Abbas Chokor
email: abbas.chokor@utc.fr

Reine Talj
email: reine.talj@hds.utc.fr

Ali Charara
email: ali.charara@hds.utc.fr

Hassan Shraim
email: hassan.shraim@gmail.com

Clovis Francis
email: cfrancis@ul.edu.lb

Active Suspension Control to Improve Passengers Comfort and Vehicle's Stability

whether they are published or not. The documents may come



Abstract-This paper aims to control roll, pitch and vertical displacement of the suspended mass in a passenger vehicle during on-road drive by means of controlling the active suspension system. One of the goals of this control is to isolate the suspended mass in the vehicle from any road perturbation to ensure a big amount of comfort to the passengers (less roll, less pitch and less vibrations). Two strategies are used: the first one is based on the adaptive super-twisting second order sliding mode control, while the second one lies on the Lyapunov theory. We validate both strategies on a double lane change test, and a slalom steering test at 100km/h. For this purpose, in addition to the active suspension control, we develop a full-vehicle model that presents reliable results in many situations during maneuver like road-bends, bumps, braking and accelerations.

I. INTRODUCTION

A vehicle may maneuver in normal situation (straight road, bends, bumps) or faces critical situations like avoiding other vehicles and hard cornering. In all these cases the driver may behave in a wrong way; sudden or hard steering will create a lateral acceleration on the vehicle's body which leads to an aggressive roll rotation of the suspended mass; sudden braking or hard accelerating will create a longitudinal acceleration which leads to a pitch rotation of the suspended mass. Besides these actions, and in addition to the road perturbations a vertical displacement may occur on the suspended mass. The latter is then exposed to roll, pitch and vertical displacement.

The suspension system acts on each corner of the vehicle to attenuate the negative effects of these behaviors and to guarantee a specific performance of the suspended mass. The choice of the kind and the parameters of the suspension system can modify this performance. There are three kinds of suspension system, the passive, semi-active and active suspension system which are described in [START_REF] Poussot-Vassal | Robust linear parameter varying multivariable automotive global chassis control[END_REF].

In this paper, we will discuss the control of the active suspension system formed by a spring, a damper and an actuator to develop an added active force in order to balance the undesirable loads transfer to reject roll, pitch and vertical displacement of the suspended mass in a passenger vehicle, which leads to isolate the chassis from any perturbation and improves passengers comfort. To do that, we will develop in Section II a credible full vehicle model that matches with many situations during maneuver like road-bends, bumps, braking and accelerations. This model will be used to validate the performances of our controllers developed in Section III, the first controller is based on the adaptive super-twisting second order sliding mode control; and the second one is based on Lyapunov's equations. The goal of these controllers is to isolate the chassis from any road perturbation by ensuring less roll, pitch and vertical displacement of the suspended mass. In Section IV we will validate these controllers using Matlab/Simulink. Finally, conclusion and perspectives are presented in Section V.

II. FULL VEHICLE MODEL

A. Vertical model 1) Quarter vehicle vertical model

Fig. 1 shows the quarter vehicle vertical model, where 𝑚 𝑠 , 𝑚 𝑢𝑠 , 𝑘 𝑡 , 𝑐 𝑡 , 𝑘 𝑠 , 𝑐 𝑠 , U, 𝐹 𝑠 , and 𝐹 𝑧 , are respectively, the quarter vehicle suspended mass, unsuspended mass, tire stiffness coefficient, tire damping coefficient, suspension stiffness coefficient, suspension damping coefficient, actuator force of the active suspension system (active force), total suspension force and vertical force on the wheel. 𝑧 𝑠 𝑎𝑛𝑑 𝑧 𝑢𝑠 are respectively the vertical displacement of the suspended mass and the unsuspended mass, 𝑧 𝑟 𝑎𝑛𝑑 𝑔 are respectively the vertical profile of the road and the gravitational constant.

Around the equilibrium, the equations that describe the system variations are: 𝐹 𝑠 = -𝑘 𝑠 (𝑧 𝑠 -𝑧 𝑢𝑠 ) -𝑐 𝑠 (𝑧̇𝑠 -𝑧̇𝑢 𝑠 ) + 𝑈

(1) 𝑧̈𝑢 𝑠 = -𝑚 𝑢𝑠 𝑔 -𝐹 𝑠 + 𝐹 𝑧

(2) 𝐹 𝑧 = -𝑘 𝑡 (𝑧 𝑢𝑠 -𝑧 𝑟 ) -𝑐 𝑡 (𝑧̇𝑢 𝑠 -𝑧̇𝑟)

Active Suspension Control to Improve Passengers Comfort and Vehicle's Stability

𝐴. 𝐶ℎ𝑜𝑘𝑜𝑟 1 , 𝑅. 𝑇𝑎𝑙𝑗 1 , 𝐴. 𝐶ℎ𝑎𝑟𝑎𝑟𝑎 1 , 𝐻. 𝑆ℎ𝑟𝑎𝑖𝑚 2 and 𝐶. 𝐹𝑟𝑎𝑛𝑐𝑖𝑠 2 .

Fig.1 quarter vehicle vertical model

To express 𝑧 𝑠 and 𝑧̇𝑠, we need to develop the full vehicle vertical model

2) Full vehicle vertical model

In order to express each corner vertical displacement, and to express the roll, pitch and center of gravity vertical displacement, we will develop, in the following, a full vehicle vertical model (see Fig. 2).

Equations ( 1), ( 2) and ( 3) are valid for each corner, let 𝑖 = {𝑓, 𝑟} index for front and rear, and 𝑗 = {𝑟, 𝑙} index for right and left respectively.

where 𝑡 𝑓 , 𝑡 𝑟 , 𝑙 𝑓 and 𝑙 𝑟 are respectively half front track, half rear track, wheelbase to the front and wheelbase to the rear. 𝑧 𝑠 ̇𝑖𝑗 are calculated by time derivative the four above equations.

𝑧 𝑠 ̇𝑓𝑟 = 𝑧̇-𝑡 𝑓 𝜃 ̇𝑐𝑜𝑠 𝜃 -𝑙 𝑓 𝜑̇𝑐𝑜𝑠 𝜑 (11)

𝑧 𝑠 ̇𝑓𝑙 = 𝑧̇+ 𝑡 𝑓 𝜃 ̇𝑐𝑜𝑠 𝜃 -𝑙 𝑓 𝜑̇𝑐𝑜𝑠 𝜑 (12) 𝑧 𝑠 ̇𝑟𝑟 = 𝑧̇-𝑡 𝑟 𝜃 ̇𝑐𝑜𝑠 𝜃 + 𝑙 𝑟 𝜑̇𝑐𝑜𝑠 𝜑 (13) 𝑧 𝑠 ̇𝑟𝑙 = 𝑧̇+ 𝑡 𝑟 𝜃 ̇𝑐𝑜𝑠 𝜃 + 𝑙 𝑟 𝜑̇𝑐𝑜𝑠 𝜑 (14)
The dynamics 𝜃 ̈, 𝜑̈ 𝑎𝑛𝑑 𝑧̈ of the suspended mass are determined using the suspensions' forces 𝐹 𝑠 𝑖𝑗 , the effect of the lateral acceleration 𝑎 𝑦 and the longitudinal acceleration 𝑎 𝑥 . 𝑀 𝑠 , ℎ θ 𝑎𝑛𝑑 ℎ 𝜑 are respectively the total suspended mass, the distance between the center of gravity of the suspended mass and the roll rotation center and the distance between the center of gravity of the suspended mass and the pitch rotation center. 𝐼 𝑥 and 𝐼 𝑦 are respectively the moment of inertia of the suspended mass around x axis and y axis. To calculate 𝑎 𝑥 𝑎𝑛𝑑 𝑎 𝑦 , we need to develop the model of the lateral and longitudinal vehicle dynamics.

θ

B. Lateral and longitudinal vehicle models

Let 𝛿 𝑓𝑙 𝑎𝑛𝑑 𝛿 𝑓𝑟 the front wheels steer angles (left and right respectively), let 𝐹 𝑥 𝑖𝑗 𝑎𝑛𝑑 𝐹 𝑦 𝑖𝑗 respectively the longitudinal and lateral forces applied to the wheel ij (see Fig. 3), 𝐹 𝑥 𝑖𝑗 𝑎𝑛𝑑 𝐹 𝑦 𝑖𝑗 depend on 𝐹 𝑧 𝑖𝑗 and are calculated from the Dugoff tire forces model presented in [START_REF] Rajamani | Vehicle dynamics and control[END_REF]. where 𝑎 𝑥 , 𝑎 𝑦 𝑎𝑛𝑑  ̈ are respectively the longitudinal acceleration, the lateral acceleration, and the yaw acceleration, M is the total vehicle mass and 𝐼 𝑧 is the moment of inertia of the vehicle around z axis.

𝑥̈= 𝑦 ̇+ (𝐹

III. CONTROLLERS DESIGN

In order to control roll, pitch and vertical displacement, we notice that the control inputs in (15), ( 16) and ( 17) are inside 𝐹 𝑠 𝑖𝑗 , they are the active forces (actuators outputs 𝑈 𝑖𝑗 ) described Then, the system to be controlled, given by the equations ( 15), ( 16) and ( 17), can be rewritten as:

{ θ ̈= {𝐸 𝜃 + 𝑀 𝑠 (ℎ θ + 𝑧)𝑎 𝑦 } 𝐼𝑥 ⁄ 𝜑̈= {𝐸 𝜑 + 𝑀 𝑠 (ℎ 𝜑 + 𝑧)𝑎 𝑥 } 𝐼𝑦 ⁄ 𝑧̈= {𝐸 𝑧 -𝑀 𝑠 𝑔}/𝑀 𝑠 (27)
Where 𝐸 𝜃 , 𝐸 𝜑 𝑎𝑛𝑑 𝐸 𝑧 are the new control inputs to the system. In section (III.A.1) and (III.A.2) we will develop two higher level controllers, respectively, the adaptive super twisting second order sliding mode control and a controller based directly on the Lyapunov's theory, where 𝐸 𝜃 , 𝐸 𝜑 𝑎𝑛𝑑 𝐸 𝑧 are the higher level control inputs to control the state variables θ, 𝜑 𝑎𝑛𝑑 𝑧 𝑠 and let them follow respectively the desired reference variables θ 𝑑𝑒𝑠 , 𝜑 𝑑𝑒𝑠 𝑎𝑛𝑑 𝑧 𝑑𝑒𝑠 . In section (III.B) we will develop the low level controller where 𝑈 * 𝑖𝑗 are the control inputs to the system. Let 𝑒 𝜃 = 𝜃 -𝜃 𝑑𝑒𝑠 (28)

𝑒 𝜑 = 𝜑 -𝜑 𝑑𝑒𝑠 (29)
𝑒 𝑧 = 𝑧 -z 𝑑𝑒𝑠 (30)
the errors between the three states and their corresponding desired variables.

A. Higher level controllers 1) Adaptive super twisting second order sliding mode control

This controller aims at reducing the roll, pitch and vertical displacement errors and their time derivatives, their corresponding sliding variables are chosen as follows:

𝑠 θ = 𝑒̇𝜃 + 𝑘 𝜃 𝑒 𝜃 (31)
𝑠 𝜑 = 𝑒̇𝜑 + 𝑘 𝜑 𝑒 𝜑 (32)

𝑠 z = 𝑒̇𝑧 + 𝑘 𝑧 𝑒 𝑧 (33)
where 𝑘 𝜃 , 𝑘 𝜑 𝑎𝑛𝑑 𝑘 𝑧 are positive control parameters to be adjusted.

The control inputs 𝐸 θ , 𝐸 𝜑 𝑎𝑛𝑑 𝐸 𝑧 are in the form:

𝐸 θ =
The super-twisting sliding mode controller is known for its robustness against parameters uncertainties and disturbances. It converges to the sliding surface in finite time. However, it could cause chattering once the equilibrium is reached. For that reason, the controller is made adaptive by multiplying the gains 𝛼 𝑖 𝑎𝑛𝑑 𝛽 𝑖 by

|𝑠 𝑖 | |𝑠 𝑖 |+𝜀 𝑖
, where 𝜀 𝑖 are small positive constants. The aim of such term is to attenuate the controller gains near to the equilibrium, and then reduce chattering; while maintaining the gains far away from the equilibrium to guarantee fast convergence.

2) Controller based on the Lyapunov theory

This controller aims at reducing the roll, pitch and vertical displacement errors, and their time derivatives by ensuring exponential convergences. Let 𝑉 𝑖 =

C. Actuators limitations

The model of the actuator is not developed in this paper. However, in order to make the forces developed by the controller reliable, we need to take into consideration the actuator's limitations [START_REF] Chamseddine | Méthodologie de diagnostic et tolérances aux défauts de systèmes complexes[END_REF]:

 Time response (0.1 sec).  Maximum/minimum force values ( +/-9800N)

IV. SIMULATION AND VALIDATION

In this section, we will validate the two controllers developed in the previous section using two scenarios. All vehicle's and controllers numerical values parameters are defined in [START_REF] Chokor | [END_REF], here in Table 1 we cite the values (SI units) of some important parameters. The first scenario is a double lane change maneuver test after 10 seconds from starting the test (in the first 10 seconds a motor torque is applied to accelerate the vehicle to a desired speed), which is considered as a frequent situation that hardly solicits the lateral dynamics of the vehicle. The second scenario is a slalom test at 100 km/h as starting speed, which is an aggressive situation that could drive the vehicle outside the stability region.

The goal is to isolate the suspended mass from any perturbation, then the desired variables are 𝜃 𝑑𝑒𝑠 = 0; 𝜑 𝑑𝑒𝑠 = 0; 𝑧 𝑑𝑒𝑠 = 0.

A. Double lane change maneuver

The double lane change test is a widespread test, the front wheels steering angle is shown in Fig. 5. Fig. 6 shows the motor torque applied to the rear wheels which can make a considerable acceleration to the body of the vehicle with a big amount of jerk. The speed of the vehicle is shown in Fig. 7, it increases quickly in the first 10 seconds due to high motor torque and then slowly due to low torque. As an influence on the body of the vehicle, we can notice the following. In the first 10 seconds, the suspended mass is exposed to a considerable pitch rotation in the open loop operation mode due to the motor torque applied to accelerate the vehicle (see Fig. 10). We can see clearly that the controllers improve the performance of the suspended mass by eliminating jerk effect, they suppress almost entirely the pitch rotation. In the next 10 seconds, the suspended mass is exposed to a considerable roll rotation in the open loop operation mode due to the double lane change maneuver (see Fig. 9). We can see clearly also the improvement provided by the controllers that drastically decrease the lateral acceleration effect induced by the steering operation and suppress almost entirely the roll rotation. Fig. 11 shows also the vertical displacement in controlled and open loop operation. This vertical displacement is a combination of all perturbations during maneuver and is considered as vibrations to be attenuated in order to ensure passengers comfort. Since the test solicits especially the roll motion, the vertical displacement is not highly disturbed, however, we can see clearly that with the proposed controllers, these vibrations are highly attenuated with respect to the open loop operation. Fig. 8 shows the additive active forces of the suspension system on each corner, developed by the actuators to isolate the suspended mass from any perturbation. The developed forces are far from the saturation limit imposed by the actuators. To explain the controllers output chattering in the first 10 seconds, we notice that the vehicle suspended mass has very weak solicitations in this duration (roll, pitch and vertical displacement dynamics are near to zero), then the active forces demanded by the actuators should be small, but in fact, these forces are calculated by doing the pseudoinverse matrix described in (47) to a vector who has small values near to zero, which will cause problems in the internal dynamics and create these chattering. The slalom steering test is considered as an aggressive test to make sure that the suspension system is still efficient. In our paper we use it to validate that the active suspension system can isolate the suspended mass even under hard situations and at high speed. From this test we can also deduce that if the driver steers suddenly at a high speed avoiding an obstacle, the vehicle could remain stable and under control.

Fig. 12 shows the front wheels steering angle with an increasing amplitude at a frequency of 0.5 Hz which is considered as an aggressive steering. The speed of the vehicle decreases with time due to the aggressive slalom steering without motor torque besides road friction, and is shown in Fig. 13. Fig. 14 shows the additive active forces of the suspension system on each corner, developed by the actuators to isolate the suspended mass from any perturbation. We can notice that both controllers provide almost similar control inputs.

Fig. 15 shows the roll response in the open-loop and closed-loop with both controllers. It is obvious that without controllers the suspended mass has a high roll rotation that reaches 6 degrees. However in the controlled operation mode the roll rotation of the vehicle is highly reduced.

Fig. 16 shows the pitch response of the suspended mass with and without controllers. Note that the pitch angle in the open loop operation case is not highly solicited in this maneuver. However, controlling the pitch angle may be critical in some other scenarios and the two controllers developed show a significant reduction of the pitch rotation with respect to the open loop case.

Fig. 17 shows the vertical displacement of the suspended mass with and without controllers which is not very affected even in the open loop operation mode. The goal of this paper is to isolate the suspended mass from any perturbation by controlling the roll, pitch and vertical displacement of the chassis. Two different controllers have been proposed in this paper, the first is based on the supertwisting sliding mode approach, and the second is based on Lyapunov's theory. The objective is highly reached with both proposed control strategies, which have been validated on two test scenarios that solicit the vehicle in hard situations. We noticed that the active suspension control has high accuracy, it can suppress almost entirely roll, pitch and vertical displacement of the suspended mass in a passenger vehicle. Concerning a comparison between these two approaches, we can deduce that the controller based on Lyapunov theory has more accuracy, due to the chattering coming from manual adjusted gains in the Sliding mode controller. In our next work, we will validate these operated approaches using Scaner Studio simulator, we will also focus on how to improve the lateral dynamics (avoid oversteering and understeering) during maneuver on hard cornering by making a specific roll angle to compensate the loads transfers. We will emphasize the coordination between the active suspension control system and other stabilizing controllers as Active front steering, differential braking, etc., in order to improve their behaviors in a "Global Chassis Control" context.

  𝐹 𝑠 𝑖𝑗 = -𝑘 𝑠 𝑖𝑗 (𝑧 𝑠 𝑖𝑗 -𝑧 𝑢𝑠 𝑖𝑗 ) -𝑐 𝑠 𝑖𝑗 (𝑧̇𝑠 𝑖𝑗 -𝑧̇𝑢 𝑠 𝑖𝑗 ) + 𝑈 𝑖𝑗 (4) 𝑧̈𝑢 𝑠 𝑖𝑗 = -𝑚 𝑢𝑠 𝑖𝑗 𝑔 -𝐹 𝑠 𝑖𝑗 + 𝐹 𝑧 𝑖𝑗 (5) 𝐹 𝑧 𝑖𝑗 = -𝑘 𝑡 𝑖𝑗 (𝑧 𝑢𝑠 𝑖𝑗 -𝑧 𝑟 𝑖𝑗 ) -𝑐 𝑡 𝑖𝑗 (𝑧̇𝑢 𝑠 𝑖𝑗 -𝑧̇𝑟 𝑖𝑗 ) (6) Let 𝜃, 𝜑 𝑎𝑛𝑑 𝑧 be respectively the roll, pitch and vertical displacement of the suspended mass, then each corner vertical displacement 𝑧 𝑠 𝑖𝑗 can be written as a function of these three states variables (𝜃, 𝜑 𝑎𝑛𝑑 𝑧). 𝑧 𝑠 𝑓𝑟 = 𝑧 -𝑡 𝑓 𝑠𝑖𝑛 𝜃 -𝑙 𝑓 𝑠𝑖𝑛 𝜑 (7) 𝑧 𝑠 𝑓𝑙 = 𝑧 + 𝑡 𝑓 𝑠𝑖𝑛 𝜃 -𝑙 𝑓 𝑠𝑖𝑛 𝜑 (8) 𝑧 𝑠 𝑟𝑟 = 𝑧 -𝑡 𝑟 𝑠𝑖𝑛 𝜃 + 𝑙 𝑟 𝑠𝑖𝑛 𝜑 (9) 𝑧 𝑠 𝑟𝑙 = 𝑧 + 𝑡 𝑟 𝑠𝑖𝑛 𝜃 + 𝑙 𝑟 𝑠𝑖𝑛 𝜑

Fig. 3

 3 Fig. 3 Horizontal plan vehicle representation

Fig. 5

 5 Fig. 5 double lane change steer

Fig. 12

 12 Fig. 12 Slalom steer front wheels angle

  ̈= { (-𝐹 𝑠 𝑓𝑟 + 𝐹 𝑠 𝑓𝑙 ) 𝑡 𝑓 + (-𝐹 𝑠 𝑟𝑟 + 𝐹 𝑠 𝑟𝑙 )𝑡 𝑟

	+𝑀 𝑠 (ℎ θ + 𝑧)𝑎 𝑦	} 𝐼 𝑥 ⁄	(15)

𝜑̈= { -(𝐹 𝑠 𝑓𝑟 + 𝐹 𝑠 𝑓𝑙 ) 𝑙 𝑓 + (𝐹 𝑠 𝑟𝑟 + 𝐹 𝑠 𝑟𝑙 )𝑙 𝑟 +𝑀 𝑠 (ℎ 𝜑 + 𝑧)𝑎 𝑥 } 𝐼 𝑦 ⁄ (16) 𝑧̈= {𝐹 𝑠 𝑓𝑟 + 𝐹 𝑠 𝑓𝑙 + 𝐹 𝑠 𝑟𝑟 + 𝐹 𝑠 𝑟𝑙 -𝑀 𝑠 𝑔} /𝑀 𝑠 (17)

  𝑥 𝑓𝑙 𝑐𝑜𝑠 𝛿 𝑓𝑙 + 𝐹 𝑥 𝑓𝑟 𝑐𝑜𝑠 𝛿 𝑓𝑟 + 𝐹 𝑥 𝑟𝑙 + 𝐹 𝑥 𝑟𝑟 -𝐹 𝑦 𝑓𝑙 𝑠𝑖𝑛 𝛿 𝑓𝑙 -𝐹 𝑦 𝑓𝑟 𝑠𝑖𝑛 𝛿 𝑓𝑟 )/𝑀 𝑦̈= -𝑥 ̇+ (𝐹 𝑥 𝑓𝑙 𝑠𝑖𝑛 𝛿 𝑓𝑙 + 𝐹 𝑥 𝑓𝑟 𝑠𝑖𝑛 𝛿 𝑓𝑟 + 𝐹 𝑦 𝑟𝑙 + 𝐹 𝑦 𝑟𝑟 +𝐹 𝑦 𝑓𝑙 𝑐𝑜𝑠 𝛿 𝑓𝑙 + 𝐹 𝑦 𝑓𝑟 𝑐𝑜𝑠 𝛿 𝑓𝑟 )/𝑀  ̈= {-𝑡 𝑓 (𝑐𝑜𝑠 𝛿 𝑓𝑙 𝐹 𝑥 𝑓𝑙 -𝑐𝑜𝑠 𝛿 𝑓𝑟 𝐹 𝑥 𝑓𝑟 + 𝑠𝑖𝑛 𝛿 𝑓𝑟 𝐹 𝑦 𝑓𝑟 -𝑠𝑖𝑛 𝛿 𝑓𝑙 𝐹 𝑦 𝑓𝑙 ) + 𝑙 𝑓 (𝑠𝑖𝑛 𝛿 𝑓𝑙 𝐹 𝑥 𝑓𝑙 + 𝑠𝑖𝑛 𝛿 𝑓𝑟 𝐹 𝑥 𝑓𝑟 + 𝑐𝑜𝑠 𝛿 𝑓𝑙 𝐹 𝑦 𝑓𝑙 + 𝑐𝑜𝑠 𝛿 𝑓𝑟 𝐹 𝑦 𝑓𝑟 ) -𝑙 𝑟 (𝐹 𝑦 𝑟𝑙 + 𝐹 𝑦 𝑟𝑟 ) -

		(18)
	𝑎 𝑥 = 𝑥̈-𝑦 ̇	(19)
		(20)
	𝑎 𝑦 = 𝑦̈+ 𝑥 ̇	(21)
	𝑡 𝑟 (𝐹 𝑥 𝑟𝑙 -𝐹 𝑥 𝑟𝑟 )}/𝐼 𝑧	(22)

  ). The system is over-actuated since θ, 𝜑 𝑎𝑛𝑑 𝑧 are the state variables to be controlled and 𝑈 𝑓𝑟 , 𝑈 𝑓𝑙 , 𝑈 𝑟𝑟 𝑎𝑛𝑑 𝑈 𝑟𝑙 are the control inputs.We define intermediate control inputs as follows:𝐸 𝜃 = (-𝐹 𝑠 𝑓𝑟 + 𝐹 𝑠 𝑓𝑙 ) 𝑡 𝑓 + (-𝐹 𝑠 𝑟𝑟 + 𝐹 𝑠 𝑟𝑙 )𝑡 𝑟

								(23)
	𝐸 𝜑 = -(𝐹 𝑠 𝑓𝑟 + 𝐹 𝑠 𝑓𝑙 ) 𝑙 𝑓 + (𝐹 𝑠 𝑟𝑟 + 𝐹 𝑠 𝑟𝑙 )𝑙 𝑟	(24)
		𝐸 𝑧 = 𝐹 𝑠 𝑓𝑟 + 𝐹 𝑠 𝑓𝑙 + 𝐹 𝑠 𝑟𝑟 + 𝐹 𝑠 𝑟𝑙		(25)
	These equations can be written in the form
	(	𝐸 𝜃 𝐸 𝜑 𝐸 𝑧	) = ( ⏟ -𝑡 𝑓 𝑡 𝑓 -𝑡 𝑟 -𝑙 𝑓 -𝑙 𝑓 𝑙 𝑟 1 1 1 𝐴	𝑡 𝑟 𝑙 𝑟 1	)	(	𝐹 𝑠 𝑓𝑟 𝐹 𝑠 𝑟𝑙 ) 𝐹 𝑠 𝑓𝑙 𝐹 𝑠 𝑟𝑟	(26)

  (𝛼 𝑖 𝑎𝑛𝑑 𝛽 𝑖 are also controller gains to be adjusted). and 𝐸 𝑒𝑞θ , 𝐸 𝑒𝑞𝜑 𝑎𝑛𝑑 𝐸 𝑒𝑞𝑧 are respectively 𝐸 θ , 𝐸 𝜑 and 𝐸 𝑧 when 𝑠̇𝑖 = 0. After some simple calculations, we can find:𝐸 𝑒𝑞θ = -𝑀 𝑠 (ℎ θ + 𝑧)𝑎 𝑦 + 𝐼 𝑥 𝜃 ̈𝑑𝑒𝑠 -𝑘 𝜃 𝐼 𝑥 (𝜃 ̇-𝜃 ̇𝑑𝑒𝑠 ) (38)𝐸 𝑒𝑞𝜑 = -𝑀 𝑠 (ℎ φ + 𝑧)𝑎 𝑥 + 𝐼 𝑦 𝜑̈𝑑 𝑒𝑠 -𝑘 𝜑 𝐼 𝑦 (𝜑̇-𝜑̇𝑑 𝑒𝑠 ) (39) 𝐸 𝑒𝑞z = 𝑀 𝑠 𝑔 + 𝑀 𝑠 𝑧̈𝑑 𝑒𝑠 -𝑘 𝑧 𝑀 𝑠 (𝑧̇-𝑧̇𝑑 𝑒𝑠 )

		𝐸 1θ + 𝐸 2θ + 𝐸 𝑒𝑞θ	(34)
		𝐸 𝜑 = 𝐸 1𝜑 + 𝐸 2𝜑 + 𝐸 𝑒𝑞𝜑	(35)
		𝐸 𝑧 = 𝐸 1𝑧 + 𝐸 2𝑧 + 𝐸 𝑒𝑞𝑧	(36)
		𝐸 1i = -𝛼 𝑖 |𝑠 𝑖 | 1/2 𝑠𝑖𝑔𝑛(𝑠 𝑖 )
	where	{ 𝐸 ̇2i = -𝛽 𝑖 𝑠𝑖𝑔𝑛(𝑠 𝑖 )	(37)
		𝑖 = 𝜃 ; 𝜑; 𝑧.

  𝐸 θ = -(𝛼 𝜃 + 𝑘 1𝜃 )𝐼 𝑥 (𝜃 ̇-𝜃 ̇𝑑𝑒𝑠 ) -(𝛼 𝜃 𝑘 1𝜃 + 𝑘 2𝜃 )𝐼 𝑥 (𝜃 -𝜃 𝑑𝑒𝑠 ) -𝛼 𝜃 𝑘 2𝜃 𝐼 𝑥 ∫(𝜃 -𝜃 𝑑𝑒𝑠 )𝑑𝑡 -𝑀 𝑠 (ℎ θ + 𝑧)𝑎 𝑦 + 𝐼 𝑥 𝜃 ̈𝑑𝑒𝑠 (44) 𝐸 𝜑 = -(𝛼 𝜑 + 𝑘 1𝜑 )𝐼 𝑦 -(𝛼 𝜑 𝑘 1𝜑 + 𝑘 2𝜑 )𝐼 𝑦 (𝜑 -𝜑 𝑑𝑒𝑠 ) -𝛼 𝜑 𝑘 2𝜑 𝐼 𝑦 ∫(𝜑 -𝜑 𝑑𝑒𝑠 )𝑑𝑡 -𝑀 𝑠 (ℎ 𝜑 + 𝑧)𝑎 𝑥 + 𝐼 𝑦 𝜑̈𝑑 𝑒𝑠 (45) 𝐸 𝑧 = -𝑀 𝑠 (𝛼 𝑧 + 𝑘 1𝑧 )(𝑧̇-𝑧̇𝑑 𝑒𝑠 ) -𝛼 𝑧 𝑘 1𝑧 𝑀 𝑠 (𝑧 -𝑧 𝑑𝑒𝑠 ) + 𝑀 𝑠 𝑔 + 𝑀 𝑠 𝑧̈𝑑 𝑒𝑠 (46) B. Low level controller Once 𝐸 𝜃 , 𝐸 𝜑 𝑎𝑛𝑑 𝐸 𝑧 are obtained from one of the controllers discussed before, we can deduce 𝐹𝑠 𝑖𝑗 by doing the pseudo-inverse matrix to (26), then we obtain:Finally, by substituting 𝐹𝑠 𝑖𝑗 by their values in (4), we find the control outputs: 𝑈 * 𝑖𝑗 = 𝐹 𝑠 𝑖𝑗 + 𝑘 𝑠 𝑖𝑗 (𝑧 𝑠 𝑖𝑗 -𝑧 𝑢𝑠 𝑖𝑗 ) + 𝑐 𝑠 𝑖𝑗 (𝑧̇𝑠 𝑖𝑗 -𝑧̇𝑢 𝑠 𝑖𝑗 ) (48) We denoted by 𝑈 * 𝑖𝑗 instead of 𝑈 𝑖𝑗 the controllers outputs because the actuators have some limitations.

	(	𝐹 𝑆 𝑓𝑟 𝐹 𝑆 𝑓𝑙 𝐹 𝑆 𝑟𝑙 ) 𝐹 𝑆 𝑟𝑟	= (𝐴 𝑇 . 𝐴) -1 . 𝐴 𝑇 (	𝐸 𝜃 𝐸 𝑧 𝐸 𝜑	)	(47)
							1 2	𝑠 𝑖	2 positive
							𝑠 θ = 𝑒̇𝜃 + 𝑘 1𝜃 𝑒 𝜃 + 𝑘 2𝜃 ∫ 𝑒 𝜃 𝑑𝑡	(41)
							𝑠 𝜑 = 𝑒̇𝜑 + 𝑘 1𝜑 𝑒 𝜑 + 𝑘 2𝜑 ∫ 𝑒 𝜑 𝑑𝑡	(42)
							𝑠 z = 𝑒̇𝑧 + 𝑘 1𝑧 𝑒 𝑧	(43)
						𝑠̇θ = 𝑒̈𝜃 + 𝑘 1𝜃 𝑒̇𝜃 + 𝑘 2𝜃 𝑒 𝜃 = -𝛼 𝜃 (𝑒̇𝜃 + 𝑘 1𝜃 𝑒 𝜃 +
						𝑘 2𝜃 ∫ 𝑒 𝜃 𝑑𝑡),	𝑠̇𝜑 = 𝑒̈𝜑 + 𝑘 1𝜑 𝑒̇𝜑 + 𝑘 2𝜑 𝑒 𝜑 = -𝛼 𝜑 (𝑒̇𝜑 +
						𝑘 1𝜑 𝑒 𝜑 + 𝑘 2𝜑 ∫ 𝑒 𝜑 𝑑𝑡),
						𝑠̇z = 𝑒̈𝑧 + 𝑘 1𝑧 𝑒̇𝑧 = -𝛼 𝑧 (𝑒̇𝑧 + 𝑘 1𝑧 𝑒 𝑧 ),
						After some simple calculations, we can find the high level
						control inputs as follows:

definite Lyapunov functions, their time derivative 𝑉 𝑖 ̇= 𝑠 𝑖 𝑠 𝑖 ̇ should be negative, then 𝑠̇𝑖 = -𝛼 𝑖 𝑠 𝑖 , where 𝛼 𝑖 are positive gains to be adjusted, we add to the equations of roll and pitch their error time integrator to reduce steady state error.

Table 1

 1 Vehicle parameters

	M	1286.4	𝑡 𝑓 = 𝑡 𝑟	0.773
	𝑚 𝑢𝑠 𝑖𝑗	40	𝑙 𝑓 & 𝑙 𝑟	1 & 1.6
	𝐼 𝑥	534	𝑘 𝑠 𝑓𝑙 = 𝑘 𝑠 𝑓𝑟	22639
	𝐼 𝑦	1860	𝑘 𝑠 𝑟𝑙 = 𝑘 𝑠 𝑟𝑟	12548
	𝐼 𝑧	1970	𝑐 𝑠 𝑖𝑗	700

ACKNOWLEDGMENT

This work was carried out and funded in the framework of the Labex MS2T. It has been also carried out in the framework of the French regional project SYSCOVI. SYSCOVI is jointly financed by the European Union. Europe engage in Picardie with the European Fund of Regional Development FEDER. It was supported by the French Government, through the program "Investments for the future" managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02), also by Idex Sorbonnes Universités: Kit Master Nord-Sud, by the French Lebanese Research program CEDRE (32918PA).


A L Charley
email: charleyal@chartreuse.cea.fr

A Lagrange

O Lartigue

J Simon

P Thony

P Schiavone

Hyper high NA achromatic interferometer for immersion lithography at 193nm

An apparatus for immersion interferometric lithography is described here where the interfering beams are created by illuminating a first diffraction grating followed by a second diffraction grating recombining the diffracted beams onto the photo-resist plane.

The main advantage of this system is to be achromatic: thus it is possible to use a basic commercial ArF excimer laser as the exposure source. We present here the calculations made to evaluate the different parameters that can influence the depth-of-focus in the immersion configuration.

As the set-up is mainly based on the two diffraction gratings, it matters to properly design it. The purpose of this paper is to show the optimization made on the diffraction gratings in taking into account their fabrication process since they are fabricated using the capabilities of the silicon line available in our laboratory. On one hand, calculations have been done to determine the second grating period as a function of the first grating period and the "immersion NA". By simply adding a fluid to a "dry" system, we will indeed be able to improve the depth of focus but not the resolution. In playing with the diffraction grating periods, we are able to benefit from the introduction of the immersion fluid. We have performed simulations in order to optimize the grating diffraction efficiency as a function of the etch depth and the fractional linewidth.

Finally we report on first results obtained with the achromatic immersion interferometer.

The apparatus was used with a 193 nm GAM excimer laser to print resist patterns having a period of 100 nm with excellent contrast.

INTRODUCTION

As immersion lithography becomes increasingly important for next generation device fabrication, benefits and drawbacks of the introduction of an immersion fluid between the imaging optics and the wafer have to be studied. In this context, it is of high interest to be able to understand the properties of hyper high NA imaging and to do so, to have access to a technique which can mimic an industrial immersion stepper by having the possibility to tune the polarization state and numerical aperture. Interferometric lithography has proven to be a useful method to study projection imaging at high numerical aperture (NA), polarization state impact on imaging, photoresists materials and defectivity due to the introduction of water and high index fluids [START_REF] Smith | Proc. SPIE[END_REF].

In two waves interference as it is in interferometric lithography, the image printed grating period Λ is given by:

NA n . 2 sin . . 2 λ θ λ = = Λ (1)
where λ is the laser wavelength, θ half of the angle between the two beams, n the fluid refractive index and NA the numerical aperture.

In standard holographic interferometric lithography at 193 nm where the beam is split and recombined by mirrors, the image field is limited to a few microns when using a commercial ArF excimer lasers since temporal and spatial coherences of such a laser are very poor. By evaluating the variations in phase difference between the two arms as a result of variations in wavelength and incidence angle, we can define the maximum image field size beyond which the grating lines wash out. Our resolution criterion is a phase difference smaller the π/2.

From ref.

2, the expression of the maximum number of resolved lines N is the following :

θ θ λ λ ∆ + ∆ = 1 N ( 2
)
where ∆λ is the spectral bandwidth, ∆θ the angular width, λ the laser wavelength and θ the incidence angle.

Hence, using a typical ArF excimer laser having ∆λ ≈ 0.4 nm and ∆θ ≈ 0.4 mrad, the maximum number of grating lines achievable is N= 300 for 300-nm-period lines, 420 for 100nm-period lines and 404 for 80-nm-period lines since N depends on θ, therefore on Λ. This Electronic mail : charleyal@chartreuse.cea.fr corresponds to field sizes of respectively 90, 42 and 32 µm which is too small to be used in practice. Since researchers have focused on the study of immersion lithography at 193 nm, several setups have been built around the world. They have been faced those laser limitations, hence the optical components have to be adapted to obtain high-contrast-fringes. This has been achieved by either playing on the laser [START_REF] Hoffnagle | [END_REF][START_REF] Raub | Proc. SPIE[END_REF] with the use of solid or frequency-doubled laser or on the optical setup with the design of a high-precision compact Talbot prism lens [START_REF] Bourov | Proc. SPIE[END_REF].

Unfortunately such setups are not without disadvantages since they require expensive lasers and optical components difficult to fabricate. We have built up a setup that has the main advantage to be achromatic: thus, temporal coherence is irrelevant and spatial coherence limits only the depth-of-focus within which interference can be obtained. Therefore it is possible to use a basic commercial ArF excimer laser as the exposure source. The other advantage of this interferometer is that the main optical elements, namely the two diffraction gratings, are fabricated using the capabilities of the silicon line available in our laboratory.

Therefore, because of the simplicity and ease of fabrication of the basic optical elements of the setup, we are free to switch to various applications such as high index fluids testing and contact holes printing in a very simple way.

In this paper we report on the design of the immersion achromatic interferometer, a tolerance study and experimental results obtained with immersion achromatic interferometer.

II. DESIGN OF THE OPTICAL SETUP

Our interferometer for 193 nm immersion lithography has been adapted from a setup studied by researchers at MIT to fabricate large area, spatially coherent gratings with periods of 100 nm at 193 nm [START_REF] Anderson | [END_REF]7,8]. It consists of two phase gratings made in optically flat fused silica substrates and a chuck that holds the resist-coated wafers. The incident laser beam is diffracted by the first diffraction grating. The zero-order beam is blocked by a metal foil and the two first-order beams are incident on the second diffraction grating that recombines them by second-order diffraction. The medium between the second diffraction grating and the resist coated wafer is filled with an index matching liquid that enables NA larger than one. The liquid used is water since it has low absorbance at 193 nm and has a refractive index of 1.44 at 193 nm [START_REF] Switkes | Proc. SPIE[END_REF]. Figure 1 is a schematic of our immersion lithography configuration.

This interferometer has the advantage of recording high contrast gratings in resist over an area as large as the grating diffractive area since it is not limited by the laser spatial coherence, hence it enables large exposure field. It can be used for study of printability at NA greater than 1.4 with high index liquids. For liquids with index greater than 1.56 (index of Electronic mail : charleyal@chartreuse.cea.fr fused silica at 193 nm), NA is limited to 1.56 because NA is limited to the lowest refractive index value between fused silica and the liquid. By turning the last diffraction grating up-side down, we can get rid off this phenomenon and benefit of the increase of refractive index whereas the refractive index difference between fused silica and the liquid becomes sufficient to provide high enough diffraction efficiency.

In this part we will calculate the periods of the two diffractive elements in order to design the optical setup to achieve numerical apertures of 0.32, 0.97, 1.2 and 1.4. In normal incidence, thanks to diffraction equations in first order, we describe the diffraction by the first grating:

1 sin sin P g g λ β γ = = (3)
And by diffraction by the second grating reached by an incident beam with an angle γ g :

λ λ γ λ β γ - = - = = (4)
where γ g , β g, γ e and β e are the angles of the +1 and -1 orders in the air gap and of the -2 and +2 orders in the water gap respectively (see figure 1).

We define here the equivalent grating pitch Q as

1 2 1 2 1 P P Q - = (5)
Q takes into account the first grating P 1 that diffracts in first orders in an arbitrary direction and the second grating P 2 that diffracts in second orders in the opposite direction. It can be viewed as an equivalent grating that replaces the pair P 1 -P 2 .

From the Eq. ( 1) we see that the final period of the resist pattern is given by 2

Q = Λ (6)
Finally the numerical aperture of the system is equal to

Q NA λ = (7)
From Eq. ( 7) the NA only depends on the equivalent grating pitch Q and can be increased by decreasing Q. The NA can reach values greater than 1 by choosing P 1 and P 2 so that Q Electronic mail : charleyal@chartreuse.cea.fr becomes smaller than the wavelength. But in the case of dry lithography, when Q<193 nm, the second-order beams present in the fused silica substrate are reflected back at the fused silica-air interface because of total internal reflection. Eq. ( 8) is the necessary condition on the last medium refractive index N to enable light transmission from fused silica to resist:

Q N λ ≥ (8)
We can work with a dry system until Q reaches the wavelength value; when Q>193 nm it is mandatory to work in immersion.

Fig. 2. shows NAs for different combinations of P 1 and P 2 . It can be seen that, for the chosen ranges of P 1 and P 2 , NA between 0.3 and 1.8 can be achieved. Moreover, different combinations of gratings P 1 and P 2 can provide the same NA.

For symmetry considerations, we would like to have P 1 =P 2 =Q. But the higher the NA, the smaller the diffraction gratings periods, hence it would necessitate gratings with periods down to 193nm for NA=1. Since diffraction gratings with such periods become difficult to manufacture, it is easier to make the system asymmetrical.

Several couples of gratings was designed for achieving different NA, hence different image pitch values, as shown in table 1. In order to make the manufacturing easier an effort was put to keep a grating in common between settings, enabling minimal change to tune the NA.

III. FRINGE FORMATION ANALYSIS

In this section, a complete tolerance study is done in order to evaluate the influence of the different elements on the fringe formation. Starting from the analysis made for the achromatic holographic configuration [START_REF] Yen | [END_REF], we will do a tolerance analysis for the immersion configuration. To do so, we will calculate the phase difference between the fields interfering at the top of the resist and look at its variation with variations of incidence angles and wavelength.

Referring to Fig. 1, the thicknesses of the different media are designed as d 1 ±δd 1 , g±δg, d 2 ±δd 2 , e±δe and are defined with an incertitude term that depends on gap variations and substrate polishing. n and N are the refractive index of fused silica and water respectively. γ 1 , β 1 , γ g and β g are the angles made by the +1 and -1 orders in the first fused silica substrate and in air respectively. γ 2 , β 2 , γ e and β e are the angles made by the -2 and +2 orders in the second fused silica substrate and in water respectively.

Electronic mail : charleyal@chartreuse.cea.fr

The fields amplitudes E1 and E2 following respectively the left and right optical paths and going successively through fused silica, air, fused silica and water, can be expressed as:

) cos ) ( 2 exp( ) cos ) ( 2 exp( ) 2 2 exp( ) cos ) ( 2 exp( ) cos ) ( . 2 exp( ) 2 exp( 2 2 2 2 2 1 1 1 1 1 1 e g e e N i d d n i x P i B g g i d d n i x P i A E β δ λ π β δ λ π π γ δ λ π γ δ λ π π ⋅ ± ⋅ × ⋅ ± ⋅ × - × ⋅ ± × ⋅ ± × = - (9) and ) cos ) ( 2 exp( ) cos ) ( 2 exp( ) 2 2 exp( ) cos ) ( 2 exp( ) cos ) ( . 2 exp( ) 2 exp( 2 2 2 2 2 1 1 1 1 1 2 e g e e N i d d n i x P i B g g i d d n i x P i A E γ δ λ π γ δ λ π π β δ λ π β δ λ π π ⋅ ± ⋅ × ⋅ ± ⋅ × × ⋅ ± × ⋅ ± × - = - ( 10
)
Where A 1 , A -1 come respectively from the transmittance of the +1 and -1 orders of the first phase grating and B 2 , B -2 come respectively from the transmittance of the +2 and -2 orders of the second phase grating.

Our interest is on the light intensity since photoresist is sensitive to the intensity and not to the electric field. The light intensity reaching the resist is equal to:

) cos( ) ( ) ( ) ( ) ( 2 1 2 1 2 1 2 2 1 2 2 1 * 2 1 2 1 ϕ ϕ - ⋅ ⋅ ⋅ ⋅ + ⋅ + ⋅ = + ⋅ + = - - - - B A B A B A B A E E E E I ( 11
)
where * denotes the complex conjugate and ϕ 1 and ϕ 2 the phase of E 1 et E 2 respectively.

The light intensity is therefore proportional to cos(φ 1 -φ 2 ). Hence it becomes interesting to look at the phase difference between E 1 and E 2 , φ 1 -φ 2 , that can be written as:

) cos (cos ) ( 2 ) cos (cos ) ( 2 ) cos (cos ) ( 2 ) cos (cos ) ( . 2 4 2 2 2 2 1 1 1 1 2 1 e e g g e e N d d n g g d d n Q x β γ δ λ π β γ δ λ π β γ δ λ π β γ δ λ π π ϕ ϕ - ⋅ ± ⋅ + - ⋅ ± ⋅ + - ⋅ ± - - ⋅ ± - ⋅ = - = ∆Φ (12)
By geometrical considerations, the various angles can be expressed in function of P 1 , Q, λ and θ (see equations 1 to 8 in appendix).

Since we tend to have the laser beam normally incident on the first grating, θ is small and sinθ =θ.

Finally:

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                  + - -         - - ⋅ ± +                   + - -         - - ⋅ ± +                   + - -         - - ⋅ ± -                   + - -         - - ⋅ ± - ⋅ = 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 1 2 2 1 2 1 1 Q N Q N ) e e ( 2 Q n Q n ) d d ( 2 P 1 P 1 ) g g ( 2 P n P n ) d d ( 2 Q x 4 λ θ λ θ δ λ π λ θ λ θ δ λ π λ θ λ θ δ λ π λ θ λ θ δ λ π π ∆Φ (13)
The intensity pattern at the resist has a period one-half that of the equivalent grating period Q since it is a two waves interference system. Because of the additional terms that depends on thicknesses, medium refractive index, λ and θ, the phase difference varies with variations of λ and θ. To be truly achromatic, it is necessary to make this sum of terms independent of λ and θ. The differential of ∆Φ compared to λ and θ is calculated in the appendix (see equation 9). By considering the ideal case of δd 1 =δd 2 =δg=δe=0, it is possible to choose e, so that d∆φ cancels, hence ∆φ becomes independent of λ and θ.

The different terms δd 1 , δd 2 , δg and δe are incertitude terms that depend on gap variations and substrate polishing. Their random variation along the gratings can introduce a phase difference greater than π/2. The phase difference should therefore remain lower than π/2 otherwise this would degrade the fringe contrast. The condition usually used is:

2 d π ∆Φ ≤ (14)
Since δd 1 , δd 2 and δg are fixed by substrate polishing and by the air gap variation, the remaining unknown is δe. The previous result gives us an interval [0, δe max ] in which δe has to be included to assure high contrast fringes. This interval will be the criterion used to define the depth of focus of the present imaging system. Indeed, in lithography, the depth of focus is the range of lens-wafer distances over which linewidths are maintained within specifications and resist profiles are adequate. In our case, since recorded lines will be well resolved when the resist coated wafer is far from the last grating of a distance e±δe max , the depth of focus is equal to 2δe max .

( ) ( ) ( ) ( ) ( )           ∆         - + ∆         -         ∆         - + ∆         - -           ∆         - + ∆         -           ∆         - + ∆         - -           ∆         - + ∆         - - = θ λ λ λ λθ θ λ λ λ λθ δ θ λ λ λ λθ θ λ λ λ λθ δ θ λ λ λ λθ δ π . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 2 2 2 2 1 2 2 3 2 2 1 2 2 2 2 2 3 2 2 2 2 2 2 1 2 2 3 2 2 1 2 2 2 1 2 3 2 2 1 2 2 1 2 2 3 2 2 1 2 1 P N P N Q n Q n d P N P N P P g P n P n d DOF (15)
For λ=193 nm, ∆θ=0.4 mrad, ∆λ=0.04 nm, θ=0.1°, n=1.561, N=1.43 and δd 1 =δd 2 =δg=1 µm, the depths of focus are summarized in table 2.

From Eq. 15, the depth of focus is dependent on the various refractive indices, the spectral bandwidth ∆λ, the angular width ∆θ and the thickness incertitude terms. ∆λ and ∆θ are fixed by the laser specifications and by using a coherent laser. By changing the last medium from air to water, the refractive is increased and therefore the DOF is enhanced by a factor of 4 at an NA of 0.97. The thickness incertitude terms are relevant in the evaluation of the DOF. But since it becomes challenging to polish fused silica substrates with a flatness tolerance better than 1 µm on a whole 100-mm plate, the DOF will be limited to the values summarized in table 2. The higher the NA, the lower the DOF that is why we will see a decrease in DOF with increasing NA even though it remains much higher than the DOF available on with an industrial projection tool at same NA.

IV. PHASE GRATINGS EFFICIENCIES

The gratings depths have been optimized to achieve high diffraction efficiency into first order at normal incidence for the first grating and into second order at oblique angle for the second grating. Efficiencies have to be as high as possible since it determines the light intensity reaching the top of the resist, hence the exposure time. The latter has to be as short as possible to minimize problems induced by vibrations.

RCWA simulations have been performed to determine the diffraction efficiencies in function of fractional linewidth and grating depth. Moharam et al. [11] have reviewed a formulation for the implementation of the RCWA for one-dimensional, rectangular-groove binary gratings for both TE and TM polarization. This formulation has been shown to be stable and efficient since energy conservation and convergence to the proper solution with an increasing number of field harmonics are achieved with all the grating and the incident-wave parameters. Fig. 3 shows plots of diffraction efficiencies as functions of fractional linewidth and groove depth for TE and TM polarization. The incidence angle for the first diffraction gratings is 0° and is dependent on the first grating period for the second grating. This figure shows that diffraction efficiency is much lower for the second diffraction order and that the larger the groove depth the higher the diffraction efficiency. The diffraction efficiency in the first order is higher for groove depth around 170 nm.

The same simulations have been performed for the gratings enabling 0.97 and 1.2 NAs. From those simulations we have determined the grating groove depth considering fabrication processes and diffraction efficiencies. For fabrication process purpose, the phase gratings were chosen with an equal line-to-space ratio for both P 1 and P 2 gratings. In table 3, the chosen groove depth for each grating are summarized. They were chosen to have sufficient grating efficiencies in TE mode while not scaling down too much the depth so as not to put too much constraints on the fabrication technology.

Simulations have been done for both TE and TM modes because this setup will be used for the study of polarization effects at oblique angles. Because it becomes really challenging to fabricate gratings with diffraction efficiency optimized in TM mode, groove depths have been chosen to have sufficient efficiency in TE mode.

V. EXPERIMENTAL RESULTS

The first grating pairs enabling NA of 0.32 and 0.97 was used for the first trials to demonstrate immersion interference imaging with the double diffraction grating immersion interferometer. A GAM excimer laser at 193nm with a spatial coherence of 500 µm and a temporal coherence of 900 µm was used enabling a DOF up to 352 µm (table 2). Before reaching the first diffraction grating the beam was spatially filtered to improve the field uniformity. A polarizer was used in all experiments unless specified to polarize the light in TE mode (s-polarization) in order to assure high contrast fringes. Fig. 4 shows the interferometric setup and its environment. A Fizeau interferometer is placed above the gratings to adjust the gratings parallelism. The deionized water dispensed between the last grating and the wafer is filtered and temperature controlled.

After compacting the setup to suppress the effect of vibrations on line profiles, high contrast lines have been obtained in immersion. XP1020B photoresist was used for all experiments with thickness of 190 nm and 70 nm for respectively the 150-nm and the 50-nm half-pitch lines. AR40 was used as a bottom anti-reflective coating with a thickness of 80 nm and at normal incidence on a 600-nm-period grating. b) Second-order diffraction efficiency for TE polarization at 18.8° incidence on a 600-nm-period grating. c) First-order diffraction efficiency for TM polarization at normal incidence on a 600-nm-period grating. d) Secondorder diffraction efficiency for TM polarization at 18.8° incidence on a 600-nm-period grating. Electronic mail : charleyal@chartreuse.cea.fr Electronic mail : charleyal@chartreuse.cea.fr Electronic mail : charleyal@chartreuse.cea.fr

( ) ( ) ( ) ( ) θ λ δ π λ λ λθ δ π θ λ δ π λ λ λθ δ π θ λ δ π λ λ λθ δ π θ λ δ π λ λ λθ δ π ∆Φ d . Q N 2 ) de e .( 2 d . Q N 2 ) de e .( 2
d . Q n 2 ) d d .( 2 d . Q n 2 ) d d .( 2
) d d .( 2 d 2 2 2 2 3 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 1 2 3 2 2 1 2 2 1 2 1 1 2 3 2 2 1 2 1 1           - ± +           - ± +           - ± +           - ± +           - ± -           - ± -           - ± -           - ± - = (9)

Fig. 1 .

 1 Fig.1. Schematic of the immersion achromatic interferometric configuration and diagram for calculating the optical paths of two arms of the double diffraction gratings interferometer.

Fig. 2 .

 2 Fig.2. Numerical apertures function of the gratings period P 1 and P 2 of the double diffraction grating interferometer.

Fig. 3 .

 3 Fig.3. Diffraction efficiencies in percent of fused silica gratings at 193 nm as a function of groove depth and fractional linewidth. a) First-order diffraction efficiency for TE polarization

Fig. 4 .

 4 Fig. 4. Picture showing the experimental setup of the double diffraction grating immersion interferometer.

Fig. 5 .

 5 Fig. 5. Resist images using water immersion lithography. a) 150 nm half-pitch corresponding to 0.32 NA. b) 50 nm half-pitch corresponding to 0.97 NA.

Fig. 1 :

 1 Fig. 1:

Fig. 2 :

 2 Fig. 2:

Fig

  Fig. 3 :

Fig. 4 :Fig

 4 Fig. 4:

TABLE CAPTION Table 1 .

 CAPTION1 Gratings designs created for the immersion double diffraction interferometer.

Table 2 .

 2 Depth of focus values for different NA configurations of the double diffraction grating interferometer.

Table 3 .

 3 Grating groove depth of each grating.

Table 1 :

 1 NA0.32 0.97 1.2 P1 (nm) 600 220 300 P2 (nm) 600 210 210 Electronic mail : charleyal@chartreuse.cea.fr

Table 2 :

 2 NA

		0.32 0.97	0.97	1.2
		(dry)	(immersion)
	DOF (µm)	352 30	126	74

XP1029 was used as a topcoat with a thickness of 38 nm to suppress the effects of contamination. All materials were supplied by Rohm&Haas. Figure 5 shows results obtained with the immersion achromatic interferometer. High-contrasted lines are obtained with nice straight profiles even at small dimensions. Resolution from 150 to 50 nm has been demonstrated in both dry and wet lithography.

VI. SUMMARY

We have completely designed an immersion interferometer to generate resist patterns with period down to 60 nm. The great advantages of this setup compared to others already existing is that it is possible to use a basic commercial ArF excimer laser as the exposure source and that the main optical elements are home-fabricated enabling us to switch to various applications such as high index fluids testing and contact holes printing in a very simple way.

Calculations have been performed to evaluate the depth of focus of the system and to determine the gratings groove depths that enable sufficient diffraction efficiency. However the depth of focus is affected by the small coherence length of the laser, even though it remains much higher than in industrial projection tools since the lowest DOF achievable in the immersion configuration is of 74 µm for 80 nm pitch gratings. Imaging at 0.32 and 0.97 NA in both dry and immersion was demonstrated and polarization effects at oblique incidence angles have been shown.

VII. APPENDIX

Diffraction angles in function of the refractive index, the grating period, the incidence angle and the wavelength:

Calculation of d∆φ compared to λ and θ:


Angélique Amelot
email: angelique.amelot@icp.inpg.fr

Solange Rossato
email: solange.rossato@icp.inpg.fr

Velar movements for the feature [±nasal] for two French speakers

This study deals with the velar movements involved in (i) the realisation of the phonemic [±nasal] contrast on vowels and (ii) contextual nasalization. We measure velar positions with an electromagnetic articulatograph (EMA) for two French speakers. Our results confirm that (i) nasal vowels are produced with a lower velum height than nasal consonants; (ii) velum height targets for nasal and oral segments can overlap, especially sequences of nasal consonant and contextually nasalized oral vowels or /l/ and /K/ consonants (iii) contextually nasalization in French is rather a carryover of the lowering of the velum than an anticipation of this movement.

Introduction

The nasality feature is widely spread in the world's languages. 96.5% of the 451 languages from UPSID database (UCLA Phonological Segment Inventory Database [START_REF] Maddieson | UCLA Phonological Segment Inventory : Data and Index[END_REF] [START_REF] Vallée | Systèmes vocaliques : de la typologie aux prédictions[END_REF] [START_REF] Boë | The nature of vowel structure[END_REF] use this feature in their consonantal system. It is also the most frequent secondary dimension used in vocalic systems (22.4% of UPSID languages), on a par with the dimension of length. This binary feature [±nasal] is produced by the lowering of the velum, which connects the nasal passage to the oral tract. For nasal vowels, the acoustic consequences are quite complex; they depend not only on velum position but also on the articulatory configuration of the vocal tract, as shown by simulation studies [START_REF] House | Analog Studies of the Nasalization of Vowels[END_REF][START_REF] Fujimura | Sweep-Tone Measurements of Vocal-Tract Characteristics[END_REF][START_REF] Feng | Modélisation acoustique et traitement du signal de parole[END_REF][START_REF] Maeda | Acoustics of Vowel Nasalization and Articulatory Shifts in French Nasal Vowels[END_REF]. [START_REF] Maeda | Acoustics of Vowel Nasalization and Articulatory Shifts in French Nasal Vowels[END_REF] synthesized a continuum from an oral vowel to a nasal one by an articulatory model and perceptual tests showed that a small opening of the velopharyngeal port (on the order of 0.4 cm 2 ) was sufficient to perceive a nasalized [i], whereas [a] was still perceived as oral, a coupling greater than 1.6 cm 2 being necessary to perceive nasalization on an [a]. Different velar positions have been observed during the production of oral vowels. [START_REF] Clumeck | Patterns of Soft Palate Movements in Six Languages[END_REF] used photodetection of velar activity on several subjects across six languages and clearly established a link between the height of the oral vowels and the velum position: /a/ is produced with a lower velum than other oral vowels. The electromyographic data recorded by [START_REF] Iwashita | Electromyographic Study on the Articulation Mechanism of Japanese Speech Spounds[END_REF] for Japanese vowels showed that velum position was correlated with lingual position. The correlation between vowel height and velar position may depend on the activity of the palatoglossus [START_REF] Moll | Velopharyngeal Closure on Vowels[END_REF], following the hypothesis according to which this muscle could serve a palate-lowering function when the activity of the levator muscle is suppressed. However, [START_REF] Bell-Berti | Patterns of Palatoglossus Activity and their Implications for Speech Organization[END_REF] find differences in the pattern of palatoglossus activity, measured by electromyography, in vowel-nasal and nasal-vowel combinations: the palatoglossus function for velum lowering may exist for speakers of some languages, but this function does not occur for all speakers and all languages. Therefore, the palatoglossus activity is not sufficient to explain the relationship between the lowering of the velum and the height of the vowel. No biomechanical link can explain the fact that low vowels are produced with a low velum height. It has been noticed that the velar position observed in low vowels could even be so low as to imply an open velopharyngeal port. For example, Durand (1953 p.34), on the basis of radiographic data on French, observed that /a/ in oral context ("il l'a" [il:a]) was produced with a velum 10 mm distant from the back pharyngeal wall. Very importantly, such a slightly open velopharyngeal port may not be sufficient to induce a nasal percept [START_REF] Maeda | Acoustics of Vowel Nasalization and Articulatory Shifts in French Nasal Vowels[END_REF]. The opening and closing phases of the velopharyngeal port involved in the production of nasal consonants also tend to nasalize the adjacent vowels. The degree of coarticulated nasality is language-specific [START_REF] Clumeck | Patterns of Soft Palate Movements in Six Languages[END_REF][START_REF] Cohn | Phonetic and Phonological Rules of Nasalization, Appear to Working Papers in Linguistics n°76[END_REF]. [START_REF] Clumeck | Patterns of Soft Palate Movements in Six Languages[END_REF] reported a strong variation across languages in the timing of anticipatory velopharyngeal port opening in vowels preceding a nasal consonant; this variation was not closely related to distinctiveness of vowel nasalization. Some phonemes did not allow the spreading of the nasality feature. [START_REF] Benguerel | Velar Height and its Timing in French: a Fiberscopic Study[END_REF], on the basis of fiberscopic data on French, observed that the plosive consonants involve more contextual constraints and less spreading of nasalization. They only allow a slight velopharyngeal port opening of less than 0.1 cm 2 [START_REF] Warren | Aerodynamics of Nasalization[END_REF], therefore the nasal vowel can be produced partly as an oral vowel.

Between partly oralized nasal vowels and partly nasalized oral vowels, how the phonological contrast can be maintained on an articulatory level? What are the specific targets for oral and nasal segments and how sensitive are these targets to contextual variations? To investigate the intrinsic height of the velum for oral/nasal phonemic contrast either on consonants or on vowels and its variations due to neighboring phonemes, we measured the velar movements in VCV sequences for all phonemes in French. The contrast between nasalized oral vowel and nasal vowel is analyzed more precisely through a specially designed part of the corpus.

Articulatory measurements

Corpus

The first part of the corpus is made up of /#VCV#/ sequences, in order to measure velar position for each French consonant (/p t k b d g f s  v z   l m n/), whatever the vocalic context: the five oral vowels /i e a o u/ and all French nasal vowels (/ 9  2 /). A single repetition of the 160 sequences was recorded. These sequences also allow for the analysis of velar position during oral and nasal vowels whatever the consonantal left or right context. The second part of the corpus focuses on the effect of contextual nasalization on oral vowels in French as compared to nasal vowels with sequences like /C n V/, /VC n / and /C n VC n /. C n is the nasal consonant /n/, as the alveolar occlusion is clearly visible on the apex coil of EMA. V is one of the French vowels /i y u e o E 9 O  oe   /. These sequences were embedded in a carrier sentence: "Vous dites ___ trois fois"; 'You say ___ three times'. The other parts of the corpus are made up of /CV n C/ sequences, words with nasal plus oral consonant clusters, and sentences, which are still under analysis.

Instrumentation

The articulatory movements were recorded with an electromagnetic midsagittal articulograph (EMA Carstens AG100). Six coils were attached with the medical glue used by dentists. A Cyanoveneer glue was used for the skin and mucous membrane, and a CG Fuji 1 glue for the tongue.

Results

The statistical tests used in this study were based on the variance analysis ANOVA with a significance level p<0.001 by default.

Velum height for oral and nasal segments

Figure 3 presents the velum height distribution for four classes: oral consonants (OC; n=118 for S1; n=141 for S2), oral vowels (OV, n=160 for S1; n=191 for S2), nasal consonants (NC; n=18 for S1; n=20 for S2) and nasal vowels (NV; n=114 for S1; n=129 for S2). The velum height is the Y coordinate of the velum coil at the middle of the segment, extracted for each phoneme of VCV sequences. Both oral vowels and consonants are produced with a similar range of velum height, no significant differences appear (S1: F(1,276)=2.131, p=0.1455; S2: F(1,330)=1.358, p=0.2447). Nasal phonemes are produced with the lowest velum height for both subjects; they are clearly different from oral phonemes, with a significant difference for the oral/nasal phonemic contrast (S1: F(1,408)=1310.851, p<0.001; S2: F(1,479)=1824.096, p<0.001). Nevertheless, there is some overlap between oral and nasal segments. Some oral segments are produced with an open velopharyngeal port: this concerns some oral vowels following or preceding a nasal consonant, and productions of the oral consonants /l/ or /K/ in nasal vowel contexts. A similar velum height is measured during the production of nasal consonants between oral vowels, and of some tokens of nasal vowels following plosives and liquids for S2. Whilst no significant difference has been observed between oral consonants and vowels, there is a significant difference between nasal vowels and nasal consonants

Velum height and vowel height

The velum height for the five vowels [i u e o a] is presented in figure 5. The number of items for S1 is /i/=31, /u/=30, /e/=33, /o/=34 and /a/=32; for S2, the numbers are: /i/=64, /u/=32, /e/=32, /o/=32 and /a/=31. We observe that the vowel produced with the lowest velum is the same for the two speakers: [a] is realized with a lower velum than /e i u o/ (S1: mean(/a/)=0.87 cm, mean deviation=0.053, mean(/iueo/)=0.978 cm, mean deviation=0.054; S2: mean(/a/=1.42 cm, mean deviation=0.063, mean(/iueo/)=0.978 cm, mean deviation=0.042). We find a significant difference for /a/ compared to /e i u o/ (S1: F(1,158)=59.631, p<0.001; S2: F(1,189)=60.647, p<0.001). In order to observe the link between vowel height and velum height, we grouped vowels into three groups: the two high vowels /i u/, the two mid vowels /e o/, and the low vowel /a/. The results show a significant difference depending on vowel height for S1 (F(1,157)=35.772, p<0.001), and no significant difference between mid and high vowels for S2 (F(1,158)=1.751, p=0.1876). Our results do not show consistent patterns of correlation between vowel height and velum height.

The velum height for the consonants

Figure 6 shows velum height for all consonants. We group the consonants into six classes: unvoiced stops (n=24 for S1; n=30 for S2), voiced stops (n=23 for S1; n=30 for S2), unvoiced fricatives (n=24 for S1; n=30 for S2), voiced fricatives (n=30 for S1; n=31 for S2), /l/ and // (n=17 for S1; n=20 for S2) and nasal consonants (n=18 for S1; n=20 for S2). As for oral vowels, we observed a speaker-specific behavior. Nasal consonants are clearly produced with the lowest velum position. For the two speakers, some productions of /l/ or // can be produced with a velum height equivalent to that of the nasal consonants, whereas this is not the case for any of the other oral consonants. For S1, paired tests reveal that all the pairs are significantly different except unvoiced fricatives which show no significant difference with voiced fricatives and unvoiced stops (F(1,52)=2. 575, p=0.1146, F(1,46)=1.274, p=0.2648). For S2, there is a significant difference between nasal consonants and the others (S2: F(1,159)=163.168, p<0.001 for each group compared with the nasal consonants), and no significant differences are found between fricatives and plosives.

Contextual nasalization

The velum height of contextually nasalized oral vowels (as compared to nasal ones) is obtained from the analysis of the second part of the corpus, measuring the position of the velum at the middle of each vowel for the three sequences /C n V/, /VC n / and /C n VC n / where C n =/n/ (S1: n=44; S2: n=44). Sequences /C n V/ offer evidence of carry-over of nasalisation, /VC n / of anticipation, and symmetrical sequences /C n VC n / quite predictably involve the greatest amount of contextual nasalisation on oral vowels. Figure 7 shows the results depending on those positions. The oral/nasal contrast for vowels is always maintained (S1: F(1,42)=37.97, p<0.001; S2: F(1,42)=219.184, p<0.001), nasalization of the vowel due to the context never reaches the target of the nasal vowels. The velum position is higher for oral vowels followed by the nasal consonant /VC n / than for oral vowels preceded by the nasal consonant /C n V/, indicating that there is more carryover than anticipation. No significant difference appears between vowels preceded by the nasal consonant and vowels surrounded by the nasal consonant (S1: F(1,27)=0.211, p=0.6493; S2: F(1,27)=0.388, p=0.5387). It is interesting to note that for S1's nasal vowels, velum height is higher in the context /C n VC n / than in the other contexts, indicating that the symmetrical nasal consonant context tends to raise the velum during the nasal vowels.

Discussion

Targets for nasal and oral segments are clearly separated, even though some overlap is observed. Nasal vowels show the lowest velum positions, nasal consonants reaching similar values only in nasal vowel context. Those results confirm a preliminary EMA recording on a single male French speaker [START_REF] Rossato | Velar Movements in French: An Articulatory and Acoustical Analysis of Coarticulation[END_REF]. The overlap between nasal and oral segments is mainly observed for contextually nasalized oral vowels. Results on the contextual nasalization on oral vowels reveal that velum height during the vowel is more affected by the preceding nasal consonant (or by a symmetrical nasal context) than by the following one, though without reaching the target of nasal vowels, hereby maintaining an articulatory difference between nasal vowel and nasalized ones. Several studies show that nasalization processes are language-specific. Our data on French yield some evidence in favor of a greater amount of carry-over than of anticipation. Some of the oral consonants /l/ and // are produced with a slightly lowered velum position in nasal vowel context, suggesting that they are produced with an open velopharyngeal port. This has been observed for both subjects.

Figure 1 .

 1 Figure 1. Position of the six coils fixed on the articulators.

Figure 2 .

 2 Figure 2. Acoustic signal with labels, sonogram and velum height for /imi/ sequence produced by S2.

Figure 3 .

 3 Figure 3. Normalized rank of velum height measured for each class (Left: S1 and right: S2).

  (S1: F(1,130)=11.495, p=0.009; S2: F(1,147)=13.442, p=0.003). Nasal vowels are produced with a lower velum height than nasal consonants. The lowest velum positions for nasal consonants are reached in nasal vowel context. Nasal consonants show a clearly bimodal distribution. The figure 4 presents the mean value and standard deviation of the velum height in the four classes.

Figure 4 .

 4 Figure 4. Mean value and standard deviation for oral consonants, oral vowels, nasal consonants and nasal vowels for the two speakers.

Figure 5 .

 5 Figure 5. The velar position of the oral vowels for the two speakers.

Figure 6 .

 6 Figure 6. Mean and standard deviation for the intrinsic height of the velum for the consonants for the two speakers.

Figure 7 .

 7 Figure7. Velum height depending on oral and nasal vowels followed, preceded or surrounded by the nasal consonant /n/ for the two speakers.

Acknowledgments

We would like to thank our subjects for their cooperation, Christophe Savariaux for making the data available, Louis-Jean Boë for useful discussions and suggestions, and Alexis Michaud for proofreading the final version. This work is supported by the French Agence Nationale pour la Recherche (ANR "Dynamique de la Nasalité").