {{h}_{j}}} \right],$$
(10)

where the following notation is introduced:

$${{n}_{j}} \equiv \sqrt {n_{x}^{2} - n_{{{\text{pl}}}}^{2}~{\text{si}}{{{\text{n}}}^{2}}{{{{\Phi }}}_{j}}} ,$$
(11)
$${{h}_{j}} = {{h}_{j}}({{\theta }}) \equiv h + ({{X}_{Q}} - {{R}_{1}}{\text{tan}}\,{{{{\Phi }}}_{j}}){{\theta }}{\text{.}}$$
(12)

Here nx is the refractive index of the medium filling the cut, and the meaning of distances XQ and R1 is clear from Fig. 2. Angles Φj are dictated by the unperturbed device and are calculated by the formula

$${{{{\Phi }}}_{j}} = \frac{{{{b}_{{{\text{in}}}}}}}{{{{R}_{{{\text{in}}}}}}}j,$$
(13)

where bin is the period of arrangement of the waveguide channels at the input.

Further we consider the region of parameters where the condition

$${{h}_{j}} \geqslant 0$$
(14)

is fulfilled.

This inequality will be verified below. In this region, the last term in (10) disappears, and we obtain

$${{\delta }}{{H}_{j}} = h({{n}_{j}} - {{n}_{{{\text{pl}}}}}\,{\text{cos}}\,{{{{\Phi }}}_{j}}) + {{n}_{j}}({{X}_{Q}} - {{R}_{1}}{\text{tan}}\,{{{{\Phi }}}_{j}})\theta .$$
(15)

In addition, we accept the following approximation. Let the angles Φj satisfy the condition

$$\Phi _{j}^{2} \ll 1,$$
(16)

which is equivalent to the following requirement:

$${{\left[ {\frac{{N{{b}_{{{\text{in}}}}}}}{{2{{R}_{{{\text{in}}}}}}}} \right]}^{2}} \ll 1$$
(17)

(note that condition (16) is appreciably less strict than the requirement \(\left| {{{\Phi }_{j}}} \right| \ll 1\)). Then, rejecting terms of the order of \(\Phi _{j}^{3}\) and higher in (15), we obtain

$$\begin{gathered} {{\delta }}{{H}_{j}} = \left[ {h({{n}_{x}} - {{n}_{{{\text{pl}}}}}) + \theta {{n}_{x}}{{X}_{Q}}} \right] - \theta {{n}_{x}}{{R}_{1}}{{\Phi }_{j}} \\ + \,\,\frac{{{{n}_{{{\text{pl}}}}}}}{{2{{n}_{x}}}}\left[ {h({{n}_{x}} - {{n}_{{{\text{pl}}}}}) - \theta {{n}_{{{\text{pl}}}}}{{X}_{Q}}} \right]\Phi _{j}^{2}. \\ \end{gathered} $$
(18)

It is seen that the choice of the value XQ = 0 substantially simplifies further analysis. It is this case that we consider below. Thus, formula (18) takes the form

$${{\delta }}{{H}_{j}} = h({{n}_{x}} - {{n}_{{{\text{pl}}}}}) - \theta {{n}_{x}}{{R}_{1}}{{{{\Phi }}}_{j}} + \frac{{{{n}_{{{\text{pl}}}}}}}{{2{{n}_{x}}}}h({{n}_{x}} - {{n}_{{{\text{pl}}}}})\Phi _{j}^{2}.$$
(19)

The term in (19) that is proportional to Φj causes a shift of the spectral characteristic along the frequency scale (see below), and the term proportional to \(\Phi _{j}^{2}\) distorts its shape. Therefore, we investigate the conditions under which the last term in (19) can be ignored. That is, we require that the following inequality be fu-lfilled:

$$\frac{{{{n}_{{{\text{pl}}}}}}}{{2{{n}_{x}}}}h\left| {{{n}_{x}} - {{n}_{{{\text{pl}}}}}} \right|\Phi _{j}^{2} \ll \left| {\theta {{n}_{x}}{{R}_{1}}{{\Phi }_{j}}} \right|.$$
(20)

By substituting into it the maximum value |Φj| ≈ Nbin/2Rin, we obtain the requirement on the working range of rotation angles

$$\left| \theta \right| \gg \frac{{{{n}_{{{\text{pl}}}}}\left| {{{n}_{x}} - {{n}_{{{\text{pl}}}}}} \right|}}{{n_{x}^{2}}}\frac{{Nh{{b}_{{{\text{in}}}}}}}{{4{{R}_{1}}{{R}_{{{\text{in}}}}}}}.$$
(21)

Under these assumptions, formula (19) is simplified and reduced to the relation

$$\begin{gathered} {{\delta }}{{H}_{j}} = h({{n}_{x}} - {{n}_{{{\text{pl}}}}}) - \theta {{n}_{x}}{{R}_{1}}{{{{\Phi }}}_{j}} \\ = h({{n}_{x}} - {{n}_{{{\text{pl}}}}}) - \theta {{n}_{x}}\frac{{{{R}_{1}}}}{{{{R}_{{{\text{in}}}}}}}{{b}_{{in}}}j \\ = h({{n}_{x}} - {{n}_{{{\text{pl}}}}}) - \theta {{b}_{{{\text{ef}}}}}j, \\ \end{gathered} $$
(22)

where the following notation is introduced:

$${{b}_{{{\text{ef}}}}} \equiv {{n}_{x}}\frac{{{{R}_{1}}}}{{{{R}_{{{\text{in}}}}}}}{{b}_{{{\text{in}}}}}.$$
(23)

4 SPECTRAL CHARACTERISTIC OF THE DEMULTIPLEXER

Transmittance of the demultiplexer at the optical frequency ν (its spectral characteristic) is defined by addition of N = (2M + 1) waves and is given by the relation (q is the number of the receiving waveguide at the output)

$${{A}_{q}}(\nu ) = {{\left| {\sum\limits_{j = - M}^M {{{\Gamma }_{j}}\exp \left( {\frac{{2\pi \nu }}{c}{{H}_{{jq}}}} \right)} } \right|}^{2}},$$
(24)

where the coefficient Γj describes the loss in the jth channel waveguide, and c is the speed of light in vacuum. Further we consider a case of uniform power distribution in channels, i.e., we assume that all Γj = Γ. For a demultiplexer without a cut, substitution of (5) into (24) using (6) and (7) leads to a well-known result

$${{A}_{q}}(\nu ) = {{{{\Gamma }}}^{2}}{{\left[ {\frac{{{\text{sin(}}2M{{\psi }_{q}}{\text{)}}}}{{\sin {{\psi }_{q}}}}} \right]}^{2}},$$
(25)

where the notation

$${{\psi }_{q}} \equiv \frac{\pi }{{~c}}{{(\Delta H)}_{q}}\nu $$
(26)

is introduced.

It is evident from (25) that Aq(ν) is a periodic function having sharp transmission peaks at the frequencies

$$\nu _{{qm}}^{{(0)}} = \frac{{mc}}{{{{{(\Delta H)}}_{q}}}},$$
(27)

where m are integers (order of interference).

According to (9) and (22), a cut (h ≠ 0) and a rotation (θ ≠ 0) of a part of the input planar waveguide will cause a change in the optical length of the channels

$$\begin{gathered} {{H}_{{jq}}} = H_{{0q}}^{{(0)}} + {{(\Delta H)}_{q}}j + h({{n}_{x}}{\kern 1pt} - {{n}_{{{\text{pl}}}}}) - \theta {{b}_{{{\text{ef}}}}}j \\ = H_{{0q}}^{{(0)}} + h({{n}_{x}}{\kern 1pt} - {{n}_{{{\text{pl}}}}}) + \left[ {{{{(\Delta H)}}_{q}}{\kern 1pt} - \theta {{b}_{{{\text{ef}}}}}} \right]j. \\ \end{gathered} $$
(28)

Substituting (28) into (24) and calculating the sum, we obtain an analogue of formula (25)

$${{A}_{q}}(\nu ) = {{{{\Gamma }}}^{2}}{{\left[ {\frac{{{\text{sin}}\left( {2M{{\psi }_{q}}(\theta )} \right)}}{{\sin {{\psi }_{q}}(\theta )}}} \right]}^{2}}.$$
(29)

But instead of (26) we now have a new definition

$${{\psi }_{q}}(\theta ) \equiv \frac{\pi }{{~c}}\left[ {{{{(\Delta H)}}_{q}} - {{b}_{{{\text{ef}}}}}\theta } \right]\nu .$$
(30)

The last two relations clearly show that interference resulted in a shift of the spectral characteristic along the frequency scale. A new position of the transmission peaks is easily obtained from these equations

$${{\nu }_{{qm}}} = \nu _{{qm}}^{{\left( 0 \right)}}\left[ {1 + \frac{{{{b}_{{{\text{ef}}}}}}}{{{{{(\Delta H)}}_{q}}}}\theta } \right].$$
(31)

5 CONDITION FOR THERMAL COMPENSATION OF THE SPECTRAL CHARACTERISTIC

Obviously, when the temperature T varies, the quantities involved in relation (31) will also vary. Let us assume that the external compensating action also changes the cut angle θ

$$\theta = {{\theta }_{{{\text{com}}}}}(T)$$
(32)

and try to find conditions that allow the position of the transmission frequency peaks (31) to be kept unchanged using the above compensation, i.e., we require fulfillment of the condition

$$\frac{\partial }{{\partial T}}{{\nu }_{{qm}}}(T) = \frac{\partial }{{\partial T}}\left\{ {\nu _{{qm}}^{{(0)}}\left[ {1 + \frac{{{{b}_{{{\text{ef}}}}}}}{{{{{(\Delta H)}}_{q}}}}{{\theta }_{{{\text{com}}}}}(T)} \right]} \right\} = 0.$$
(33)

It is known that for the initial (i.e., at h = 0, θ = 0) demultiplexer the relative frequency peak shift is defined by the relation that can be obtained from (27)

$$\frac{1}{{\nu _{{qm}}^{{(0)}}}}\frac{\partial }{{\partial T}}\nu _{{qm}}^{{(0)}}(T) = - \frac{{{{n}_{{{\text{ch}}}}}}}{{n_{{{\text{ch}}}}^{{\text{g}}}}}\left[ {{{\gamma }_{{{\text{ch}}}}} + \frac{1}{{{{n}_{{{\text{ch}}}}}}}\frac{{\partial {{n}_{{{\text{ch}}}}}}}{{\partial T}}} \right],$$
(34)

where γch is the coefficient of thermal expansion of the channel wave guide, and \(n_{{{\text{ch}}}}^{{\text{g}}}\) is the group waveguide refractive index of the channel.

After calculating the derivative (33) and rejecting negligibly small quantities, one gets the requirement on the angle of compensation

$$\frac{\partial }{{\partial T}}{{\theta }_{{{\text{com}}}}} = \frac{{{{{(\Delta H)}}_{q}}}}{{{{b}_{{{\text{ef}}}}}}}\frac{{{{n}_{{{\text{ch}}}}}}}{{n_{{{\text{ch}}}}^{{\text{g}}}}}\left[ {{{\gamma }_{{{\text{ch}}}}} + \frac{1}{{{{n}_{{{\text{ch}}}}}}}\frac{{\partial {{n}_{{{\text{ch}}}}}}}{{\partial T}}} \right].$$
(35)

Substituting into it relation (23) for bef and assuming that, according to (7), \({{(\Delta H)}_{q}} \cong {{n}_{{{\text{ch}}}}}\Delta L\), we obtain the final result

$$\begin{gathered} \frac{{\partial {{\theta }_{{{\text{com}}}}}}}{{\partial T}} = \frac{{{{n}_{{{\text{ch}}}}}\Delta L{{R}_{{{\text{in}}}}}}}{{{{n}_{x}}{{b}_{{{\text{in}}}}}{{R}_{1}}}}\frac{{{{n}_{{{\text{ch}}}}}}}{{n_{{{\text{ch}}}}^{{\text{g}}}}}\left[ {{{\gamma }_{{{\text{ch}}}}} + \frac{1}{{{{n}_{{{\text{ch}}}}}}}\frac{{\partial {{n}_{{{\text{ch}}}}}}}{{\partial T}}} \right] \\ = \chi \left[ {{{\gamma }_{{{\text{ch}}}}} + \frac{1}{{{{n}_{\text{ch}}}}}\frac{{\partial {{n}_{{{\text{ch}}}}}}}{{\partial T}}} \right], \\ \end{gathered} $$
(36)

where the following dimensionless factor is introduced:

$$\chi \equiv \frac{{n_{{{\text{ch}}}}^{2}\Delta L{{R}_{{{\text{in}}}}}}}{{{{n}_{x}}n_{{{\text{ch}}}}^{{\text{g}}}{{b}_{{{\text{in}}}}}{{R}_{1}}}}.$$
(37)

Using (34), we can write (36) as

$$\frac{{\partial {{\theta }_{{{\text{com}}}}}}}{{\partial T}} = - \chi \frac{1}{{\nu _{{qm}}^{{(0)}}}}\frac{\partial }{{\partial T}}\left[ {\nu _{{qm}}^{{(0)}}(T)} \right].$$
(38)

Thus, the desired derivative of the angle of compensation with respect to temperature is defined by the relative rate of variation with temperature of the peak maximum of the initial demultiplexer multiplied by the geometric factor (37).

6 NUMERICAL ESTIMATES

Let us illustrate estimation of the above results using the values from Table 1. The characteristics of the materials in this table are taken from [21, 22].

Table 1. Technical and physical AWG parameters used for numerical estimations
Full size table

The calculated spectral characteristic for these parameter values is shown in Fig. 4.

Fig. 4.
figure 4

Example of the spectral characteristic of the arrayed waveguide grating. The channel transmission band is (FW) ≈ 100 GHz; the free spectral region is (FSR) ≈ 6.4THz.

Full size image

We check condition (14) that should be fulfilled to ensure applicability of subsequent calculations. Considering definition (12), it reduces to the following requirement:

$$\theta \leqslant \frac{{2{{R}_{{{\text{in}}}}}h}}{{N{{R}_{1}}{{b}_{{{\text{in}}}}}}} = \frac{{2 \times 6.0 \times 0.040}}{{129 \times 4.4 \times 0.015}} \approx 0.056\,\,{\text{rad,}}$$
(39)

which is easily fulfilled in the working range of angles θ.

Now we estimate the factor χ entering into (36) and defined by (37). An example of the refractive index of the medium filling the cut will the value nx = 1.4. Then the factor χ takes the form

$$\chi = \frac{{n_{{{\text{ch}}}}^{2}\Delta L{{R}_{{{\text{in}}}}}}}{{{{n}_{x}}n_{{{\text{ch}}}}^{{\text{g}}}{{b}_{{{\text{in}}}}}{{R}_{1}}}} = \frac{{{{{1.455}}^{2}} \times 30.6 \times 6.0}}{{1.4 \times 1.473 \times 15 \times 4.4}} \approx 3.24$$
(40)

and correspondingly

$$\begin{gathered} \frac{{\partial {{\theta }_{{{\text{com}}}}}}}{{\partial T}} = 3.24\left[ {4 \times {{{10}}^{{ - 6}}} + \frac{1}{{1.455}}1.0 \times {{{10}}^{{ - 5}}}} \right] \\ \approx 3.5 \times {{10}^{{ - 5}}}~\,\,{{{\text{rad}}} \mathord{\left/ {\vphantom {{{\text{rad}}} {\text{K}}}} \right. \kern-0em} {\text{K}}}. \\ \end{gathered} $$
(41)

Note that formula (38) is more suitable for practical application than (36), because the relative frequency shift (34) can be measured before interreference in the demultiplexer, and this experiment is simpler than measurements of several quantities entering into (36). When choosing the medium to fill the gap, one should also remember that the condition nx ≈ npl ≈ nch should be followed to minimize the Fresnel reflection loss in the cut. A similar choice also makes it easier to fulfil requirement (21) that the rotation angles of the entire working range should satisfy. In our example it means that

$$\begin{gathered} \left| \theta \right| \ll \frac{{1.45 \times 0.05 \times 129 \times 0.040 \times 0.015}}{{{{{1.4}}^{2}} \times 4 \times 4.4 \times 6.0}} \\ \approx 2.7 \times {{10}^{{ - 5}}}~\,\,{\text{rad}}. \\ \end{gathered} $$
(42)

It should be noted that, according to (31), the choice of a nonzero initial (i.e., at room temperature) angle θ0 will actually shift the transmission peak frequency, which should be taken into account in developing the demultiplexer.

7 CONCLUSIONS

A method for passive thermal compensation of an AWG demultiplexer using a cut in the input planar waveguide and controllable rotation of a part of this waveguide has been investigated. A possibility of implementing this thermal compensation is shown, restrictions on the choice of demultiplexer parameters to be observed for demultiplexer functioning are obtained, and the corresponding recommendations are given. A simple analytical expression for the derivative of the compensation angle with respect to temperature is obtained, which is needed for the development of the multiplexer. In the given numerical example, this derivative is 3.5 × 10–5 rad/K.

APPENDIX

The optical path length Hjq of the wave propagating through the channel waveguide with number j and arriving at the output receiving waveguide with number q is composed of several terms. In Fig. 2, it is seen that

$$\begin{gathered} {{H}_{{jq}}} = {{n}_{{{\text{pl}}}}}\left| {E{{D}_{j}}} \right| + {{n}_{x}}\left| {{{D}_{j}}{{A}_{j}}} \right| \\ + \,\,{{n}_{{{\text{pl}}}}}\left| {{{A}_{j}}{{T}_{j}}} \right| + {{n}_{{{\text{ch}}}}}{{L}_{j}} + {{n}_{{{\text{pl}}}}}{{R}_{{{\text{ex}}}}}{{f}_{{jq}}}, \\ \end{gathered} $$
(A1)

where |EDj|, |DjAj|, and |AjTj| are the lengths of the corresponding segments.

The use of refraction laws at the boundaries of the cut and an assumption that perturbation of refraction angles caused by the cut (h ≠ 0) and the rotation (θ ≠ 0) is small lead (after rather cumbersome calculations) to the following results:

$$\begin{gathered} \left| {E{{D}_{j}}} \right| + \left| {{{A}_{j}}{{T}_{j}}} \right| = {{R}_{{{\text{in}}}}} - \left( {\cos {{\Phi }_{j}} + \frac{{{{n}_{{{\text{pl}}}}}}}{{{{n}_{j}}}}{{{\sin }}^{2}}{{\Phi }_{j}}} \right)h \\ - \,\,\theta \left( {{{X}_{Q}} - {{R}_{1}}\tan {{{{\Phi }}}_{j}}} \right)\frac{{{{n}_{{{\text{pl}}}}}}}{{{{n}_{j}}}}{{\sin }^{2}}{{\Phi }_{j}}, \\ \end{gathered} $$
(A2)
$$\left| {{{D}_{j}}{{A}_{j}}} \right| = \frac{{{{n}_{x}}}}{{{{n}_{j}}}}\left| {h + ({{X}_{Q}} - {{R}_{1}}{\text{tan}}\,{{\Phi }_{j}})\theta } \right|,$$
(A3)

where

$${{n}_{j}} \equiv \sqrt {n_{x}^{2} - n_{{{\text{pl}}}}^{2}\,{{{\sin }}^{2}}{{{{\Phi }}}_{j}}} .$$
(A4)

After substitution of (A2) and (A3) into (A1) and some simple transformations we get

$$\begin{gathered} {{H}_{{jq}}} = {{n}_{{{\text{pl}}}}}\left[ {{{R}_{{{\text{in}}}}} - h\cos {{\Phi }_{j}}} \right] + {{n}_{j}}{{h}_{j}}(\theta ) \\ + \,\,\frac{{n_{x}^{2}}}{{{{n}_{j}}}}\left[ {\left| {{{h}_{j}}(\theta )} \right| - {{h}_{j}}(\theta )} \right] + {{n}_{{{\text{ch}}}}}{{L}_{j}} + {{n}_{{{\text{pl}}}}}{{R}_{{{\text{ex}}}}}{{f}_{{jq}}}, \\ \end{gathered} $$
(A5)

where

$${{h}_{j}} \equiv h + \left( {{{X}_{Q}} - {{R}_{1}}\tan {{\Phi }_{j}}} \right)\theta .$$
(A6)