Field-widened Michelson interferometer for spectral discrimination in high-spectral-resolution lidar: theoretical framework

Zhongtao Cheng; Dong Liu; Jing Luo; Yongying Yang; Yudi Zhou; Yupeng Zhang; Lulin Duan; Lin Su; Liming Yang; Yibing Shen; Kaiwei Wang; Jian Bai

doi:10.1364/OE.23.012117

1. Introduction

High-spectral-resolution lidars (HSRLs) are increasingly being developed for atmospheric aerosol remote sensing applications due to the straightforward and independent retrievals of aerosol extinction and backscatter without reliance on assumptions about lidar ratio [1–8]. In HSRL technique, spectral discrimination between scattering from molecules and aerosol particles is one of the most critical processes, which needs to be accomplished by means of a narrowband spectroscopic filter. Note that, the molecular backscatter spectrum consists of a central peak (called Cabannes scattering) and sidebands (called the rotational Raman scattering) [8]. Generally, most HSRLs only need to take the Cabannes scattering into account as the rotational Raman scattering can be rejected easily by an optical pre-filter. To separate the broad spectrum of the molecular Cabannes scattering from the narrowband aerosol Mie scattering component, iodine absorption filter has been successfully deployed in many HSRLs because of its robust, stable filtering characteristic and high rejection to aerosol scattering signal [3, 5, 6, 8]. However, the gaseous absorption lines do not exist at many convenient laser wavelengths, limiting the developments of multi-wavelength HSRL instruments. Interferometric filter has attracted many attentions to extend the spectral domain of HSRL technique, among which, Fabry-Perot interferometer (FPI) plays an important role for its simplicity in realization [2, 7]. In addition, Mach-Zehnder interferometer (MZI) is also proposed to act as the spectral discrimination filter in HSRL [4].

Nonetheless, the aforementioned interferometric filters have the disadvantage of small acceptable field of view (FOV), thus the photon efficiency of the instrument is not so satisfactory. A field-widened Michelson interferometer (FWMI) was conceived by NASA Langley Research Center to overcome this problem in their HSRL-2 project [9]. In principle, this scenario would be very brilliant since FWMI can perform well at relatively large off-axis angles resulting from the field-widening design, while it also keeps the superiority of wavelength flexibility. In [9], the realization of the FWMI originates from the methodology proposed by G.G.Shepherd, et al. [10, 11], where, an analogous wide-angle Michelson interferometer (WAMI) is employed for measurement of wind and temperature on the upper atmosphere. In the following parts of this paper, we would like to mention the two acronyms “FWMI” and “WAMI” simultaneously for indicating these two different applications concerning this special kind of Michelson interferometer (MI), in spite of that they both have similar literal meanings. The fact is that the roles the FWMI and WAMI play in their respective realms are distinct obviously, thus we should make more in-depth researches on the theoretical foundations of FWMI. As of today, there exist no literatures that can provide systematic illustrations about this new application of the FWMI, such as its design, parameter calibration, performance estimation, etc., which stimulates urgent needs for us to investigate the exclusive features of the FWMI used in HSRL applications.

In our previous works, we have studied the HSRL technique thoroughly from the perspective of spectral discrimination [12–14]. Many new concepts were proposed to interpret the principle of HSRL. On the basis of these works, we recognize the superiority of the FWMI and develop a prototype interferometer aimed to be used as a spectral discriminator in HSRL for profiling aerosol optical properties. In this paper, we will mainly focus on the independent theoretical framework through which we solve the confronted problems during the development of the FWMI, such as the designing, parameter calibration, prior performance budget, posterior performance estimation, and so on. This framework includes two main parts, i.e., the design foundation and the comprehensive performance estimation model about the FWMI. These two parts should be related with each other tightly and constitute a general theoretical system which is expected to provide common guidelines for the FWMI application in HSRL instrument. It will be a valuable contribution to the lidar community to develop a new generation of HSRLs based on the FWMI spectroscopic filters. The experimental analysis will be presented in our next publication.

In section 2, we illustrate theoretical foundations for designing the FWMI, which is motivated by the comparative analysis about the distinctions between HSRL technique and wind or temperature detection technique. The mathematical description is introduced to embody the designing process. In section 3, a general model desired to evaluate the comprehensive performance of the FWMI is established and the modeling processes are described considering some specific practical imperfections or conditions. The applications and results of the proposed theoretical framework are exemplified in section 4. Some concluding remarks are summarized in section 5.

2. Theoretical foundations for designing the FWMI

2.1 General optical configuration of the FWMI

To begin with, Fig. 1(a) shows a general schematic diagram of the optical configuration for the FWMI in high-spectral-resolution receiver channel of our HSRL, and Fig. 1(b) exhibits its interior optical path, which will be helpful to understand the calculation of its optical path difference (OPD), as will be presented below. It should be emphasized that the FWM here is placed with a small tilted angle with respect to the normally incident light rays. This will produce two practical advantages. On one hand, the output beam from the reflection channel of the FWMI is not coaxial with the input signal in this case, thus it can be guided out of the incident direction conveniently, which can avoid the unnecessary cross-talk between other receiver channels and the FWMI channel. On the other hand, the reflected radiation can also be used as analytical signal to retrieve the atmospheric parameters along with the one from the transmission channel, if needed, as are often the cases for FPI and MZI. In our current development of HSRL instrument, the reflection channel is kept not used, and only the signal from the transmission channel is recorded by a detector such as photomultiplier tube (PMT) [13]. Note that, Fig. 1 shows a general construction scheme of FWMI, in which, the interference arm consisting of a glass and an air gap is called the hybrid arm and the other arm is referred as the pure-glass arm. This structure of the FWMI is called the hybrid-structure FWMI. If we set the glass length in the hybrid arm to zero, then it came back to the configuration of the NASA's FWMI, which is called as the pure-structure FWMI here. Additionally, anti-reflection (AR) coatings are employed on the interfaces of different materials to improve the photon efficiency, and a piezoelectric transducer (PZT) is connected to mirror1 (M₁) for wavelength tuning the interferometer.

Fig. 1 (a) Schematic diagram of the optical configuration for the FWMI in the high-spectral-resolution receiver channel of our HSRL and (b) its interior optical path.

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2.2 Physical motivations

Actually, the conception of the FWMI is indeed enlightened by the basic characteristic of the WAMI but with completely different operation principles. The WAMI is proposed to enhance the radiance response in many instruments developed for atmospheric wind and temperature detections, such as wind imaging interferometer (WINDII) [11, 15] and ground-based airglow imaging interferometer (GBAII) [16]. In these applications, the basic idea is that a single Gaussian airglow line produces a cosiusoidal signal within a Gaussian envelope at the output of an MI. Based on the phase-stepping interferograms recorded on the CCD along with some calibration processes, the wind information can be measured from the phase of the signal, and the temperature can be retrieved from the fringe visibility. In essence, it is the Fourier transform ability of an MI that makes the characteristics of the Gaussian airglow line, e.g., the half-intensity width and central wavenumber, obtainable, thus the atmospheric temperature and wind velocity are also very easy to be inferred according to the corresponding relations between these spectral characteristics and target parameters. The advantages of WAMI in wind and temperature detections are reflected as several aspects: (1) it has an OPD that varies slowly with incident angle, allowing the collection of light over a large solid angle and providing a large sensitivity; (2) its wide FOV can be kept at several wavelengths by chromatic compensation designing, which makes multi-lines observation possible; (3) its OPD is stable at a range of operating temperatures through the thermal compensation designing.

However, the cases become completely discrepant when FWMI is used as a spectral discriminator in HSRL, which are demonstrated as follows:

The function of FWMI here is to spectrally separate the aerosol scattering signal and molecular scattering signal, rather than to act as the Fourier transform spectroscopy to analyze the scattering spectrum as is the case for WAMI. In HSRL, the power detector such as PMT needs to be adopted instead of the image detector CCD. When the signal spectrum propagates through the FWMI and impinges on the PMT, the recorded signal is the integral of the scattering spectrum weighted by FWMI transmission function. Generally, FWMI should be properly designed and controlled to ensure that the valley of its transmission function is at the centroid of aerosol signal spectrum. In this sense, FWMI performs more like a spectroscopic filter, as is used in the Michelson Doppler Imager on the Solar and Heliospheric Observatory mission [17].
In multi-channel HSRL, it is better to tilt the FWMI at a small angle with respect to the incident beam to keep the back-propagating light from contaminating the signal that is received by other channel and to allow for sampling of the signal from the reflection channel of the FWMI, if needed. Obviously, the basic requirement on the field-widening angle is determined by the divergence of the input beam, and the tilted placement condition would increase this requirement. The traditional realization for the field-widening angle of WAMI is to expand the expression of OPD with respect to the normally incident angle and then set the coefficient of the squared term to zero by properly determining the lengths of arms. To avoid consuming a part of the field-widening angle on the tilt configuration, a new approach for enlarging the FOV of FWMI should be proposed, which will be introduced later.
The chromatic compensation is not necessary since FWMI only needs to work at a single chosen laser wavelength in HSRL. Consequently, the glass selection procedure as done in the design of WAMI can be skipped over. Thus more candidate glasses would appear.
The considerations for determining a basic parameter of this type of interferometer, i.e., the fixed OPD (FOPD), are changed remarkably. For WAMI, FOPD should be selected based on the target spectral lines and detection sensitivity, while that is dependent of the spectral discrimination requirement and transmittance requirement for the useful signal as for FWMI. We will describe this process in detail in Section 4.

In the whole, these above-mentioned distinguishes between FWMI and WAMI ask for a particular designing and modeling theory on the basis of traditional field-widening concept, in order to optimize the utility of FWMI as a spectral discriminator in HSRL.

2.3 Mathematical equations for the required conditions

From Fig. 1(b), the relationship between the generalized OPD of the FWMI and the incident angle $θ$ of light beam can be described by

O P D (θ) = 2 (n_{1} d_{1} \cos θ_{1} - n_{2} d_{2} \cos θ_{2} - n_{3} d_{3} \cos θ_{3}),

where

θ_{k}, k = 1, 2, 3

, is the incident angle at each of the three interfaces of FWMI as shown in Fig. 1(b);

n_{1}

and

n_{2}

are the refractive indices of the glass in the pure-glass arm and hybrid arm, respectively,

n_{3}

is the refractive index of the air gap;

d_{k}

is the length of the corresponding interference arm denoted in Fig. 1(b). Equation (1) can be rewritten through expressing

θ_{k}

as a function of

θ

by Snell’s law as

O P D (θ) = 2 [n_{1} d_{1} {(1 - \frac{\sin^{2} θ}{n_{1}^{2}})}^{1 / 2} - n_{2} d_{2} {(1 - \frac{\sin^{2} θ}{n_{2}^{2}})}^{1 / 2} - n_{3} d_{3} {(1 - \frac{\sin^{2} θ}{n_{3}^{2}})}^{1 / 2}] .

As is discussed, we need to tilt the FWMI at a fixed angle $θ_{t}$ . If the half divergent angle of the incident beam is $θ_{d}$ , then light rays incident on the FWMI are within the angular range $θ_{t} - θ_{d} < θ < θ_{t} + θ_{d}$ . Here, we expand the expression of the OPD with regard to the central incident angle $θ_{t}$ , yielding

\begin{array}{l} O P D (θ) = O P D (θ_{t}) + ω (θ_{t}) (\sin^{2} θ - \sin^{2} θ_{t}) \\ + ψ (θ_{t}) {(\sin^{2} θ - \sin^{2} θ_{t})}^{2} + Ο [{(\sin^{2} θ - \sin^{2} θ_{t})}^{2}], \end{array}

where,

Ο [\cdot]

denotes the high order infinitesimal quantity;

ω (θ_{t}) = - (\frac{d_{1}}{\sqrt{n_{1}^{2} - \sin^{2} θ_{t}}} - \frac{d_{2}}{\sqrt{n_{2}^{2} - \sin^{2} θ_{t}}} - \frac{d_{3}}{\sqrt{n_{3}^{2} - \sin^{2} θ_{t}}}),

and

ψ (θ_{t}) = - \frac{1}{4} [\frac{d_{1}}{{(n_{1}^{2} - \sin^{2} θ_{t})}^{3 / 2}} - \frac{d_{2}}{{(n_{2}^{2} - \sin^{2} θ_{t})}^{3 / 2}} - \frac{d_{3}}{{(n_{3}^{2} - \sin^{2} θ_{t})}^{3 / 2}}] .

The field-widening condition is accomplished by setting the second term of Eq. (3) to zero:

ω (θ_{t}) = 0.

Then, the OPD is only dependent of a small quantity

{(\sin^{2} θ - \sin^{2} θ_{t})}^{2}

, and can be very stable with respect to incident angle. The OPD at the central incident angle

θ_{t}

is called the fixed OPD (FOPD), an important parameter determining the performance of the FWMI. The thermal compensation condition refers to setting the derivative of FOPD with respect to temperature to zero. In our approach, the FOPD is

O P D (θ_{t})

, hence it requires

\begin{array}{l} \partial O P D (θ_{t}) / \partial T = 2 {[α_{1} d_{1} {(n_{1}^{2} - \sin^{2} θ_{t})}^{1 / 2} + β_{1} n_{1} d_{1} {(n_{1}^{2} - \sin^{2} θ_{t})}^{- 1 / 2}] \\ - [α_{2} d_{2} {(n_{2}^{2} - \sin^{2} θ_{t})}^{1 / 2} + β_{2} n_{2} d_{2} {(n_{2}^{2} - \sin^{2} θ_{t})}^{- 1 / 2}] \\ - [α_{3} d_{3} {(n_{3}^{2} - \sin^{2} θ_{t})}^{1 / 2} + β_{3} n_{3} d_{3} {(n_{3}^{2} - \sin^{2} θ_{t})}^{- 1 / 2}]} = 0, \end{array}

where,

α_{k} = \frac{\partial d_{k} / \partial T}{d_{k}}

, which is called the thermal expansion coefficients of the glass material, and

β_{k} = \partial n_{k} / \partial T

is the temperature coefficient of refractive indices.

Revisiting Eqs. (6) and (7), it is easy to note that the field-widening condition and the thermal compensation condition become identical with the traditional ones as used in WAMI when $θ_{t}$ is equal to zero. The proposed manner can be particularly adaptable to the design of a tilted FWMI employed in HSRL.

3. Practical performance estimation model

3.1 General estimation for the spectral discrimination characteristic of an FWMI

The above analysis introduces a theoretical concept for the design of an FWMI. For the development of a practical FWMI, a model is apparently in urgent need to provide significant guidance for manufacturing tolerance estimation, performance budget and pragmatic operation of this type of interferometer. To establish such a model, one of the most important preconditions is to choose a relevant metric to quantize the performance of the FWMI. We have made sufficient research about the accuracy dependence of the HSRL on its spectroscopic filter from a general perspective. A finding is obtained that the spectral discrimination ratio (SDR), which is defined as the ratio of the molecular transmittance to the aerosol transmittance of the spectroscopic filter, is directly associated with the retrieval accuracy of the HSRL system based on both mathematical proof and numerical simulation [12]. This finding can be applicable to most multi-channel HSRL systems whatever the employed spectroscopic filter is. Therefore, we propose to employ the SDR as a comprehensive and uniform performance metric for the FWMI development. Additionally, the model must provide a convenient interface to incorporate as much practical parameters as possible to reflect the most real condition of the FWMI.

Let us start with deriving a general estimation for the SDR of an FWMI. As is known well, the universal expression for synthesized intensity of dual-beam interference can be described as [18]

F (υ, θ) = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} \cos [2 π υ \cdot O P D (θ) / c],

where,

I_{1}

and

I_{2}

are the two intensities of radiation getting to the output after travelling through each interference arm of an FWMI,

c

is the light speed and

O P D (θ)

is formatted by Eq. (2). From Eq. (8), it can be seen that the transmission functions of the FWMI for radiation with different incident angles are also different. Before the operation of the FWMI, the PZT connected to mirror 1 (see Fig. 1) should be adjusted through a frequency locking system until the center of the whole interferogram is darkest (in this case, the aerosol scattering signal has the lowest transmittance in the fringe center). If this adjustment is realized ideally, the OPD of the central incidence would satisfy the destructive interference condition, that is

O P D (θ_{t}) = (m + 1 / 2) c / υ_{0},

where,

m

is an arbitrary integer and

υ_{0}

is central frequency of emitted laser.

Now we firstly write Eq. (8) as

F (υ - υ_{0}, θ) = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} \cos [2 π (υ - υ_{0}) \cdot O P D (θ) / c + 2 π υ_{0} \cdot O P D (θ) / c] .

Substituting

υ

and

O P D (θ)

in Eq. (10) by

υ_{0} + Δ υ

and

O P D (θ_{t}) + Δ O P D (θ)

respectively and neglecting the high-order infinitesimal

2 π \cdot Δ υ \cdot Δ O P D (θ) / c

, then the transmission function can be simplified as combining Eq. (9)

F (υ - υ_{0}, θ) = I_{1} + I_{2} - 2 \sqrt{I_{1} I_{2}} \cos [2 π (υ - υ_{0}) / F S R (θ_{t}) + Δ ϕ],

where,

Δ ϕ (θ) = 2 π υ_{0} \cdot Δ O P D (θ) / c

is the phase difference between the phase of the light with incident angle

θ

and that with central incidence

θ_{t}

, and

F S R (θ_{t}) = c / O P D (θ_{t})

is called the free spectral range (FSR) of the FWMI. The formation of Eq. (11) separates the transmission function of the FWMI into three independent parts, i.e., the contribution from the radiation amplitude, the contribution from the spectral frequency and the contribution from the incident angle. This fact will give us extreme convenience for the development of a practical model.

The backscatter signal spectrums impinging on the FWMI can be represented approximately by Gaussian lineshape as [2, 4]

S_{i} (υ - υ_{0}) = \exp [- {(υ - υ_{0})}^{2} / γ_{i}^{2}] / γ_{i} \sqrt{π},

where,

γ_{i}

is the half-width at the

1 / e

height,

υ

is the frequency and the subscript

i = a, m

indicates parameters for aerosol (

i = a

) and molecular (

i = m

) signals. Now we can get the local transmittance of the FWMI to aerosol and molecular backscattering signals with incident angle

θ

ℓ_{i} (θ) = \int_{- \infty}^{\infty} S_{i} (υ - υ_{0}) F (υ - υ_{0}, θ) d υ / \int_{- \infty}^{\infty} S_{i} (υ - υ_{0}) d υ .

Inserting Eqs. (11) and (12) into Eq. (13), it takes the form

ℓ_{i} (θ) = I_{1} + I_{2} - 2 \sqrt{I_{1} I_{2}} \exp [- {(\frac{π r_{i}}{F S R (θ_{t})})}^{2}] \cos [Δ ϕ (θ)] .

The overall transmittance of the FWMI to all the incident light within the divergence range, $T_{i}$ , can be estimated by the mean local transmittance over the whole detection plane, which can be expressed by double integral with the form of polar coordinates as

T_{i} = \int_{- π}^{π} d φ \int_{0}^{f θ_{d}} ℓ_{m a p_{i}} (ρ, φ) ρ d ρ / π f^{2} θ_{d}^{2},

where,

θ_{d}

is the half divergent angle of the incident beam,

f

is the focal length of the lens used to converge the beam into the detector and

ℓ_{m a p_{i}} (ρ, φ)

is a mapping function from the local transmission function

ℓ_{i} (θ)

which can be described by

ℓ_{m a p_{i}} (ρ, φ) = ℓ_{i} [arc \cos (\frac{2 f \cos^{2} θ_{t} - ρ \sin (2 θ_{t}) \cos φ}{2 \sqrt{f^{2} + ρ^{2}} \cos θ_{t}})] .

Equations (15) and (16) provide general estimation for the overall transmittance of the FWMI once the tilted angle $θ_{t}$ and divergent angle $θ_{d}$ are given, which are indispensable calibration parameters in HSRL retrieval. Also, they yield the direct calculation for the SDR

S D R = T_{m} / T_{a} .

It should be noted that the

T_{i}

and SDR are not dependent of the focal length

f

although the

f

appears in Eqs. (15) and (16) explicitly. Actually, if we complete the integral in Eq. (15) through a numerical approach, the same result would be obtained whatever the

f

is set to. This fact indirectly reveals the correctness of these proposed mathematical equations since the independence of the overall transmittances of the FWMI on the

f

should be obvious according to the physical concept.

3.2 Effects from practical conditions or imperfections

The above model provides a general analytical methodology to estimate the transmittance and SDR of the FWMI. One can notice that the spectral discrimination characteristic of the FWMI can be changed by the spectrum and divergent angle of the backscatter signal, the transmission function of the interferometer, and the spectral relationship between the backscatter signal and the transmission function of the interferometer. Any practical imperfection or condition would affect the spectral discrimination performance of the FWMI through affecting at least one of these three aspects.

3.2.1 Spectrum and divergent angle of the backscatter signal

The spectrum of the backscatter signal contains the contributions from the aerosol scattering and atmospheric molecules scattering. These two types of backscatter signals can both be approximated by Gaussian spectrum as is expressed by Eq. (12). The amplitudes of their spectrums will not attach any influence to the transmittances of the FWMI since the definition of the transmittance is normalized. The spectral width of the aerosol scattering signal is nearly the same as the transmitted laser because of the nature of elastic scattering while that of the molecular scattering signal varies with different atmospheric temperatures and pressures, which can be evaluated from the standard atmospheric model with high accuracy.

The divergent angle of the backscatter signal is a key factor that we should consider in the model. Although the purpose of the development of the FWMI is to resist the influence from the divergent beam, this resistance is yet limited with the increasing of the divergent angle. Therefore, estimating the performance of the FWMI over varieties of divergent illuminations takes over an important position in this model. This is indeed the reason why we emphasize the angular dependence in the equations in section 3.1.

3.2.2 Transmission function of the interferometer

It is apparent that the transmission function of the FWMI itself determines the spectral discrimination performance directly. Many practical imperfections will affect the transmission function by virtue of affecting the amplitudes and the phase difference of the two interference signals from Eq. (14). Here, we will describe some typical conditions that may change the transmission function of the FWMI, and illustrate the usages of our model for the performance estimation in these cases.

a) Anti-reflection (AR) coating imperfection and the absorption of glass

The interfaces between different mediums of the FWMI have AR coatings to make the system more photon efficient. A perfect AR coating will transmit the entire incident signal with no reflection. However, practical AR coating will have a non-zero reflectance such that part of the incident signal will be reflected back and interferes with the primary signals. Because of the dramatic difference in the intensities, the reflected signals by AR coatings are likely to act as background noises that degrade the visibility of the interference fringes. Also, ideal glass will transmit light without loss but real glass does have absorption which will cause a transmission loss. The imperfection of the AR coating as well as the absorption of the glasses will be introduced into our mode to evaluate the transmission characteristic of the FWMI.

We define reflectance of the AR coating as $r_{A R}$ and the absorption coefficient of glass as $α_{g}$ . When light travels through the interferometer, a portion of amplitude of $r_{A R}$ will be reflected every time it hits the AR coatings and the irradiance will be reduced to $\exp (- α_{g} d)$ of the original when it travels a distance $d$ through the glass. Based on this simple fact, an iterative procedure similar to ray tracing is required to determine the outputted background intensity due to the multi-reflection induced by the AR coating and the glass absorption. As for the general FWMI structure described in Fig. 1, there exist five interfaces that need to have AR coating. Assuming that the incident amplitude hitting the FWMI is unity, we can firstly substitute the primary output intensities from the two arms of the FWMI (the output intensities that get to the output port straightforwardly) into Eq. (11) to get the transmission function resulting from the optical interference. Then we need to calculate the background intensity superimposed on the interference fringe which is caused by multi-reflection of the AR coatings. The ultimate transmission function of the FWMI would be the sum of interference intensity, which is a function of light frequency, and the background intensity, which is a DC term without dependence on the optical frequency component. Obviously, this DC term would deteriorate the frequency selectivity of the interferometer considerably since it contains the aerosol signal and molecular signal simultaneously. Having obtained the new transmission function, the transmission characteristic of the FWMI can be gotten following the procedures described by Eqs. (14)-(17).

b) Dimension deviation due to glass fabrication and temperature drift

The OPD is one of the most important parameters which determine the transmission function of the FWMI. Many factors can make the OPD deviate from the designed one such as glass fabrication error, temperature drift, etc.

The transmission function variation due to the dimension deviation is very straightforward from Eq. (2). Therefore, we will not describe this situation in more detail. The temperature drift can induce variations of the glass length and refractive index, thus change the OPD of the FWMI. The thermal dependence of the glass arm length can be expressed as

L = L_{0} (1 + α \cdot Δ T),

where,

L_{0}

is the original length,

α

is the thermal expansion coefficient, and

Δ T

is the temperature change. Additionally, we can evaluate the temperature dependence of the refractive index of glass by the empirical formula below [19]:

Δ n_{g} = \frac{n_{g}^{2} - 1}{2 n_{g}} [D_{0} Δ T + D_{1} Δ T^{2} + D_{2} Δ T^{3} + \frac{E_{0} Δ T + 2 E_{1} Δ T^{2}}{λ^{2} - λ_{t k}^{2}}],

where,

D_{i}

and

E_{i}

are dispersion constants of the glass that are easily available from the glass manufactures;

Δ T

is the temperature deviation from the reference temperature (

20^{°} C

). As for the refractive index of air, it can be modeled as [19]

n_{a i r} = 1 + \frac{(6432.8 + \frac{2949810 λ^{2}}{146 λ^{2} - 1} + \frac{25540 λ^{2}}{41 λ^{2} - 1}) 1.0 \times 10^{- 8} P}{1 + (T - 15) \cdot (3.4785 \times 10^{- 3})},

where,

T

and

P

are the temperature and pressure of the air, respectively, and the wavelength

λ

and the pressure

P

are in

μ m

and

a t m

, respectively. Once these new dimensions and refractive indices of glasses are evaluated in respect of a given temperature drift, the new transmission function of the FWMI will be also obtained directly from Eq. (11).

c) Cumulative wavefront error

We use the term cumulative wavefront error to refer to the total wavefront errors contributed by many mixed wavefront imperfections such as the figure of each optical surface, inhomogeneity of refractive index, and even the parallelism between the two arms, etc. When there is no cumulative wavefront error, the interference fringe produced by the FWMI can be perfectly dark. However, the cumulative wavefront error will attach different OPD distortion values to rays that pass through different aperture coordinates. In this case, the local transmission function is the mean transmittance over the whole wavefront aperture, that is,

ℓ_{i} (θ) = 〈 ℓ_{p_{i}} (θ, x, y) 〉,

where,

〈 \cdot 〉

denotes the average operation over the whole wavefront aperture;

ℓ_{p_{i}} (θ, x, y)

is the new local transmission function which is dependent of incident angle and wavefront aperture, and it takes the form

ℓ_{p_{i}} (θ, x, y) = I_{1} + I_{2} - 2 \sqrt{I_{1} I_{2}} \exp [- {(\frac{π r_{i}}{F S R (θ_{t})})}^{2}] \cos [Δ ϕ (θ) + Δ W (x, y)],

where,

Δ W (x, y)

is the cumulative wavefront error. Introducing Eq. (22) into Eq. (21), we have

\begin{array}{l} ℓ_{i} (θ) = I_{1} + I_{2} - 2 \sqrt{I_{1} I_{2}} \exp [- {(\frac{π r_{i}}{F S R (θ_{t})})}^{2}] \times \\ [\cos Δ ϕ (θ) 〈 \cos Δ W (x, y) 〉 - \sin Δ ϕ (θ) 〈 \sin Δ W (x, y) 〉] . \end{array}

Equation (23) gives the rectified expression of the local transmission function considering the cumulative wavefront error. It deserves mentioning that in our previous work [14], we occasionally find that the transmittances of the FWMI only depend on the root-mean-square (RMS) of the wavefront error in the case of normal incidence but nearly show no relationship with the specific form of $Δ W (x, y)$ according to some numerical evaluations. Now we can extend this conclusion to more general cases mathematically from Eq. (23). Since $Δ W$ is often very small, $〈 \cos Δ W 〉 \approx 〈 1 - Δ W^{2} / 2 〉 = 1 - R M S^{2} / 2$ and $〈 \sin Δ W 〉 \approx 〈 Δ W 〉 = 0$ . So the independence of $ℓ_{i} (θ)$ on the specific distribution of $Δ W$ is obvious. It is the RMS of $Δ W$ that determines the transmittance of the FWMI.

d) Angular error of tilted placement

As is indicated above, our current FWMI is intended to operate with a small tilted angle $θ_{t}$ to avoid cross-talk between different receiver channels due to the light from the reflection channel. We have improved the design method specially to take the tilted angle into consideration in section 2. However, there must be angular placement error relative to the theoretically specified one because of the inevitable adjustment error. This error should be incorporated into our model in order to provide some guidelines for determining the operation condition of the FWMI application.

3.2.3 Spectral relationship between the backscatter signal and the interferometer

In practical application of the FWMI, it is important to match the frequency at the valley transmission of interferometer to the central frequency of the transmitted laser to suppress the aerosol spectrum as much as possible. In the derivation of Eq. (11), we assume that these two frequencies are completely matched, which is revealed by Eq. (9). More often than not, an electro-optic servo loop needs to be employed to achieve satisfactory frequency match. However, due to the optical noise, electronic noise and the ambient turbulent, it is not possible to lock the interferometer to the laser completely. A frequency displacement $Δ υ_{L}$ will exist between the valley of FWMI spectral function and the laser central frequency. In this case, a new term $2 π \cdot Δ υ_{L} / F S R (θ_{t})$ should be added to $Δ ϕ (θ)$ in Eq. (11). Then, the effects from this frequency mismatch on the performance of the FWMI can be readily evaluated based on the method introduced in section 3.1.

4. Application of the proposed theoretical framework and result

In section 2, we have described the particular design theory motivated by the operation principle of the FWMI in HSRL system. Furthermore, we introduce a comprehensive model which provides a general mathematical interface to evaluate the spectral discrimination parameters of the FWMI in section 3. Into this model, many practical conditions or imperfections can be easily integrated to embody the model’s coverage and usage. These two parts compose a complete and independent theoretical framework that is expected to be capable of solving most of the theoretical or engineering problems encountered in the FWMI application. We will give some typical examples to illustrate the implementation of the developed theoretical framework on the study of the FWMI in this section.

4.1 Systematic guideline for the design and fabrication of FWMI

One of the most important tasks of this theoretical framework is to provide systematic guideline for the design and fabrication of a practical FWMI. One can see from section 2 that, the FOPD should be determined before choosing the glass materials and calculating the lengths of arms. Since the FOPD is directly related to the FSR of the FWMI from Eq. (11), its determination must be based on our basic requirements for the spectral discrimination.

Assume that all the imperfections that can degrade the performance of the FWMI are absent completely and do not consider the tilted operation for the time being. Thus, the overall transmittance can be simplified as from Eq. (15)

T_{i} = \frac{1}{2} - \frac{1}{2} \exp [- \frac{π^{2} γ_{i}^{2}}{{(c / F O P D)}^{2}}] .

We will illustrate the process for designing the FWMI that operates in the HSRL system using laser wavelength of 532nm. In this case, the half width of the signal spectrum scattered by atmospheric molecules from the altitudes between 1Km and 10Km is among the range from about 1.3GHz to 1.5GHz [20]. Without loss of generality, we can choose

γ_{m} = 1.4

GHz for the design of the FWMI working at 532nm, and

γ_{a} = 0.05

GHz for our laser line width of 100 MHz. Tendencies of the molecular transmittance and SDR with respect to the FOPD can be calculated by Eq. (24) as shown in Fig. 2. Obviously, the molecular transmittance and the SDR exhibit inverse variation when the FOPD changes. The fact is that, we would like a large molecular transmittance to ensure a high signal-to-noise ratio (SNR) and a large SDR to ensure a good spectral discrimination simultaneously. This implies that the FOPD should be carefully determined to reach a compromise between these two quantities. According to our previous research, we suggest considering a suitable SDR for higher molecular transmittance instead of using unnecessarily high SDR when designing a spectral discrimination filter for HSRL [12]. To this end, we prefer to adopt an FOPD of 100mm as a trade-off, yielding a theoretically best SDR of about 322 and a molecular transmittance of about 44% (the limited molecular transmittance is 50%). Often, the molecular transmittance is very stable under different conditions once the FOPD is determined because the molecular spectrum width is comparable to the FSR of the FWMI (this conclusion can also be verified easily by the proposed model). Any practical imperfection would deteriorate the performance of the FWMI in different levels, manifested by the decrease of the SDR. Therefore, the FOPD of 100mm can be considered to produce a very good transmission to the molecular signal and make a great allowance for tolerance budget at the same time.

Fig. 2 Variations of SDR and molecular transmittance with respect to FOPD under the best operation condition with no imperfections.

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The next step is to choose the glass materials and determine their dimensions. For the design of the hybrid-structure FWMI, Eqs. (2),(6) and (7) should be used. These three linear equations with three unknowns $[d_{1}, d_{2}, d_{3}]$ would be easily solved for any given glass pair. For designing the pure-structure FWMI, only Eqs. (2) and (6) need be adopted since this structure does not hold the characteristic of thermal compensation. From this point, we can see that the design of the FWMI needs not to pre-select the glass types as is often necessary for the design of the WAMI. Of course, maybe there also exist some realistic limitations for the glass choosing such as the availability of the materials, the process technology for a specified glass, and the upper limitation for the glass dimension. Table 1 shows several typical results for the designs of the hybrid-structure FWMI and the pure-structure FWMI by enumerating the glasses in Schott Glass Library (SGL). In the designs, the tilted angle is chosen to be 1.5 degree (this angle can be determined flexibly based on the specific optical configuration of HSRL receiver system, and we recommend 1.5 degree as a good one for most cases). To present an intuitive understanding for these results, their OPD variations with the incident angle and temperature drift are plotted in Fig. 3. We can see from Figs. 3(a) and 3(c) that the OPD variations among the angular range $(- 5^{°}, 5^{°})$ are less than 0.3 $λ$ for all the five schemes listed in Tables 1, resulting from the field widening design. Comparing Fig. 3(b) with Fig. 3(d), one can find that the OPD variation of the hybrid-structure FWMI induced by temperature drift of one Celsius is only with the order of about 10^-3 $λ$ while that for the pure-structure FWMI is considerable (~ $5 λ$ ). Apparently, this is the advantage of thermal compensation design.

Table 1. Three typical design results for hybrid-structure FWMI with glass materials from SGL are numbered with H*. The H1 scheme has the shortest arm lengths, the H2 one is featured with the largest field-widened angle, and the H3 holds the best thermal stability among all the possible hybrid-structure FWMIs from SGL. Typical design results for pure-structure FWMI are denoted with P*. The P1 scheme has the shortest arm lengths and the largest field-widened angle simultaneously, and the P2 one is featured with the best thermal stability among all the possible pure-structure FWMIs from SGL.

View Table

Fig. 3 OPD variations with respect to the incident angle and temperature drift for the FWMI configurations in Table 1. (a) and (b) are the results for the hybrid-structure FWMIs, and (c) and (d) are that for the pure-structure FWMIs.

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Having determined the glass materials and the corresponding dimensions of the interference arms, the following procedure is to estimate the tolerance budget in order to establish some necessary demands for the fabrication of a practical interferometer. Without loss of generality, let us take the scheme H1 and P2 as examples. The tolerance estimations on the glass dimension, the cumulative wavefront error RMS and the reflectivity of the AR coating can be obtained according to the relevant descriptions in section 3; results are presented in Fig. 4. It can be seen that the FWMIs do not put forward too strict challenge for the glass dimension machinery from Figs. 4(a) and 4(b). A machining accuracy of 0.01mm is enough to ensure the SDRs to stay above 250. Figure 4(c) shows the influence from three different cumulative wavefront errors on the SDR of FWMI. Obviously, the SDR only depends on the RMS rather than the specified form of the wavefront error, as is predicted by Eq. (23). From Fig. 4(c), we find that this error source exerts prominent deterioration on the performance of the FWMI. For instance, cumulative wavefront error with RMS of 0.05 $λ$ would make the SDR decrease to about 12, which is intolerable in practical case. Fortunately, it does not mean that the demand for the glass surface figure is also be of such high level. As is illustrated before, the cumulative wavefront is the total wavefront affected by many imperfections. Wavefront errors from different error sources may cancel out to some extent, and the total wavefront error is hopeful to get to very satisfactory condition. In fact, the glass arms of the FWMI can be fabricated into cylinder which allows us to rotate the aperture to realize the best surface figure match. Also, the PZT can be adjusted to facilitate the wavefront error compensation. All these approaches can be carried out iteratively until the RMS reaches a tolerable result indicated by the model. In our FWMI prototype, the ultimate cumulative wavefront RMS can be less than 0.009 $λ$ when the incident divergence is 0.5 degree (the experimental results about this FWMI will be presented in our future publication). Figure 4(d) demonstrates the SDR variation with respect to the reflectivity of the AR coating under three different glass absorption coefficients. Apparently, the imperfection of AR coating is another dominate factor that would degrade the performance of the FWMI while the effect from the glass absorption can be negligible because the plots corresponding to different $ρ$ are overlapped together. The AR coatings with low reflectivity (such as <0.05%) are suggested. At last, it is obvious but still necessary to emphasize that the tolerance budgets for the hybrid-structure FWMI and the pure-structure FWMI are very coincident according to Fig. 4.

Fig. 4 Tolerance estimations for the glass dimension, the cumulative wavefront error RMS and the reflectivity of the AR coating. (a) and (b) show the SDR variation with respect to dimension deviation for the H1 scheme and P2 scheme respectively, (c) is the SDR variation with respect to three different cumulative wavefront errors for both the H1 and P2 schemes, and (d) is that with respect to the reflectivity of the AR coating.

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4.2 Pragmatic reference for the operation of FWMI

After the FWMI is fabricated, its operation conditions, such as the divergent angle of the incident beam, the ambient temperature stability, the frequency locking accuracy, etc., also impact the practical spectral discrimination performance. Following the design examples in section 4.1, we will also take the H1 and P2 schemes into analysis in this part to illustrate the function of the proposed theoretical framework.

Figure 5(a) presents the relationship between the SDR and the half divergent angle of the incident beam. It can be found that both the two schemes have the similar field widening performance from the perspective of the SDR, which has already been demonstrated intuitively in Fig. (3). Also, it is noticeable that the SDR keeps very stable when the divergent angle is less than 1 degree. Based on this plot, it is convenient to determine the allowed divergent angle for the incident beam. For example, we can consider increasing the signal divergence to about 0.5 degree yet without any degradation of the SDR by employing the FWMI (larger divergence is allowed but not necessary in reality). Note that, the field widening characteristic is the basic distinction between FWMI and ordinary MI. Therefore, we actually adopted a fixed divergent angle of 0.5 degree to implement tolerance budgets in section 4.1, and we will continue this condition in the following analysis to make conclusions more significant.

Fig. 5 Influence analysis of some practical imperfections and conditions on the SDR of the FWMI based on the proposed framework. (a)-(d) are the corresponding results for the divergent angle of the incident beam, the temperature drift, the frequency locking error and the angular error of tilted placement respectively.

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Figure 5(b) shows the thermal dependence of the SDR for both H1 scheme and P2 scheme. The phenomenon there would be very easy to be interpreted. Due to the thermal compensation from the two different glass arms, the SDR of the hybrid-structure FWMI nearly stays unchanged when the working temperature drifts by 0.5 Celsius whereas that of the pure-structure FWMI decreases sharply. From this analysis, conclusion can be made that for the application of the pure-structure FWMI, temperature stability should be controlled severely (such as within 0.05 Celsius). However, temperature control is much looser for the application of the hybrid-structure FWMI.

The performance of the FWMI with respect to the frequency locking error is shown in Fig. 5(c). We can see that the frequency locking error results in nearly the same performance degradation for the hybrid-structure and the pure-structure FWMIs. What’s more, it can be found that the frequency locking error impairs the performance of the FWMI seriously. For instance, a locking error of 0.03GHz would make the SDR downgrade to about 182 (forty percent decrease related to the theoretically best one). This discussion is beneficial for us to determine the basic requirement for the design of frequency locking system of the FWMI.

Another problem we should consider is the tilted placement angle for the FWMI operation since it determines the angular range of the incident beam. As is discussed before, the FWMI is designed based on a pre-chosen tilted angle of 1.5 degree. However, it is inevitable to introduce some placement deviation in reality due to the adjustment error. Figure 5(d) indicates the impact of the tilted angle on the SDR of the FWMI. Obviously, the two types of FWMI hold the same sensitivity to the tilted placement error. It is interesting that the negative deviation of the tiled angle almost imposes no influence on the SDR whereas the positive one conspicuously makes the SDR decrease. This is because the positive tilted angle deviation moves the practical incident range far from the normally incident condition while the negative one among a small range would make more rays enter the FWMI near the normal condition. It can be expected that if the negative angular deviation becomes larger than some threshold, the SDR would also start to fall down. Fortunately, this characteristic of the FWMI in response to the angular error of tilted placement provides a good tolerance for the adjustment accuracy. For example, the deviation of one degree would not change the SDR apparently.

4.3 Powerful tool for the comprehensive performance estimation of the FWMI

When taking only one factor that can affect the performance of the FWMI into the model in a single analysis, we can study the specified imperfection thoroughly and thus, reasonable performance budget can be made which is an indispensable procedure in the FWMI design and operation. When the performance budgets for all the concerned imperfections or conditions have been made, it is also necessary to validate the total performance or re-allocate the budgets (if needed) by estimating the SDR from the comprehensive effects induced by many factors simultaneously. Since the proposed framework explicitly incorporates so many practical factors, it is a powerful tool for this comprehensive performance estimation of the FWMI. We can even program this framework into a versatile script, by which, many practical imperfections or conditions can be analyzed jointly according to our requirements. Herein, we will exemplify this ability of the proposed theoretical framework through simultaneously discussing three of the above-mentioned imperfections that degrade the SDR of the FWMI mostly, that is, the cumulative wavefront error, the frequency locking error and the reflectivity of the AR coatings; ultimate results are presented in Fig. 6. We point out that the data in Fig. 6 is only associated with the H1 scheme in Table 1, and that for the P2 scheme will not be shown here again since they are expected to be very similar based on the above discussions.

Fig. 6 Comprehensive estimations of the SDR considering the cumulative wavefront error, the frequency locking error and the AR coating imperfection simultaneously.

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From Fig. 6, it is obvious that any condition that deviates from the perfect point (the original point of the 3D coordinate system) would degrade the SDR of the FWMI. There exists a specific area where the SDR is in satisfactory levels. This area gives the suggestion for the basic requirements for these three imperfections. For example, if the frequency locking error can be keep within 10MHz, the RMS of the cumulative wavefront is ensured less than 0.01 $λ$ and the reflectivity of the AR coatings is less than 0.05%, the SDR can stay above 130, which is still a very good result according to our previous study in [12]. Generally, these would not be challenging requirements. From this illustration, it can be seen that this theoretical framework can be a powerful tool for us to re-adjust or compromise the performance budgets in order to guarantee a comprehensively acceptable SDR considering our fabrication ability and available technique. Certainly, more aspects can be further added in according to the pragmatic needs.

5. Conclusion

Due to the widened FOV, the FWMI is expected to be a good candidate as the spectroscopic filter in HSRL application, which would enhance the photon efficiency of the instruments. An independent theoretical framework about this application of the FWMI, which includes the foundation for parameters design and a performance estimation model, is described in this paper. The parameter determination process for the FWMI is motivated by comparisons on the operation principles with its counterpart in atmospheric wind and temperature detection, that is, the WAMI. Mathematical formulas are given to concretize the design. In the performance estimation model, a general method to evaluate the transmittances and the SDR of the FWMI is proposed. This model can be employed to analyze the comprehensive performance of the FWMI even under many practical imperfections or conditions. Discussions incorporating such realistic factors which may degrade the performance of the FWMI as the divergent angle of the backscatter signal, the AR coating imperfection and the absorption of glass, the cumulative wavefront error, the frequency locking error, etc., are made to illustrate the implementation of the modeling. Applications of the proposed theoretical framework are exemplified in detail. It can been found that the theoretical framework presents a complete and powerful tool for solving most of theoretical or engineering problems encountered in the FWMI development such as the designing, parameter calibration, prior performance budget, posterior performance estimation, and so on. It will be a valuable contribution to the lidar community to develop new generation of HSRLs based on the FWMI spectroscopic filter.

Acknowledgment

This work was partially supported by the National Natural Science Foundation of China (41305014, 11275172, 61475141), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20130101120133), the Aviation Science Funds (20140376001), the Fundamental Research Funds for the Central Universities (2013QNA5006), the Zhejiang Department of Education Research Program (Y201329660), the Zhejiang Key Discipline of Instrument Science and Technology (JL130113), and the State Key Lab. of Modern Optical Instrumentation Innovation Program (MOI2015QN01).

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NO.	Arms	Pure-Glass Arm	Hybrid Arm
H1	Material	N-SF66	Air gap	P-SF68
H1	Length/mm	53.2230	19.1608	16.7970
H2	Material	N-PK52A	Air gap	N-SF66
H2	Length/mm	87.2380	13.6854	86.2420
H3	Material	N-LASF46A	Air gap	SF57
H3	Length/mm	91.4330	16.3548	58.4220
P1	Material	P-SF68	Air gap	\
P1	Length/mm	32.7670	16.2143	\
P2	Material	N-SF66	Air gap	\
P2	Length/mm	35.1860	18.1622	\

Field-widened Michelson interferometer for spectral discrimination in high-spectral-resolution lidar: theoretical framework

Abstract

1. Introduction

2. Theoretical foundations for designing the FWMI

2.1 General optical configuration of the FWMI

2.2 Physical motivations

2.3 Mathematical equations for the required conditions

3. Practical performance estimation model

3.1 General estimation for the spectral discrimination characteristic of an FWMI

3.2 Effects from practical conditions or imperfections

3.2.1 Spectrum and divergent angle of the backscatter signal

3.2.2 Transmission function of the interferometer

a) Anti-reflection (AR) coating imperfection and the absorption of glass

b) Dimension deviation due to glass fabrication and temperature drift

c) Cumulative wavefront error

d) Angular error of tilted placement

3.2.3 Spectral relationship between the backscatter signal and the interferometer

4. Application of the proposed theoretical framework and result

4.1 Systematic guideline for the design and fabrication of FWMI

4.2 Pragmatic reference for the operation of FWMI

4.3 Powerful tool for the comprehensive performance estimation of the FWMI

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (6)

Tables (1)

Equations (24)

Optics Express